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+{"text":"\\section{Introduction}\n\\label{intro}\nAccording to the Constituent Quark Model (CQM) proposed by Gell-Mann~\\cite{Gell-Mann:1964ewy} and Zweig~\\cite{Zweig:1964ruk} baryons are formed from three-quarks $(qqq)$ and  mesons are consisted of quark and anti-quark pairs $(\\bar{q}q)$. This model has been very successful in classifying the hadrons. Many new hadrons predicted by this model have subsequently been observed with the advances of the accelerators' technologies. However, many states of the hadrons predicted by the quark model are still waiting to be discovered and form the major research area of the hadron spectroscopy. In 2003, BELLE collaboration announced the discovery of a new exotic hadron $X(3872)$ in the decay channel of $B^0 \\rightarrow J\/\\Psi \\pi^+ \\pi^- K$ whose properties could not be explained by the CQM~\\cite{Belle:2007hrb}. This discovery was later verified by BABAR~\\cite{BaBar:2004oro}, CDF~\\cite{CDF:2003cab}, D0~\\cite{D0:2004zmu}, LHCb~\\cite{LHCb:2011zzp} and CMS~\\cite{CMS:2013fpt} collaborations. This unexpected observation increased the interest in exotic hadrons, and more than twenty exotic hadronic states have been discovered at accelerator and flavor factories till now ~\\cite{Zyla:2020zbs}. These exotic states have been observed either as tetraquarks (two quarks and antiquark pairs $qq\\bar{q}\\bar{q}$) or as pentaquarks (four quarks and an antiquark $qqqq\\bar{q}$). All the exotic hadronic states discovered up to now contained a pair of heavy valence quarks (either $\\bar{c} c $ or $\\bar{b}b$). No exotic state with a single heavy quark has been observed yet~\\cite{Chen:2016spr}. \n\nThe theoretical attempts for the explanation of the unexpected states can be categorized into two main approaches~\\cite{Ali:2017jda,Godfrey:2008nc}. The exotic states can be tightly bound color-singlet tetraquark state ($\\bar{Q}Q \\bar{q}{q}$) formed by two heavy ($\\bar{Q}Q$) and light ($\\bar{q}q$) quark-antiquark pair states bound by a gluon. This framework is named as diquark model in the literature~\\cite{Maiani:2004vq,Maiani:2004uc,Jaffe:2003sg}. Another idea is that these exotic states are weakly bound molecular states of two mesons (for tetraquarks) or a meson and a baryon (pentaquarks)~\\cite{Godfrey:2008nc,Swanson:2006st,Karliner:2015ina}. The mass and decay widths of the exotic hadrons can be calculated in diquark and molecular state models, which needs to be confirmed with the experiments. The properties of the exotic states both from the theoretical and experimental perspectives are discussed widely in the literature (see reviews ~\\cite{Guo:2017jvc,Agaev:2020zad,Godfrey:2008nc,Lebed:2016hpi,Esposito:2016noz}). \n\nOne of these exotic hadrons, namely, very narrow $T_{cc}^+$ tetraquark state in the $D^0 D^0 \\pi^+$ spectrum recently has been observed by LHCb Collaboration~\\cite{LHCb:2021vvq,LHCb:2021auc}. This is the first experimental evidence of the open double charmed tetraquark with $cc\\bar{u}\\bar{d}$ quark configuration. The spin-parity of $T_{cc}^+$ state is determined as $J^P = 1^+$ and the measured mass of the tetraquark $T_{cc}^+$ is located at $(-273 \\pm 61 \\pm 5_{-14}^{+11})~keV$ just below the $D^0 D^{+ \\ast}$ mass threshold. For this reason, the molecular picture is quite attractive for studying the properties of the $T_{cc}^+$ state~\\cite{Li:2012ss,Xu:2017tsr,Li_2021,Ren:2021dsi,Chen:2021vhg,Wu:2021kbu,Chen:2021tnn,Wang:2020rcx}.\n\nThe interactions between $D^0 D^{*+}$ and  $D^+ D^{*0}$ are practically the same. Hence, if  $T_{cc}^+$ were described as the $D^0 D^{*+}$ molecule, there should also exist the other partner  molecule $D^+ D^{*0}$. It is a well-known fact that the mixing takes place if two states have the same total angular momentum and parity, i.e., $J^P$. Since the states, $D^0 D^{*+}$ and $D^+ D^{*0}$ carry the same $J^P$, mixing between these two states is expected. A similar argument can also be made about the existence of mixing angles between the $QQ\\bar{q}_1 \\bar{q}_2$ states, where $Q$ is the heavy $c(b)$ quark, $q_1$ and $q_2$ are the light $u,~d$ and $s$ quarks by anticipating the existence of $T_{bb}$ states.\n\nIn the present work, we calculate the mixing angles between  \n$(\\bar{q}_1 \\gamma_5 Q) (\\bar{q}_2 \\gamma_\\mu Q)$ and \n$(\\bar{q}_2 \\gamma_5 Q)  (\\bar{q}_1 \\gamma_\\mu Q)$ states within the QCD\nsum rules method by following the approach introduced in \\cite{Aliev:2010ra}, assuming that these states are molecular states. Possible measurement of the mixing angle can indirectly mimic the nature of the tetraquark state as a hadronic molecule.\n\nThe paper is organized as follows. In Section \\ref{sec:2}, the theoretical calculations on the mixing angle between tetraquark states are performed. The Section~\\ref{sec:3} is devoted to the numerical analysis of this quantity, and the last section contains our discussion and conclusion.\n\\section{Determination of the mixing angles between the $(\\bar{q}_1 \\gamma_5 Q)\n(\\bar{q}_2 \\gamma_\\mu Q)$ and $(\\bar{q}_2 \\gamma_5 Q)\n(\\bar{q}_1 \\gamma_\\mu Q)$ states}\n\\label{sec:2}\n\nIn determination of the mixing angles between the \n$(\\bar{q}_1 \\gamma_5 Q) (\\bar{q}_2 \\gamma_\\mu Q)$ and\n$(\\bar{q}_2 \\gamma_5 Q)  (\\bar{q}_1 \\gamma_\\mu Q)$ states\nin the framework of the QCD sum rules method, we start by considering the\nfollowing correlation function,\n\\begin{eqnarray}\n\\label{ema01}\n\\Pi_{\\mu\\nu} = i \\int d^4x e^{ipx} \\left< 0 \\left| T\\left\\{ J_{\\mu}^{(1)} (x) \\bar{J}_{\\nu}^{(2)} (0)\n\\right\\} \\right| 0 \\right>~.\n\\end{eqnarray}\nThis correlation function can be written in terms of two independent invariant functions as,\n\\begin{equation}\n  \\label{eq:a1}\n  \\Pi_{\\mu \\nu}(p^2) = (g_{\\mu \\nu} - \\frac{p_\\mu p_\\nu}{p^2}) \\Pi_1 (p^2) + \\frac{p_\\mu p_\\nu}{p^2} \\Pi_2(p^2)\n\\end{equation}\nwhere the first and second structures describe the contribution of spin-1 and spin-0 states, respectively. Here $J_{1\\mu}$ and $J_{2\\nu}$ are the interpolating currents of the corresponding physical states that can be written as linear combinations of unmixed states as\n\\begin{eqnarray}\n\\label{ema02}\nJ_{\\mu}^{(1)} (x)  &=&  \\cos\\theta j_\\mu^{(1)} + \\sin\\theta  j_\\mu^{(2)}~, \\nonumber \\\\\nJ_{\\nu}^{(2)} (x)  &=&  - \\sin\\theta j_\\nu^{(1)} + \\cos\\theta  j_\\nu^{(2)}~,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nj_{\\mu}^{(1)} (x)  &=&  (\\bar{q}_1^a \\gamma_5 Q^a) (\\bar{q}_2^b \\gamma_\\mu Q^b)~,\\nonumber \\\\\nj_{\\nu}^{(2)} (x)  &=&  (\\bar{q}_2^a \\gamma_5 Q^a) (\\bar{q}_1^b \\gamma_\\nu Q^b)~. \\nonumber\n\\end{eqnarray}\ncorrespond to unmixed states. Here $a$ and $b$ are the color indices.\n\nNow let us introduce the correlation function corresponding to the unmixed states, i.e,\n\\begin{equation}\n  \\label{eq:a2}\n  \\tilde{\\Pi}_{\\mu \\nu}^{i j} = i \\int d^4 x e^{i p x } \\langle 0 | j_\\mu^{i} j_\\nu^{j} |0 \\rangle\n\\end{equation}\nwhere $i$ and $j$ runs from 1 to 2. Again separating the contribution of spin-0 and spin-1 states, this correlation function can be written as\n\\begin{equation}\n  \\label{eq:a3}\n  \\tilde{\\Pi}^{i j}_{\\mu \\nu}(p^2) = (g_{\\mu \\nu} - \\frac{p_\\mu p_\\nu}{p^2}) \\tilde{\\Pi}_1^{i j} (p^2) + \\frac{p_\\mu p_\\nu}{p^2} \\tilde{\\Pi}_2^{i j}(p^2)\n\\end{equation}\nIn the following discussions, we will only consider the coefficient of the structure $(g_{\\mu \\nu} - \\frac{p_\\mu p_\\nu}{p^2})$ since all the considered states are assumed to have quantum numbers $J^{P} = 1^+$. \n\nThe sum rules for the quantity under consideration can be obtained \nby calculating the correlation function in two\ndifferent regions, i.e., in terms of hadrons and in terms of\nquarks--gluons in the deep Euclidean domain, and matching the results \nof the two representations of the correlation function.\nThe phenomenological part of the correlation function can be\nobtained by saturating it with hadron states carrying the same quantum\nnumbers as the interpolating current and then isolating the ground state\ncontributions. The currents $J_{1\\mu}$ and $J_{2\\mu}$ are created from the\nvacuum states of the corresponding mesons, respectively, and hence the\nphenomenological part of the correlation function should be equal to zero.\nIn other words, the mixing angle is solely determined in terms of the quark\nand gluon degrees of freedom. As a result, the mixing angle is free from the\nuncertainties coming from the hadronic part.\n\nWe now turn our attention to the calculation of the theoretical part of the\ncorrelation function. \n\nUsing Eqs.(\\ref{ema01}) and (\\ref{ema02}), we get,\n\\begin{eqnarray}\n\\label{ema03}\n-\\sin\\theta \\cos\\theta \\,\\widetilde{\\Pi}_{\\mu\\nu}^{(11)} + \\cos^2\\!\\theta \\,\\widetilde{\\Pi}_{\\mu\\nu}^{(12)}\n-\\sin^2\\!\\theta \\, \\widetilde{\\Pi}_{\\mu\\nu}^{(21)} + \\sin\\theta \\cos\\theta \\, \\widetilde{\\Pi}_{\\mu\\nu}^{(22)}\n= 0~.\n\\end{eqnarray}\nChoosing the coefficient of the structure  $(g_{\\mu \\nu} - \\frac{p_\\mu p_\\nu}{p^2})$, which only contains the contribution\nof $J^P = 1^+$ state, we get from Eq.(\\ref{ema03}),\n\\begin{eqnarray}\n\\sin\\theta \\cos\\theta \\left (\\widetilde{\\Pi}^{(22)} - \\widetilde{\\Pi}^{(11)}\n\\right) + \\cos^2\\!\\theta \\,\\widetilde{\\Pi}^{(12)}\n- \\sin^2\\!\\theta \\,\\widetilde{\\Pi}^{(21)} = 0~. \\nonumber\n\\end{eqnarray}\nDividing both sides to $\\cos^2\\!\\theta$ (assuming that $\\cos\\theta \\neq 0$),\nand solving the quadratic equation for $\\tan\\theta$ we get,\n\\begin{eqnarray}\n\\label{ema04}   \n\\tan\\theta = { \\widetilde{\\Pi}^{(22)} - \\widetilde{\\Pi}^{(11)} \\pm\n\\sqrt{ \\left( \\widetilde{\\Pi}^{(22)} - \\widetilde{\\Pi}^{(11)} \\right)^2 +\n4 \\widetilde{\\Pi}^{(12)} \\widetilde{\\Pi}^{(21)} }  \\over\n2 \\widetilde{\\Pi}^{(21)} }~.\n\\end{eqnarray}\nWe already noted that for obtaining the sum rules for the relevant quantity, the correlation function (Eq.\\ref{ema01}) should be calculated in terms of quarks and\ngluons in the deep Euclidean region $p^2 \\ll 0$ by using the operator\nproduct expansion (OPE). Its expression can be obtained by substituting Eq.(\\ref{ema02}) into Eq.(\\ref{ema01}) and then using the Wick theorem. As a result, we obtain the correlation function in terms of the light\nand heavy quark propagators and the light quark condensates. So, to express the correlation function in terms of the quark and gluon degrees of freedom, we need the expressions of the heavy and light quark\npropagators.   \n\nThe light-quark propagators, to first order in the light quark mass, is\ncalculated in \\cite{Ioffe:1983ju,Chiu:1986cf}, whose expression is given as\n\\begin{eqnarray}\n\\label{ema06}\nS_q^{ab}(x)  &=&  {i \\rlap\/x\\over 2\\pi^2 x^4} \\delta^{ab} - \n{m_q\\over 4 \\pi^2 x^2} \\delta^{ab} -\n{\\la \\bar q q \\ra \\over 12} \\left(1 - i {m_q\\over 4} \\rlap\/x \\right) \\delta^{ab} -\n{x^2\\over 192} m_0^2 \\la \\bar q q \\ra   \\left( 1 -\ni {m_q\\over 6}\\rlap\/x \\right) \\delta^{ab}\\nonumber \\\\\n&+& {i\\over 32 \\pi^2 x^2} g_s G_{\\mu\\nu}^{ab} (\\sigma^{\\mu\\nu} \\rlap\/x + \\rlap\/x\n\\sigma^{\\mu\\nu}) - {4 \\pi \\over 3^9\\, 2^{10}}  \\la \\bar q q \\ra \\langle g_s^2 G^2 \\rangle x^2 \\delta^{ab} + \\cdots\n\\end{eqnarray}\nThe heavy-quark propagator in x--representation is given as \\cite{Huang:2012ti}, \n\\begin{eqnarray}\n\\label{ema07}\nS^{ab}_Q(x)  &=&  {m_Q^2 \\delta^{ab} \\over (2 \\pi)^2} \\left[\ni \\rlap\/x { K_2(m_Q\\sqrt{-x^2}) \\over \\ (\\sqrt{-x^2})^2 } +\n{ K_1(m_Q\\sqrt{-x^2}) \\over \\sqrt{-x^2} } \\right] \\nonumber \\\\\n&-&  {m_Q  g_s G_{\\mu\\nu}^{ab} \\over 8 (2\\pi)^2}\n\\left[ i (\\sigma^{\\mu\\nu} \\rlap\/x + \\rlap\/x\n\\sigma^{\\mu\\nu}) { K_1(m_Q\\sqrt{-x^2}) \\over \\sqrt{-x^2} }\n+ 2 \\sigma^{\\mu\\nu} K_0(m_Q\\sqrt{-x^2}) \\right] \\nonumber \\\\\n&-&   {\\langle g_s^2 G^2 \\rangle \\delta^{ab} \\over (3^2\\,2^8 \\pi)^2} \\left[\n(i m_Q \\rlap\/x - 6) { K_1(m_Q\\sqrt{-x^2}) \\over \\sqrt{-x^2} }\n+ m_Q x^4 { K_2(m_Q\\sqrt{-x^2}) \\over (\\sqrt{-x^2})^2 } \\right]~,\n\\end{eqnarray}\nwhere $G^{\\mu\\nu}$ is the gluon field strength tensor, $g_s$ is the strong coupling constant and $K_0$, $K_1$ and \n$K_2$ are the modified Bessel functions of the second kind.\n\nThe invariant functions $\\widetilde{\\Pi}^{(ij)}$ can be related to their imaginary part (spectral density)\nwith the help of the dispersion relation,\n\\begin{eqnarray}\n\\label{ema08}\n\\widetilde{\\Pi}^{(ij)} = \\int_{s_{min}}^\\infty {\\rho^{(ij)}(s) \\over s-p^2} ds~,\n\\end{eqnarray}\nwhere $s_{min} = (2 m_Q + m_{q_1} + m_{q_2})^2$. The spectral densities $\\rho^{(12)}(s)$ and $\\rho^{(21)}(s)$ are calculated in this work and their explicit expressions are presented in Appendix~\\ref{appendix:a}. The spectral densities $\\rho^{(11)}(s)$ and $\\rho^{(22)}(s)$ are already calculated in \\cite{Aliev:2021dgx}. \nPerforming the Borel transformation over the variable $-p^2$, and assuming\nthe quark hadron duality, we get from Eq.(\\ref{ema08}),\n\\begin{eqnarray}\n\\label{ema09}\n\\widetilde{\\Pi}^{(ij)(B)} =  \\int_{s_{min}}^{s_0} \\rho^{(ij)}(s) e^{-s\/M^2} ds~,\n\\end{eqnarray}\nwhere $M^2$ is the Borel mass parameter and $s_0$ is the continuum\nthreshold. Substituting Eq.(\\ref{ema09}) into Eq.(\\ref{ema04}) we obtain the\nexpression of the mixing angle in terms of the quark and gluon degrees of\nfreedom.\n\n\\section{Numerical Analysis}\n\\label{sec:3}\nHaving obtained the expression for the mixing angle, we are ready to perform\nthe numerical analysis in the framework of the QCD sum rules. For this purpose,\nwe need the values of some input parameters, which are presented \nin Table~\\ref{tab:1}. For the heavy quark-masses, $\\overline{\\mbox{MS}}$ values are used.\n\n\\begin{table*}[h]\n  \\centering\n  \\renewcommand{\\arraystretch}{1.4}\n  \\setlength{\\tabcolsep}{7pt}\n  \\begin{tabular}{lr}\n    \\toprule\n     Parameters             & Value         \\\\\n    \\midrule\n    $\\overline{m}_u(2~GeV)$          & $ (2.2_{-0.4}^{+0.6})~MeV$         ~\\cite{PhysRevD.98.030001} \\\\\n    $\\overline{m}_d(2~GeV)$          & $ \u00a7(4.7_{-0.4}^{+0.8})~MeV$        ~\\cite{PhysRevD.98.030001} \\\\\n    $\\overline{m}_s(1~GeV)$          & $(0.114 \\pm 0.021)~GeV$            ~\\cite{PhysRevD.98.030001} \\\\ \n    $\\overline{m}_c(\\overline{m}_c)$ & $(1.28 \\pm 0.03)~GeV$              ~\\cite{PhysRevD.98.030001} \\\\\n    $\\overline{m}_b(\\overline{m}_b)$ & $(4.18 \\pm 0.03)~GeV$              ~\\cite{PhysRevD.98.030001} \\\\ \n    $\\la \\bar u u \\ra (1~GeV)$                    & $(-246_{-19}^{+28}~MeV)^3$         ~\\cite{Gelhausen:2014jea} \\\\\n    $\\la \\bar d d \\ra (1~GeV)$                    & $(1+\\gamma) \\la \\bar u u \\ra~GeV^3$             ~\\cite{Ioffe:1981kw} \\\\\n    $m_0^2$                          & $(0.8 \\pm 0.1)~GeV^2$              ~\\cite{Ioffe:2002ee} \\\\\n    $\\langle g_s^2 G^2 \\rangle$                            & $ 4 \\pi^2 (0.012 \\pm 0.006)~GeV^4$ ~\\cite{Ioffe:2002ee} \\\\\n    $\\la \\bar s s \\ra$                            & $(0.8 \\pm 0.2) \\la \\bar u u \\ra$                ~\\cite{Ioffe:2002ee} \\\\\n    $\\gamma$                         & $ -0.003 \\div 0.01$                ~\\cite{Jin:1994jz} \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{The values of the input parameters used in our calculations.}\n  \\label{tab:1}\n\\end{table*}\n\n\nThe sum rules contain two auxiliary parameters, namely, Borel mass square $M^2$, and the continuum threshold $s_0$. Therefore the so-called working regions of these two parameters must be determined in a way that the mixing angle exhibits good stability with respect to the variation of these parameters, respectively.\n\nThe lower and upper bounds of the Borel mass parameter $M^2$ are determined by requiring that the OPE should be convergent and pole contribution is dominant with respect to the continuum one. In other words, the upper bound of $M^2$ is obtained from the condition that the pole contribution should be  more than $50 \\%$, i.e,\n\\begin{equation}\n  \\label{eq:pole}\n  \\text{pole contribution} = \\frac{\\int_{s_{\\text{min}}}^{s_0} \\rho(s) e^{-s\/M^2} ds }{\\int_{s_{\\text{min}}}^{\\infty} \\rho(s) e^{-s\/M^2} ds} > 0.5~. \n\\end{equation}\n\nTo obtain the lower bound for $M^2$, we restrict the total condensate contributions to be less than $30 \\%$ of the result, i.e.,\n\\begin{equation}\n  \\label{eq:lower}\n  \\frac{\\sum \\Pi_i^{\\text{condensates}}}{\\Pi_{\\text{total}}} < 30 \\%\n\\end{equation}\n\nThese condition lead us to the working regions of $M^2$ that are presented in Table~\\ref{tab:2}.\n\nOn the other hand, the threshold value $s_0$ is determined by requiring that the variation in the obtained mass value of the considered hadron should be minimum. Using the working region of $M^2$, we find that mass sum rules exhibits very good stability on variation of the $s_0$ that are presented in Table~\\ref{tab:2}. \n\n\n\\begin{table*}[h]\n\\centering\n\\renewcommand{\\arraystretch}{1.4}\n\\setlength{\\tabcolsep}{7pt}\n\\begin{tabular}{lcc}\n    \\toprule\n                                & $s_0~(GeV^2)$     & $M^2~(GeV^2)$  \\\\\n\\midrule\n$D^+ D^{0\\ast}   (D^0 D^{+\\ast})$   &  $20 \\div 21$   &  $2.4 \\div  2.9$    \\\\  \n$D^+ D_s^{+\\ast} (D_s^+ D^{+\\ast})$ &  $20 \\div 21$   &  $2.4 \\div  2.9$    \\\\ \n$D^0 D_s^{+\\ast} (D_s^+ D^{0\\ast})$ &  $20 \\div 21$   &  $2.4 \\div  2.8$    \\\\ \n$B^0 B^{-\\ast}   (B^- B^{0\\ast})$   & $115 \\div 120$  & $8 \\div 11$    \\\\ \n$B^0 B_s^{0\\ast} (B_s^0 B^{0\\ast})$ & $120 \\div 125$  & $8 \\div 12$    \\\\\n$B^- B_s^{0\\ast} (B_s^0 B^{-\\ast})$ & $120 \\div 125$  & $8 \\div 12$    \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{The working regions the Borel mass parameter $M^2$, and the\ncontinuum threshold $s_0$ for different tetraquark states in molecular\npicture} \\label{tab:2}\n\\end{table*}\n\nHaving the values of input parameters and working regions of $M^2$ and $s_0$, we can perform numerical analysis for the mixing angles. In Figure~\\ref{fig:1}, we present the dependency of the mixing angle between $D^0 D_s^{+*}$ and $D_s^+ D^{0*}$ on Borel mass square at the fixed values of $s_0$. Similar analyses are performed for all other mixing angles, and the results are collected in Table~\\ref{tab:3}.\n\\begin{table*}[h]\n\\centering\n\\renewcommand{\\arraystretch}{1.4}\n\\setlength{\\tabcolsep}{7pt}\n\\begin{tabular}{lcc}\n\\toprule\n                                & $\\Delta \\theta^{\\degree} = |\\theta^{\\degree} - 45^{\\degree}| $      \\\\\n\\midrule\n$\\Delta \\theta_{ D^+ D^{0\\ast}   \\leftrightarrow D^0 D^{+\\ast} }  $  &  $0.20 \\pm 0.05$   \\\\\n$\\Delta \\theta_{ D^+ D_s^{+\\ast} \\leftrightarrow D_s^+ D^{+\\ast} }$  &  $2.8 \\pm 0.8$   \\\\\n$\\Delta \\theta_{ D^0 D_s^{+\\ast} \\leftrightarrow D_s^+ D^{0\\ast} }$  &  $3.1 \\pm 0.9$   \\\\\n$\\Delta \\theta_{ B^0 B^{-\\ast}   \\leftrightarrow B^- B^{0\\ast} }  $  &  $0.019 \\pm 0.006$   \\\\\n$\\Delta \\theta_{ B^0 B_s^{0\\ast} \\leftrightarrow B_s^0 B^{0\\ast} }$  &  $0.36 \\pm 0.03$    \\\\\n$\\Delta \\theta_{ B^- B_s^{0\\ast} \\leftrightarrow B_s^0 B^{-\\ast} }$  &  $0.37 \\pm 0.03$   \\\\\n\n\\bottomrule\n\\end{tabular}\n\\caption{The values of the mixing angles between possible tetraquark states.}\n\\label{tab:3}\n\\end{table*}\n\nThe mixing angle between $D^+ D^{0\\ast}$ and $D^0 D^{+\\ast}$ system has been estimated within one boson exchange framework \\cite{Chen:2021vhg}. However, the obtained value $\\theta = \\pm 30.8^{\\degree}$ is considerably smaller than our result.\nIt should be noted that the deviation from $\\tan\\theta = \\pm 1$ is due to the\nisospin symmetry breaking, and in our case, this violation is small for the\n$D^+ D^{\\ast 0}$ and  $D^0 D^{\\ast +}$ systems. Our results show that\nthe mixing angles that deviate relatively considerable from $\\theta= \\pm 45^0$ are only for the \n$D_s D^{\\ast }$ and  $D D_s^{\\ast}$ tetraquark systems.\n\\begin{figure}[h]\n  \\centering\n\\includegraphics[width=0.7\\textwidth]{fig1.eps}\n\\caption{The dependency of the mixing angles between $D^0 D_s^{+*}$ and $D_s^+ D^{0*}$ on Borel mass square at the fixed values of $s_0$.}\n\\label{fig:1}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nRecently, LHCb Collaboration announced the observation of a new type of hadronic state, $T_{cc}$, containing two charmed and anti-$u$ and anti$-d$ quarks in the $D^0 D^0 \\pi^+$ mass spectrum slightly below the $D^0D^{*+}$ threshold with quantum numbers $J^P = 1^+$. Analysis conducted in~\\cite{Wang:2020rcx,Li:2012ss,Xu:2017tsr} show that the molecular picture can successfully describe this exotic state. Moreover, the quark model predicts the existence of similar states with two heavy quarks and the same quantum numbers. It is a well-known fact that the states having the same quantum numbers in principle can be mixed. Inspired by this fact, the mixing angles between tetraquark systems with two heavy quarks in the molecular picture are calculated in the framework of the QCD sum rules method. Inspired by the discovery of $T_{cc}$ state, we also studied the mixing angles for B-meson molecules for the possible $T_{bb}$ state that has not been observed yet. Our predictions on the mixing angles show that the violation of isospin symmetry leads to a very small deviation from $45^{\\degree}$, which corresponds to the exact isospin symmetry. The deviation from $45^{\\degree}$ is relatively large  especially for $D_s D^{\\ast}$ and  $D D_s^{\\ast}$ systems.  Hopefully, these findings will be tested in future LHCb experiments as well as flavor factories and provide useful information for understanding the inner structures of the tetraquark systems with two heavy quarks.     \n\\bibliographystyle{utcaps_mod}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}}\n   \n\\newcommand{\\:\\mbox{\\sf Z} \\hspace{-0.82em} \\mbox{\\sf Z}\\,}{\\:\\mbox{\\sf Z} \\hspace{-0.82em} \\mbox{\\sf Z}\\,}\n\\newcommand{\\mbox{\\scriptsize \\sf Z}  \\! \\! \\mbox{\\scriptsize \\sf Z}}{\\mbox{\\scriptsize \\sf Z}  \\! \\! \\mbox{\\scriptsize \\sf Z}}\n\\def\\Bins#1#2{\\left[{#1 \\textstyle \\atop #2}\\right]}\n\\def\\Bin#1#2{\\biggl[{#1 \\atop #2}\\biggr]}\n\\def\\Mults#1#2#3#4{\\left[{#1 \\textstyle \\atop #2}\\right]^{(#3)}_{#4}}\n\\def\\Mult#1#2#3#4{\\biggl[{#1 \\atop #2}\\biggr]^{(#3)}_{#4}}\n\\def\\dMults#1#2#3#4{\\left\\{{#1 \\textstyle \\atop #2}\\right\\}^{(#3)}_{#4}}\n\\def\\dMult#1#2#3#4{\\biggl\\{{#1 \\atop #2}\\biggr\\}^{(#3)}_{#4}}\n\\def\\mults#1#2{\\left({\\textstyle {#1 \\atop #2} } \\right)}\n\\def\\mult#1#2{\\biggl({#1 \\atop #2}\\biggr)}\n\\def\\mbox{e}{\\mbox{e}}\n\\def\\lfloor{\\lfloor}\n\\def\\rfloor{\\rfloor}\n\\def\\mbox{\\scriptsize e}{\\mbox{\\scriptsize e}}\n\\def\\case#1#2{{\\textstyle{#1\\over #2}}}\n\\def\\varepsilon{\\varepsilon}\n\\def\\lambda{\\lambda}\n\\def\\mod#1{\\; (\\bmod \\: #1)}\n\\def\\scriptstyle{\\scriptstyle}\n\n\\begin{document}\n\n\\title{The Andrews-Gordon identities \\\\\nand \\\\\n$q$-multinomial coefficients}\n \n\\author{S.~Ole Warnaar\\thanks{\ne-mail: {\\tt warnaar@maths.mu.oz.au}}\n\\\\\nMathematics Department\\\\\nUniversity of Melbourne\\\\\nParkville, Victoria 3052\\\\\nAustralia}\n  \n\\date{January, 1996 \\\\ \\hspace{1mm}\n \\\\\nPreprint No. 01-96}\n   \n\\maketitle\n   \n\\begin{abstract}\nWe prove polynomial boson-fermion\nidentities for the generating function\nof the number of partitions of $n$ of the form \n$n=\\sum_{j=1}^{L-1} j f_j$,\nwith $f_1\\leq i-1$, $f_{L-1} \\leq i'-1$ and\n$f_j+f_{j+1}\\leq k$.\nThe bosonic side of the identities involves $q$-deformations\nof the coefficients of $x^a$ in the expansion of\n$(1+x+\\cdots+ x^k)^L$.\nA combinatorial interpretation for these $q$-multinomial\ncoefficients is given using Durfee dissection partitions.\nThe fermionic side of the polynomial identities \narises as the partition function of a one-dimensional\nlattice-gas of fermionic particles.\n\nIn the limit $L\\to\\infty$, our identities reproduce the\nanalytic form of Gordon's generalization of the Rogers--Ramanujan\nidentities, as found by Andrews.\nUsing the $q \\to 1\/q$ duality, identities are obtained\nfor branching functions corresponding to cosets of type\n$({\\rm A}^{(1)}_1)_k \\times ({\\rm A}^{(1)}_1)_{\\ell} \/\n({\\rm A}^{(1)}_1)_{k+\\ell}$ of fractional level $\\ell$.\n\\end{abstract}\n     \n\\newpage\n      \n\\nsection{Introduction}\\label{intro}\nThe Rogers--Ramanujan identities can be stated\nas the following $q$-series identities.\n\\begin{theorem}[Rogers--Ramanujan]\nFor $a=0,1$ and $|q|<1$,\n\\begin{equation}\n\\sum_{n = 0}^{\\infty} \\frac{q^{n(n+a)}}{(1-q)(1-q^2)\\ldots (1-q^n)}\n=  \\prod_{j=0}^{\\infty}\n(1-q^{5j+1+a})^{-1}\n(1-q^{5j+4-a})^{-1}.\n\\label{RR}\n\\end{equation}\n\\end{theorem}\nSince their independent discovery\nby Rogers~[1-3], Ramanujan~\\cite{Ramanujan} and \nalso Schur~\\cite{Schur}, many beautiful generalizations have been\nfound, mostly arising from partition-theoretic or Lie-algebraic\nconsiderations, see refs.~\\cite{Andrews76,LP} and references\ntherein. \n\nMost surprising, in 1981 Baxter rediscovered the Rogers--Ramanujan\nidentities (\\ref{RR}) in his calculation \nof the order parameters of the hard-hexagon\nmodel~\\cite{Baxter81}, a lattice gas of hard-core particles\nof interest in statistical mechanics.\nIt took however another ten years to fully realize the power \nof the (solvable) lattice model approach to finding $q$-series identities.\nIn particular, based on a numerical study of the\neigenspectrum of the critical three-state Potts model~\\cite{KM,DKMM}\n(yet another lattice model in statistical mechanics),\nthe Stony Brook group found an amazing variety of new $q$-series identities \nof Rogers--Ramanujan type~\\cite{KKMMa,KKMMb}.\nAlmost none of these identities had been encountered\npreviously in the context of either partition theory\nor the theory of infinite dimensional Lie algebras.\n\nMore specific, in the work of refs.~\\cite{KKMMa,KKMMb}\nexpressions for Virasoro characters were given through\nsystems of fermionic quasi-particles. Equating these\n{\\em fermionic} character forms with the well-known \nRocha-Caridi type {\\em bosonic} expressions~\\cite{RochaCaridi},\nled to many $q$-series identities for Virasoro characters,\ngeneralizing  the Rogers--Ramanujan identities \n(which are associated to the $M(2,5)$ minimal model).\n\n\\vspace*{5mm}\nThe proof of the Rogers--Ramanujan identities by means\nof an extension to polynomial identities whose degree is determined by\na fixed integer $L$, was initiated by Schur~\\cite{Schur}.\nBefore we elaborate on this approach, we need the \ncombinatorial version of the Rogers--Ramanujan identities\nstating that\n\\begin{theorem}[Rogers--Ramanujan]\\label{RRC}\nFor $a=0,1$,\nthe partitions of $n$ into parts congruent to $1+a$ or $4-a$\n$\\mod{5}$ are equinumerous with the partitions of $n$ in\nwhich the difference between any two parts is at least 2 and\n1 occurs at most $1-a$ times.\n\\end{theorem}\nDenoting the number of occurences of the part $j$ in a partition\nby $f_j$, the second type of partitions in the above theorem\nare those partitions\nof $n=\\sum_{\\j \\geq 1} j f_j$\nwhich satisfy the following {\\em frequency conditions}:\n\\begin{equation}\nf_j+f_{j+1} \\leq 1 \\; \\; \\forall j\n\\qquad {\\rm and} \\qquad f_1 \\leq 1-a.\n\\end{equation}\nSchur notes that imposing the additional condition\n$f_j=0$ for $j \\geq L+1$, the generating\nfunction of the ``frequency partitions'' satisfies the\nrecurrence \n\\begin{equation}\ng_L =  g_{L-1} + q^L g_{L-2}.\n\\label{Schurr}\n\\end{equation}\nTogether with the appropriate initial conditions,\nSchur was able to solve these recurrences,\nto obtain an alternating-sign type solution, now called a\nbosonic expression.\nTaking $L\\to\\infty$ in these bosonic polynomials yields\n(after use of Jacobi's triple product identity) the\nright-hand side of (\\ref{RR}). Since this indeed\ncorresponds to the generating function of the ``$\\mod{5}$''\npartitions, this proves theorem~\\ref{RRC}.\nMuch later, Andrews~\\cite{Andrews70} obtained a solution to the\nrecurrence relation as a finite \n$q$-series with manifestly positive integer coefficients, \nnow called a fermionic expression.\nTaking $L\\to\\infty$ in these fermionic polynomials yields\nthe left-hand side of (\\ref{RR}).\n\nRecently much progress has been made in proving\nthe boson-fermion identities of~\\cite{KKMMa,KKMMb}\n(and generalizations thereof), by following the Andrews--Schur\napproach.\nThat is, for many of the Virasoro-character identities, finitizations \nto polynomial boson-fermion identities have been found, which\ncould then be proven either fully recursively (\\`a la Andrews)\nor one side combinatorially and one side \nrecursively (\\`a la Schur), see refs.~[15--30].\n\n\n\\vspace*{5mm}\nIn this paper we consider polynomial identities which imply\nthe Andrews--Gordon generalization of the Rogers--Ramanujan identities.\nFirst, Gordon's theorem~\\cite{Gordon},\nwhich provides a combinatorial generalization\nof the Rogers--Ramanujan identities, reads\n\\begin{theorem}[Gordon]\\label{GI}\nFor all $k\\geq 1$, $1\\leq i\\leq k+1$,\nlet $A_{k,i}(n)$ be the number of partitions of $n$\ninto parts not congruent to $0$ or $\\pm i \\mod{2k+3}$\nand let $B_{k,i}(n)$ be the number of\npartitions of $n$ of the form\n$n = \\sum_{j\\geq 1} j f_j$, with $f_1\\leq i-1$ and\n$f_j+f_{j+1}\\leq k$ (for all $j$). \nThen $A_{k,i}(n)=B_{k,i}(n)$.\n\\end{theorem}\nSubsequently the following analytic counterpart of this result \nwas obtained by Andrews~\\cite{Andrews74}, \ngeneralizing the analytic form (\\ref{RR}) of the\nRogers--Ramanujan identities.\n\\begin{theorem}[Andrews]\nFor all $k\\geq 1$, $1\\leq i\\leq k+1$ and $|q|<1$,\n\\begin{equation}\n\\renewcommand{\\arraystretch}{0.8}\n\\sum_{n_1,n_2,\\ldots,n_{k}\\geq 0}\n\\frac{ q^{N_1^2 + \\cdots + N_k^2 + N_i + \\cdots + N_k}}\n{(q)_{n_1} (q)_{n_2} \\cdots (q)_{n_k} } \n=  \\prod_{\n\\begin{array}{c}\n\\scriptstyle j=1 \\\\\n\\scriptstyle j \\not\\equiv 0,\\pm i \\mod{2k+3}\n\\end{array}}^{\\infty} (1-q^j)^{-1}\n\\label{An}\n\\end{equation}\nwith\n\\begin{equation}\nN_j = n_j + \\cdots + n_{k}\n\\label{Nj}\n\\end{equation}\nand $(q)_a=\\prod_{k=1}^a (1-q^k)$ for $a>0$ and $(q)_0=1$.\n\\end{theorem}\nApplication of Jacobi's triple product identity admits for a rewriting\nof the right-hand side of (\\ref{An}) to\n\\begin{equation}\n\\frac{1}{(q)_{\\infty}}\n\\sum_{j=-\\infty}^{\\infty} (-)^j \nq^{j\\bigl( (2k+3)(j+1)-2i\\bigr)\/2}.\n\\label{Anb}\n\\end{equation}\n\nEquating (\\ref{Anb}) and the left-hand side of (\\ref{An}),\ngives an example of a boson-fermion identity.\nHere we consider, in the spirit of Schur,\na ``natural'' finitization of Gordon's \nfrequency condition such that this boson-fermion identity \nis a limiting case of polynomial identities.\nIn particular, we are interested in the quantity\n$B_{k,i,i';L}(n)$, counting the number of partitions\nof $n$ of the form\n\\begin{equation}\nn = \\sum_{j=1}^{L-1} j f_j\n\\end{equation}\nwith frequency conditions\n\\begin{equation}\nf_1\\leq i-1, \\qquad f_{L-1} \\leq i'-1 \\qquad {\\rm and}\n\\qquad \nf_j+f_{j+1}\\leq k \\quad \\mbox{for $j=1,\\ldots,L-2$}.\n\\end{equation}\nIf we denote the generating function of\npartitions counted by $B_{k,i,i';L}(n)$ by\n$G_{k,i,i';L}(q)$, then clearly\n$\\lim_{L\\to\\infty}\nG_{k,i,i';L}(q)=G_{k,i}(q)$, with\n$G_{k,i}$ the generating function\nassociated with $B_{k,i}(n)$ of theorem~\\ref{GI}.\nAlso note that $G_{k,i,1;L}=G_{k,i,k+1;L-1}$.\nOur main results can be formulated as the following\ntwo theorems for $G_{k,i,i';L}$.\n\nLet $\\Bin{L}{a}$ be the Gaussian polynomial or $q$-binomial coefficient\ndefined by\n\\begin{equation}\n\\Bin{L}{a}= \\Bin{L}{a}_q = \\left\\{\n\\begin{array}{ll}\n\\displaystyle \\frac{(q)_L}\n{(q)_{a}(q)_{L-a}} \\qquad & 0\\leq a \\leq L \\\\[3mm]\n0& \\mbox{otherwise.}\n\\end{array}\n\\right.\n\\label{Gpoly}\n\\end{equation}\nFurther, let ${\\cal I}_k$ be the incidence matrix\nof the Dynkin diagram of A$_k$ with an additional\ntadpole at the $k$-th node: \n\\begin{equation}\n({\\cal I}_k)_{j,\\ell}\n=\\delta_{j,\\ell-1}+\\delta_{j,\\ell+1}+\n\\delta_{j,\\ell} \\delta_{j,k} \\qquad\n\\qquad j,\\ell=1,\\ldots,k,\n\\end{equation}\nand let $C_k$ be the corresponding Cartan-type\nmatrix, $(C_k)_{j,\\ell}=2 \\delta_{j,\\ell}-\n({\\cal I}_k)_{j,\\ell}$.\nFinally let $\\vec{n}$, $\\vec{m}$ and $\\vec{\\mbox{e}}_j$ be\n$k$-dimensional (column)-vectors with entries\n$\\vec{n}_j=n_j$, $\\vec{m}_j=m_j$\nand $(\\vec{\\mbox{e}}_j)_{\\ell}=\n\\delta_{j,\\ell}$. Then\n\\begin{theorem}\\label{ft}\nFor all $k\\geq 1$, $1\\leq i,i'\\leq k+1$ and\n$kL\\geq 2k-i-i'+2$, \n{\\rm \n\\begin{equation}\nG_{k,i,i';L}(q)=\n\\sum_{n_1,n_2,\\ldots,n_k\\geq 0}\nq^{\\displaystyle \\, \\vec{n}^T C^{-1}_k (\\vec{n} \n+\\vec{\\mbox{e}}_k -\\vec{\\mbox{e}}_{i-1})}\n\\prod_{j=1}^k\n\\Bin{n_j+m_j}{n_j},\n\\label{Fermion}\n\\end{equation}}\nwith $(m,n)$-system \\cite{Berkovich} given by\n{\\rm \n\\begin{equation}\n\\vec{m}+\\vec{n}=\n\\case{1}{2}\\Bigl({\\cal I}_k \\, \\vec{m}\n+(L\\!-\\! 2) \\, \\vec{\\mbox{e}}_k + \\vec{\\mbox{e}}_{i-1}\n+\\vec{\\mbox{e}}_{i'-1}\\Bigr).\n\\label{mn}\n\\end{equation}}\n\\end{theorem}\nWe note that $(C^{-1}_k)_{j,\\ell}=\\min(j,\\ell)$\nand hence that, using the variables $N_j$ of \n(\\ref{Nj}), we can rewrite the quadratic\nexponent of $q$ in (\\ref{Fermion}) as\n$N_1^2+ \\cdots + N_k^2+N_i+\\cdots+ N_k$.\nFor $k\\geq 2$, the ``finitization'' (\\ref{Fermion})-(\\ref{mn}) of the\nleft-hand side of (\\ref{An}) is new. \nFor $k=1$ it is the already mentioned\nfermionic solution to the\nrecurrence (\\ref{Schurr}) as found by Andrews~\\cite{Andrews70}. \nAnother finitization,\nwhich does not seem to be related to a finitization\nof Gordon's frequency conditions,\nhas recently been proposed in refs.~[17-19]\n(see also ref.~\\cite{BM94}).\nA more general expression, which\nincludes (\\ref{Fermion})-(\\ref{mn}) and that of [17-19] as special cases,\nwill be discused in section~\\ref{secdis}.\n\nOur second result, which is maybe of more interest\nmathematically since it involves new generalizations of the\nGaussian polynomials, can be stated as follows.\nLet $\\Mults{L}{a}{p}{k}$ be the $q$-multinomial\ncoefficient defined in equation (\\ref{qM}) of the \nsubsequent section. \nAlso, define \n\\begin{equation}\nr=k-i'+1\n\\label{defr}\n\\end{equation}\nand \n\\begin{equation}\ns = \\left\\{\\begin{array}{lcl}\ni \\qquad & \\mbox{for} & i=1,3,\\ldots,2\\lfloor\\frac{k}{2}\\rfloor +1 \n\\\\[1mm]\n2k+3-i & \\mbox{for} & i=2,4,\\ldots,2 \\lfloor\\frac{k+1}{2} \\rfloor,\n\\end{array} \\right.\n\\label{defs}\n\\end{equation}\nso that $r=0,1,\\ldots,k$ and $s=1,3,\\ldots,2k+1$.\nThen\n\\begin{theorem}\\label{bt}\nFor all $k>0$, $1\\leq i,i'\\leq k+1$ and\n$kL\\geq 2k-i-i'+2$, \n\\begin{eqnarray}\nG_{k,i,i';L}(q)&=&\n\\sum_{j=-\\infty}^{\\infty} \\left\\{\nq^{j\\bigl((2j+1)(2k+3)-2s\\bigr)}\n\\Mult{L}{\\case{1}{2}(k L+k-s-r+1)+(2k+3)j}{r}{k} \n\\right.  \\nonumber \\\\\n& & \\hspace{10mm} \\left.\n-q^{\\bigl(2j+1\\bigr)\\bigl((2k+3)j+s\\bigr)}\n\\Mult{L}{\\case{1}{2}(k L+k+s-r+1)+(2k+3)j}{r}{k}\n\\right\\}\n\\label{Boson1}\n\\end{eqnarray}\nfor $r\\equiv k(L+1) \\mod{2}$ and\n\\begin{eqnarray}\nG_{k,i,i';L}(q)&=&\n\\sum_{j=-\\infty}^{\\infty} \\left\\{\nq^{j\\bigl((2j+1)(2k+3)-2s\\bigr)}\n\\Mult{L}{\\case{1}{2}(k L-k+s-r-2)-(2k+3)j}{r}{k} \n\\right.  \\nonumber \\\\\n& & \\hspace{10mm} \\left.\n-q^{\\bigl(2j+1\\bigr)\\bigl((2k+3)j+s\\bigr)}\n\\Mult{L}{\\case{1}{2}(k L-k-s-r-2)-(2k+3)j}{r}{k}\n\\right\\}\n\\label{Boson2}\n\\end{eqnarray}\nfor $r\\not\\equiv k(L+1) \\mod{2}$.\n\\end{theorem}\nFor $k\\geq 3$,\nthe finitizations (\\ref{Boson1}) and (\\ref{Boson2}) of the\nright-hand side of (\\ref{An}) are new.\nFor $k=1$ (\\ref{Boson1}) and (\\ref{Boson2}) are Schur's bosonic polynomials.\nFor $k=2$, $\\Mults{L}{a}{p}{2}$ being a $q$-trinomial coefficient, \n(\\ref{Boson1}) and (\\ref{Boson2}) were (in a slightly different\nrepresentation) first obtained in ref.~\\cite{AB}.\nAn altogether different alternating-sign expression for $G_{k,i,i';L}$\nin terms of $q$-binomials has been found in ref.~\\cite{DJKMO}.\nA different finitization of the right-hand side\nof (\\ref{An}) involving $q$-binomials has been given \nin~\\cite{Andrews70,ABF}.\nA more general expression, which includes (\\ref{Boson1}), (\\ref{Boson2}) \nand that of ref.~\\cite{Andrews70,ABF} as special cases,\nwill be discused in section~\\ref{secdis}.\n\nEquating (\\ref{Fermion}) and (\\ref{Boson1})--(\\ref{Boson2})\nleads to non-trivial polynomial identities, which in the\nlimit $L\\to\\infty$ reduce to Andrews' analytic form of\nGordon's identity. For $k=1$ these are the polynomial identities\nfeaturing in the Andrews--Schur proof\nof the Rogers--Ramanujan identities (\\ref{RR})~\\cite{Andrews70}.\n\n\\vspace*{5mm}\nThe remainder of the paper is organized\nas follows. In the next section we\nintroduce the $q$-multinomial coefficients\nand list some $q$-multinomial identities needed\nfor the proof of theorem~\\ref{bt}.\nThen, in section~\\ref{secC}, a combinatorial interpretation\nof the $q$-multinomials is given using Andrews' Durfee dissection\npartitions. In section~\\ref{pt4} we give a recursive \nproof of theorem~\\ref{bt}\nand in section~\\ref{pt3} we prove theorem~\\ref{ft} combinatorially, \ninterpreting the restricted frequency partitions as configurations of a \none-dimensional lattice-gas of fermionic particles.\nWe conclude this paper with a discussion of our results,\na conjecture generalizing theorems~\\ref{ft} and~\\ref{bt},\nand some new identities for the branching functions of\ncosets of type\n$({\\rm A}^{(1)}_1)_k \\times ({\\rm A}^{(1)}_1)_{\\ell} \/\n({\\rm A}^{(1)}_1)_{k+\\ell}$ with fractional level $\\ell$.\nFinally, proofs of some of the $q$-multinomial identities \nare given in the appendix.\n\n\\nsection{$q$-multinomial coefficients}\\label{secqB}\nBefore introducing the $q$-multinomial coefficients,\nwe first recall some\nfacts about ordinary multinomials.\nFollowing ref.~\\cite{AB}, we define $\\mults{L}{a}_k$ \nfor $a=0,\\ldots,kL$ as\n\\begin{equation}\n(1+x+\\cdots + x^k)^L = \\sum_{a=0}^{k L} \\:\n\\mult{L}{a}_k \\: x^a.\n\\end{equation}\nMultiple use of the binomial theorem yields\n\\begin{equation}\n\\mult{L}{a}_k =\n\\sum_{j_1+ \\cdots + j_k = a}\n\\mult{L}{j_1}\n\\mult{j_1}{j_2}\n\\cdots\n\\mult{j_{k-1}}{j_k},\n\\label{multq1}\n\\end{equation}\nwhere $\\mults{L}{a}=\\mults{L}{a}_1$ is the usual\nbinomial coefficient.\n \nSome readily established properties of $\\mults{L}{a}_k$ are\nthe symmetry relation\n\\begin{equation}\n\\mult{L}{a}_k = \\mult{L}{kL-a}_k\n\\label{sym}\n\\end{equation}\nand the recurrence\n\\begin{equation}\n\\mult{L}{a}_k =\n\\sum_{m=0}^k\n\\mult{L-1}{a-m}_k .\n\\label{rec}\n\\end{equation}\n  \nFor our subsequent working it will be convenient to define $k+1$\ndifferent $q$-deformations of the\nmultinomial coefficient (\\ref{multq1}).\n\\begin{definition}\nFor $p=0,\\ldots,k$ we set\n\\begin{equation}\n\\Mult{L}{a}{p}{k} =\n\\sum_{j_1+ \\cdots + j_k = a}\nq^{\\;\\displaystyle\n\\sum_{\\ell=1}^{k-1} (L-j_{\\ell})j_{\\ell+1}-\n\\sum_{\\ell=k-p}^{k-1} j_{\\ell+1}}\n\\Bin{L}{j_1}\n\\Bin{j_1}{j_2}\n\\cdots\n\\Bin{j_{k-1}}{j_k},\n\\label{qM}\n\\end{equation}\nwith $\\Bins{L}{a}$ the standard $q$-binomial coefficients\nof (\\ref{Gpoly}).\n\\end{definition}\nNote that $\\Mults{L}{a}{p}{k}$ is unequal to zero for \n$a=0,\\ldots,kL$ only. Also note the initial condition\n\\begin{equation}\n\\Mults{0}{a}{p}{k}  = \\delta_{a,0}.\n\\label{Lis0}\n\\end{equation}\n\nIn the following we state a number of $q$-deformations\nto (\\ref{sym}) and (\\ref{rec}).\nAlthough our list is certainly not exhaustive, we\nhave restricted ourselves to those identities\nwhich in our view are simplest, and to those \nneeded for proving theorem~\\ref{bt}.\nMost of these identities are generalizations of known\n$q$-binomial and $q$-trinomial identities which, for example,\ncan be found in refs.~\\cite{AB,Andrews94,BMO,BM95}.\n\nFirst we put some simple symmetry properties\ngeneralizing (\\ref{sym}), in a lemma.\n\\begin{lemma}\\label{lemsym}\nFor $p=0,\\ldots,k$ the following symmetries hold:\n\\begin{equation}\n\\Mult{L}{a}{p}{k} = q^{(k-p)L-a} \n\\Mult{L}{kL-a}{k-p}{k} \n\\qquad {\\mbox and}\n\\qquad\n\\Mult{L}{a}{0}{k} = \n\\Mult{L}{kL-a}{0}{k}.\n\\label{qs}\n\\end{equation}\n\\end{lemma}\nThe proof of this lemma is given in the appendix.\n\nTo our mind the simplest way of $q$-deforming\n(\\ref{rec}) (which was communicated to us by\nA. Schilling) is\n\\begin{proposition}[Fundamental recurrences; Schilling]\\label{pfr}\nFor $p=0,\\ldots,k$, the $q$-multinomials\nsatisfy\n\\begin{equation}\n\\Mult{L}{a}{p}{k}=\n\\sum_{m=0}^{k-p} q^{m(L-1)} \\Mult{L-1}{a-m}{m}{k}+\n\\sum_{m=k-p+1}^k q^{L(k-p)-m} \\Mult{L-1}{a-m}{m}{k}.\n\\label{frec}\n\\end{equation}\n\\end{proposition}\nIn the next section we give a combinatorial proof of this important\nresult for the $p=0$ case.\nAn analytic proof for general $p$ has been given by Schilling\nin ref.~\\cite{Schillingb}.\n\nWe now give some equations, proven in the appendix,\nwhich all reduce to the tautology $1=1$ in the $q\\to 1$ limit.\n\\begin{proposition}\\label{ptaut}\nFor all $p=-1,\\ldots,k-1$, we have\n\\begin{equation}\n\\Mult{L}{a}{p}{k}\n+ q^L \\Mult{L}{kL-a-p-1}{p+1}{k}\n=\n\\Mult{L}{kL-a-p-1}{p}{k}\n+q^L \\Mult{L}{a}{p+1}{k} ,\n\\label{taut}\n\\end{equation}\nwith $\\Mults{L}{a}{-1}{k}=0$.\n\\end{proposition}\nThe power of these ($q$-deformed) tautologies is that they allow \nfor an endless number of different rewritings of the fundamental \nrecurrences.\nIn particular, as shown in the appendix, they allow for the \nnon-trivial transformation of (\\ref{frec}) into\n\\begin{proposition}\\label{pqr2}\nFor all $p=0,\\ldots,k$, we have\n\\begin{eqnarray}\n\\Mult{L}{a}{p}{k} &=&\n\\renewcommand{\\arraystretch}{0.6}\n\\sum_{\\begin{array}{c}\n\\scriptstyle m=0 \\\\\n\\scriptstyle m\\equiv p+k \\mod{2}\n\\end{array} }^{k-p}\nq^{m(L-1)}\n\\Mult{L-1}{a-\\case{1}{2}(m-p+k)}{m}{k}\n\\nonumber \\\\\n&+&\n\\renewcommand{\\arraystretch}{0.6}\n\\sum_{\\begin{array}{c}\n\\scriptstyle m=0 \\\\\n\\scriptstyle m\\not\\equiv p+k \\mod{2}\n\\end{array} }^{k-p-1}\nq^{m(L-1)}\n\\Mult{L-1}{kL-a-\\case{1}{2}(m+p+k+1)}{m}{k}\n\\nonumber \\\\\n&+&\n\\renewcommand{\\arraystretch}{0.6}\n\\sum_{\\begin{array}{c}\n\\scriptstyle m=k-p+2 \\\\\n\\scriptstyle m\\equiv p+k \\mod{2}\n\\end{array} }^k\nq^{\\case{1}{2}\\bigl((2L-1)(k-p)-m\\bigr)}\n\\Mult{L-1}{a-\\case{1}{2}(m-p+k)}{m}{k}\n\\nonumber \\\\\n&+&\n\\renewcommand{\\arraystretch}{0.6}\n\\sum_{\\begin{array}{c}\n\\scriptstyle m=k-p+1 \\\\\n\\scriptstyle m\\not\\equiv p+k \\mod{2}\n\\end{array} }^k\nq^{kL+\\case{1}{2}\\bigl((2L+1)(k-p)-m+1\\bigr)-2a}\n\\Mult{L-1}{kL-a-\\case{1}{2}(m+p-k-1)}{m}{k}.\n\\label{qrec}\n\\end{eqnarray}\n\\end{proposition}\nIt is thanks to these rather unappealing\nrecurrences that we can prove theorem~\\ref{bt}.\n\nBefore concluding this section on the $q$-multinomial\ncoefficients let us make some further remarks.\nFirst, for $k=1$ and $k=2$ we reproduce the well-known\n$q$-binomial and $q$-trinomial coefficients.\nIn particular, \n\\begin{equation}\n\\Mult{L}{a}{0}{1} = \\Bin{L}{a}\n\\end{equation}\nand \n\\begin{equation}\n\\Mult{L}{a}{p}{2} = \\mult{L;L-a-p;q}{L-a}_2 \\qquad\n\\mbox{ for } p=0,1,\n\\label{trin}\n\\end{equation}\nwhere on the right-hand side of (\\ref{trin}) we have\nused the $q$-trinomial notation introduced\nby Andrews and Baxter~\\cite{AB}.\n\nSecond, in \\cite{AB}, several\nrecurrences involving $q$-trinomials with\njust a single superscript $(p)$ are given.\nWe note that such recurrences follow from (\\ref{frec})\nby taking the difference between various values\nof $p$.\nIn particular we have for all $r=0,\\ldots,p$\n\\begin{eqnarray}\n\\Mult{L}{a}{p}{k} = \\Mult{L}{a}{p-r}{k}\n&+& q^{L(k-p)-a} \\sum_{m=0}^{p-r-1} \\left(1-q^{rL}\\right)\nq^{m(L-1)} \\Mult{L-1}{kL-a-m}{m}{k}\n\\nonumber \\\\\n&+& q^{L(k-p)-a} \\sum_{m=p-r}^{p-1} \\left(1-q^{(p-m)L}\\right)\nq^{m(L-1)} \\Mult{L-1}{kL-a-m}{m}{k}.\n\\label{pr}\n\\end{eqnarray}\nThis can be used to eliminate all multinomials\n$\\Mults{..}{..}{m}{k}$ for $m=0,\\ldots,p-1,p+1,\\ldots,k$\nin favour of $\\Mults{..}{..}{p}{k}$. \nThe price to be paid for this is that the resulting\nexpressions tend to get very complicated if $k$ gets large.\n\nA further remark we wish to make is that to \nour knowledge the general $q$-deformed multinomials\nas presented in (\\ref{qM}) are new.\nThe multinomial $\\Mults{L}{a}{0}{k}$ however was already\nsuggested as a ``good'' $q$-multinomial by Andrews in\n\\cite{Andrews94}, where the following \ngenerating function for $q$-multinomials was proposed\nfor all $k>1$:\n\\begin{equation}\np_{k,L}(x) = \\sum_{a=0}^L x^a q^{\\mults{a}{2}} \n\\Bin{L}{a} \\; p_{k-1,a}(xq^L),\n\\end{equation}\nwith $p_{0,L}(x)=1$.\nClearly,\n\\begin{equation}\np_{k,L} = \\sum_{a=0}^{kL} x^a q^{\\mults{a}{2}} \\Mult{L}{a}{0}{k}.\n\\end{equation}\nAlso in the work of Date {\\em et al.} the $\\Mult{L}{a}{0}{k}$\nmakes a brief appearance, see ref.~\\cite{DJKMO} eqn.~(3.29).\n\nThe more general $q$-multinomials of equation (\\ref{qM}) have been\nintroduced independently by Schilling~\\cite{Schillingb}.\n(The notation used in ref.~\\cite{Schillingb} and that of the present\npaper is almost identical apart from the fact that\n$\\Mults{L}{a}{p}{k}$ is replaced by\n$\\Mults{L}{kL\/2-a}{p}{k}$.)\n\n\\nsection{Combinatorics of $q$-multinomial coefficients}\\label{secC}\nIn this section a combinatorial\ninterpretation of the $q$-multinomials coefficients is given\nusing Andrews' Durfee dissections~\\cite{Andrews79}. \nWe then show how the fundamental recurrences\n(\\ref{frec}) with $p=0$ follow as an immediate consequence of this\ninterpretation.\n\nAs a first step it is convenient to change\nvariables from $q$ to $1\/q$.\nUsing the elementary transformation property of the\nGaussian polynomials\n\\begin{equation}\n\\Bin{L}{a}_{1\/q}  = \nq^{-a(L-a)}\\Bin{L}{a}_{q}, \n\\end{equation}\nwe set\n\\begin{definition}\\label{dd}\nFor $p=0,\\ldots,k$\n{\\rm \n\\begin{eqnarray}\n\\dMult{L}{a}{p}{k} &:=&\n\\left. q^{-a L} \\Mult{L}{a}{p}{k}  \\; \\right|_{q\\to 1\/q}\n\\nonumber \\\\\n&=& \n\\sum_{N_1+ \\cdots + N_k = a}\nq^{N_1^2 + \\cdots + N_k^2\n+N_{k-p+1}+ \\cdots + N_k}\n\\Bin{L}{N_1}\n\\Bin{N_1}{N_2}\n\\cdots\n\\Bin{N_{k-1}}{N_k} \n\\label{dqM} \\\\\n&=& \n\\sum_{\\vec{n}^T C^{-1}_k \\vec{\\mbox{\\scriptsize e}}_k=a}\nq^{\\displaystyle \\, \\vec{n}^T C^{-1}_k (\\vec{n} \n+\\vec{\\mbox{e}}_k -\\vec{\\mbox{e}}_{k-p})}\n\\frac{(q)_L}{(q)_{L-\\vec{n}^T C^{-1}_k \\vec{\\mbox{\\scriptsize e}}_1} (q)_{n_1}\n(q)_{n_2} \\ldots (q)_{n_k}}\\; .\n\\end{eqnarray}}\n\\end{definition}\n\n\\subsection{Successive Durfee squares and Durfee dissections}\nAs a short intermezzo, we review some of the ideas introduced by\nAndrews in ref.~\\cite{Andrews79}, needed for our interpretation of\n(\\ref{dqM}).\nThose already familiar with such concepts as ``$(k,a)$-Durfee dissection\nof a partition'' and ``$(k,a)$-admissible partitions'' may wish to skip\nthe following and resume in section~\\ref{next}.\nThroughout the following a partition and its corresponding\nFerrers graph are identified.\n\n\\begin{definition}\nThe Durfee square of a partition is the maximal square of nodes\n(including the upper-leftmost node).\n\\end{definition}\nThe {\\em size} of the Durfee square is the number of rows\nfor which $r_{\\ell}\\geq \\ell$,\nlabelling the rows (=parts) of a partition by $r_1\\geq r_2 \\geq \\ldots$.\nCopying the example from ref.~\\cite{Andrews79}, the Ferrers graph and\nDurfee square of the partition $\\pi_{\\rm ex}=9+7+5+4+4+3+1+1$ is shown \nin Figure~\\ref{DS}(a).\n\nThe portion of a partition of $n$ below its Durfee square defines \na partition of $m<n$. For this ``smaller'' partition one can again draw\nthe Durfee square.\nContinuing this process of drawing squares,\nwe end up with the successive Durfee squares of a partition.\nFor the partition $\\pi_{\\rm ex}$ this is shown in Figure~\\ref{DS}(b).\nIf a partition $\\pi$ has $k$ successive Durfee squares,\nwith $N_{\\ell}$ the size of the $\\ell$-th square,\nthen $\\pi$ has exactly $N_1+\\cdots + N_k$ parts with\n$N_1+\\cdots+N_{\\ell}$ parts $\\geq N_{\\ell}$\nfor all $\\ell=1,\\ldots,k$. \n\nFollowing Andrews we now slightly generalize the previous notions.\n\\begin{definition}[Durfee rectangle]\nThe Durfee rectangle of a partition is the maximal rectangle\nof nodes\nwhose height exceeds its width by precisely one row.\n\\end{definition}\nThe Durfee rectangle of the partition $\\pi_{\\rm ex}$ is shown in\nFigure~\\ref{DS}(c).\nThe {\\em size} of the Durfee rectangle is its width.\n\nOne can now combine the Durfee squares and rectangles to define\n\\begin{definition}[Durfee dissection]\nThe $(k,i)$-Durfee dissection of a partition\nis obtained by drawing $i-1$ successive Durfee\nsquares followed by $k-i+1$ successive Durfee rectangles.\n\\end{definition}\nIn the following it will be convenient to adopt a slightly\nunconventional labelling. In particular,\nwe label the Durfee squares from 1 to $i-1$\nand the rectangles from $i$ to $k$. Correspondingly,\n$N_{\\ell}$ is the size of the Durfee square or rectangle\nlabelled by $\\ell$.\nWe note that in the $(k,i)$-dissection of a partition corresponding to\nDurfee squares and rectangles of respective sizes\n$N_1\\geq N_2 \\geq \\ldots \\geq N_k$, all  \nthe $N_{\\ell}$ beyond some fixed $\\ell'$ may actually be zero.\n\nFinally we come to the most important definition of this section.\n\\begin{definition}[$(k,i)$-admissible]\nLet $N_1\\geq N_2 \\geq \\ldots \\geq N_k$ be\nthe respective sizes of the Durfee squares and rectangles\nin the $(k,i)$-Durfee dissection  of a partition $\\pi$.\nThen $\\pi$  is $(k,i)$-admissible if\n\\begin{itemize}\n\\item\n$\\pi$ has no parts below its last successive Durfee\nrectangle (or square if $i=k+1$.)\n\\item\nFor $\\ell=i,\\ldots,k$,\nthe last row of the Durfee rectangle labelled by $\\ell$ has\n$N_{\\ell}$ nodes.\n\\end{itemize}\n\\end{definition}\nThe first condition is equivalent to stating\nthat the number of parts of $\\pi$ equals\n$N_1+\\cdots + N_k + \\max(\\ell'-i+1,0)$,\nwhere $\\ell'$ labels the number of Durfee squares and rectangles\nof non-zero size; $N_{\\ell}>0$ for $\\ell \\leq \\ell'$ and\n$N_{\\ell}=0$ for $\\ell>\\ell'$.\nThe second condition is equivalent to stating  that the last\nrow of each Durfee rectangle is actually a part of $\\pi$.\n\\begin{figure}[t]\n\\centerline{\\epsffile{DS.ps}}\n\\caption{(a) Durfee square of the partition \n$\\pi_{\\rm ex}=9+7+5+4+4+3+1+1$. (b) The four successive Durfee\nsquares of $\\pi_{\\rm ex}$. (c) The Durfee rectangle of $\\pi_{\\rm ex}$.}\n\\label{DS}\n\\end{figure}\n\n\\subsection{$(k,i;L,a)$-admissible partitions and $q$-multinomial\ncoefficients}\\label{next}\nUsing the previous definitions we are now prepared for the\ncombinatorial interpretation of (\\ref{dqM}).\n\n\\begin{definition}[$(k,i;L,a)$-admissible]\nLet $N_1\\geq N_2 \\geq \\ldots \\geq N_k$ be\nthe respective sizes of the Durfee squares and rectangles\nof a $(k,i)$-admissible partition $\\pi$.\nThen $\\pi$ is said to be\n$(k,i;L,a)$-admissible if \nthe largest part of $\\pi$ is less or equal to $L$ and\n$N_1+\\cdots + N_k=a$.\n\\end{definition}\nFor a $(k,i;L,a)$-admissible partition $\\pi$, the portion $\\pi_{\\ell}$\nof $\\pi$ to the right of the Durfee square or rectangle labelled\nby $\\ell$ (and below the Durfee square or rectangle \nlabelled $N_{\\ell-1}$), \nis a partition with largest part $\\leq N_{\\ell-1}-N_{\\ell}$\n(where $N_0=L$) and number of parts $\\leq N_{\\ell}$.\nRecalling that the Gaussian polynomial (\\ref{Gpoly}) \nis the generating function of partitions with largest part $\\leq L-a$ and\nnumber of parts $\\leq a$~\\cite{Andrews76},\nwe thus find that the generating function of\n$(k,i;L,a)$-admissible partitions is given by\n\\begin{equation}\n\\sum_{N_1+ \\cdots + N_k = a}\nq^{N_1^2} \\Bin{L}{N_1} \\cdots\nq^{N_{i-1}^2} \\Bin{N_{i-2}}{N_{i-1}} \nq^{N_i(N_i+1)} \\Bin{N_{i-1}}{N_i} \\cdots\nq^{N_k(N_k+1)} \\Bin{N_{k-1}}{N_k} = \n\\dMult{L}{a}{k-i+1}{k}.\n\\end{equation}\nDenoting an arbitrary partition \nof $n$ with largest part $\\leq L$ and number of parts $\\leq a$ by a\nrectangle of width $L$ and height $a$, the $(k,i;L,a)$-admissible\npartitions can be represented graphically as shown \nin Figure~\\ref{combin} for the case $k=2$.\n\\begin{figure}[t]\n\\epsfxsize = 13cm\n\\centerline{\\epsffile{combint.ps}}\n\\caption{Graphical representation of \nthe $(2,i;L,a)$-admissible partitions,\ngenerated by $\\dMults{L}{a}{3-i}{2}$.\nThe respective values of $N_1$ and $N_2$ are free to vary,\nonly their sum taken the fixed value $a$.\nNote that the number of parts in the second and third figure\nare actually not fixed, but vary between $a$ and $a-i+3$, depending\non the number of Durfee rectangles of non-zero size.}\n\\label{combin}\n\\end{figure}\n\nEquipped with the above interpretation we return to the\nrecurrence relation (\\ref{frec}) for $p=0$.\nUsing definition~\\ref{dd} to rewrite this in terms of \n$\\dMults{L}{a}{p}{k}$, gives\n\\begin{equation}\n\\dMult{L}{a}{0}{k}=\nq^a \\sum_{m=0}^{k} \\dMult{L-1}{a-m}{m}{k}.\n\\end{equation}\nThis is obviously true if the following combinatorial statements\nhold.\n\\begin{lemma}\\label{s2}\n\\hfill\n\\begin{itemize}\n\\item\nAdding a column of $a$ nodes to the left \nof a $(k,k-m+1;L-1,a-m)$-admissible partition with\n$m\\in \\{0,1,\\ldots,k\\}$, yields\na $(k,k+1;L,a)$-admissible partition.\n\\item\nRemoving the first column (of $a$ nodes) from a \n$(k,k+1;L,a)$-admissible partition yields a\n$(k,k-m+1;L-1,a-m)$-admissible partition \nfor some $m\\in \\{0,1,\\ldots,k\\}$.\n\\end{itemize}\n\\end{lemma}\nTo show the first statement,\nwe note that a partition is $(k,k+1;L,a)$-admissible\nif it has exactly $a$ parts, has largest part $\\leq L$ and has at most\n$k$ successive Durfee squares.\nA $(k,k-m+1;L-1,a-m)$-admissible partition  has at most $a$ parts and\nhas largest part $\\leq L-1$. Hence adding a column of $a$ nodes\nto the left of such a partition, yields a partition $\\pi$\nwhich has $a$ parts and largest part $\\leq L$.\nRemains to show that $\\pi$ has at most\n$k$ successive Durfee squares.\nTo see this first assume that \nthe $(k,k-m+1;L-1,a-m)$-admissible partition only consists of\nDurfee squares and rectangles. That is, we have a partition of\n$N_1^2+\\cdots + N_k^2 + N_{k-m+1} + \\cdots + N_k$, with\n$N_1+ \\cdots + N_k=a-m$.\nAdding a column of $a$ dots trivially yields a partition $\\pi$\nwith $k$ successive Durfee squares with respective sizes\n\\begin{equation}\nN_1 \\geq N_2 \\geq ... \\geq N_{k-m} \\geq N_{k-m+1}+1 \\geq \\ldots\n\\geq N_k+1>0,\n\\end{equation}\nwith $\\pi$ having a column of $N_{\\ell}$  \nnodes to the right of the $\\ell$-th successive Durfee square\nfor each $\\ell \\leq k-m$.\nNow note that we in fact have treated the ``worst'' possible cases.\nAll other $(k,k-m+1;L-1,a-m)$-admissible partitions \ncan be obtained from the ``bare'' ones just treated by adding\npartitions with largest part \n$\\leq N_{\\ell-1}-N_{\\ell}$\n(where $N_0=L$) and number of parts $\\leq N_{\\ell}$\nto the right of the Durfee square or rectangle labelled by $\\ell$ for all\n$\\ell$.\nLet $\\pi$ be such a ``dressed'' partition, obtained from a bare\n$(k,k-m+1;L-1,a-m)$-admissible partition $\\pi_b$, and let the images\nof $\\pi$ and $\\pi_b$ after adding a column of $a$ dots be $\\pi'$ and $\\pi_b'$.\nFurther, let $N_{\\ell}$ and $M_{\\ell}$ be the size of the \n$\\ell$-th successive Durfee\nsquare of $\\pi_b'$ and $\\pi'$, respectively. \nSince $\\pi$ is obtained from $\\pi_b$ by adding additional nodes\nto its rows, we have $M_1+\\cdots M_{\\ell} \\geq N_1+\\cdots+N_{\\ell}$\nfor all $\\ell$. \nFrom the fact that $\\pi_b'$ has at most $k$ successive Durfee squares\nit thus follows that this is also true for $\\pi'$.\n\nTo show the second statement of the lemma, note that \nfrom (\\ref{rec}) we see that the map implied by\nthe first statement is in fact a map onto the set\nof $(k,k+1,L,a)$-admissible partitions.\nSince for $m\\neq m'$,\nthe set of $(k,k-m+1;L-1,a-m)$-admissible partitions is distinct from\nthe set of $(k,k-m'+1;L-1,a-m')$-admissible partitions,\nthe second statement immediately follows.\n\nTo prove (\\ref{frec}) is true for general $p$, we need to establish\n\\begin{equation}\n\\dMult{L}{a}{p}{k}=\nq^a \\sum_{m=0}^{k-p} \\dMult{L-1}{a-m}{m}{k}\n+q^a \\sum_{m=k-p+1}^k q^{L(p-k+m)} \\dMult{L-1}{a-m}{m}{k}.\n\\end{equation}\nUnfortunately, a generalization of lemma~\\ref{s2}\nwhich would imply this more general result has so far eluded us.\n\nBefore concluding our discussion of $q$-multinomial\ncoefficients we note that if the restriction\non $L$ is dropped in the $(k,i;L,a)$-admissible\npartitions, their generating function reduces to\n\\begin{equation}\n\\renewcommand{\\arraystretch}{0.7}\n\\lim_{L\\to\\infty} \\dMult{L}{a}{k-i+1}{k}=\n\\sum_{\n\\begin{array}{c}\n\\scriptstyle N_1+ \\cdots + N_k = a \\\\\n\\scriptstyle n_1,\\ldots,n_k \\geq 0\n\\end{array} }\n\\frac{q^{N_1^2 + \\cdots + N_k^2\n+N_i+ \\cdots + N_k}}\n{(q)_{n_1} (q)_{n_2} \\ldots (q)_{n_k}},\n\\end{equation}\nwhich, up to a factor $(q)_a$, is the representation of\nthe Alder polynomials \\cite{Alder} as found in ref.~\\cite{Andrews74}.\n\n\\nsection{Proof of theorem~6}\\label{pt4}\nWith the results of the previous two sections,\nproving theorem~\\ref{bt} is elementary.\nFirst we define $S_{k,i,i';L}$ as the set of partitions\nof $n$ of the form $n=\\sum_{j=1}^{L-1} j f_j$ satisfying\nthe frequency conditions $f_1\\leq i-1$,\n$f_{L-1} \\leq i'-1$ and\n$f_j+f_{j+1}\\leq k$ for $j=1,\\ldots,L-2$.\nLet $\\pi$ be a partition in $S_{k,i,i';L}$,\nwith $\\ell$ rows of length $L-1$.\nUsing the frequency condition this implies\n$f_{L-2}\\leq k-\\ell$. Hence, by removing the\nfirst $\\ell$ rows, $\\pi$ maps onto a partition\nin $S_{k,i,k-\\ell+1;L-1}$.\nConversely, by adding $\\ell$ rows at the top\nto a partition in $S_{k,i,k-\\ell+1;L-1}$,\nwe obtain a partition in $S_{k,i,i';L}$.\nSince in the above $\\ell$ can take the values\n$\\ell=0,\\ldots,i'-1$, the following recurrences\nhold:\n\\begin{equation}\nG_{k,i,i';L}(q) = \\sum_{\\ell=0}^{i'-1} q^{\\ell(L-1)}\nG_{k,i,k-\\ell+1;L-1}(q) \\qquad {\\rm for} \\quad i'=1,\\ldots,k+1.\n\\label{Grec}\n\\end{equation}\nIn addition to this we have the initial condition\n\\begin{equation}\nG_{k,i,i';2}(q) = \\sum_{\\ell=0}^{\\min(i'-1,i-1)} q^{\\ell}.\n\\end{equation}\nUsing the recurrence relations, it is in fact an easy matter\nto verify that this is consistent with the condition\n\\begin{equation}\nG_{k,i,i';0}(q) = \\delta_{i,i'}.\n\\label{Ginit}\n\\end{equation}\n\nRemains to verify that (\\ref{Boson1}) and (\\ref{Boson2}) satisfy the\nrecurrence (\\ref{Grec}) and initial condition (\\ref{Ginit}).\nSince in these two equations we have used the variables $r$ and $s$\ninstead of $i'$ and $i$, let us first rewrite (\\ref{Grec}) and\n(\\ref{Ginit}).\nSuppressing the $k$, $s$ and $q$ dependence, setting\n$G_{k,i,i';L}(q)=G_L(r)$, we get \n\\begin{equation}\nG_L(r) = \\sum_{\\ell=0}^{k-r} q^{\\ell(L-1)}\nG_{L-1}(\\ell) \\qquad {\\rm for} \\quad r=0,\\ldots,k\n\\label{Grec2}\n\\end{equation}\nand\n\\begin{equation}\nG_0(r) = \\left\\{\\begin{array}{lcl}\n\\delta_{s+r,k+1} \\quad &{\\rm for }&\\;\ns=1,3,\\ldots,2\\lfloor \\frac{k}{2} \\rfloor + 1 \\\\\n\\delta_{s-r,k+2} &{\\rm for } & \\;\ns=2\\lfloor \\frac{k}{2} \\rfloor + 3,\\ldots,2k+1.\n\\end{array} \\right.\n\\label{Ginit2}\n\\end{equation}\n\nTo verify that (\\ref{Boson1}) and (\\ref{Boson2}) satisfy the\ninitial condition (\\ref{Ginit2}), we set $L=0$ and use the\nfact that $r=0,1,\\ldots,k$ and $s=1,3,\\ldots,2k+1$.\nFrom this and equation (\\ref{Lis0})\none immediately sees that the only non-vanishing term in\n(\\ref{Boson1}) is given by $\\Mults{0}{(k-s-r+1)\/2}{r}{k}\n=\\delta_{s+r,k+1}$. Similarly the only non-vanishing term in\n(\\ref{Boson2}) is $\\Mults{0}{(-k+s-r-2)\/2}{r}{k}\n=\\delta_{s-r,k+2}$. \nNow recall that (\\ref{Boson1}) with $L=0$ is \n$G_0(r)$ for $r\\equiv k$.\nFrom the allowed range of $r$ this implies \n$s=1,3,\\ldots,2\\lfloor \\frac{k}{2} \\rfloor + 1$, in\naccordance with the top-line of (\\ref{Ginit2}).\nAlso, since (\\ref{Boson2}) with $L=0$ is\n$G_0(r)$ for $r\\not\\equiv k$, and because of the range of\n$r$, we get \n$s=2\\lfloor \\frac{k}{2} \\rfloor + 3,\\ldots,2k+1$, in accordance with the\nsecond line in (\\ref{Ginit2}).\n\nChecking that (\\ref{Boson1}) and (\\ref{Boson2}) \nsolve the recurrence relation (\\ref{Grec2}) splits into\nseveral cases due to the parity dependence of $G_L(r)$ and\nof the $q$-multinomial recurrences (\\ref{qrec}).\nAll of these cases are completely analogous and we\nrestrict our attention to $k$ and $r$ being even, so\nthat $G_L(r)$ is given by equation (\\ref{Boson1}).\nSubstituting recurrences (\\ref{qrec}), the first and\nsecond sum in (\\ref{qrec}) immediately give\nthe right-hand side of (\\ref{Grec2}).\nConsequently, the other two terms in (\\ref{qrec}) give rise\nto unwanted terms that have to cancel in order for (\\ref{Grec2})\nto be true.\nDividing out the common factor $q^{(2L-1)(k-r)\/2}$  and\nmaking the change of variables $m\\to m-1$ in the last\nsum of (\\ref{qrec}), the unwanted terms read\n\\begin{eqnarray}\n\\lefteqn{\n\\renewcommand{\\arraystretch}{0.6}\n\\sum_{\\begin{array}{c}\n\\scriptstyle m=k-r+2 \\\\\n\\scriptstyle m \\mbox{\\scriptsize even}\n\\end{array} }^k\nq^{-\\frac{1}{2}m}\n\\sum_{j=-\\infty}^{\\infty} \n\\left\\{\nq^{j\\bigl((2j+1)(2k+3)-2s\\bigr)}\n\\Mult{L-1}{\\case{1}{2}(k L-s-m+1)+(2k+3)j}{m}{k} \\right. }\n\\nonumber \\\\[-3mm]\n&&\\hspace{31mm}- \nq^{\\bigl(2j+1\\bigr)\\bigl((2k+3)j+s\\bigr)}\n\\Mult{L-1}{\\case{1}{2}(k L+s-m+1)+(2k+3)j}{m}{k}\n\\nonumber \\\\\n&&\\hspace{31mm}+\nq^{\\bigl(2j-1\\bigr)\\bigl((2k+3)j-s\\bigr)}\n\\Mult{L-1}{\\case{1}{2}(k L+s-m+1)-(2k+3)j}{m-1}{k} \n\\nonumber \\\\\n&&\\hspace{29mm}- \\left.\nq^{j\\bigl((2j-1)(2k+3)+2s\\bigr)}\n\\Mult{L-1}{\\case{1}{2}(k L-s-m+1)-(2k+3)j}{m-1}{k} \\right\\}.\n\\end{eqnarray}\nAfter changing the summation variable $j\\to -j$ in the second and\nfourth term, this becomes\n\\begin{eqnarray}\n\\lefteqn{\n\\renewcommand{\\arraystretch}{0.6}\n\\sum_{\\begin{array}{c}\n\\scriptstyle m=k-r+2 \\\\\n\\scriptstyle m \\mbox{\\scriptsize even}\n\\end{array} }^k\nq^{-\\frac{1}{2}m}\n\\sum_{j=-\\infty}^{\\infty} \nq^{j\\bigl((2j+1)(2k+3)-2s\\bigr)} \\times} \\\\[-2mm]\n\\lefteqn{\\hspace{-5mm}\\left\\{\n\\Mult{L-1}{\\case{1}{2}(k L-s-m+1)+(2k+3)j}{m}{k}\n- q^{s-2(2k+3)j}\n\\Mult{L-1}{\\case{1}{2}(k L+s-m+1)-(2k+3)j}{m}{k} \\right.}\n\\nonumber \\\\\n\\lefteqn{ \\hspace{-5mm} \\left.\n+ q^{s-2(2k+3)j}\n\\Mult{L-1}{\\case{1}{2}(k L+s-m+1)-(2k+3)j}{m-1}{k} \n-\\Mult{L-1}{\\case{1}{2}(k L-s-m+1)+(2k+3)j}{m-1}{k} \\right\\}.}\n\\nonumber \n\\end{eqnarray}\nWe now show that the term within the curly braces vanishes\nfor all $m$ and $j$.\nTo establish this, we apply the symmetry (\\ref{qs}) to all\nfour $q$-multinomials within the braces and divide by \n$q^{(k-m)(L-1)-\\frac{1}{2}(kL-s-m+1)-(2k+3)j}$.\nAfter replacing $L$ by $L+1$ and $m$ by $k-p$, this gives\n\\begin{eqnarray}\n\\lefteqn{\n\\Mult{L}{\\case{1}{2}(k L+s-p-1)-(2k+3)j}{p}{k}\n-\\Mult{L}{\\case{1}{2}(k L-s-p-1)+(2k+3)j}{p}{k}}\n\\nonumber \\\\\n\\lefteqn{+ q^L\n\\Mult{L}{\\case{1}{2}(k L-s-p-1)+(2k+3)j}{p+1}{k} \n-q^L\n\\Mult{L}{\\case{1}{2}(k L+s-p-1)-(2k+3)j}{p+1}{k} .}\n\\end{eqnarray}\nRecalling the tautology (\\ref{taut}) with $a=\\frac{1}{2}(kL+s-p-1)\n-(2k+3)j$ this indeed gives zero.\n\n\n\\nsection{Proof of theorem~5}\\label{pt3}\n\\subsection{From partitions to paths}\nTo prove expression (\\ref{Fermion}) of theorem~\\ref{ft}, \nwe reformulate the problem of calculating the generating\nfunction $G_{k,i,i';L}(q)$ into a lattice path\nproblem. Hereto we represent each partition $\\pi$\nin $S_{k,i,i';L}$ as a restricted lattice path $p(\\pi)$,\nsimilar in spirit to the lattice path formulation of\nthe left-hand side of (\\ref{An}) by \nBressoud~\\cite{Bressoud}.\\footnote{Finitizing \nBressoud's lattice paths by fixing\nthe length of his paths to $L$,\nresults in the left-hand side\nof (\\ref{fq}) of the next section. \nHence the lattice paths introduced here are intrinsically\ndifferent from those of ref.~\\cite{Bressoud} and in fact\ncorrespond to a finitization of the paths of ref.~\\cite{RV}.}\n\nTo map a partition $\\pi$ of $n=\\sum_{j=1}^{L-1} j f_j$\nonto a lattice path $p(\\pi)$, draw a horizontal \nline-segment in the $(x,y)$-plane\nfrom $(j-\\frac{1}{2},f_j)$ to $(j+\\frac{1}{2},f_j)$ \nfor each $j=1,\\ldots,L-1$. \nAlso draw vertical line-segments  from $(j+\\frac{1}{2},f_j)$ to\n$(j+\\frac{1}{2},f_{j+1})$ for all $j=0,\\ldots,L-1$, \nwhere $f_0=f_{L-1}=0$.\nAs a result $\\pi$ is represented by a lattice path (or\nhistogram) from $(\\case{1}{2},0)$ to $(L-\\case{1}{2},0)$.\nThe frequency condition $f_j+f_{j+1}\\leq k$\ntranslates into the condition that the sum of the {\\em heights}\nof a path at $x$-positions $j$ and $j+1$ does not exceed $k$.\nThe restrictions $f_1 \\leq i-1$ and $f_{L-1} \\leq i'-1$\ncorrespond to the restrictions that the heights at $x=1$ and $x=L-1$\nare less than $i$ and $i'$, respectively.\nAn example of a lattice path for $k\\geq 8$, $i\\geq 3$ and\n$i'\\geq 1$, is shown in Figure~\\ref{latpath}.\n\\begin{figure}[t]\n\\epsfxsize = 13cm\n\\centerline{\\epsffile{histogram.ps}}\n\\caption{A lattice path of the partition $(f_1,\\ldots,f_{L-1})=\n(2,4,3,3,5,3,2,4,1,0,1,3,0,0,7,0,$\n$1,1,2,4,3,3,0,1,2,1,1,3,4,4,2,2,0,0)$. \nThe shaded regions correspond to the two\nparticles with largest charge (=8), as described below.}\n\\label{latpath}\n\\end{figure}\n\nThe above map clearly is reversible, and any lattice path\nsatisfying the above height conditions maps onto a partition\nin $S_{k,i,i';L}$. From now on we let $P_{k,i,i';L}$ denote\nthe set of restricted lattice paths corresponding to \nthe set of partions $S_{k,i,i';L}$.\n\nFrom the map of partitions onto paths, the\nproblem of calculating the generating function \n$G_{k,i,i';L}(q)$ can be reformulated as\n\\begin{equation}\nG_{k,i,i';L}(q) = \n\\sum_{\\textstyle p\\!\\in\\!\\! P_{k,i,i';L}} W(p) \n\\label{Gpath}\n\\end{equation}\nwith Boltzmann weight $W(p)=\\prod_{j=1}^{L-1} q^{j f_j}$.\n\nBefore we actually compute the above sum, we remark \nthat in the following $k,i$ and $i'$ will always be fixed.\nHence, to simplify notation, we use $G_L$ and $P_L$ to denote\n$G_{k,i,i';L}$ and $P_{k,i,i';L}$, respectively.\n\n\n\\subsection{Fermi-gas partition function; $i=i'=k+1$}\nTo perform the sum (\\ref{Gpath}) over the\nrestricted lattice path, we follow a procedure similar to\nthe one employed in our proof of Virasoro-character identities\nfor the unitary minimal models~\\cite{Wa,Wb}.\nThat is, the sum (\\ref{Gpath}) is interpreted as the\ngrand-canonical partition function of a one-dimensional\nlattice-gas of fermionic particles.\n\nThe idea of this approach is to view each lattice path\nas a configuration of particles on a one-dimensional lattice.\nSince not all lattice paths correspond to the same\n{\\em particle content} $\\vec{n}$, this gives rise to a natural\ndecomposition of (\\ref{Gpath}) into\n\\begin{equation}\nG_L(q)  = \\sum_{\\vec{n}} Z_L(\\vec{n};q),\n\\label{GZ}\n\\end{equation}\nwith $Z_L$ the canonical partition function,\n\\begin{equation}\nZ_L(\\vec{n};q) = \n\\sum_{\\textstyle p\\!\\in\\!\\! P_L(\\vec{n})} W(p).\n\\end{equation}\nHere $P_L(\\vec{n})\\subset P_L$ is the set of paths corresponding to a\nparticle configuration with content $\\vec{n}$.\nTo avoid making the following description of the lattice gas unnecessarily\ncomplicated, we assume $i=i'=k+1$ in the remainder of this section.\nSubsequently we will briefly indicate how to modify the calculations\nto give results for general $i$ and $i'$.\n\nTo describe how to interpret each path in $P_L=P_{k,k+1,k+1;L}$\nas a particle configuration,\nwe first introduce a special kind of paths from which all \nother paths can be constructed.\n\\begin{definition}[minimal paths]\nThe path shown in Figure~\\ref{min} is called \nthe minimal path of content $\\vec{n}$.\n\\end{definition}\n\\begin{definition}[charged particle]\nIn a minimal path, each column with non-zero height $t$ corresponds to a \nparticle of charge $t$.\n\\end{definition}\n\\begin{figure}[t]\n\\epsfxsize = 15cm\n\\centerline{\\epsffile{min.ps}}\n\\caption{The minimal path of content $\\vec{n}=(n_1,\\ldots,n_k)^T$.}\n\\label{min}\n\\end{figure}\nNote that in the minimal path the particles are ordered according to\ntheir charge and that adjacent particles are separated by\na single empty column.\nThe number of particles of charge $t$ is denoted $n_t$ and $\\vec{n}=\n(n_1,\\ldots,n_k)^T$.\nFor later use it will be convenient to give each particle a label,\n$p_{t,\\ell}$ denoting the $\\ell$-th particle of charge $t$,\ncounted from the right.\n\nSince the length of a path is fixed by $L$,\nthere are only a finite number of minimal paths.\nIn particular, we have $2(n_1+\\cdots + n_k) \\leq L$, so that there\nare $\n\\mults{\\lfloor L\/2 \\rfloor +k}{k}\n$\ndifferent minimal paths.\n\nIn the following we show that all non-minimal paths in $P_L$ can \nbe constructed out of one (and only one) minimal path using a set \nof elementary moves.\nHereto we first describe how various local configurations may be \nchanged by moving a particle.\\footnote{In moving a particle we always\nmean motion from left to right.}\nTo suit the eye, the particle being moved in each example has been shaded.\n\nTo describe the moves we first consider the simplest type of\nmotion, when the two columns immediately to the right of a particle \nare empty.\n\\begin{definition}[free motion]\nThe following sequence of moves is called free motion:\n\\[ \\hspace{3cm} \\epsfxsize = 9cm\n\\epsffile{free.ps} \\]\n\\end{definition}\nClearly, a particle of charge $t$ in free motion takes \n$t$ moves to fully shift position by one unit.\n\nNow assume that in moving a particle of charge $t$,\nwe at some stage encounter the local configuration shown in \nFigure~\\ref{ts}(a). We then allow the particle to make $t-s$ more\nmoves following the rules of free motion, to end up with\nthe local configuration shown in (b).\nIf instead of (a) we encounter the configurations (c) or (d),\nthe particle can make no further moves.\n\\begin{figure}[h]\n\\epsfxsize = 12cm\n\\centerline{\\epsffile{moves1.ps}}\n\\caption{}\n\\label{ts}\n\\end{figure}\n\nIn case of the configuration of Figure~\\ref{ts}(b), there are\nthree possibilities. Either we have one of \nthe configurations shown in Figure~\\ref{ts2}(a) and (b),\nin which case the particle cannot move any further,\nor we have the configuration shown in Figure~\\ref{ts2}(c) \n(with $0\\leq u < t-s$), \nin which case we can make $t-u-s$ moves, going from\n(c) to (e).\n\\begin{figure}[h]\n\\epsfxsize = 15cm\n\\centerline{\\epsffile{moves2.ps}}\n\\caption{}\n\\label{ts2}\n\\end{figure}\nIgnoring the for our rules irrelevant column \nimmediately to the left of the\nparticle, the configuration of Figure~\\ref{ts2}(e)\nis essentially the same as that of Figure~\\ref{ts}(b).\nTo further move the particle we can thus refer to the rules\ngiven in Figure~\\ref{ts2}. That is, if the column immediately\nto the right of the white column of height $u$ has height $t-u$ \n(corresponding to the configuration of Figure~\\ref{ts2}(a) with $s\\to u$)\nthe particle cannot move any further.\nSimilarly, if the column immediately\nto the right of the white column of height $u$ has height $v>t-u$ \n(corresponding to the configuration of \nFigure~\\ref{ts2}(b) with $u\\to v$ and $s\\to u$)\nthe particle cannot move any further.\nHowever, if the height of the column immediately to the right\nof the white column of height $u$ is $0\\leq v<t-u$\n(corresponding to the configuration of Figure~\\ref{ts2}(c) \nwith $u\\to v$ and $s\\to u$) we can make another $t-u-v$ moves.\n\nHaving introduced all necessary moves we come to the main propositions\nof this section\n\\begin{proposition}[rules of motion]\\label{prm}\nEach non-minimal configuration can be obtained from one and only one\nminimal configuration by letting the particles carry out elementary moves\nin the following order:\n\\begin{itemize}\n\\item\nParticle $p_{t,\\ell}$ moves prior to $p_{s,\\ell'}$ if $t<s$.\n\\item\nParticle $p_{t,\\ell'}$ moves prior to $p_{t,\\ell}$ if $\\ell'<\\ell$.\n\\end{itemize}\n\\end{proposition}\nTo prove this let us assume that we have completed the motion\nof all particles of charge less than $t$ and all particles $p_{t,\\ell'}$ \nwith $\\ell'<\\ell$, and that we are\ncurrently moving the particle $p_{t,\\ell}$.\nTherefore, for $x\\leq 2n_k+\\cdots + 2n_{t+1} + 2(n_t-\\ell):=x_{\\min}$,\nthe lattice path still corresponds to the minimal path.\n\nNow note that in moving the particle $p_{t,\\ell}$,\nwe never create a local configuration in which the sum of the height\nof two consecutive columns is greater than $t$, see\nthe free motion and the Figures~\\ref{ts}(a)(b) and \\ref{ts2}(c)--(e).\nSince we have not moved any of the particles of charge greater than\n$t$, this means that to the right of $x_{min}$ no two\nconsecutive columns have summed heights greater than $t$.\\footnote{\nThis also means that in moving a particle of charge\n$t$, the configurations of Figure~\\ref{ts}(d) and Figure~\\ref{ts2}(b)\nin fact never arise.}\nAlso note that as soon as $p_{t,\\ell}$ meets two columns immediately\nto its right whose summed heights equal $t$, $p_{t,\\ell}$ cannot\nmove any further, see Figures~\\ref{ts}(c) and ~\\ref{ts2}(a).\nConsequently, $p_{t,\\ell}$ always corresponds to the\nleftmost two consecutive columns right of $x_{\\min}$\nwhose summed height equals $t$. (In fact, it corresponds to\nthe leftmost two consecutive columns right of $x_{min}$ with maximal\nsummed heights.)\n\nNow we define reversed moves by reading all the previous \nfigures in a mirror. Using this motion we can move\n$p_{t,\\ell}$ all the way back to its minimal position but\nnot any further.\nTo see this we note that the only situations in which $p_{t,\\ell}$ \ncannot be moved further back is if it meets two \nconsecutive columns to its left whose summed heights\nare greater or equal to $t$. Since we have just argued that such\na configuration cannot occur between $x_{\\min}$ and $p_{t,\\ell}$\nwe can indeed move $p_{t,\\ell}$ back to its minimal position using\nthe reversed moves. Once it is back in its minimal position\nwe either have the mirror image of Figure~\\ref{ts}(c) (in case $\\ell<n_t$)\nor (d) (in case $\\ell=n_t$). Neither of these configurations allows\nfor further reversed moves. \n\nThe above, however, gives a general procedure for reducing each non-minimal\npath to a minimal one by simply reversing the rules of motion\nin the proposition.\nThat is, we first scan the path for all particles of charge $k$,\nby locating all occurrence of two consecutive columns of\nsummed heights $k$. From left to right these label the particles $p_{k,n_k}$\nto $p_{k,1}$. Applying the previous reasoning with $t=k$, we can\nfirst move $p_{k,n_k}$ back using reversed moves, than  \n$p_{k,n_k-1}$, et cetera, until  all particles of charge $k$\nhave taken their ``minimal position''.\nRepeating this for the particles of charge $k-1$, then\nthe particles of charge $k-2$, et cetera, each non-minimal\npath reduces to a unique minimal path.\n\nAs an example to the above,\nfor the path of Figure~\\ref{latpath} the shaded regions\nmark the (two) particles with largest charge (=8). Moving them\nback using the reversed motion, the leftmost particle being\nmoved first, we end up with the path shown in Figure~\\ref{redpath},\nin which now the (three) particles with next-largest charge \nhave been marked. We leave it to the reader to further reduce\nthe path to obtain the minimal path of content $(n_1,n_2,\\ldots,n_8)=\n(2,1,2,1,3,1,3,2)$.\n\\begin{figure}[t]\n\\epsfxsize = 13cm\n\\centerline{\\epsffile{redhis.ps}}\n\\caption{The lattice path obtained from Figure~1 after\nmoving the largest particles to their minimal position.\nThe shaded regions mark the three next-largest particles.}\n\\label{redpath}\n\\end{figure}\n\nThe elementary moves and the reversed moves are clearly reversible.\nIf a particle of charge $t$ has made an elementary move changing a path\nfrom $p$ to $p'$, we can always carry a reversed move going \nfrom $p'$ back to $p$.\nSince each path can be reduced to a unique minimal path using the\nreversed moves by carrying out the rules of motion of proposition~\\ref{prm}\nin reversed order, we have thus established that using the rules of motion\nwe can generate each non-minimal path uniquely from a minimal path.\nHence the proposition is proven.\n\nWe now have established the decomposition of the sum (\\ref{Gpath}) into\n(\\ref{GZ}), where $Z_L(\\vec{n};q)$ is the generating function of the\npaths generated by the minimal path labelled by $\\vec{n}$,\nor, in other words, $Z_L(\\vec{n};q)$ is the partition function of\na lattice gas of fermions with particle content $\\vec{n}$.\nThe fermionic nature being that, unlike particles of different charge,\nparticles of equal charge cannot exchange position.\n\nOur next result concerns the actual computation of  the\npartition function.\n\\begin{proposition}\\label{pZ}\nThe partition function $Z_L$ is given by\n\\begin{equation}\nZ_L(\\vec{n};q) = \nq^{\\displaystyle \\, \\vec{n}^T C^{-1}_k \\vec{n}} \n\\prod_{t=1}^k\n\\Bin{n_t+m_t}{n_t},\n\\label{ZLn}\n\\end{equation}\nwith {\\rm $\n\\vec{m}+\\vec{n}=\n\\case{1}{2}({\\cal I}_k \\, \\vec{m}\n+L \\, \\vec{\\mbox{e}}_k)$}.\n\\end{proposition}\nTo prove this we first determine\nthe contribution to $Z_L$ of the minimal path of content $\\vec{n}$,\n\\begin{eqnarray}\nW(p_{\\min})\/\\ln q  &=& \n\\sum_{t=1}^k t \\sum_{\\ell=1}^{n_t} \n\\Bigl(2\\ell-1 + 2 \\sum_{s=t+1}^k n_s\\Bigr)\n=\n\\sum_{t=1}^k t n_t\n\\Bigl(n_t + 2 \\sum_{s=t+1}^k n_s\\Bigr) \\nonumber \\\\\n&=& \n\\sum_{r=1}^k \\sum_{t=r}^k n_t \n\\Bigl(n_t + 2 \\sum_{s=t+1}^k n_s\\Bigr) \n= \n\\sum_{r=1}^k N_r^2 = \n\\vec{n}^T C^{-1}_k \\vec{n} .\n\\label{Wmin}\n\\end{eqnarray}\n\nTo obtain the contribution to $Z_L$ of the non-minimal\nconfigurations, we apply the rules of motion of proposition~\\ref{prm}.\nIf $e_{\\ell}$ denotes the number of elementary moves carried out by\n$p_{t,\\ell}$, the generating function of moving the particles\nof charge $t$ reads\n\\begin{equation}\n\\sum_{e_1=0}^{m_t} \n\\sum_{e_2=0}^{e_1}\n\\ldots\n\\sum_{e_{n_t}=0}^{e_{n_t-1}}\nq^{e_1+e_2 + \\cdots + e_{n_t}} = \\Bin{m_t + n_t}{n_t}.\n\\label{respart}\n\\end{equation}\nHere we have used the fact that each elementary move generates a factor $q$\nand that $p_{t,\\ell}$ cannot carry out more elementary moves than\n$p_{t,\\ell-1}$. (If $p_{t,\\ell}$ has made as many moves as $p_{t,\\ell-1}$\nwe obtain either the local configuration of Figure~\\ref{ts}(c) or\nFigure~\\ref{ts2}{a}, prohibiting any further moves.)\nThe number $m_t$ in (\\ref{respart}) is the maximal number of\nelementary moves $p_{t,1}$ can make and remains to be determined.\nIf the content of the minimal path is $\\vec{n}$,\nthe $x$-position of $p_{t,1}$ in $p_{\\min}$ is $2(n_k+\\cdots + n_t)-1:=x_0$.\nTo fix $m_t$, let us assume that after the motion of the particles \nof charge less than $t$ has been completed, the nontrivial part of the \nlattice path is encoded by the sequence of heights\n$(f_{x_0},\\ldots,f_{L-1})$, with $f_{x_0}=t$, $f_{x_0+1}=0$ and\n$f_j+f_{j+1}<t$ for $j>x_0$.\nThe particle $p_{t,1}$ can now move all the way to $x=L-1$ \nmaking\n\\begin{eqnarray}\nm_t &=&\n(t-f_{x_0+2}) +\n(t-f_{x_0+2}-f_{x_0+3}) +\n(t-f_{x_0+3}-f_{x_0+4}) \\nonumber \\\\ \n& & \\qquad \\qquad + \\cdots\n+ (t-f_{L-2}-f_{L-1}) + (t-f_{L-1}) \\nonumber \\\\\n&=& t(L-x_0-1) -2\\sum_{j=x_0+2}^{L-1} f_j\n\\label{mt}\n\\end{eqnarray}\nelementary moves.\nTo simplify this, note that the sum on the right-hand side is\nnothing but twice the sum of the heights of the columns\nright of $x=x_0$, which is $2 \\sum_{s=1}^{t-1} s n_s$.\nSubstituting this in (\\ref{mt}) and using the definition of\n$x_0$, results in \n\\begin{equation}\nm_t  = t L - 2\\sum_{s=1}^{t-1} s n_s - 2 t \\sum_{s=t}^k n_t\n= t L -2\\sum_{s=1}^k \\min(s,t) n_s\n= L(C^{-1}_k)_{t,k}\n-2\\sum_{s=1}^k (C^{-1}_k)_{t,s} n_s\n\\label{mt2}\n\\end{equation}\nin accordance with proposition~\\ref{pZ}.\n\nPutting together the results (\\ref{Wmin}), (\\ref{respart})\nand (\\ref{mt2}) completes the proof of proposition~\\ref{pZ}.\nSubstituting the form (\\ref{ZLn}) of the partition function \ninto (\\ref{GZ}) proves expression\n(\\ref{Fermion}) of theorem~\\ref{ft}, for $i=i'=k+1$.\n\n\n\\subsection{Fermi-gas partition function; general $i$ and $i'$}\nModifying the proof of theorem~\\ref{ft} for $i=i'=k+1$ to all\n$i$ and $i'$ is straightforward and few details will be given.\nIt is in fact interesting to note that unlike our proof for the\ncharacter identities of the unitary minimal models~\\cite{Wa,Wb}, \nthe general case here does not require the introduction of \nadditional ``boundary particles''. \n\nFirst let us consider the general $i'$ case, with $i=k+1$.\nThis implies that the height $f_{L-1}$ of the last column\nof the lattice paths is no longer free to take any of the values\n$1,\\ldots,k$, but is bound by $f_{L-1}\\leq i'-1$.\nFor the particles of charge less or equal to $i'-1$ this does\nnot impose any new restrictions on the maximal number of moves\n$p_{t,1}$ can make. For $t>i'-1$ however, $m_t$ in (\\ref{mt2})\nhas to be decreased by $t-i'+1$.\nThus we find that $m_t$ of (\\ref{mt2}) has to be replaced by\n$m_t - \\max(0,t-i'+1)= m_t -t + \\min(t,i'-1)$.\nRecalling $(C^{-1}_k)_{s,t}=\\min(s,t)$, this yields\n\\begin{equation}\nm_t = (L-1)(C^{-1}_k)_{t,k} \n+ (C^{-1}_k)_{t,i'-1} \n-2\\sum_{s=1}^k (C^{-1}_k)_{t,s} n_s\n\\end{equation}\nand therefore,  $m_t+n_t = \n\\case{1}{2}(\\sum_{s=1}^k ({\\cal I}_k)_{t,s}  m_s\n+(L-1) \\delta_{t,k} +\\delta_{t,i'-1})$\nwhich is in accordance with proposition~\\ref{pZ}, for $i=k+1$.\n\nSecond, consider $i$ general, but $i'=k+1$, so that\n$f_1\\leq i-1$, $f_{L-1}\\leq k$.\nNow the modification is slightly more involved\nsince the actual minimal paths change from those of Figure~\\ref{min}\nto those of Figure~\\ref{min2}.\nThis leads to a change in the calculation of $W(p_{\\min})$ to\n\\begin{eqnarray}\nW(p_{\\min})\/\\ln q  &=& \n\\sum_{t=1}^k \\sum_{\\ell=1}^{n_t} \n\\Bigl(2\\ell t - \\min(t,i-1) + 2t \\sum_{s=t+1}^k n_s\\Bigr)\n\\nonumber \\\\\n&=&\n\\sum_{t=1}^k t n_t\n\\Bigl(n_t + 2 \\sum_{s=t+1}^k n_s\\Bigr)\n+\\sum_{t=i}^k (t-i+1) n_t \\nonumber \\\\\n&=& \n\\vec{n}^T C^{-1}_k \\vec{n} + \\sum_{t=1}^k t n_t -\n\\sum_{t=1}^k \\min(t,i-1) n_t \\; = \\; \n\\vec{n}^T C^{-1}_k (\\vec{n} + \\mbox{e}_k -\\mbox{e}_{i-1}),\n\\end{eqnarray}\nwhich is indeed the general form of the quadratic exponent\nof $q$ in (\\ref{Fermion}).\nAlso $m_t$ again requires modification, which is in fact\nsimilar to the previous case:\n$m_t \\to m_t - \\max(0,t-i+1)$.\nTo see this note that it takes $\\max(0,t-i+1)$\nelementary moves to move $p_{t,1}$ from its\nminimal position in Figure~\\ref{min} to its minimal\nposition in Figure~\\ref{min2}.\n\\begin{figure}[t]\n\\epsfxsize = 15cm\n\\centerline{\\epsffile{min2.ps}}\n\\caption{The minimal path of content $\\vec{n}=(n_1,\\ldots,n_k)^T$\nfor general $i$. The dashed lines are drawn to mark the\ndifferent particles.}\n\\label{min2}\n\\end{figure}\n\nFinally, combining the previous two cases, and using the fact\nthat the modifications of $m_t$ due to $f_1\\leq i-1$ and \n$f_{L-1}\\leq i'-1$ \nare independent, we immediately arrive at the general form of\n(\\ref{Fermion}) with $(m,n)$-system (\\ref{mn}).\n\n\\nsection{Discussion}\\label{secdis}\nIn this paper we have presented polynomial identities \nwhich arise from finitizing Gordon's frequency partitions.\nThe bosonic side of the identities involves \n$q$-deformations of the coefficient of $x^a$ in the\nexpansion of $(1+x+\\cdots +x^k)^L$.\nThe fermionic side follows from interpreting\nthe generating function of the frequency partitions,\nas the grand-canonical partition function of a one-dimensional\nlattice gas.\n\nInterestingly, in recent publications, Foda and Quano, and Kirillov,\nhave given different polynomial identities which imply \n(\\ref{An})~[17-19].\nIn the notation of section~\\ref{intro} these identities \ncan be expressed as\n\\begin{theorem}[Foda--Quano, Kirillov]\nFor all $k\\geq 1$, $1\\leq i\\leq k+1$ and $L\\geq k-i+1$,\n\\begin{eqnarray}\n\\lefteqn{\n\\sum_{n_1,\\ldots,n_k\\geq 0}\nq^{N_1^2 + \\cdots + N_k^2 + N_i + \\cdots + N_k}\n\\prod_{j=1}^{k}\n\\Bin{L-N_j-N_{j+1}-2\\sum_{\\ell=1}^{j-1} N_{\\ell}\n-\\alpha_{i,j}}{n_j} } \\nonumber \\\\\n&&\\qquad = \n\\sum_{j=-\\infty}^{\\infty} (-)^j \nq^{j\\bigl( (2k+3)(j+1)-2i\\bigr)\/2}\n\\Bin{L}{\\lfloor \\frac{L-k+i-1-(2k+3)j}{2} \\rfloor},\n\\label{fq}\n\\end{eqnarray}\nwith $N_{k+1}=0$ and $\\alpha_{i,j}=\\max(0,j-i+1)$.\n\\end{theorem}\n\nAn explanation for this different finitization of (\\ref{An})\ncan be found in a theorem due to Andrews~\\cite{Andrews72}:\n\\begin{theorem}[Andrews]\nLet $Q_{k,i}(n)$ be the number of partitions of $n$\nwhose successive ranks lie in the interval\n$[2-i,2k-i+1]$.\nThen $Q_{k,i}(n)=A_{k,i}(n)$.\n\\end{theorem}\nIt turns out that it is the (natural) finitization of these\nsuccessive rank partitions which gives rise to the\nabove alternative polynomial finitization. That is,\n(\\ref{fq}) is an identity for the generating function\nof partitions with largest part $\\leq \n\\lfloor (L+k-i+2)\/2 \\rfloor$, number of parts\n$\\leq \\lfloor (L-k+i-1)\/2 \\rfloor$, whose successive ranks \nlie in the interval $[2-i,2k-i+1]$.\n\nLet us now reexpress (\\ref{fq}) into a form similar\nto equations (\\ref{Fermion}), (\\ref{Boson1}) and (\\ref{Boson2}). \nHereto we eliminate $i$ in the right-hand side of\n(\\ref{fq}) in favour of the variable $s$ of equation\n(\\ref{defs}) and split the result into two cases.\nThis gives\n\\begin{eqnarray}\n\\lefteqn{\\mbox{RHS}(\\ref{fq})=\n\\sum_{j=-\\infty}^{\\infty} \\left\\{\nq^{j\\bigl((2j+1)(2k+3)-2s\\bigr)}\n\\Mult{L}{\\case{1}{2}(L+k-s+1)+(2k+3)j}{0}{1}\n\\right. } \\nonumber \\\\\n& &  \\qquad \\qquad \\qquad \\left.\n-q^{\\bigl(2j+1\\bigr)\\bigl((2k+3)j+s\\bigr)}\n\\Mult{L}{\\case{1}{2}(L+k+s+1)+(2k+3)j}{0}{1}\n\\right\\}\n\\end{eqnarray}\nfor $L+k$ even, and\n\\begin{eqnarray}\n\\lefteqn{\\mbox{RHS}(\\ref{fq})=\n\\sum_{j=-\\infty}^{\\infty} \\left\\{\nq^{j\\bigl((2j+1)(2k+3)-2s\\bigr)}\n\\Mult{L}{\\case{1}{2}(L-k+s-2)-(2k+3)j}{0}{1}\n\\right. } \\nonumber \\\\\n& &  \\qquad \\qquad \\qquad \\left.\n-q^{\\bigl(2j+1\\bigr)\\bigl((2k+3)j+s\\bigr)}\n\\Mult{L}{\\case{1}{2}(L-k-s-2)-(2k+3)j}{0}{1}\n\\right\\}\n\\end{eqnarray}\nfor $L+k$ odd.\nThis we recognize to be exactly (\\ref{Boson1}) and (\\ref{Boson2}) with\n$r=0$, $k L$ replaced by $L$ and $\\Mults{\\ldots}{\\ldots}{0}{k}$\nreplaced by $\\Mults{\\ldots}{\\ldots}{0}{1}$.\n\nSimilarly, if we express the left-hand side of (\\ref{fq})\nthrough an $(n,m)$-system, we find \nprecisely (\\ref{Fermion}) but with\n\\begin{equation}\n\\vec{m}+\\vec{n}=\n\\case{1}{2}\\Bigl({\\cal I}_k \\, \\vec{m}\n+L \\, \\vec{\\mbox{e}}_1 + \\vec{\\mbox{e}}_{i-1} -\\vec{\\mbox{e}}_k \\Bigr).\n\\label{mnFQ}\n\\end{equation}\nThis is just (\\ref{mn}) with $r=0$ and \n$L\\, \\mbox{e}_k$ replaced by $L\\, \\mbox{e}_1$.\n\nFrom the above observations it does not require much insight to\npropose more general polynomial identities which have\n(\\ref{fq}) and those implied by the theorems~\\ref{ft} and\n\\ref{bt} as special cases.\nIn particular, we have confirmed the following conjecture \nby extensive series expansions.\n\\begin{conjecture}\\label{conj}\nFor all $k\\geq 1$, $1\\leq \\ell \\leq k$, $1\\leq i \\leq k+1$,\n$1\\leq i'\\leq \\ell+1$ and $\\ell L\\geq k+\\ell-i-i'+2$\n{\\rm\n\\begin{equation}\n\\sum_{n_1,n_2,\\ldots,n_k\\geq 0}\nq^{\\displaystyle \\, \\vec{n}^T C^{-1}_k (\\vec{n}\n+\\vec{\\mbox{e}}_k -\\vec{\\mbox{e}}_{i-1})}\n\\prod_{j=1}^k\n\\Bin{n_j+m_j}{n_j}\n\\end{equation}}\nwith $(m,n)$-system given by\n{\\rm\n\\begin{equation}\n\\vec{m}+\\vec{n}=\n\\case{1}{2}\\Bigl({\\cal I}_k \\, \\vec{m}\n+(L\\!-1) \\, \\vec{\\mbox{e}}_{\\ell} + \\vec{\\mbox{e}}_{i-1}\n+\\vec{\\mbox{e}}_{i'-1}-\\vec{\\mbox{e}}_k \\Bigr)\n\\label{mnGen}\n\\end{equation}}\nequals\n\\begin{eqnarray}\n\\lefteqn{\n\\sum_{j=-\\infty}^{\\infty} \\left\\{\nq^{j\\bigl((2j+1)(2k+3)-2s\\bigr)}\n\\Mult{L}{\\case{1}{2}(\\ell L+k-s-r+1)+(2k+3)j}{r}{\\ell} \n\\right. } \\nonumber \\\\\n& & \\hspace{2mm} \\left.\n-q^{\\bigl(2j+1\\bigr)\\bigl((2k+3)j+s\\bigr)}\n\\Mult{L}{\\case{1}{2}(\\ell L+k+s-r+1)+(2k+3)j}{r}{\\ell}\n\\right\\}\n\\end{eqnarray}\nfor $r\\equiv \\ell L+k \\mod{2}$ and\n\\begin{eqnarray}\n\\lefteqn{\n\\sum_{j=-\\infty}^{\\infty} \\left\\{\nq^{j\\bigl((2j+1)(2k+3)-2s\\bigr)}\n\\Mult{L}{\\case{1}{2}(\\ell L-k+s-r-2)-(2k+3)j}{r}{\\ell} \n\\right. } \\nonumber \\\\\n& & \\hspace{2mm} \\left.\n-q^{\\bigl(2j+1\\bigr)\\bigl((2k+3)j+s\\bigr)}\n\\Mult{L}{\\case{1}{2}(\\ell L-k-s-r-2)-(2k+3)j}{r}{\\ell}\n\\right\\}\n\\end{eqnarray}\nfor $r\\not\\equiv \\ell L+k \\mod{2}$.\nHere $s$ is defined as in (\\ref{defs}) and \n\\begin{equation}\nr=\\ell-i'+1\n\\end{equation}\nso that $r=0,\\ldots,\\ell$.\n\\end{conjecture}\nFor later reference, let us denote these more general polynomials\nas $G_{k,i,i';L}^{(\\ell)}(q)$. Then $\\ell=k$ corresponds\nto the polynomials considered in this paper and\n$\\ell=1$ to those of Foda, Quano and Kirillov.\n\nThe above conjecture leads one to wonder whether there are in fact\n(at least) $k$ different partition theoretical interpretations \nof (\\ref{An}), each of which has a natural finitization\ncorresponding to the polynomials $G_{k,i,i';L}^{(\\ell)}(q)$\nwith $\\ell=1,\\ldots,k$.\n\nIntimately related to the conjecture and perhaps even more surprising\nis the following observation, originating from the work\nof Andrews and Baxter~\\cite{AB}.\nFor $k\\geq 0$ and $1\\leq i \\leq k+1$,\ndefine a $k$-variable generating function\n\\begin{equation}\nf(x_1,\\ldots,x_k) = \n\\sum_{n_1,n_2,\\ldots,n_{k}\\geq 0}\n\\frac{q^{N_1^2 + \\cdots + N_k^2+N_i+\\cdots+N_k} \n\\, x_1^{2N_1} x_2^{2(N_1+N_2)} \n\\cdots x_k^{2(N_1+\\cdots + N_k)}}\n{(x_1)_{n_1+1} (x_2)_{n_2+1} \\cdots (x_k)_{n_k+1} } \\: ,\n\\label{fk}\n\\end{equation}\nwhere $(x)_n = \\prod_{k=0}^{n-1} (1-xq^k)$.\nObviously, $(1-x_1) \\cdots (1-x_k) f(1,\\ldots,1)$\ncorresponds to the left-hand side of (\\ref{An}). \nNow define the polynomials $P(\\ell_1,\\ldots,\\ell_k):=P(\\vec{\\ell})$\nas the coefficients in the series expansion of $f$,\n\\begin{equation}\nf(x_1,\\ldots,x_k) = \\sum_{\\ell_1,\\ldots,\\ell_k} \nP(\\vec{\\ell}) \\,x_1^{\\ell_1} \\cdots x_k^{\\ell_k}.\n\\end{equation}\nFrom the readily derived functional equations for $f$\nand the recurrences (\\ref{Grec}) with (\\ref{Ginit})\none can deduce that\n\\begin{equation}\nP(\\vec{m}+2 C_k^{-1} \\vec{n}) = G_{k,i,i';L}(q)\n\\end{equation}\nwith $\\vec{m}$ and $\\vec{n}$ given by (\\ref{mn}).\nSimilarly the polynomials of Foda, Quano and Kirillov arise\nagain as $P(\\vec{m}+2 C_k^{-1} \\vec{n})$ where $\\vec{m}$ and\n$\\vec{n}$ now satisfy (\\ref{mnFQ}).\nAgain we found numerically that also the polynomials featuring the\nconjecture appear naturally. That is,\n\\begin{equation}\nP(\\vec{m}+2 C_k^{-1} \\vec{n}) = G^{(\\ell)}_{k,i,i';L}(q),\n\\end{equation}\nwhere now the generalized $(m,n)$-system (\\ref{mnGen})\nshould hold (so that $\\vec{m}+2 C_k^{-1} \\vec{n}=\nC^{-1}_k ((L\\!-1) \\, \\vec{\\mbox{e}}_{\\ell} + \\vec{\\mbox{e}}_{i-1}\n+\\vec{\\mbox{e}}_{i'-1}-\\vec{\\mbox{e}}_k)$).\n\n\\vspace*{5mm}\n\nAlthough all the polynomial identities implied by \nconjecture~\\ref{conj} reduce to Andrews' identity~(\\ref{An})\nin the $L$ to infinity limit,\nthey still provide a powerful tool for generating\nnew $q$-series results.\nThat is, if we first replace $q\\to 1\/q$ and then take $L\\to\\infty$,\nnew identities arise.\nTo state these, we need some more notation.\nThe inverse Cartan matrix of the Lie algebra A$_{\\ell-1}$ is denoted\nby $B_{\\ell-1}$, and $\\vec{\\mu}$ and $\\vec{\\varepsilon}_j$ are\n$(\\ell-1)$-dimensional (column) vectors with entries\n$\\vec{\\mu}_j=\\mu_j$ and $(\\vec{\\epsilon}_j)_{m}=\\delta_{j,m}$.\nFurthermore, we need the $k$-dimensional vector\n\\begin{equation}\n\\vec{Q}_{i,i',\\ell} = \n\\vec{\\mbox{e}}_i + \\vec{\\mbox{e}}_{i+2} + \\cdots +\n\\vec{\\mbox{e}}_{i'} + \\vec{\\mbox{e}}_{i'+2} + \\cdots +\n\\vec{\\mbox{e}}_{\\ell+1} + \\vec{\\mbox{e}}_{\\ell+3} + \\cdots ,\n\\end{equation}\nwith $\\vec{\\mbox{e}}_j = \\vec{0}$ for $j \\geq k+1$.\nUsing this notation, we are led to the following conjecture.\n\\begin{conjecture}\nFor all $k\\geq 1$, $1\\leq \\ell \\leq k$, $1\\leq i \\leq k+1$,\n$1\\leq i'\\leq \\ell+1$ and $|q|<1$, the $q$-series\n{\\rm\n\\begin{equation}\n\\renewcommand{\\arraystretch}{0.7}\nq^{(i'+i-2)\/4} \\hspace{-4mm}\n\\sum_{\\begin{array}{c}\n\\scriptstyle m_1,m_2,\\ldots,m_k\\geq 0\\\\\n\\scriptstyle m_j \\equiv (\\vec{Q}_{i,i',\\ell})_j \\mod{2}\n\\end{array}}\n\\hspace{-4mm}\n\\frac{q^{\\displaystyle \\, \\case{1}{4} \\, \\vec{m}^T C_k (\\vec{m}\n+ 2\\,\\vec{\\mbox{e}}_k - 2\\,\\vec{\\mbox{e}}_{i-1})}}{(q)_{m_{\\ell}}}\n\\prod_{\\begin{array}{c}\n\\scriptstyle j=1\\\\\n\\scriptstyle j \\neq \\ell\n\\end{array} }^k\n\\Bin{\\case{1}{2}\\bigl({\\cal I}_k \\, \\vec{m}\n+ \\vec{\\mbox{e}}_{i-1}\n+\\vec{\\mbox{e}}_{i'-1}-\\vec{\\mbox{e}}_k \\bigr)}{m_j}\n\\label{Fdual}\n\\end{equation}}\nequals\n{\\rm \\begin{eqnarray}\n\\renewcommand{\\arraystretch}{0.7}\n\\lefteqn{\nq^{(k+r-s+1)(k-r-s+1)\/(4\\ell)}\n\\; \\frac{1}{(q)_{\\infty}} \\;\n\\sum_{n=0}^{\\ell-1}\n\\hspace{-2mm}\n\\sum_{\\begin{array}{c}\n\\scriptstyle \\mu_1,\\ldots,\\mu_{\\ell-1}\\geq 0 \\\\\n\\scriptstyle n + \\ell (B_{\\ell-1} \\vec{\\mu})_1\n\\equiv 0 \\mod{\\ell}\n\\end{array}}\n\\hspace{-4mm}\n\\frac{q^{\\displaystyle \\, \\vec{\\mu}^T B_{\\ell-1} (\\vec{\\mu}\n - \\vec{\\epsilon}_r)}}{(q)_{\\mu_1} \\cdots (q)_{\\mu_{\\ell-1}}}} \\nonumber \\\\\n& &\\renewcommand{\\arraystretch}{0.7}\n\\mbox{} \\times \\biggl\\{\n\\sum_{\\begin{array}{c}\n\\scriptstyle j=-\\infty \\\\\n\\scriptstyle n+(k-s-r+1)\/2+(2k+3)j\\equiv 0 \\mod{\\ell}\n\\end{array}}^{\\infty}\nq^{j\\bigl((2k-2\\ell+3)(2k+3)j+\n(2k+3)(k-\\ell+1)-(2k-2\\ell+3)s\\bigr)\/\\ell} \\biggr. \\nonumber \\\\\n& &\\renewcommand{\\arraystretch}{0.7}\n\\biggl. \\qquad\n- \\hspace{-4mm}\n\\sum_{\\begin{array}{c}\n\\scriptstyle j=-\\infty \\\\\n\\scriptstyle n+(k+s-r+1)\/2+(2k+3)j\\equiv 0 \\mod{\\ell}\n\\end{array}}^{\\infty}\nq^{\\bigl((2k-2\\ell+3)j+(k-\\ell+1)\\bigr)\n\\bigl((2k+3)j+s\\bigr)\/\\ell}\n\\biggr\\}\n\\label{rk}\n\\end{eqnarray}}\nfor $r\\equiv k \\mod{2}$, and equals\n{\\rm \\begin{eqnarray}\n\\renewcommand{\\arraystretch}{0.7}\n\\lefteqn{\nq^{(k+r-s+2)(k-r-s+2)\/(4\\ell)}\n\\; \\frac{1}{(q)_{\\infty}} \\; \n\\sum_{n=0}^{\\ell-1}\n\\sum_{\\begin{array}{c}\n\\scriptstyle \\mu_1,\\ldots,\\mu_{\\ell-1}\\geq 0 \\\\\n\\scriptstyle n + \\ell (B_{\\ell-1} \\vec{\\mu})_1\n\\equiv 0 \\mod{\\ell} \\end{array}}\n\\frac{q^{\\displaystyle \\, \\vec{\\mu}^T B_{\\ell-1} (\\vec{\\mu}\n - \\vec{\\epsilon}_r)}}{(q)_{\\mu_1} \\cdots (q)_{\\mu_{\\ell-1}}}} \\nonumber \\\\\n& &\\renewcommand{\\arraystretch}{0.7}\n\\mbox{} \\times \\biggl\\{\n\\sum_{\\begin{array}{c}\n\\scriptstyle j=-\\infty \\\\\n\\scriptstyle n-(k-s+r+2)\/2-(2k+3)j\\equiv 0 \\mod{\\ell}\n\\end{array}}^{\\infty}\nq^{j\\bigl((2k-2\\ell+3)(2k+3)j+\n(2k+3)(k-\\ell+2)-(2k-2\\ell+3)s\\bigr)\/\\ell} \\biggr. \\nonumber \\\\\n& &\\renewcommand{\\arraystretch}{0.7}\n\\biggl. \\qquad\n- \\hspace{-4mm}\n\\sum_{\\begin{array}{c}\n\\scriptstyle j=-\\infty \\\\\n\\scriptstyle n-(k+s+r+2)\/2-(2k+3)j\\equiv 0 \\mod{\\ell}\n\\end{array}}^{\\infty}\nq^{\\bigl((2k-2\\ell+3)j+(k-\\ell+2)\\bigr)\n\\bigl((2k+3)j+s\\bigr)\/\\ell}\n\\biggr\\}\n\\label{rnk}\n\\end{eqnarray}}\nfor $r\\not\\equiv k \\mod{2}$. \n\\end{conjecture}\nSince conjecture~\\ref{conj} is proven for\n$\\ell=1$ and $k$, we can for these particular values\nclaim the above as theorem. \nIn fact, for $\\ell=1$, the above was first conjectured in ref.~\\cite{KKMMb}\nand proven in~\\cite{FQ94}.\nIn refs.~\\cite{KMQ,BNY} expressions for the branching functions of\nthe $({\\rm A}^{(1)}_1)_{M} \\times ({\\rm A}^{(1)}_1)_{N} \/\n({\\rm A}^{(1)}_1)_{M+N}$\ncoset conformal field theories were given similar \nto (\\ref{rk}) and (\\ref{rnk}).\nThis similarity suggests that (\\ref{Fdual}), (\\ref{rk}) and (\\ref{rnk})\ncorrespond to the branching functions of the coset \n$({\\rm A}^{(1)}_1)_{\\ell} \\times ({\\rm A}^{(1)}_1)_{k-\\ell-1\/2} \/\n({\\rm A}^{(1)}_1)_{k-1\/2}$ of fractional level.\n\nA very last comment we wish to make is that\nthere exist other polynomial identities\nthan those discussed in this paper which\nimply the Andrews--Gordon identity~\\ref{An} and which\ninvolve the $q$-multinomial coefficients.\n\\begin{theorem}\\label{t7}\nFor all $k \\leq 1$ and $1 \\leq i \\leq k+1$,\n\\begin{equation}\n\\sum_{a=0}^L \\dMult{L}{a}{k-i+1}{k} =\n\\sum_{j=-L}^L\n(-)^j\nq^{j\\bigl( (2k+3)(j+1)-2i\\bigr)\/2} \\frac{(q)_L}{(q)_{L-j}(q)_{L+j}} \\: .\n\\label{e7}\n\\end{equation}\n\\end{theorem}\nNote that for $i=k+1$ the left-hand side is the generating\nfunction of partitions with at most $k$ successive Durfee\nsquares and with largest part $\\leq L$.\n\nThe proof of theorem~\\ref{t7} follows readily using the\nBailey lattice of refs.~\\cite{AAB}.\nFor $k=1$ (\\ref{e7}) was first obtained by Rogers~\\cite{Rogers94}.\nFor other $k$ it is implicit in refs.~\\cite{AAB,Andrews84}.\n\n\\section*{Acknowledgements}\nI thank Anne Schilling for helpful and stimulating\ndiscussions on the $q$-multinomial coefficients.\nEspecially her communication of equation (\\ref{frec})\nhas been indispensable for proving proposition~\\ref{pqr2}.\nI thank Alexander Berkovich for very constructive discussions \non the nature of the fermi-gas of section~\\ref{pt4}.\nI wish to thank Professor G.~E.~Andrews for drawing my attention\nto the relevance of equation (\\ref{fk}) and\nBarry McCoy for electronic lectures on\nthe history of the Rogers--Ramanujan identities.\nFinally, helpful and interesting discussions with\nOmar Foda and Peter Forrester are greatfully acknowledged.\nThis work is supported by the Australian Research Council.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:introduction}\n\nActive galactic nuclei (AGN) are believed to be powered by the release of gravitational potential energy when matter falls onto supermassive black holes in the centres of galaxies. Some AGN have broad emission lines that are thought to be Doppler broadened emission from gas orbiting the central black hole. The broad lines vary in response to the continuum emission, suggesting that they are powered by ionizing radiation originating in the immediate vicinity of the black hole \\citep{1969ApJ...155.1077B,1972ApJ...171..213D}. The time delay between the variations in the continuum and emission lines can be used to measure the structure and characteristic size of the broad emission line region (BLR) by the method of reverberation mapping \\citep{1972ApJ...171..467B,1982ApJ...255..419B,1993PASP..105..247P}.\n\nThe radius of the BLR combined with the width of the emission line provides a measurement of the mass of the central black hole \\citep{1998ApJ...501...82P,2000ApJ...533..631K}. Reverberation masses have been found to correlate with properties of the host galaxy, such as stellar velocity dispersion \\citep[e.g.][]{2010ApJ...716..269W} and bulge mass \\citep[e.g.][]{2009ApJ...694L.166B,2013ARA&A..51..511K}, as has been found for black holes in inactive galaxies \\citep[e.g.][]{1995ARA&A..33..581K,1998AJ....115.2285M,2000ApJ...539L...9F,2000ApJ...539L..13G} and indicating a connection between supermassive black hole growth and galaxy evolution. In addition, BLR radii have been found to correlate with the UV\/optical luminosity of the AGN \\citep[e.g.][]{,2000ApJ...533..631K,2005ApJ...629...61K,2009ApJ...697..160B,2013ApJ...767..149B}, suggesting that AGN can be used as independent standard candles for cosmology \\citep{2011ApJ...740L..49W,2014MNRAS.441.3454K}. The radius-luminosity relation also enables black hole masses to be estimated out to large redshift ($z>6$) by measuring the AGN luminosity and broad emission line width in a single spectrum, yielding single-epoch black hole mass estimates \\citep[e.g.][]{2002ApJ...571..733V,2002MNRAS.337..109M,2006ApJ...641..689V,2013BASI...41...61S}.\n\nBecause of limited knowledge of the structure of the BLR, it is necessary to introduce a scaling factor in traditional reverberation mapping black hole mass measurements, such that the black hole mass is determined by\n\\begin{equation} \\label{eq:ffactor}\n    M_\\textrm{BH} = f \\frac{\\Delta V^2 R_\\textrm{BLR}}{G},\n\\end{equation}\nwhere $\\Delta V$ is the velocity width of the varying part of the emission line, $G$ is the gravitational constant, and $f$ is the order-unity scaling factor that encapsulates the unknown details regarding the BLR gas and kinematics \\citep{1999ApJ...526..579W,2004ApJ...615..645O}. The $f$-factor is generally calibrated using the $M_\\textrm{BH} - \\sigma_\\star$ relation between the black hole mass and the stellar velocity dispersion of the bulge in the host galaxy. This relies on determining $M_\\textrm{BH}$ independently in quiescent galaxies using stellar and gas dynamics (see \\citealt{2013ARA&A..51..511K} for a review), and then determining the average scaling factor $\\langle f \\rangle$ that brings the ensemble of active galaxies into agreement with the quiescent $M_\\textrm{BH} - \\sigma_\\star$ relation. This process only corrects for the average offset, but it results in black hole masses that are uncertain for any particular AGN by a factor of $2-3$. In addition, $\\langle f \\rangle$ is determined using \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace, but often applied to other emission lines such as \\ifmmode {\\rm Mg}\\,{\\sc ii \\xspace} \\else Mg\\,{\\sc ii}\\fi \\xspace $\\lambda 2799$ and \\ifmmode {\\rm C}\\,{\\sc iv \\xspace} \\else C\\,{\\sc iv}\\fi \\xspace $\\lambda 1549$, although the structure of the BLR for these emission lines is very uncertain \\citep{2006ApJ...647..901M,2008AJ....135.1849W,2007ApJ...659..997K,2014ApJ...795..164T}.\n\nThe main uncertainty in measuring black hole masses using reverberation mapping thus comes from our limited knowledge of the kinematics and geometry of the BLR. A precise measurement of the kinematic and geometric structure of the BLR would enable the determination of $f$ for any individual AGN. This could help reduce scatter in black hole mass determinations for all single epoch mass estimates and their applications \\citep[][]{2013ApJ...764...45K}. A better understanding of changes in BLR structure with luminosity and redshift will also reduce systematic errors possibly affecting current scaling relations. To make a full map of the BLR we need to determine not just its characteristic radius, but also the full velocity-resolved transfer function that describes the relation between the continuum emission and the emission line response.\n\nTo first order, the problem of reverberation mapping can be formulated as a deconvolution problem in which the flux in the emission-line light curve, at a given wavelength $\\lambda$, $F_l(t,\\lambda)$ is given by a convolution of the AGN continuum light curve, over some wavelength range, $F_c(t)$ with a transfer function $\\Psi(t,\\lambda)$ that encodes the physics and geometry of the BLR,\n\\begin{equation} \\label{eq:continous_transferequation}\n    F_l(t,\\lambda) = \\int_{-\\infty}^\\infty \\Psi(\\tau,\\lambda) F_c(t-\\tau)d\\tau.\n\\end{equation}\nThe transfer function, as a function of time delay and wavelength $\\Psi(\\tau, \\lambda)$, is called the velocity--delay map \\citep{1991ApJ...379..586W}. In this approximation, the transfer function is assumed to be linear. Even though the detailed physics is likely to be more complicated, the linear approximation is currently sufficient given that observational datasets have only recently become good enough to probe the full velocity-resolved transfer function. Thus, the transfer function introduced here represents an observed projection of the underlying structure of the AGN. A sound inference of the transfer function will need to account for any residual mismatches between the assumed model and the data, such as non-linearities.\n\nBecause transfer functions represent projections of the underlying physical structure, physical and geometrical models are required to interpret them. Several groups have gone through the exercise of predicting transfer functions based on underlying physical models for the BLR structure \\citep[e.g.][]{1996ApJ...466..704C,1997ApJ...479..200B,2012MNRAS.426.3086G,1992MNRAS.256..103P}. These studies provide a valuable catalogue of transfer functions that can be consulted when interpreting results from reverberation mapping.\n\nMuch effort has gone into developing efficient methods for estimating the BLR size and transfer function $\\Psi(\\tau,\\lambda)$. Early attempts relied on estimating the time delay by-eye \\citep{1973ApL....13..165C}, but many sophisticated methods have been developed since. \\cite{1982ApJ...255..419B} were the first to calculate transfer functions directly using the convolution theorem of Fourier transforms and provided a comprehensive catalogue of transfer functions for a number of idealised BLR structures. Unfortunately, the requirement for very high quality data, as well as difficulties in dealing with measurements errors, mean that the method has seen little application since its publication. This may change with future high cadence, high signal-to-noise reverberation mapping campaigns.\n\nAnother type of inversion procedure is the maximum entropy method that finds the solution for the transfer function that has the highest entropy, while still providing a good fit to the data \\citep{1984MNRAS.211..111S,1991ApJ...371..541K,1991ApJ...367L...5H,1994ASPC...69...23H}. Maximum entropy has been successful in estimating transfer functions and velocity--delay maps in a number of AGN \\citep[e.g.][]{1991ApJ...371..541K,1996MNRAS.283..748U,2010ApJ...720L..46B,2013ApJ...764...47G}. The downsides of maximum entropy are that it is computationally expensive, it relies on a number of assumptions about the shape of the transfer function, and it is difficult to carry out rigorous error analysis and model comparisons.\n\nAnother class of method is dynamical modelling in which a full physical model of the BLR is constructed, and its parameters are inferred within the framework of Bayesian statistics. The statistical framework allows for rigorous error analysis as well model selection \\citep{2011ApJ...730..139P}. Furthermore, dynamical modelling circumvents the need for interpreting transfer functions by providing direct estimates of physical model parameters such as inclination and the black hole mass, which in turn allows for the $f$-factor to be calculated directly. The main challenges of dynamical modelling methods are that they require long computation times and the assumption that the model is flexible enough to provide a good description of the BLR.\n\nDynamical modelling and maximum entropy both provide useful and complementary constraints on the BLR structure. They also rely on a number of assumptions about the allowed shape of the transfer function and are fairly computationally expensive. It is therefore worthwhile to consider alternative methods for analysing reverberation mapping data that are less computationally expensive and allow for greater flexibility in estimating velocity-resolved transfer functions.\n\nHere we develop a method for reverberation mapping based on regularized linear inversion \\citep[RLI;][]{1994PASP..106.1091V,1995ApJ...440..166K}, which we extend by including statistical modelling of the AGN continuum light curve light curves. The method is complementary to other reverberation mapping techniques, and has the advantage that it provides an analytical expression for the transfer function with very few assumptions about its shape. RLI is a flexible, free-form method, allowing the data to suggest the form of the inferred transfer function. This makes RLI an ideal tool for exploring BLR physics beyond the framework of current BLR models. At the same time RLI is one of the fastest reverberation methods that provides a direct estimate of the transfer function. We extend the RLI method to include error in the light curves by a combination of Gaussian process modelling and bootstrap resampling, thereby providing a robust estimate of the transfer function and its uncertainties.\n\nAs a first application, we apply our method to photometric and spectroscopic light curves of five nearby AGN measured by the Lick AGN Monitoring Project 2008 collaboration \\citep[LAMP 2008;][]{2008ApJ...689L..21B}. The main purpose of this project was to measure masses of supermassive black holes in 13 nearby ($z<0.05$) Seyfert 1 galaxies \\citep{2009ApJ...705..199B}. Besides improving black hole mass estimates, LAMP 2008 has produced a medley of scientific results including: AGN variability characteristics \\citep{2009ApJS..185..156W}, an update of the $M_\\textrm{BH} - \\sigma_\\star$ relation with reverberation mapped AGN \\citep{2010ApJ...716..269W}, detailed reverberation mapping studies \\citep{2010ApJ...716..993B,2010ApJ...720L..46B}, probing the $R_\\textrm{BLR}-L$ relation in the X-rays \\citep{2010ApJ...723..409G}, recalibrating single-epoch black hole masses \\citep{2012ApJ...747...30P} and dynamical modelling of the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace BLR \\citep{2014MNRAS.445.3073P}.\n\nWe analyse five objects from LAMP 2008: Arp 151 (Mrk 40), Mrk 1310, NGC 5548, NGC 6814 and SBS 1116+583A, providing integrated response functions, time delays and velocity--delay response maps for the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace emission line in each object, as well as \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace and \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace in Arp 151. We begin in Section \\ref{sec:data} by describing how we obtained light curves from the LAMP 2008 dataset. In Section \\ref{sec:methods} we outline the RLI method and show results from simulations. Section \\ref{sec:results} presents the main results of our analysis. The results are discussed in Section \\ref{sec:discussion} and finally we provide a short summary and conclusions in Section \\ref{sec:summary-and-conclusion}.\n\nAll time delay-axes in the figures are in the observed frame. All time delays quoted in the text and in Table 1 are in the rest frame of the AGN.\n\n\\section{Data} \\label{sec:data}\n\nWe use data from LAMP 2008\\footnote{Data available at \\url{http:\/\/www.physics.uci.edu\/~barth\/lamp.html}}, a dedicated spectroscopic reverberation mapping campaign that ran for 64 nights at the Lick Observatory 3-m Shane telescope. The spectroscopic data were supplemented by photometric monitoring using smaller telescopes, including the 30-inch Katzman Automatic Imaging Telescope (KAIT), the 2-m Multicolor Active Galactic Nuclei Monitoring telescope, the Palomar 60-inch telescope, and the 32-inch Tenagra II telescope. A detailed description of the LAMP 2008 observing campaign and initial results are published in \\cite{2009ApJS..185..156W} and \\cite{2009ApJ...705..199B}.\n\n\\subsection{Continuum light curves}\nFor our analysis, we use Johnson $B$ and $V$ broad band continuum light curves from LAMP 2008. The bands were chosen to improve dynamical modelling results and resolve variability features. The fluxes are measured using standard aperture photometry (see \\citealt{2009ApJS..185..156W} for a complete discussion). The light curves are the same as those used in dynamical modelling by \\cite{2014MNRAS.445.3073P}. The $B$ and $V$ band light curves are very similar for all objects analysed \\citep[see][]{2009ApJS..185..156W}, and the choice of continuum light should not significantly affect our results.\n\n\\subsection{Emission line spectra and light curves} \\label{sec:emission-line-lightcurves}\nEmission-line light curves are generated from flux-calibrated spectra from the LAMP 2008 campaign. The \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace emission line is isolated in all objects using spectral decomposition, by modelling all line and continuum components individually, and subtracting away everything but the emission line of interest \\citep{2012ApJ...747...30P}. We decide to keep the narrow component of \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace to avoid introducing additional error at the centre of the line. This should not affect our results, as the narrow-line component is constant on the time-scale of the observing campaign \\citep[see][]{2012ApJ...747...30P}, and because we only consider variations around the mean flux in the line. In Arp 151 we also analyse the \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace and \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace lines, where \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace is isolated using spectral decomposition in a way similar to that for \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace \\citep{2011ApJ...743L...4B}. Because no spectral decomposition is available for \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace we follow the procedure of \\cite{2010ApJ...716..993B}, isolating \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace by subtracting a straight line fit to two wavelength windows on either side of the line.\n\nThe resulting emission line spectra have the same spectral resolution as the original LAMP 2008 data. The spectral dispersions are provided in \\cite{2009ApJ...705..199B} and range from $11.6 - 14.7$\\,\\AA\\,\\,(FWHM), corresponding to $5.9 - 7.5$ pixels per resolution element.\n\nA few of the spectra in the LAMP 2008 dataset have been identified as unreliable due to low spectral quality or suspicious features above the noise \\citep{2012ApJ...747...30P}. We tested the effect of removing these spectra in the analysis and found that it made little difference to the overall results. Because of this, and to be able to compare directly with recent results from dynamical modelling \\citep{2014MNRAS.445.3073P}, we decided to include the unreliable spectra in the RLI analysis presented here. The emission line light curves for \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace are thus identical to those used for the dynamical modelling analysis in \\cite{2014MNRAS.445.3073P}.\n\n\\section{Regularized linear inversion} \\label{sec:methods} \\label{sec:rli}\n\nRegularized linear inversion seeks to solve the transfer equation (Equation \\ref{eq:continous_transferequation}) for the transfer function $\\Psi(\\tau,\\lambda)$ analytically and without any assumptions regarding its functional form (only that it is a bounded linear operator). This approach is potentially very powerful in that it relies solely on the data when deriving the transfer function. This in turn allows for very little freedom for the method to select a proper solution. In the presence of noise, or if the system deviates slightly from the linear assumption, this lack of freedom means that a solution cannot be found at all. Therefore, instead of looking for a unique solution that fits the data, the method determines the solution that minimizes the $\\chi^2$. A simple $\\chi^2$ minimization will generally over-fit the data and make the inversion unstable. This is overcome by a regularization in which the first-order derivative of the solution is to be minimized along with the $\\chi^2$, such that the solution is smoothed at the level of the noise \\citep{phillips1962technique,tikhonov1963solution}. The result is comparable to maximizing the entropy under a $\\chi^2$ constraint as in the case of maximum entropy methods. RLI has several advantages over other methods, specifically it 1) makes no assumption about the shape or positivity of the transfer function, 2) can be solved analytically, and 3) has very few free parameters (the regularization scale as well as the transfer function window and resolution). Note that while RLI can in principle fit any shape of transfer function, in practice the result will be limited by the sampling and noise in the data, which means that a minimum scale exists below which the response function will be unresolved.\n\nBy measuring the continuum ($F_c$) and emission line ($F_l$) light curves we can in principle solve the transfer equation \\eqref{eq:continous_transferequation} for $\\Psi(t,\\lambda)$. In reality, data is always discrete, so we rewrite the transfer equation as a linear matrix equation of the form\n\\begin{equation} \\label{eq:discrete_transferequation}\n    \\mathbf{L}_{\\Delta \\lambda} = \\mathbf{\\Psi}_{\\Delta \\lambda} \\mathbf{C},\n\\end{equation}\nwhere $\\mathbf{L}_{\\Delta \\lambda}$ is the emission line light curve integrated over the wavelength range (spectral bin) $\\Delta \\lambda$, $\\mathbf{C}$ is a matrix of continuum light curves (see below), and $\\mathbf{\\Psi}_{\\Delta \\lambda}$ is the transfer function corresponding to the given wavelength range. While we allow the transfer function to depend on wavelength, symmetries in the BLR as well as observational projections will tend to correlate transfer functions in the time and wavelength domains. The set of transfer equations over a range of frequencies across an emission-line we call the velocity--delay map for the given emission line \\citep{1991ApJ...379..586W}.\n\nFor perfect noise-free data solving the discrete transfer equation \\eqref{eq:discrete_transferequation} would be a simple matter of inverting $\\mathbf{C}$ to obtain the transfer function $\\mathbf{\\Psi}$. Because of noise we cannot hope to find an exact solution to the linear inversion problem. Instead we seek a solution that minimizes the $\\chi^2$ together with a smoothing condition that ensures that we are not fitting the noise.\n\nInterpreting the transfer function to make statements about the structure of the BLR relies on a number of assumptions. First, we assume that the variations in the AGN continuum bands are correlated with the AGN ionizing continuum (this is not necessarily true, see e.g. \\citealt{1998ApJ...500..162C,2014MNRAS.444.1469M,2015ApJ...806..129E} who find evidence for a time delay between the UV and optical continuum in NGC 7469 and NGC 5548). Second, we assume that the continuum emission originates from a region negligible in size compared to the BLR. Third, we assume that the continuum ionizing radiation is emitted isotropically. Last, we assume that the BLR structure is constant and the response linear for the duration of the campaign, such that a single linear transfer function can be calculated from the full light curves.\n\n\\subsection{Solving for the response function}\nRather than solving for the transfer function, which includes constant emissivity components of the BLR, we consider only variations about the mean of the light curves. By doing this, we are measuring the response of the line emission to a change in the continuum emission that is ionizing the BLR gas. Hence the quantity we are solving for, $\\Psi(\\tau,\\lambda)$, is the response function \\citep{1991ApJ...371..541K,1993MNRAS.263..149G}.\n\nFollowing the notation of \\cite{1995ApJ...440..166K}, and considering only variation about the mean of the light curves, we write $\\chi^2$ as\n\\begin{equation} \\label{eq:chi2}\n    \\chi^2 = \\sum_{i=M}^N \\frac{1}{\\sigma_l^2(t_i)} \\left [ \\delta F_l(t_i) - \\sum_{j=1}^M [F_c(t_i - \\tau_j) - \\langle F_c \\rangle]\\Psi(t_j) \\right ]^2.\n\\end{equation}\nThis expression can be recast to matrix notation,\n\\begin{equation} \\label{eq:chi2matrix}\n    \\chi^2 = (\\mathbf{L} - \\mathbf{C}\\mathbf{\\Psi})^2,\n\\end{equation}\nwhere the light curves enter as\n\\begin{align}\n    \\mathbf{C}_{ij} &= [F_c(t-\\tau_j) - \\langle F_c \\rangle] \/ \\sigma_l(t_i) \\label{eq:C}\\\\\n    \\mathbf{L} &= \\delta F_l (t_i) \/ \\sigma_l(t_i), \\label{eq:L}\n\\end{align}\nand the variation about the mean in the emission-line light curve is defined as\n\\begin{equation}\n    \\delta F_l (t_i) = F_l(t_i) - \\langle F_l \\rangle.\n\\end{equation}\nWe choose a time delay resolution of $1\\,\\textrm{day}$, as this corresponds to the highest resolution of the data. By running the analysis with different resolutions we have confirmed that the choice of resolution, within reasonable values, does not change our results. Contrary to \\cite{1995ApJ...440..166K}, we calculate a constant mean of the continuum light curve data points such that $\\langle F_c \\rangle$ is not a function of the time delay $\\tau$. This is done in order to allow meaningful comparisons between different continuum realizations (see Section \\ref{sec:cont-errors}). Minimizing $\\chi^2$ in Equation \\eqref{eq:chi2matrix} leads to the linear equation\n\\begin{equation}\n    \\mathbf{C^T C \\Psi} = \\mathbf{C^T L}.\n\\end{equation}\nAlthough this expression looks simple, it turns out to be ill-conditioned. To remedy this, we put an extra constraint on the problem, namely that the solution should be smooth at the scale of the noise (this effectively avoids fitting the noise). To guarantee smoothness of the solution, we introduce a differencing operator $\\mathbf{H}$ acting on $\\mathbf{\\Psi}$, and require that the first-order difference (the discrete version of the first-order differential) be minimized together with the ordinary $\\chi^2$. To control the weights between the $\\chi^2$ and the first-order difference, a scaling parameter $\\kappa$ is introduced that sets the scale of the regularization. Thus, the expression to minimize becomes\n\\begin{equation} \\label{eq:regularization}\n    (\\mathbf{C^T C} + \\kappa \\mathbf{H^T H}) \\mathbf{\\Psi} = \\mathbf{C^T L}.\n\\end{equation}\nThis expression is more stable under inversion, while sacrificing detail by emphasizing smoothness of the solution. The question is then how to choose a suitable scale $\\kappa$ for the regularization.\n\nThe best choice of regularization scale depends on the signal-to-noise in the data, as well as the level of uncorrelated systematic uncertainties that result in deviations from the assumption of linear response. As a starting point for selecting a regularization scale $\\kappa$, we follow the recommendation of \\cite{press1992numerical} in which $\\kappa = \\kappa_0$ is chosen to provide equal weights to the two left-hand-side terms in equation \\eqref{eq:regularization},\n\\begin{equation} \\label{eq:lambda0}\n    \\kappa_0 = \\frac{\\textrm{Tr}(\\mathbf{C^T C})}{\\textrm{Tr}(\\mathbf{H^T H})}.\n\\end{equation}\n\\cite{1995ApJ...440..166K} suggest using the largest value of $\\kappa$ which gives an acceptable $\\chi^2$, while \\cite{1994PASP..106.1091V} suggest using a value of $\\kappa$ which provides the ``best'' trade-off between resolution and noise. By running tests on simulated data, we find that good results are generally achieved for $\\kappa = \\kappa_0$. For this reason, and to reduce the number of free parameters in the method, we fix $\\kappa = \\kappa_0$ for all our analysis (see Section \\ref{sec:changing-the-regularization-scale} for a test of the effect of changing the regularization scale).\n\nWe calculate all response functions in the time delay interval $0 - 30$\\,days. The lower bound is chosen to impose causality in the solution. The upper bound is chosen to be well beyond the time-scale where we expect a significant response based on previous measurements and expectations for Seyfert galaxies \\citep{2013ApJ...767..149B}. We tested the effect of changing the time delay windows and found that it had only minor effects on the results, as longs as the window is long enough to include the main response. When presenting the results, we show only the first part of the response function from $0 - 15$\\,days, as we found the significant response power to be located at these scales.\n\n\\subsection{Continuum light curve errors} \\label{sec:cont-errors}\n\nContinuum flux errors are not included explicitly in the RLI formalism presented above. Instead, we include continuum errors and interpolation by modelling the continuum light curves using Gaussian processes \\citep{2011ApJ...730..139P}. Studies of AGN variability, using sampling intervals of days, have suggested that AGN continuum variability is well modelled by a damped random walk or Ornstein--Uhlenbeck (O--U) process \\citep{2009ApJ...698..895K,2010ApJ...708..927K,2010ApJ...721.1014M,2013ApJ...765..106Z}. Formally this is a CAR(1) or continuous-time first-order autoregressive process, which is a stationary Gaussian process with a power spectral density (PSD) slope of $-2$. The CAR(1) process is often used in reverberation mapping methods for modelling the continuum light curve and enable efficient interpolation and error estimation \\citep[e.g.][]{2011ApJ...735...80Z,2011ApJ...730..139P}.\n\nRecent high cadence observations of AGN using the \\emph{Kepler} spacecraft \\citep{2011ApJ...743L..12M,2012ApJ...749...70C,2014ApJ...795....2E,2015MNRAS.451.4328K}, have found steep PSD slopes ($< -2$) inconsistent with a CAR(1) process. The effect of having a steeper PSD slope is to suppress small scale variability in the light curves. Using a CAR(1) process for modelling the continuum light curves may therefore result in over-fitting spurious features, due to noise at short time scales. This introduces artificial structure at the level of the noise, that can lead to underestimated errors being propagated form the continuum model to other reverberation mapping parameters.\n\nThe magnitude of the error introduced by assuming the wrong continuum model depends on the sampling rate and errors in the data, as well as how it is implemented in the reverberation mapping method. Regularized linear inversion, as implemented here, mitigates the effects noise in the light curves by smoothing the solution at the shortest time scales. This results in a solution that is dominated by the longer multi-day variability features. For this reason, the response functions derived should not be affected by the details of the assumed continuum model, as long as multi-day time scale features are accurately reproduced. Even so, the shallower slope of the CAR(1) process, compared to \\emph{Kepler} AGN light curves, may still lead to underestimated errors.\n\nTo determine how the choice of continuum model PSD affects our results, we run RLI using a number of different continuum models with varying degrees of small scale structure, corresponding to varying PSD slopes. We do this by changing the power $\\alpha$ in the covariance function for the Gaussian process. The covariance is given by \\citep{2011ApJ...730..139P}\n\\begin{equation} \\label{eq:covariance}\n\tC(t_1, t_2) = \\sigma^2 \\exp \\left [ - \\left ( \\frac{|t_2-t_1|}{\\tau} \\right )^\\alpha \\right ],\n\\end{equation}\nwhere $\\sigma$ is the long-term standard deviation, $\\tau$ is the typical time scale of variations and the power $\\alpha$ take on values in the interval $[1,2]$. A power of $\\alpha = 1$ corresponds to the CAR(1) model, whereas $\\alpha > 1$ produces smoother continuum models, with PSD slopes $< -2$. Using data from LAMP 2008 for Arp 151 (see results in section \\ref{sec:results-arp151}), we test four different continuum models with $\\alpha = 1.0$, $1.8$, $1.9$ and $2.0$ respectively (see Figure \\ref{fig:psd-slopes}). We find that the emission line fits and response functions are not strongly affected by the assumed continuum model. For this reason, and to be consistent with other reverberation mapping methods such as {\\sc javelin \\xspace} and dynamical modelling, we choose to use a CAR(1) process as a prior for the Gaussian process continuum modelling. Future analyses of high-cadence reverberation mapping data should likewise be careful in considering the effects of the assumed continuum model.\n\n\\begin{figure}\n    \\includegraphics[width=1.0\\columnwidth]{fig1.eps}\n    \\caption{The effect of using continuum models with varying levels of small scale structure. The upper panels show the CAR(1), $\\alpha=1.0$ in Equation \\eqref{eq:covariance}, continuum model used for the results in this paper. The panels below show results for continuum models with increasing $\\alpha$, resulting in decreasing structure at short time scales, corresponding to progressively steeper, or more negative, power spectral density slopes. The left panels show the continuum light curve of Arp 151 from LAMP 2008 as grey error bars. In each case, the best fit (median and 1$\\sigma$ percentiles) continuum model is shown as a blue band and a single realization of the best fit continuum model is shown as a solid black line. The middle panels show the Arp 151 \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace emission line light curve (black error bars) along with the fit from RLI (green line and band) using the corresponding continuum models on the left. The right panels show the resulting response functions from RLI. The excellent agreement between response functions derived using different continuum models leads us to conclude that the level of short time scale structure in the continuum model does not affect the RLI results derived from the LAMP 2008 dataset.}\n    \\label{fig:psd-slopes}\n\\end{figure}\n\nThe finite sampling and duration of the light curves hampers our ability to model structure on scales close to or below the sampling time scale, and on scales longer than the duration of the light curve. At short time scales the light curves will likely be dominated by observational errors due to the small fractional variability at these scales. Regularized linear inversion deals with this by smoothing the solution at the smallest time scales, resulting in a solution dominated by the longer multi-day variability features. For all the objects analysed, the time scale of interest (a few days) is well within the sampling rate of the data.\n\nFor each continuum light curve we find the best fit Gaussian process parameters using the {\\sc dnest3 \\xspace} Nested Sampling code by \\cite{2010ascl.soft10029B}. The best-fit Gaussian process is used to interpolate the continuum light curve, which we need to calculate the response for arbitrary time delays.\n\nIn addition to interpolation, we use the statistical variability of the Gaussian process to generate a number of realizations of the continuum light curve that are used to estimate statistical errors on the calculated response functions. This allows us to include measurement errors in the continuum light curves that otherwise do not explicitly enter in the RLI formalism. For each continuum light curve, we generate 1000 realizations of the best-fit Gaussian process. This library of continuum realizations is used throughout the analysis to calculate response functions and response maps.\n\n\\subsection{Emission-line light curve errors} \\label{sec:line-errors}\nEmission-line light curve errors $\\sigma_l(t)$ are included explicitly in the RLI formalism in eqs. \\eqref{eq:C} and \\eqref{eq:L}. We allow for the possibility of additional systematic errors in the emission line fluxes by bootstrap resampling of the emission line light curves in each inversion. Bootstrap resampling is done by re-sampling data points in the light curve, such that each point can be chosen zero or more times and the total number of data points is kept constant. If a point is chosen $N$ times, its error is reduced accordingly by $\\sqrt{N}$. We find that the results are only weakly affected by bootstrap resampling, indicating that systematic uncertainties affecting individual epochs in our data sample do not have strong influence on the results.\n\n\\subsection{Testing on simulated data (1D)}\nWe test our RLI method on simulated velocity-integrated (1D) light curves convolved with a selection of different response functions to produce a continuum-emission line light curve pair. To make the light curves as realistic as possible, the continuum light curve is simulated using a Gaussian process with a power spectral density similar to that found by recent Kepler observations \\citep{2014ApJ...795....2E}. The normalization of the simulated continuum light curve is chosen to match that of typical LAMP 2008 light curves \\citep{2009ApJ...705..199B}, such that the simulated continuum has a fractional variability of $F_{\\textrm{var}} \\sim 12$ per cent \\citep[$F_\\textrm{var}$;][]{2003MNRAS.345.1271V} and a signal-to-noise ratio of $\\textrm{SNR} \\sim 100$. Emission line light curves are generated by convolving the simulated continuum light curve with the chosen response function (see Fig. \\ref{fig:sim-1d}). The simulated light curves are degraded and down-sampled to a level similar to the LAMP 2008 dataset (see section \\ref{sec:data}). We test our code on five different response functions: top-hat, multimodal top-hat, Gamma distribution, sinusoidal, and a delta function. The simulated light curves and response functions are shown together with the fits from RLI in Fig. \\ref{fig:sim-1d}. We further test our method by simulating continuum light curves using a damped random walk. Regularized inversion performs slightly better in this case, especially for small values of $\\kappa$, due to the increased structure at short time scales, but the overall results are very similar to the simulated Kepler light curves shown in Fig. \\ref{fig:sim-1d}.\n\nRLI is generally successful in recovering all simulated response functions, but like all reverberation mapping methods the overall performance depends on the number of strong variability features in the light curve sample. Because of the smoothing constraint and the finite sampling of the light curves, RLI cannot fit very sharp features in the simulated response functions. This is particularly evident in the case of the top-hat and delta function response functions. For the smooth Gamma and sinusoidal distributions, we find that RLI is able to match the full shape of the response function. We tested the method on different simulations and found that the deviations at particular time delays are driven by noise in the input data and thus cannot be reduced without re-sampling, or extending, the light curve.\n\nThe smallest scale resolvable by RLI can be seen in the recovery of the delta-function in the bottom panel in Fig. \\ref{fig:sim-1d}. This smallest scale, or point spread distribution, comes from the incomplete sampling of the light curves as well as the smoothing imposed on the solution to avoid fitting noise in the input.\n\nWe test the effect of changing the regularization scale $\\kappa$ by one order of magnitude in each direction. The effect of increasing the regularization ($\\kappa\/\\kappa_0 = 10$, dashed blue line in Fig. \\ref{fig:sim-1d}) is to smooth the response functions, thereby reducing some of the fluctuations in the wings, but also significantly increasing the width of the PSF (Fig. \\ref{fig:sim-1d}, lower right panel). In the same way, a reduction in regularization scale ($\\kappa\/\\kappa_0 = 0.1$, dotted red line in Fig. \\ref{fig:sim-1d}) tends to amplify spurious features in the response functions while improving the PSF by decreasing its width. The best trade-off between high resolution and signal-driven results is a matter of preference. We find that $\\kappa = \\kappa_0$ provides a good balance between resolution and noise. For this reason, and for consistency, we fix the regularization scale to $\\kappa = \\kappa_0$ for all our analysis.\n\nIt is important to note that these simulations assume an exact linear relation between the continuum and emission line, Equation \\eqref{eq:continous_transferequation}, including only random errors and sampling gaps. The simulations and the results from RLI presented in Fig. \\ref{fig:sim-1d} thus represent an artificial best-case scenario. The main purpose of Fig. \\ref{fig:sim-1d} is to illustrate the effect of changing the regularization scale $\\kappa$, show the finite point spread distribution of the RLI method due to discrete sampling, and the ability of RLI to recover negative and multimodal response functions.\n\n\\begin{figure}\n    \\includegraphics[width=1.0\\columnwidth]{fig2.eps}\n    \\caption{Testing regularized linear inversion on simulated data by recovering response functions for simulated light curves. The top left panel shows the simulated continuum light curve, generated using a Gaussian process with a power spectral density similar to that found by recent Kepler observations \\protect\\citep{2014ApJ...795....2E}. The magenta band plotted on top of the continuum shows the standard deviation of the Gaussian process fit used in the analysis. The left panels, below the continuum, show the emission line light curves obtained by convolving the continuum with the corresponding response functions shown in the right panels. Before being analysed, the simulated light curves are degraded and down sampled to a level similar to the light curves of LAMP 2008. The response functions recovered from regularized linear inversion are plotted on top of the input response functions in the right panels. The thin black line is the input response function while the solid green (red dotted and blue dashed) lines shows the response functions calculated from regularized linear inversion with a regularization scale of $\\kappa\/\\kappa_0 = 1$ ($\\kappa\/\\kappa_0 = 0.1$ and $\\kappa\/\\kappa_0 = 10$). The response functions are, from top to bottom: top-hat, multimodal top-hat, Gamma distribution, sinusoidal, and the delta function. The grey dashed horizontal lines indicate the level of zero response.}\n    \\label{fig:sim-1d}\n\\end{figure}\n\n\\subsection{Testing on simulated data (2D)}\nIn addition to testing one-dimensional response functions, we test our RLI implementation on data simulated using the geometric and dynamical model of the BLR used in dynamical modelling of reverberation mapping data \\citep{2011ApJ...730..139P,2011ApJ...733L..33B}. The dynamical modelling method simulates the broad line region as a number of discrete point particles in a parametrized geometrical configuration. We test RLI on two different simulated BLR configurations, one in which the BLR particles are in near-circular orbits, and one with entirely inflowing orbits (see \\citealt{2014MNRAS.445.3055P} for details on the simulated datasets). The results are shown next to the true response maps in Fig. \\ref{fig:sim-2d}. We note that the data are simulated using a kinematic and geometric model for the BLR, and are not necessarily representative of true AGN response maps.\n\nThese simulations provide a qualitative indication of the 2D point-spread-function of the RLI method. Because the solution is forced to be smooth in the presence of noise, the resulting response maps will not reproduce the input maps exactly. Instead, we retrieve a smoothed version of the input response maps. The stripes in the RLI maps appear because we analyse each velocity bin individually. This means that the solution is not smoothed in the velocity (or wavelength) direction. We did this to simplify the calculations and to let any correlations between velocity bins be driven by the data only. This is contrary to maximum entropy, where the result is also smoothed (or regularized) in the velocity direction. If some velocity bins have exceptionally large errors, RLI might fail in that bin and produce a flat response.\n\nAlthough the response maps from dynamical modelling and RLI look somewhat different, they share some of the same symmetries. The effect of in-fall in the upper panel in Fig. \\ref{fig:sim-2d} is clearly recovered in the RLI map. Likewise, the lower panel in Fig. \\ref{fig:sim-2d} shows that dynamical modelling and RLI both show a symmetric response on either side of the emission-line centre, suggesting a structure where the BLR clouds are on closed orbits around the central black hole.\n\n\\begin{figure}\n    \\includegraphics[width=\\linewidth]{fig3.eps}\n    \\caption{Testing regularized linear inversion on simulated two-dimensional response maps (left column) generated using the dynamical modelling code of \\protect\\cite{2014MNRAS.445.3055P}. The response functions are calculated for each velocity bin individually using RLI, and then plotted together to produce velocity--delay maps (right column).}\n    \\label{fig:sim-2d}\n\\end{figure}\n\n\\subsection{Effect of changing the regularization scale \\texorpdfstring{$\\kappa$}{k}} \\label{sec:changing-the-regularization-scale}\n\nFig. \\ref{fig:result-changing-lambda} shows the effect of changing the regularization scale $\\kappa$ using the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace data for Arp 151 as an example (see full results in Section \\ref{sec:results-arp151}). We find that, for low vales of $\\kappa$ (i.e. more weight on the $\\chi^2$-term), the solution is very unstable to perturbations in the input, and the error in the response function increases at all time delays. The reason that the error is not a function of time delay, is that the noise is not correlated with the signal in the light curve. Therefore, the response function can draw power at any time delay to fit the noise, so long as it draws similarly less power at other scales. Because the noise is random in each continuum realization, the time delays at which the response function draws its power will be evenly distributed in delay space, and the response function will fluctuate with more or less the same amplitude at all time delays. There will still on average be more power at scales correlated with the signal, thus the median shape of the response function peaks around the true delay, but the amplitude variations due to noise produce large error bars at all time delays. This behaviour is expected and illustrates the need for regularization to achieve stable solutions.\n\nAs the regularization scale is increased, more weight is put on the smoothness of the solution. This suppresses the effects of noise in the input and emphasizes the large scale behaviour of the light curves. Because of the smoothing introduced in the regularization, there will be a minimum resolvable scale, analogous to an extended point-spread-function (see Fig. \\ref{fig:sim-1d}). Any structure below the noise level will thus be smoothed out in RLI. \n\n\\begin{figure}\n    \\includegraphics[width=\\linewidth]{fig4.eps}\n    \\caption{Effect of changing the regularization scale $\\kappa$, increasing the smoothness of the solution. Using Arp 151, \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace as an example. The left panels show the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace light curve (black points with error bars) along with the fit from RLI (solid green line and band). The right panels show the corresponding response function found using RLI. The regularization scales affects all time scales in the response function equally, which is why the error bars are more of less constant as a function of time delay for each regularization scale $\\kappa$.}\n    \\label{fig:result-changing-lambda}\n\\end{figure}\n\n\\section{Results} \\label{sec:results}\n\nHere we present results from regularized linear inversion of light curves of five local AGN from the LAMP 2008 dataset. Fig. \\ref{fig:result-arp151} to \\ref{fig:result-sbs1116} show results from regularized linear inversion of broad band photometric continuum light curves and \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace light curves from spectral decomposition. We provide fits to the light curves as well as integrated response functions and velocity--delay maps for all object. Table \\ref{table} lists derived median time delays for the integrated light curves of all AGN analysed, as well as time delays from cross-correlation and {\\sc javelin \\xspace} for comparison.\n\n\\subsection{Analysis}\n\n\\subsubsection{Light curves}\n\nThe upper left panels in Fig. \\ref{fig:result-arp151} to \\ref{fig:result-sbs1116} show the observed light curves plotted as black points with error-bars. The blue band on top of the continuum light curve shows the median and 1$\\sigma$ percentiles of 1000 realizations of the Gaussian process best-fit model. This band indicates the range of continuum light curve realizations used together with bootstrap resampling to determine errors on the derived response functions (see Sections \\ref{sec:cont-errors} and \\ref{sec:line-errors}). The middle left panels show the integrated emission-line light-curves obtained by integrating the corresponding emission-line over the wavelength range listed in Table \\ref{table}. The solid line and green band on top of the emission line light curves show the median and 1$\\sigma$ percentiles of the fits obtained from RLI when analysing all 1000 realizations of the continuum together with 100 bootstrap resamplings for each continuum realization.\n\n\\subsubsection{Integrated response functions}\n\nThe lower left panels of Fig. \\ref{fig:result-arp151} to \\ref{fig:result-sbs1116} show the best-fit integrated response functions under the smoothing constraint described in Section \\ref{sec:rli}. The response function and the associated error-bars are calculated from the median and 1$\\sigma$ percentiles of RLI using 1000 iterations of continuum light curve interpolations as well as 100 bootstrap resamplings of the emission line light curve. Because we minimize not just the $\\chi^2$, but also the first derivative of the response function, the medians and error-bars of the response functions will be naturally correlated.\n\n\\subsubsection{Velocity-resolved response functions}\n\nThe upper right panels of Fig. \\ref{fig:result-arp151} to \\ref{fig:result-sbs1116} show the velocity--delay maps from regularized linear inversion of the spectra for all epochs in the data. Each spectral bin is treated individually, such that any correlations below the pixel resolution (see Section \\ref{sec:emission-line-lightcurves}) are driven by the data. As in the case of the integrated analysis, we fix the regularization scale to $\\kappa = \\kappa_0$ for all velocity bins.\n\nThe velocity--delay maps show the response on a linear scale as a function of time delay and Doppler velocity with respect to the emission-line centre. White colour corresponds to zero response while black is maximum response (see the colour bars for each individual object). Below the response maps we plot the spectrum of the corresponding emission line. The black line is the mean spectrum across all epochs considered and the grey line is the root-mean-square variability (standard deviation about the mean) of the spectra, normalized to the same scale as the mean spectrum.\n\nThe lower right panels of Fig. \\ref{fig:result-arp151} to \\ref{fig:result-sbs1116} show a selection of response functions for three separate velocity-bins. The location of the bins are indicated by the vertical dashed lines in the response maps and spectra. The colours correspond to the relative wavelengths, such that the red response function is for the highest radial velocity bin (redshifted with respect to the line centre), the green response function is for the line centre at $0~\\rm{km}\/\\rm{s}$ and the blue response function is for the lowest radial velocity bin (blueshifted with respect to the line centre).\n\n\\subsubsection{The time delay} \\label{sec:the-time-delay}\n\nThe scalar time delay from cross correlation methods, \\ifmmode \\tau_{\\rm CCF \\xspace} \\else $\\tau_{\\rm CCF}$\\fi \\xspace, is typically calculated as the centroid of the CCF above a threshold (usually $80$ per cent of the maximum of the CCF). This effectively ignores any contributions from negative response. To compare our results to those from cross correlation, we calculate a RLI time delay, \\ifmmode \\tau_{\\rm RLI \\xspace} \\else $\\tau_{\\rm RLI}$\\fi \\xspace, as the median of the positive values of the response function. The time delays and error bars for each emission line and object, quoted in Table \\ref{table} and in the text, are then estimated as the median and 1$\\sigma$ percentiles of all time delays determined from the response functions of all continuum realizations (see Sections \\ref{sec:cont-errors} and \\ref{sec:line-errors}). All time delays quoted in the text and in Table \\ref{table} are in the rest frame of the AGN. The time delay units on the axes of the figures are in the observed frame.\n\n\\subsubsection{Comparing with other methods}\n\nWe compare our results to those of other methods, including cross-correlation, {\\sc javelin \\xspace}, maximum entropy, and dynamical modelling.\n\nThe cross-correlation method calculates the cross-correlation function (CCF) between the continuum and emission line light curve to find the time delay, generally characterized by the centroid of the CCF, of the responding gas in the BLR. First applied by \\cite{1986ApJ...305..175G}, the method has since been substantially developed \\citep[e.g.][]{1988ApJ...333..646E,1994PASP..106..879W,1998PASP..110..660P}. The CCF is related to the transfer function through the light curve auto-correlation function \\citep{1991vagn.conf..343P}. This means that the width of the CCF depends on the variability characteristics of the AGN as well as the sampling of the data. For this reason, the CCF is rarely interpreted by itself, but is rather used as a tool to estimate the characteristic size of the BLR by calculating the time delay. Table \\ref{table} compares integrated time delays from RLI to time delays determined using cross correlation methods by \\cite{2009ApJ...705..199B} for \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace and \\cite{2010ApJ...716..993B} for \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace and \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace. Fig. \\ref{fig:result-misty-bins} compares velocity-resolved time delays from RLI to velocity-resolved time delays from cross correlation of the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace line by \\cite{2009ApJ...705..199B}. The comparisons for each object are described in Section \\ref{sec:results-for-each-agn}. See the previous section for a description of how the RLI time delays are estimated from the response functions.\n\n{\\sc javelin \\xspace} finds the time delay by using a top hat model for the transfer function \\citep{2011ApJ...735...80Z}. Light curve variability is modelled using a CAR(1) process, which is the same process used to model the continuum variability in our RLI method. Like the cross-correlation method, {\\sc javelin \\xspace} provides only a scalar time delay as a measure of the BLR structure. \\cite{2013ApJ...773...90G} determined time delays using {\\sc javelin \\xspace} on the LAMP 2008 dataset. We list time delays from \\cite{2013ApJ...773...90G} in Table \\ref{table} alongside results from cross-correlation and RLI for comparison. All {\\sc javelin \\xspace} time delays $\\ifmmode \\tau_{\\rm JAVELIN \\xspace} \\else $\\tau_{\\rm JAVELIN}$\\fi \\xspace$ quoted in the text are also from \\cite{2013ApJ...773...90G}.\n\nThe dynamical modelling method \\citep{2011ApJ...730..139P,2014MNRAS.445.3055P} implements a full three-dimensional model of the BLR. The BLR emission is modelled as coming from a number of point particles that are drawn from a parametrized phase space distribution and linearly reprocess radiation from a central source. All measured parameters have a direct physical interpretation, which means that the black hole mass $M_\\textrm{BH}$ and the virial factor $f$ can be determined directly. Dynamical modelling results include fully velocity-resolved transfer functions. Because the method is based on a physical model of the BLR, the response will be correlated across velocity bins. This is contrary to our implementation of regularized linear inversion, where each velocity bin is analysed individually.\n\nAnother important difference between dynamical modelling and RLI is that, because dynamical modelling traces photons through the BLR to the observer, the resulting velocity--delay maps are transfer functions representing the absolute reprocessed emission. This is contrary to RLI which subtracts the mean component of the light curves to only calculate the response in the emission-line to variations in the continuum, thus measuring response (rather than transfer) functions. However, in the current implementation of dynamical modelling by \\cite{2011ApJ...730..139P}, the responsivity of the BLR gas is assumed to be constant throughout the BLR. In other words, the response of the BLR is correlated one-to-one with the emissivity distribution, regardless of the level of ionizing continuum. Because of this, the transfer functions and response functions will be directly proportional, and we can compare response maps from RLI to the velocity-resolved transfer functions of dynamical modelling directly.\n\nMaximum entropy methods \\citep{1984MNRAS.211..111S,1991ApJ...371..541K,1991ApJ...367L...5H,1994ASPC...69...23H} work by selecting solutions to the convolution problem (Equation \\ref{eq:continous_transferequation}) that simultaneously provide a good fit to the data while being as simple (smooth) as possible. The transfer functions are proposed based on a number of assumptions about the form of the solutions. This means that, like the dynamical modelling method, the derived response values will be dependent across velocity bins and the selection of the solution with the maximum entropy tends to result in very smooth velocity--delay maps. This is similar to RLI except that we impose no correlations between velocity bins, so any correlations between bins will be driven purely by the data. We compare our results to the response map for \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace in Arp 151 from maximum entropy \\citep{2010ApJ...720L..46B} and dynamical modelling \\citep{2014MNRAS.445.3073P} in Fig. \\ref{fig:result-comparison-2d}. The comparisons are described in Section \\ref{sec:results-arp151}.\n\n\\subsection{Results for each AGN} \\label{sec:results-for-each-agn}\n\n\\begin{table*}\\centering\n\\begin{tabular}{lcccccc}\\toprule\nObject   & Continuum & Emission  & Wavelength  & \\multicolumn{3}{|c|}{Time delay}                                \\\\\n         & Band      & Line      & Range (\\AA) & \\multicolumn{3}{|c|}{(days)}                                    \\\\\n         &           &           &             & $\\ifmmode \\tau_{\\rm CCF \\xspace} \\else $\\tau_{\\rm CCF}$\\fi \\xspace$ &           $\\ifmmode \\tau_{\\rm JAVELIN \\xspace} \\else $\\tau_{\\rm JAVELIN}$\\fi \\xspace$ &             $\\ifmmode \\tau_{\\rm RLI \\xspace} \\else $\\tau_{\\rm RLI}$\\fi \\xspace$               \\\\\n\\midrule\nArp 151  & $B$         & \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace & 6575 - 6825 & $7.8^{+1.0}_{-1.0}$ &                     & $6.8^{+0.9}_{-1.4}$ \\\\\nArp 151  & $B$         & \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace  & 4792 - 4934 & $4.0^{+0.5}_{-0.7}$ & $3.6^{+0.7}_{-0.2}$ & $4.0^{+0.7}_{-0.8}$ \\\\\nArp 151  & $B$         & \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace & 4310 - 4393 & $1.4^{+0.8}_{-0.7}$ &                     & $3.0^{+0.8}_{-0.8}$ \\\\\nMrk 1310 & $B$         & \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace  & 4815 - 4914 & $3.7^{+0.6}_{-0.6}$ & $4.2^{+0.9}_{-0.1}$ & $2.7^{+0.3}_{-0.3}$ \\\\\nNGC 5548 & $V$         & \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace  & 4706 - 5041 & $4.2^{+0.9}_{-1.3}$ & $5.5^{+0.6}_{-0.7}$ & $4.7^{+1.8}_{-1.8}$ \\\\\nNGC 6814 & $V$         & \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace  & 4776 - 4936 & $6.5^{+0.9}_{-1.0}$ & $7.4^{+0.1}_{-0.1}$ & $7.3^{+1.2}_{-1.0}$ \\\\\nSBS 1116+583A & $B$    & \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace  & 4797 - 4926 & $2.3^{+0.6}_{-0.5}$ & $2.4^{+0.9}_{-0.9}$ & $2.0^{+1.1}_{-0.6}$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[The caption]{Time delays from cross-correlation ($\\ifmmode \\tau_{\\rm CCF \\xspace} \\else $\\tau_{\\rm CCF}$\\fi \\xspace$) are reproduced from \\cite{2010ApJ...716..993B}. Time delays from {\\sc javelin \\xspace} ($\\ifmmode \\tau_{\\rm JAVELIN \\xspace} \\else $\\tau_{\\rm JAVELIN}$\\fi \\xspace$) are reproduced from \\cite{2013ApJ...773...90G}. All time delays are given in the rest frame.}\n\\label{table}\n\\end{table*}\n\n\\subsubsection{Arp 151} \\label{sec:results-arp151}\n\nResults for Arp 151 are presented in Fig. \\ref{fig:result-arp151} for \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace and Fig. \\ref{fig:result-arp151-halpha} and \\ref{fig:result-arp151-hgamma} for \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace and \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace respectively. Light curves for Arp 151 are the most variable and highest quality of the LAMP 2008 dataset, which is why we include only \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace and \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace for this object. RLI provides decent fits to the Arp 151 light curves, with some notable outliers at later epochs (e.g. \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace, Fig. \\ref{fig:result-arp151}). Because of the smoothing constraint imposed on the solution we do not expect our code to fit these points. This is similar to what is found in other analyses of the same data \\citep{2010ApJ...720L..46B,2014MNRAS.445.3073P}. Such strong variability on short time-scales is in any case not consistent with models where the BLR has only extended emission (see discussion in Section \\ref{sec:discussion-linear-response}).\n\nThe integrated response function for \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace (Fig. \\ref{fig:result-arp151}) has a plateau from 0 days out to about 7 days where it starts to decrease. The shape of the response function is broader than what is found by maximum entropy methods \\citep{2010ApJ...720L..46B}. We find a median \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace time delay of $\\ifmmode \\tau_{\\rm RLI \\xspace} \\else $\\tau_{\\rm RLI}$\\fi \\xspace = 4.0^{+0.7}_{-0.8}$\\,days, consistent with results from cross-correlation ($\\ifmmode \\tau_{\\rm CCF \\xspace} \\else $\\tau_{\\rm CCF}$\\fi \\xspace = 4.0^{+0.5}_{-0.7}$\\,days) and {\\sc javelin \\xspace} ($\\ifmmode \\tau_{\\rm JAVELIN \\xspace} \\else $\\tau_{\\rm JAVELIN}$\\fi \\xspace = 3.6^{+0.7}_{-0.2}$\\,days). The integrated response function for \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace (Fig. \\ref{fig:result-arp151-halpha}) has a flat low response out to about 6 days after which is rises to a peak response at a time delay of around 8 days, and then drops to slightly negative response after 10 days. The time delay for \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace ($\\ifmmode \\tau_{\\rm RLI \\xspace} \\else $\\tau_{\\rm RLI}$\\fi \\xspace = 6.8^{+0.9}_{-1.4}$) is consistent with that from cross correlation ($\\ifmmode \\tau_{\\rm CCF \\xspace} \\else $\\tau_{\\rm CCF}$\\fi \\xspace = 7.8^{+1.0}_{-1.0}$). The integrated response function for \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace (Fig. \\ref{fig:result-arp151-hgamma}) shows significant response at zero delay with a slight rise to a peak at 3 days after which the response drops to zero beyond 8 days. The time delay from RLI ($\\ifmmode \\tau_{\\rm RLI \\xspace} \\else $\\tau_{\\rm RLI}$\\fi \\xspace = 3.0^{+0.8}_{-0.8}$) is somewhat larger, but consistent with, the result from cross correlation ($\\ifmmode \\tau_{\\rm CCF \\xspace} \\else $\\tau_{\\rm CCF}$\\fi \\xspace = 1.4^{+0.8}_{-0.7}$).\n\nThe velocity--delay map for \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace (Fig. \\ref{fig:result-arp151}) shows that the bulk of the response is centred on the emission line with a time delay of about 5 days, similar to the median delay calculated from the integrated response. There seems to be an area of increased response redward of the line centre from $0 - 1000\\,\\textrm{km}\/\\textrm{s}$. The velocity--delay map for \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace (Fig. \\ref{fig:result-arp151-halpha}) shows a strong response around 10 days centred on the emission line and extending to lower time delays. The origin of the prompt response redward of the line centre is unclear. It may be a spurious feature due to numerical effects, a poorly subtracted continuum (the continuum for \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace was subtracted using a linear fit rather than spectral decomposition as for \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace, see Section \\ref{sec:emission-line-lightcurves}), or perhaps due to residual contamination from \\ifmmode [{\\rm N}\\,{\\sc ii}],~{\\lambda6586}\\,{\\rm \\AA} \\else [N\\,{\\sc ii}],~$\\lambda6586$\\,\\AA\\fi \\xspace. We do not find evidence for a blueward plume from 15 to 20 days (not shown) as seen in the maximum entropy maps in \\cite{2010ApJ...720L..46B}. The velocity--delay map for \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace (Fig. \\ref{fig:result-arp151-hgamma}) shows a broad response with the longest delays at the centre of the line, and progressively decreasing time delays in the wings. We find additional prompt response at the centre of \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace.\n\nThe decrease of the median time delay in Arp 151 from \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace to \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace and \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace has been previously observed (\\citealt{2010ApJ...720L..46B}, see \\citealt{2009NewAR..53..140G} for a review), and may be an effect of the varying optical depth for the Balmer lines. Because the optical depth is larger for the transitions between lower excitation states, \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace photons will have a harder time escaping the BLR without being absorbed. This results in the \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace emission being predominantly directed back towards the ionizing source at the centre, whereas the lines formed by the higher excitation states, \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace and \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace, are progressively more isotropic. That is one possible mechanism by which the median time delay can decline in higher excitation emission lines, while all the Balmer lines originate from hydrogen gas located at similar distances from the ionizing source.\n\n\\begin{figure*}\n    \\includegraphics[totalheight=230pt]{fig5.eps}\n    \\caption{Results from regularized linear inversion of light curves from LAMP 2008 of the Arp 151 \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace emission line. The left panel shows the continuum (upper left panel, black points) and integrated emission-line light curves (middle left panel, black points) along with the integrated response function (lower left panel, black points). The blue shaded region on top of the continuum light curve shows the median and 1$\\sigma$ percentiles from the Gaussian process realizations used to model the uncertainty in the data. The green line on top of the emission line light curve shows the best fitting result from RLI, with the green shaded band indicating the 1$\\sigma$ percentiles. The horizontal grey dashed line in the lower left panel indicates the location of zero response. The right panel shows the velocity--delay map (upper right panel) calculated by regularized linear inversion of light curves for each wavelength bin individually. Below the velocity--delay map, the mean (black line) and RMS (grey line) spectrum is shown as a function of velocity with respect to the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace emission line centre (middle-right panel). Normalized response functions for three velocity bins are shown below the velocity--delay map (lower right panel) with the bins indicated by dashed vertical lines in the velocity--delay map. The colours (blue, green, red) correspond to the Doppler shift with respect to the observer, with blue being negative velocity, green is the central velocity, and red is positive velocity. All time delays in the results figures are in the observed frame.}\n    \\label{fig:result-arp151}\n\\end{figure*}\n\n\\begin{figure*}\n    \\includegraphics[totalheight=230pt]{fig6.eps}\n    \\caption{Same as Fig. \\ref{fig:result-arp151} but for Arp 151, \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace.}\n    \\label{fig:result-arp151-halpha}\n\\end{figure*}\n\n\\begin{figure*}\n    \\includegraphics[totalheight=230pt]{fig7.eps}\n    \\caption{Same as Fig. \\ref{fig:result-arp151} but for Arp 151, \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace.}\n    \\label{fig:result-arp151-hgamma}\n\\end{figure*}\n\nFig. \\ref{fig:result-misty-bins} shows a comparison between RLI and velocity-resolved cross-correlation time delays by \\cite{2009ApJ...705..199B} for a number of velocity bins across the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace emission line. We find excellent agreement with cross-correlation for all velocity bins and confirm the signature of prompt response redward of the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace line centre, while the bulk of the response is at the centre of the emission line at a time delay of around 5 days.\n\nFig. \\ref{fig:result-comparison-2d} shows velocity--delay maps for \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace from dynamical modelling, maximum entropy, and RLI. All methods find prompt response in the red wing of the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace emission line. This is a characteristic feature of BLR models with free-falling gas or a disk of gas containing a hot spot \\citep[see][]{1991ApJ...379..586W,2010ApJ...720L..46B}. Some models also produce prompt red-side response for outflowing gas at particular observed orientations \\citep{1997ApJ...479..200B}. The result from RLI has a stronger response at the line centre, and a weaker prompt response in the red wing compared to dynamical modelling and maximum entropy. In addition RLI finds prompt response at the line centre, which is not seen in the maximum entropy maps.\n\n\\subsubsection{Mrk 1310}\n\nResults for Mrk 1310 are presented in Fig. \\ref{fig:result-mrk1310}. There is more scatter in the light curves of Mrk 1310 compared to Arp 151. Because of the smoothing constraint, RLI is not able to fit the emission line light curve exactly. This is to be expected, as the method mainly picks up large scale variability to avoid problems with noise on shorter time-scales. The scatter in the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace light curve towards the end of the campaign is in any case difficult to reconcile with a simple linear response to the continuum light curve variations near the same epochs.\n\nThe integrated response function for \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace is more peaked than that of Arp 151. The scatter in the derived response functions from our error analysis is also smaller, indicating that the inversion is stable under perturbations in the input light curves. The median time delay of the response function is $\\ifmmode \\tau_{\\rm RLI \\xspace} \\else $\\tau_{\\rm RLI}$\\fi \\xspace = 2.7^{+0.3}_{-0.3}$\\,days. This delay is slightly shorter than, but consistent with, the cross-correlation result ($\\ifmmode \\tau_{\\rm CCF \\xspace} \\else $\\tau_{\\rm CCF}$\\fi \\xspace = 3.7^{+0.6}_{-0.6}$\\,days), while the result from {\\sc javelin \\xspace} ($\\ifmmode \\tau_{\\rm JAVELIN \\xspace} \\else $\\tau_{\\rm JAVELIN}$\\fi \\xspace = 4.2^{+0.9}_{-0.1}$\\,days) is longer by 1 day compared to the time delay from RLI. The integrated response function dips slightly below zero at longer time delays, which could indicate a negative response in the emission-line, although we do not find this to be significant (see discussion in Section \\ref{sec:negative-response}).\n\nThe velocity-resolved response map shows a strong response at the line centre with a time delay around 3 days, similar to the integrated time delay. The width of the response is about $\\pm 1000$\\,\\ifmmode {\\rm km}\\,{\\rm s}^{-1} \\xspace \\else km\\,s$^{-1}$\\fi \\xspace, similar to Arp 151. The delay is fairly constant across the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace line. This result agrees well with velocity-resolved cross-correlation that shows a nearly flat response as a function of velocity, with slightly shorter delays in the wings at $\\pm 1000$\\,\\ifmmode {\\rm km}\\,{\\rm s}^{-1} \\xspace \\else km\\,s$^{-1}$\\fi \\xspace (Fig. \\ref{fig:result-misty-bins}). It is also consistent with dynamical modelling results, with a peak response centred on the emission line at fairly short time delays (Fig. \\ref{fig:result-comparison-2d}, middle left panel).\n\n\\begin{figure*}\n    \\includegraphics[totalheight=230pt]{fig8.eps}\n    \\caption{Same as Fig. \\ref{fig:result-arp151} but for Mrk 1310, \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace.}\n    \\label{fig:result-mrk1310}\n\\end{figure*}\n\n\\subsubsection{NGC 5548}\n\nResults for NGC 5548 are presented in Fig. \\ref{fig:result-ngc5548}. The \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace light curve of NGC 5548 has more scatter relative to variability amplitude compared to Arp 151 and Mrk 1310. Consequently, RLI has a more difficult time fitting this object.\n\nThe integrated response function increases from zero delay up to around 3 days and then has a slowly decreasing plateau out to about 8 days after which it drops off. This yields a median time delay of $\\ifmmode \\tau_{\\rm RLI \\xspace} \\else $\\tau_{\\rm RLI}$\\fi \\xspace = 4.7^{+1.8}_{-1.8}$\\,days, consistent within 1$\\sigma$ with cross-correlation ($\\ifmmode \\tau_{\\rm CCF \\xspace} \\else $\\tau_{\\rm CCF}$\\fi \\xspace = 4.2^{+0.9}_{-1.3}$\\,days) and {\\sc javelin \\xspace} ($\\ifmmode \\tau_{\\rm JAVELIN \\xspace} \\else $\\tau_{\\rm JAVELIN}$\\fi \\xspace = 5.5^{+0.6}_{-0.7}$\\,days).\n\nThe \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace emission line in NGC 5548 is significantly broader compared to the other objects analysed and the variability across the line shows more irregular structure, as is seen in the velocity-resolved response map in the upper right panel of Fig. \\ref{fig:result-ngc5548}. We find a somewhat weak response at the line centre, which peaks at around 5 days. On either side of the line centre at $\\pm 5000$\\,\\ifmmode {\\rm km}\\,{\\rm s}^{-1} \\xspace \\else km\\,s$^{-1}$\\fi \\xspace we find stronger isolated response features with peak time delays around 8 days. The velocity-resolved time delays agree well with cross-correlation (Fig. \\ref{fig:result-misty-bins}). Dynamical modelling and RLI both show a level of prompt response in the red wing of \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace, while the rest of the maps show little similarity (Fig. \\ref{fig:result-comparison-2d}).\n\n\\begin{figure*}\n    \\includegraphics[totalheight=230pt]{fig9.eps}\n    \\caption{Same as Fig. \\ref{fig:result-arp151} but for NGC 5548, \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace.}\n    \\label{fig:result-ngc5548}\n\\end{figure*}\n\n\\subsubsection{NGC 6814}\n\nResults for NGC 6814 are presented in Fig. \\ref{fig:result-ngc6814}. The light curves of NGC 6814 show an appreciable amount of variability. The continuum $V$-band light curve has a clear broad peak at $\\textrm{HJD} - 2454000 = 560\\,\\textrm{days}$ after which it drops off to a slowly rising plateau that extends to shortly after $\\textrm{HJD} - 2454000 = 600\\,\\textrm{days}$. The emission line light curve has the same over-all trend as the continuum, but instead of having a plateau after the initial peak, it rises to almost the same level as the initial peak. Because of this, RLI has a difficult time matching the first and second peaks in the emission line light curve, explaining why the fit is underestimated for the second peak.\n\nThe integrated response function peaks around 8 days and has broad flat wings to either side (Fig. \\ref{fig:result-ngc6814}, lower left panel). The median time delay of $\\ifmmode \\tau_{\\rm RLI \\xspace} \\else $\\tau_{\\rm RLI}$\\fi \\xspace = 7.3^{+1.2}_{-1.0}$\\,days is consistent with results from cross-correlation ($\\ifmmode \\tau_{\\rm CCF \\xspace} \\else $\\tau_{\\rm CCF}$\\fi \\xspace = 6.5^{+0.9}_{-1.0}$\\,days) and {\\sc javelin \\xspace} ($\\ifmmode \\tau_{\\rm JAVELIN \\xspace} \\else $\\tau_{\\rm JAVELIN}$\\fi \\xspace = 7.4^{+0.1}_{-0.1}$\\,days).\n\nThe velocity-resolved response map for NGC 6814 shows a clear isolated response peak centred on the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace emission line (Fig. \\ref{fig:result-ngc6814}, upper right panel). There is a somewhat peculiar dip in the response map just blueward ($\\sim -100$\\,\\ifmmode {\\rm km}\\,{\\rm s}^{-1} \\xspace \\else km\\,s$^{-1}$\\fi \\xspace) of the line centre. Further away at around $\\pm 500$\\,\\ifmmode {\\rm km}\\,{\\rm s}^{-1} \\xspace \\else km\\,s$^{-1}$\\fi \\xspace the response peaks on either side of the line centre and then drops off to zero beyond $\\pm 1500$\\,\\ifmmode {\\rm km}\\,{\\rm s}^{-1} \\xspace \\else km\\,s$^{-1}$\\fi \\xspace. The velocity-resolved time delays agree well with cross-correlation at centre of the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace line (Fig. \\ref{fig:result-misty-bins}). At lower velocities we find a slightly longer time delay compared to the cross-correlation result. At higher velocities, blueward of the line, RLI was unable to recover a time delay due to a lack of emission line response. Looking at the fully resolved response map from RLI in Fig. \\ref{fig:result-ngc6814} we see that RLI calculates little response in the wings beyond $\\pm 1500$\\,\\ifmmode {\\rm km}\\,{\\rm s}^{-1} \\xspace \\else km\\,s$^{-1}$\\fi \\xspace, which might explain the discrepancy with the cross-correlation results. Dynamical modelling of NGC 6814 tends to prefer shorter time delays than what is found using other methods (see Fig. \\ref{fig:result-comparison-2d}). The highest velocity bin failed to produce a time delay estimate due to a noisy response function.\n\n\\begin{figure*}\n    \\includegraphics[totalheight=230pt]{fig10.eps}\n    \\caption{Same as Fig. \\ref{fig:result-arp151} but for NGC 6814, \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace.}\n    \\label{fig:result-ngc6814}\n\\end{figure*}\n\n\\subsubsection{SBS 1116+583A}\n\nResults for SBS 1116+583A are presented in Fig. \\ref{fig:result-sbs1116}. The emission line light curve of SBS 1116+583A lacks significant features above the noise level. As a result the RLI fit does not match the emission line light curve well. The results for this object, including the velocity--delay map, should thus be interpreted with caution.\n\nThe integrated response function prefers a relatively short time delay of $\\ifmmode \\tau_{\\rm RLI \\xspace} \\else $\\tau_{\\rm RLI}$\\fi \\xspace = 2.0^{+1.1}_{-0.6}$\\,days, which is slightly shorter, but fully consistent with cross-correlation ($\\ifmmode \\tau_{\\rm CCF \\xspace} \\else $\\tau_{\\rm CCF}$\\fi \\xspace = 2.3^{+0.6}_{-0.5}$\\,days) and {\\sc javelin \\xspace} ($\\ifmmode \\tau_{\\rm JAVELIN \\xspace} \\else $\\tau_{\\rm JAVELIN}$\\fi \\xspace = 2.4^{+0.9}_{-0.9}$\\,days).\n\nThe velocity-resolved response map for SBS 1116+583A (Fig. \\ref{fig:result-sbs1116}) shows a multimodal response map with a broad response component at short time delays, around 2 days, and an additional isolated response at 10 days at the centre of the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace line. This is a good example of the flexibility of RLI to reconstruct complicated velocity--delay maps. The velocity--delay map shows some resemblance to models in which the BLR is confined to an isotropically illuminated disk \\citep[see Fig. 5 in][]{2010ApJ...720L..46B}.\n\nIt is not very meaningful to calculate a single time delay from a multimodal response function, even so we apply the same procedure for determining the time delay as for the previous objects (see Section \\ref{sec:the-time-delay}). The resulting velocity-resolved time delays agree well with cross-correlation, showing longer time delays at the centre of the emission line and shorter time delays in the wings (Fig. \\ref{fig:result-misty-bins}). We note that the time delay calculated close to the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace line centre fall between the two modes in the response map, and is thus not well constrained. We include these time delays for comparison with the velocity-resolved cross-correlation, but they should be interpreted with caution, as suggested by the large error-bars. The lowest velocity bin failed to produce a time delay estimate due to a noisy response function.\n\nThe results for SBS 1116+583A from RLI are in general agreement with dynamical modelling, although dynamical modelling does not reproduce the same strong response at $\\tau = 10$\\,days and generally prefers shorter time delays at the line centre (see Fig. \\ref{fig:result-comparison-2d}).\n\n\\begin{figure*}\n    \\includegraphics[totalheight=230pt]{fig11.eps}\n    \\caption{Same as Fig. \\ref{fig:result-arp151} but for SBS 1116+583A, \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace.}\n    \\label{fig:result-sbs1116}\n\\end{figure*}\n\n\\begin{figure*}\n    \\includegraphics[width=1.0\\linewidth]{fig12.eps}\n    \\caption{Comparison of \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace time delays from regularized linear inversion with cross-correlation results from \\protect\\cite{2009ApJ...705..199B}. The upper panels show time delays as a function of velocity with respect to the \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace emission line centre. Red filled circles with error bars show time delays from this paper. Black open squares with error bars are from \\protect\\cite{2009ApJ...705..199B}. Light curves are obtained by integrating the emission in each velocity bin indicated by the horizontal error-bars. The time delays from RLI are calculated as the median of the positive part of the response (see Section \\ref{sec:the-time-delay}). The lower panels show the mean (black) and RMS (grey) spectra. All time delays in this figure are in the observed frame.}\n    \\label{fig:result-misty-bins}\n\\end{figure*}\n\n\\begin{figure*}\n    \\includegraphics[width=1.0\\linewidth]{fig13.eps}\n    \\caption[]{Comparison of velocity--delay maps from regularized linear inversion to maximum likelihood for Arp 151 \\citep{2010ApJ...720L..46B} and dynamical modelling for all objects \\citep{2014MNRAS.445.3073P}. All three methods used the same dataset from LAMP 2008 (see Section \\ref{sec:data}). Regularized linear inversion and dynamical modelling used spectral decomposition for continuum subtraction, while the maximum likelihood analysis used a linear fit to remove the continuum level below the line.}\n    \\label{fig:result-comparison-2d}\n\\end{figure*}\n\n\\section{Discussion} \\label{sec:discussion}\n\n\\subsection{Assumption of a constant linear response} \\label{sec:discussion-linear-response}\nLong temporal baseline multi-epoch observations of AGN reveal significant changes in the broad emission line equivalent widths \\citep[e.g.][]{1990ApJ...357..338K,1992AJ....103.1084P,2003ApJ...587..123G,2004MNRAS.352..277G} and line profiles \\citep[e.g.][]{1996ApJ...466..174W,2001ApJ...554..245S,2007ApJ...668..708S} on time-scales of months to years. These effects could possibly account for some of the discrepancies found when matching longer time-scale variability patterns using linear reverberation mapping, such as the inability of RLI to match the second peak in the emission line light curve in NGC 6814 (Fig. \\ref{fig:result-ngc6814}).\n\nIn addition to long time-scale non-linearities, some light curves of the LAMP 2008 dataset show high-variability features (e.g. late epochs in Mrk 1310, Fig. \\ref{fig:result-mrk1310}). If these features are associated with the BLR, they are inconsistent with models in which the BLR emission comes from an extended region. This remains true even after exclusion of potentially unreliable spectra in the LAMP 2008 dataset, as described in section \\ref{sec:emission-line-lightcurves}. Whether these outliers are due to systematic measurement uncertainties affecting individual epochs, an indication of unknown processes in the BLR, or a combination of these, is unclear.\n\nWhatever the origin of non-linear features in the emission line light curves, current reverberation mapping techniques, such as cross-correlation, maximum entropy, and RLI, will be insufficient to describe them. Even so these methods remain valuable tools for testing models of the BLR and investigate departures from linearity in AGN light curves. Implementation of photoionization physics into dynamical modelling codes may be a way to probe non-linear processes in the BLR.\n\n\\subsection{Ionizing versus observed continuum}\nStudies suggest a non-negligible time delay between the ionizing UV continuum and the optical continuum on the order of 1 day in NGC 7469 and NGC 5548 \\citep[e.g.][]{1998ApJ...500..162C,2014MNRAS.444.1469M,2015ApJ...806..129E}. This means that the $V$ and $B$ band continuum light curves originate from a region comparable in size to some of the shorter time delays found for Balmer emission lines. This extended continuum emission region will introduce geometric smoothing in the optical continuum light curves used for reverberation mapping. If the delay between the ionizing continuum and the measured optical continuum is constant on time-scales of reverberation mapping campaigns, linear reverberation methods can still be applied, but the interpretation of the response functions will have to be revised to include not just the geometry of the BLR, but also the geometry of the extended optical continuum emission region. It will be interesting to extend this analysis in the future by performing RLI of optical emission line variability as driven by UV continuum to assess the magnitude of these systematic effects.\n\n\\subsection{Response function errors}\nThe smoothing condition used to regularize the inversion problem introduces correlations in the response functions by linking nearby points through the first-order derivative. This correlation is an implicit assumption of the method and it reflects our belief that the emission line light curve is produced by a superposition of photons emitted from a fairly homogeneous distribution of gas in the BLR. The error bars on the response functions should thus not be interpreted as errors on the individual points of the response function, but rather as an indication of the range of solutions we get by re-interpolating the continuum light curve and re-sampling the emission line light curve.\n\nIn our current implementation of RLI no correlation in response is imposed between response velocity bins. This can be witnessed by the vertical streaks in the presented response maps. Not imposing a correlation has the advantage that any correlation between velocity-bins in the final response maps must be driven by the data, or at least systematic effects correlated with the emission lines. It seems natural to assume that the BLR emissivity is correlated in position as well as velocity space, motivating a smoothing constraint in the velocity domain, as is also assumed in dynamical modelling and maximum entropy methods. Because we found the velocity-bins to be naturally correlated in the response maps, and in order to keep the method fast and simple, we decided not to impose any smoothing in the velocity-domain.\n\n\\subsection{Negative response values} \\label{sec:negative-response}\nSome models suggest that negative values occur naturally in AGN response functions \\citep{1993ApJ...404..570S,1993MNRAS.263..149G,2014MNRAS.444...43G}, and negative response values have recently been observed in X-rays \\citep{2007MNRAS.382..985M,2009Natur.459..540F}. Negative response can arise in different ways, such as if an increase in ionizing flux causes part of the responding gas to become fully ionized, or if the BLR structure changes on a dynamical time-scale, such as if the continuum decreases while new material falls into the BLR. Note that in both of these examples the BLR structure changes on time-scales of the same order as the time delay. This means that the assumption of a constant linear response breaks down and more general methods, such as dynamical modelling, are required to account for the time varying part of the response.\n\nAn advantage of RLI over other reverberation mapping methods, is that it does not require the response function to be everywhere positive. We find some preference for negative response in some of the objects analysed (e.g. Mrk 1310, Fig. \\ref{fig:result-mrk1310}). While none of the response functions show strong evidence for negative response these results may be an indication that more complicated processes are taking place in the BLR, than what can be contained in a simple constant linear response model. However, negative values occur naturally as artefacts when linear inversion methods are applied to discrete and noisy data, even if the true response is everywhere positive. This is somewhat in analogy to aliasing distortions when dealing with discrete Fourier transforms. These artefacts are seen as ringing effects, such as observed in the inversion of the simulated 1D data (see Fig. \\ref{fig:sim-1d}). It is not possible to impose positivity while keeping an analytical expression for the response function \\citep{1994PASP..106.1091V}, and since simplicity and flexibility were the main motivations for using RLI, we leave the positivity constraint to other methods such as maximum entropy and dynamical modelling. With these considerations in mind we emphasize that all the negative response values found in our analysis of the LAMP 2008 data are of low statistical significance. In particular, the integrals of the response functions are positive for all objects analysed.\n\n\\section{Summary and conclusion} \\label{sec:summary-and-conclusion}\n\nBuilding on previous work by \\cite{1995ApJ...440..166K}, we develop a new method for reverberation mapping based on regularized linear inversion that includes statistical modelling of the AGN continuum light curves. In this implementation, response function errors are evaluated using a combination of Gaussian process fitting of the continuum light curves as well as bootstrap resampling of the emission line light curves. Regularized linear inversion has the advantage over other reverberation mapping methods that it makes few assumptions about the shape of the response function, and it allows for negative values in the response function. In addition, because the method is based on an analytical solution to the transfer equation, it is very computationally efficient. The scale of regularization is a function of the data only, such that no user-input is required, except for the time baseline of the analysis. This means that flexible response functions can be calculated quickly and consistently.\n\nWe test the method on simulated data and show that it is able to recover unimodal and multimodal response functions as well as response functions with negative values (see Fig. \\ref{fig:sim-1d}). We further test the method on data simulated using dynamical modelling and show that it is able to recover 2-dimensional velocity--delay maps (see Fig. \\ref{fig:sim-2d}).\n\nWe apply RLI to light curves of five nearby Seyfert 1 galaxies, taken by the LAMP 2008 collaboration, and present time delays, integrated response functions and velocity--delay maps for the H$\\beta$ line in all objects, as well as H$\\alpha$ and H$\\gamma$ for Arp 151. This is the first time a reverberation mapping method allowing for negative response has been applied to a large dataset. Our results are in good agreement with previous studies based on cross-correlation and dynamical modelling, offering a powerful corroboration of the assumptions underlying these reverberation mapping methods.\n\nThe main physical results of this work can be summarized as follows:\n\n\\begin{enumerate}\n    \\item Despite using a very different method for calculating time delays we find that our results are in good agreement with results from cross correlation.\n    \\item We find asymmetric response of the H$\\beta$ emission line in Arp 151, with prompt response in the red wing of the emission line, consistent with models that include bulk gas flows in the BLR.\n    \\item We confirm previous studies finding that lines originating from high excitation states, such as \\ifmmode {\\rm H}\\,{\\gamma \\xspace} \\else H\\,{$\\gamma$}\\fi \\xspace, have shorter time delays compared to lines originating from lower excitation states, such as \\ifmmode {\\rm H}\\,{\\alpha \\xspace} \\else H\\,{$\\alpha$}\\fi \\xspace and \\ifmmode {\\rm H}\\,{\\beta \\xspace} \\else H\\,{$\\beta$}\\fi \\xspace.\n    \\item While some objects, such as Arp 151 and Mrk 1310, show a degree of negative response at longer time delays, we find no conclusive evidence for negative response values in any of the objects.\n\\end{enumerate}\n\nThis work demonstrates that regularized linear inversion is a valuable tool for reverberation mapping and a worthwhile complementary method for analysing high quality reverberation mapping datasets. The recent multi-wavelength reverberation mapping campaign of NGC 5548 \\citep{2015ApJ...806..128D,2015ApJ...806..129E}, including simultaneous ground-based monitoring in the optical and spectroscopy in the ultraviolet with the Cosmic Origins Spectrograph on the \\emph{Hubble Space Telescope}, will provide exceptionally high quality data and thereby a great opportunity to test the RLI method and its underlying assumptions.\n\n\\section*{acknowledgements}\n\nWe thank the referee, Rick Edelson, for comments and suggestions that led to substantial improvements of the manuscript.\nWe are grateful to Julian Krolik, Mike Goad, Kirk Korista and Brendon Brewer for enlightening input, as well as Anthea King and Darach Watson for discussions leading to the improvement of this work.\nThe Dark Cosmology Centre (DARK) is funded by the Danish National Research Foundation.\nAP acknowledges support from the NSF through the Graduate Research Fellowship Program and from the University of California Santa Barbara through a Dean's Fellowship.\nTT acknowledges support from the Packard foundation in the form of a Packard Fellowship.\nTT thanks the American Academy in Rome and the Observatory of Monteporzio Catone for their kind hospitality during the writing of this manuscript.\nResearch by TT is supported by NSF grant AST-1412693 (New Frontiers in Reverberation Mapping).\nDP acknowledges support through the EACOA Fellowship from The East Asian Core Observatories Association, which consists of the National Astronomical Observatories, Chinese Academy of Science (NAOC), the National Astronomical Observatory of Japan (NAOJ), Korean Astronomy and Space Science Institute (KASI), and Academia Sinica Institute of Astronomy and Astrophysics (ASIAA).\nResearch by AJB is supported by NSF grants AST-1108835 and AST-1412693.\nResearch by MCB is supported by NSF CAREER grant AST-1253702.\nThis work makes use of the Lick AGN Monitoring Project 2008 dataset supported by NSF grants AST-0548198 (UC Irvine), AST-0607485 and AST-0908886 (UC Berkeley), AST-0642621 (UC Santa Barbara), and AST-0507450 (UC Riverside).\nThis work makes use of the {\\sc{eigen}} header library \\citep{eigenweb}. All figures are produced in {\\sc{matplotlib}} \\citep{Hunter:2007}.\n\n\\clearpage\n\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAll available experimental data in particle physics are consistent with\nthe Standard Model (SM) of strong and electroweak interactions, provided\n[\\ref{LPHEP91}]\n\\begin{equation}\n\\label{mtop}\n91 \\; {\\rm GeV} < m_t < 180 \\; {\\rm GeV} \\;\\;\\; (95 \\% \\, {\\rm c.l.})\n\\end{equation}\nand\n\\begin{equation}\n\\label{mhiggs}\n57 \\; {\\rm GeV} < m_{\\varphi} \\;\\;\\; (95 \\% \\, {\\rm c.l.}) \\, ,\n\\end{equation}\nwhere $m_t$ and $m_{\\varphi}$ denote the masses of the top quark\nand of the SM Higgs boson, respectively. The lower limits on\n$m_t$ and $m_{\\varphi}$ are obtained from unsuccessful direct\nsearches at the Tevatron and LEP. The upper limit on $m_t$ is\nobtained as a consistency condition of the SM, after the inclusion of\nradiative corrections, with the high-precision data on electroweak\nphenomena. Strong evidence for the existence of the top quark, with\nthe quantum numbers predicted by the SM, is also provided by the\nprecise measurements of the weak isospin of the $b$-quark.\nIn the case of the Higgs boson, the situation is radically different.\nThere is no experimental evidence yet that the minimal SM Higgs\nmechanism is the correct description of electroweak symmetry breaking.\nFortunately, present and future accelerators will give decisive\ncontributions towards the experimental solution of this problem.\nIf the SM description of the Higgs mechanism is correct, LEP or the LHC\nand SSC should be able to find the SM Higgs boson and study its\nproperties.\n\nDespite its remarkable successes, the SM can only be regarded as an\neffective low-energy theory, valid up to some energy scale $\\Lambda$\nat which it is replaced by some more fundamental theory. Certainly\n$\\Lambda$ is less than the Planck scale, $M_{\\rm P} \\sim 10^{19} \\; {\\rm GeV}$, since\none needs a theory of quantum gravity to describe physics at these energies.\nHowever, there are also arguments, originating precisely from the study of\nthe untested Higgs sector \\footnote{For reviews of Higgs boson physics see,\ne.g., refs. [\\ref{higgs},\\ref{hunter}]}, which suggest that $\\Lambda$ should\nrather be close to the Fermi scale $G_{\\rm F}^{-1\/2} \\sim 300 \\; {\\rm GeV}$. The\nessence of these arguments is the following. Triviality of the $\\lambda\n\\varphi^4$ theory, absence of Landau poles and perturbative unitarity imply\nthat within the SM $m_{\\varphi} <$ 600--800 GeV. If one then tries to extend\nthe validity of the SM to energy scales $\\Lambda \\gamma\\gamma G_{\\rm F}^{-1\/2}$, one is\nfaced with the fact that in the SM there is no symmetry to justify the\nsmallness of the Higgs mass with respect to the (physical) cut-off\n$\\Lambda$. This is apparent from the fact that in the SM one-loop\nradiative corrections to the Higgs mass are quadratically divergent;\nit is known as the naturalness (or hierarchy) problem of the SM.\nMotivated by this problem, much theoretical effort has been devoted\nto finding descriptions of electroweak symmetry breaking which modify the\nSM at scales $\\Lambda \\sim G_{\\rm F}^{-1\/2}$. The likely possibility of\nsuch modifications is the reason why, when discussing the experimental\nstudy of electroweak symmetry breaking, one should not be confined to\nthe SM Higgs, but also consider alternatives to it, which might have\nradically different\nsignatures, and in some cases be more difficult to detect than the SM Higgs.\nOnly after a thorough study of these alternatives can one be\ndefinite about the validity of the so-called `no-lose theorems', stating that\nthe physics signatures of electroweak symmetry breaking cannot be missed\nat LEP or the LHC and SSC.\n\nWhen considering alternatives to the minimal SM Higgs sector, it is natural to\nconcentrate on models which are theoretically motivated, phenomenologically\nacceptable and calculationally well-defined. The most attractive possibility\nsatisfying these criteria is the Minimal Supersymmetric Standard Model (MSSM)\n[\\ref{MSSM}].\nThis possibility is theoretically motivated by the fact that low-energy\nsupersymmetry, effectively broken in the vicinity of the electroweak scale,\nis the only theoretical framework that can {\\em naturally} accommodate\n{\\em elementary} Higgs bosons. The simplest and most predictive realization of\nlow-energy supersymmetry is the MSSM, defined by 1)~minimal gauge group:\n$SU(3)_C \\times\nSU(2)_L \\times U(1)_Y$; 2) minimal particle content: three generations of\nquarks and leptons and two Higgs doublets, plus their superpartners; 3) an\nexact discrete $R$-parity, which guarantees (perturbative) baryon- and\nlepton-number conservation: $R=+1$ for SM particles and Higgs bosons,\n$R=-1$ for their superpartners; 4) supersymmetry breaking parametrized by\nexplicit but soft breaking terms: gaugino and scalar masses and trilinear\nscalar couplings.\n\nBesides the solution of the naturalness problem, there are other virtues of\nthe MSSM which are not shared by many other alternatives to the SM Higgs and\nshould also be recalled to further motivate our study. The MSSM successfully\nsurvives all the stringent phenomenological tests coming from precision\nmeasurements at LEP: in most of its parameter space, the MSSM predictions\nfor the LEP observables are extremely close to the SM predictions, evaluated\nfor a relatively light SM Higgs [\\ref{susyrad}]. This can be compared, for\nexample, with the simplest technicolor models, which are ruled out by the\nrecent LEP data [\\ref{guido}]. Again in contrast with models of dynamical\nelectroweak symmetry breaking,\nthe MSSM has a high degree of predictivity, since all masses and couplings\nof the Higgs boson sector can be computed, at the tree-level, in terms of\nonly two parameters, and radiative corrections can be kept under control:\nin particular, cross-sections and branching ratios for the MSSM Higgs bosons\ncan be reliably computed in perturbation theory. Furthermore, it is\nintriguing that the idea of grand unification, which fails in its minimal\nnon-supersymmetric implementation, can be successfully combined with\nthat of low-energy supersymmetry: minimal supersymmetric grand unification\npredicts a value of $\\sin^2 \\theta_W (m_Z)$ which is in good agreement with\nthe measured one, and a value of the grand-unification mass which\ncould explain why proton decay has escaped detection\nso far [\\ref{ross}]. Finally, as a consequence of $R$-parity, the lightest\nsupersymmetric\nparticle, which is typically neutral and weakly interacting, is absolutely\nstable, and thus a natural candidate for dark matter.\n\nAny consistent supersymmetric extension of the SM requires at least two Higgs\ndoublets, in order to give masses to all charged quarks and leptons and to\navoid gauge anomalies originated by the spin-$1\/2$ higgsinos. The MSSM\nhas just two complex Higgs doublets, with the following $SU(3)_C \\times\nSU(2)_L \\times U(1)_Y$ quantum numbers ($Q=T_{3L}+Y$):\n\\begin{equation}\nH_1 \\equiv\n\\pmatrix\n{H_1^0 \\cr H_1^- \\cr}\n\\sim ( 1, 2, -1\/2) \\, ,\n\\;\\;\\;\\;\\;\nH_2 \\equiv\n\\pmatrix\n{H_2^+ \\cr H_2^0}\n\\sim ( 1, 2, +1\/2) \\, .\n\\end{equation}\nOther non-minimal models can be constructed, but they typically increase\nthe number of parameters without correspondingly increasing the physical\nmotivation. For example, the simplest non-minimal model [\\ref{nmssm}]\nis constructed\nby adding a singlet Higgs field $N$ and by requiring purely trilinear\nsuperpotential couplings. In this model, the Higgs sector has already\ntwo more parameters than in the MSSM. Folklore arguments in favour of\nthis model are that it avoids the introduction of a supersymmetry-preserving\nmass parameter $\\mu \\sim G_{\\rm F}^{-1\/2}$ and that the homogeneity properties\nof its superpotential recall the structure of some superstring effective\ntheories. A closer look, however, shows that these statements should be\ntaken with a grain of salt. First, in the low-energy effective theory\nwith softly broken global supersymmetry, the supersymmetric mass $\\mu \\sim\nG_{\\rm F}^{-1\/2}$ could well be a remnant of local supersymmetry breaking,\nif the underlying supergravity theory has a suitable structure of\ninteractions [\\ref{muproblem}]. Moreover, when embedded in a grand-unified\ntheory, the non-minimal model with a singlet Higgs field might develop\ndangerous instabilities [\\ref{singlet}]. Also, the trilinear $N^3$\nsuperpotential coupling, which is usually invoked to avoid a massless axion,\nis typically absent in string models. We therefore concentrate in this paper\non the MSSM only.\n\nThe previous considerations should have convinced the reader that the Higgs\nsector of the MSSM is worth a systematic study in view of the forthcoming\nhadron colliders, the LHC and SSC. To perform such a study, one has to\ndeal with the rich particle spectrum of the MSSM. As discussed in more detail\nlater, the Higgs sector contains one charged ($H^\\pm$) and three neutral\n($h,H,A$) physical states. At the classical level, all\nHiggs boson masses and couplings can be expressed in terms of\ntwo parameters only, for example $m_A$ and $\\tan \\beta \\equiv\nv_2\/v_1$. This makes the discussion more complicated than in the\nSM, where the only free parameter in the Higgs sector is the Higgs\nmass, $m_{\\varphi}$.\nIn addition, when considering production and decay of Higgs\nbosons, the whole particle spectrum of the model has to be\ntaken into account. As in the SM, the top-quark mass $m_t$\nis an important parameter: barring the fine-tuned cases of a very light\nstop squark, or of charginos very close in mass to $m_Z\/2$,\nthe limits of eq. (1) are also valid in the MSSM [\\ref{susyrad}].\nIn contrast with the SM, also the supersymmetric $R$-odd particles\n(squarks, sleptons, gauginos, higgsinos) can play an important role in\nthe production and decay of supersymmetric Higgs bosons [\\ref{gunion}].\nClearly, to keep track simultaneously of all supersymmetric-particle masses\nwould be a difficult (and confusing) task. We shall therefore\nconcentrate, following the approach of ref. [\\ref{kz}], on the limiting\ncase where all supersymmetric-particle masses\nare heavy enough not to play an important role in the phenomenology\nof supersymmetric Higgs bosons. This is phenomenologically meaningful,\nsince one can argue that a relatively light\nsupersymmetric-particle spectrum is likely to give independent,\ndetectable signatures at LEP or at the LHC and SSC.\n\nAnother motivation for the present study is the recent realization\n[\\ref{pioneer}] that\ntree-level formulae for Higgs-boson masses and couplings can receive large\nradiative corrections, dominated by the exchange of virtual top and bottom\nquarks and squarks in loop diagrams. For example, tree-level formulae would\npredict the existence of a neutral Higgs boson ($h$) lighter than the $Z$.\nIf this were true, there would be a chance of testing completely\nthe MSSM Higgs sector at LEP~II, with no need for the LHC and SSC.\nHowever, $m_h$ can receive a large positive shift by radiative\ncorrections, which can push $h$ beyond the LEP II discovery reach.\nThis makes the LHC and SSC important, not only for a possible confirmation\nof a SUSY Higgs signal seen at LEP, but also for the exploration of\nthe parameter space inaccessible to LEP.\n\nThe phenomenology of the SM Higgs at the LHC\n[\\ref{zkwjsaa}--\\ref{evian}]\nand SSC [\\ref{snow},\\ref{sdcgem}] has been intensely\nstudied over the last years: a lot of effort was required to prove\n[\\ref{zkwjsaa},\\ref{froid}], at\nleast on paper, that the combination of LEP and the LHC\/SSC is sufficient to\nexplore the full theoretically allowed range of SM Higgs masses.\nHowever, those results cannot be directly applied to the neutral states of\nthe MSSM, since there are important differences in the couplings, and of\ncourse one needs to analyse separately the case of the charged Higgs.\nEven in the case in which all the $R$-odd supersymmetric particles are very\nheavy, the Higgs sector of the MSSM represents a non-trivial extension of\nthe SM case. Also several studies of the MSSM Higgs sector have already\nappeared in the literature. In particular, tree-level formulae for\nthe MSSM Higgs boson masses and couplings are available, and they have\nalready been used to compute cross-sections and branching ratios for\nrepresentative values of the MSSM parameters [\\ref{hunter}].\nHowever, the existing\nanalyses are not systematic enough to allow for a definite conclusion\nconcerning the discovery potential of the LHC and SSC, even in the simple\ncase of large sparticle masses. Also, they do not include radiative\ncorrections to Higgs-boson masses and couplings. In this paper we plan\nto help filling these two gaps.\nThe strategy for a systematic study of neutral supersymmetric Higgs\nbosons at the LHC was outlined in ref. [\\ref{kz}]: however, at that time\nradiative corrections were not available, and also the $\\gamma \\gamma$\nbranching ratio was incorrectly encoded in the computer program.\nOur goal will be to see if LEP and the LHC\/SSC can be sensitive\nto supersymmetric Higgs bosons in the whole $(m_A,\\tan \\beta)$ space.\n\nThe structure of the paper is the following. In sect.~2\nwe review the theoretical structure of the Higgs sector\nof the MSSM, including radiatively corrected formulae for\nHiggs-boson masses and couplings. In sect.~3 we survey\nthe present LEP I limits, after the inclusion of radiative corrections,\nand the plausible sensitivity of LEP II. In sect.~4\nwe present branching ratios and widths of neutral and\ncharged supersymmetric Higgs bosons. In sect.~5\nwe compute the relevant cross-sections at the LHC and\nSSC, and in sect.~6 we examine in some detail the most\npromising signals for\ndiscovery. Finally, sect.~7 contains a concluding discussion\nof our results and of prospects for further work.\n\\section{Higgs masses and couplings in the MSSM}\nFor a discussion of Higgs-boson masses and couplings in the MSSM, the\nobvious starting point is the tree-level Higgs potential [\\ref{MSSM}]\n\\begin{eqnarray}\n\\label{V0}\nV_0 &  = &  m_1^2 \\left| H_1 \\right|^2 + m_2^2 \\left| H_2\n\\right|^2 + m_3^2 \\left( H_1 H_2 + {\\rm h.c.} \\right) \\nonumber \\\\\n& &\n+ {1 \\over 8}g^2 \\left( H_2^{\\dagger} {\\vec\\sigma} H_2\n+ H_1^{\\dagger} {\\vec\\sigma} H_1 \\right)^2 +\n{1 \\over 8}g'^2 \\left(  \\left| H_2 \\right|^2 -\n\\left| H_1 \\right|^2 \\right)^2,\n\\end{eqnarray}\nwhere $m_1^2,m_2^2,m_3^2$ are essentially arbitrary mass parameters,\n$g$ and $g'$ are the $SU(2)$ and $U(1)$ coupling constants,\nrespectively, and ${\\vec\\sigma}$ are the Pauli matrices. $SU(2)$\nindices are left implicit and contracted in the obvious way. It is not\nrestrictive to choose $m_3^2$ real and negative, and then the vacuum\nexpectation values $v_1 \\equiv \\langle H_1^0 \\rangle$ and $v_2 \\equiv\n\\langle H_2^0 \\rangle$ real and positive.\n\nThe physical states of the MSSM Higgs sector are three neutral bosons\n(two CP-even, $h$ and $H$, and one CP-odd, $A$) and a charged boson,\n$H^{\\pm}$. A physical constraint comes from the fact that the combination\n$(v_1^2+v_2^2)$, which determines the $W$ and $Z$ boson masses, must\nreproduce their measured values. Once this constraint is imposed, in the Born\napproximation the MSSM Higgs sector contains only two independent parameters.\nA convenient choice, which will be adopted throughout this paper, is to take as\nindependent parameters $m_A$, the physical mass of the CP-odd neutral boson,\nand $\\tan \\beta \\equiv v_2 \/ v_1$, where $v_1$ gives mass to charged leptons\nand quarks of charge $-1\/3$, $v_2$ gives mass to quarks of charge $2\/3$.\nThe parameter $m_A$ is essentially unconstrained, even if naturalness\narguments suggest that it should be smaller than $O(500 \\; {\\rm GeV})$, whereas\nfor $\\tan \\beta$ the range permitted by model calculations\nis $1 \\le \\tan \\beta \\simlt \\frac{m_t}{m_b}$.\n\nAt the classical level, the mass matrix of neutral CP-even Higgs bosons\nreads\n\\begin{equation}\n\\label{cpeven0}\n\\left( {\\cal M}_R^0 \\right)^2 =\n\\left[\n\\pmatrix{\n\\cot\\beta & -1 \\cr\n-1 & \\tan\\beta \\cr}\n{m_Z^2 \\over 2}\n+\n\\pmatrix{\n\\tan\\beta & -1 \\cr\n-1 & \\cot\\beta \\cr}\n{m_A^2 \\over 2}\n\\right]\n\\sin 2\\beta\n\\end{equation}\nand the charged-Higgs mass is given by\n\\begin{equation}\n\\label{mch0}\nm_{H^\\pm}^2 = m_W^2 + m_A^2.\n\\end{equation}\n{}From eq. (\\ref{cpeven0}), one obtains\n\\begin{equation}\n\\label{mh0}\nm_{h,H}^2 = {1 \\over 2} \\left[\nm_A^2 + m_Z^2 \\mp \\sqrt{(m_A^2 + m_Z^2)^2\n- 4 m_A^2 m_Z^2 \\cos^2 2 \\beta}\n\\right],\n\\end{equation}\nand also celebrated inequalities such as $m_W, m_A < m_{H^{\\pm}}$,\n$m_h <  m_Z < m_H$, $m_h <  m_A < m_H$. Similarly, one can easily compute all\nthe Higgs-boson couplings by observing that the mixing angle $\\alpha$,\nrequired to diagonalize the mass matrix (\\ref{cpeven0}), is given by\n\\begin{equation}\n\\cos 2 \\alpha = - \\cos 2 \\beta \\; { { m_A^2  - m_Z^2 }\\over {\n m_H^2 - m_h^2  }} \\ ,\\ \\ - {\\pi \\over 2} < \\alpha\\  {\\leq}\\  0.\n\\end{equation}\nFor example, the couplings of the three neutral Higgs bosons\nare easily obtained from the SM Higgs couplings if\none multiplies them by the $\\alpha$- and $\\beta$-dependent factors\nsummarized in table~1.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n& & & \\\\\n&\n$\n\\begin{array}{c}\nd\\overline{d},s\\overline{s},b\\overline{b} \\\\\ne^+e^-,\\mu^+\\mu^-,\\tau^+\\tau^-\n\\end{array}\n$\n& $u\\overline{u},c\\overline{c},t\\overline{t}$\n& $W^+W^-,ZZ$ \\\\\n& & & \\\\\n\\hline\n& & & \\\\\n$h$\n& $- \\sin \\alpha \/ \\cos \\beta $\n& $  \\cos \\alpha \/ \\sin \\beta $\n& $  \\sin \\, (\\beta -\\alpha)  $ \\\\\n& & & \\\\\n\\hline\n& & & \\\\\n$H$\n& $\\cos \\alpha \/ \\cos \\beta$\n& $\\sin \\alpha \/ \\sin \\beta$\n& $\\cos \\, (\\beta -\\alpha)  $\n\\\\\n& & & \\\\\n\\hline\n& & & \\\\\n$A$\n& $-i\\gamma_5 \\tan \\beta $\n& $-i\\gamma_5 \\cot \\beta $\n& $0$\n\\\\\n& & & \\\\\n\\hline\n\\end{tabular}\n\\caption{Correction factors for the couplings of the MSSM neutral Higgs\nbosons to fermion and vector boson pairs.}\n\\label{couplings}\n\\end{center}\n\\end{table}\nThe remaining tree-level Higgs-boson couplings in the MSSM can be easily\ncomputed and are summarized, for example, in ref. [\\ref{hunter}]. An\nimportant consequence of the structure of the classical Higgs potential\nof eq. (\\ref{V0}) is the existence of at least one neutral CP-even Higgs\nboson, weighing less than or about $m_Z$ and with approximately standard\ncouplings to the $Z$. This raised the hope that the crucial experiment\non the MSSM Higgs sector could be entirely performed at LEP II (with\nsufficient centre-of-mass energy, luminosity and $b$-tagging efficiency),\nand took some interest away from the large hadron collider environment.\n\nHowever, it was recently pointed out [\\ref{pioneer}] that\nthe masses of the Higgs bosons in the MSSM are subject to large radiative\ncorrections, associated with the top quark and its $SU(2)$ and supersymmetric\npartners\\footnote{Previous studies [\\ref{previous}] either\nneglected the case of a heavy top quark, or\nconcentrated on the violations of the neutral-Higgs mass sum rule\nwithout computing corrections to individual Higgs masses.}.\nSeveral papers [\\ref{higgscorr}--\\ref{berz}] have subsequently investigated\nvarious aspects of these corrections and their implications for experimental\nsearches at LEP. In the rest of this section, we shall summarize and\nillustrate the main effects of radiative corrections on Higgs-boson\nparameters.\n\nAs far as Higgs-boson masses and self-couplings are concerned, a convenient\napproximate way of parametrizing one-loop radiative corrections is to\nsubstitute the tree-level Higgs potential of eq. (\\ref{V0}) with the one-loop\neffective potential, and to identify Higgs-boson masses and self-couplings\nwith the appropriate combinations of derivatives of the effective potential,\nevaluated at the minimum.\nThe comparison with explicit diagrammatic calculations\nshows that the effective potential approximation is more than adequate for\nour purposes. Also, inspection shows that the most important corrections\nare due to loops of top and bottom quarks and squarks.\nAt the minimum  $ \\langle H_1^0 \\rangle = v_1$,\n$\\langle  H_2^0 \\rangle = v_2$, $ \\langle H_1^- \\rangle = \\langle H_2^+\n\\rangle = 0$, and neglecting intergenerational mixing, one obtains for\nthe top and bottom quark and squark masses the familiar expressions\n\\begin{equation}\n\\label{mq}\nm_t^2 = h_t^2 v_2^2 \\, ,\n\\;\\;\\;\\;\\;\nm_b^2 = h_b^2 v_1^2 \\, ,\n\\end{equation}\n\\begin{eqnarray}\n\\label{stopeigen}\nm_{\\tilde{t}_{1,2}}^2 &=& m_t^2+\\frac{1}{2}(m_Q^2+m_U^2)+\n\\frac{1}{4}m_Z^2\\cos 2\\beta\n\\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}\\nonumber \\\\\n& & \\pm\n\\sqrt{\\left[\\frac{1}{2}(m_Q^2-m_U^2)\n+\\frac{1}{12}(8m_W^2-5m_Z^2)\\cos 2\\beta\\right]^2\n+m_t^2\\left(A_t+\\mu\\cot\\beta\\right)^2},\\nonumber \\\\\n\\hfill\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{sbottomeigen}\nm_{\\tilde{b}_{1,2}}^2 &=& m_b^2+\\frac{1}{2}(m_Q^2+m_D^2)-\n\\frac{1}{4}m_Z^2\\cos 2\\beta\n\\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}\\nonumber \\\\\n& & \\pm\n\\sqrt{\\left[\\frac{1}{2}(m_Q^2-m_D^2)\n-\\frac{1}{12}(4m_W^2-m_Z^2)\\cos 2\\beta\\right]^2\n+m_b^2\\left(A_b+\\mu\\tan\\beta\\right)^2}.\\nonumber \\\\\n\\hfill\n\\end{eqnarray}\nIn eqs. (\\ref{mq}) to (\\ref{sbottomeigen}), $h_t$ and $h_b$ are the top and\nbottom Yukawa couplings, and $m_Q,m_U,m_D$ are soft supersymmetry-breaking\nsquark masses. The parameters $A_t$, $A_b$ and $\\mu$, which determine the\namount of mixing in the stop and sbottom mass matrices, are defined\nby the trilinear potential terms\n$h_t A_t ( \\tilde{t}_L \\tilde{t^c}_L H_2^0\n- \\tilde{b}_L \\tilde{t^c}_L H_2^+ ) + \\, {\\rm h.c.}$,\n$h_b A_b ( \\tilde{b}_L \\tilde{b^c}_L H_1^0 -\n\\tilde{t}_L \\tilde{b^c}_L H_1^-) + \\, {\\rm h.c.}$\nand by the superpotential mass term $\\mu (H_1^0 H_2^0 - H_1^- H_2^+)$,\nrespectively.\n\nTo simplify the discussion, in the following we will take a\nuniversal soft supersymmetry-breaking squark mass,\n\\begin{equation}\n\\label{soft}\nm_Q^2 = m_U^2 = m_D^2 \\equiv m_{\\tilde{q}}^2\\, ,\n\\end{equation}\nand we will assume negligible mixing in the stop and sbottom mass\nmatrices,\n\\begin{equation}\n\\label{mixing}\nA_t = A_b = \\mu = 0 \\, .\n\\end{equation}\nFormulae valid for arbitrary values of the parameters can be found in\nrefs. [\\ref{erz3},\\ref{berz}], but the qualitative features corresponding\nto the parameter choice of eqs. (\\ref{soft}) and (\\ref{mixing}) are\nrepresentative of a very large region of parameter space. In the case\nunder consideration, and neglecting $D$-term contributions to the\nfield-dependent stop and sbottom masses, the neutral CP-even mass matrix\nis modified at one loop as follows\n\\begin{equation}\n\\label{cpeven1}\n{\\cal M}_R^2 = \\left( {\\cal M}_R^0 \\right)^2\n+\n\\pmatrix{\n\\Delta_1^2 & 0 \\cr\n0 & \\Delta_2^2 \\cr},\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{delta1}\n\\Delta_1^2 = {3 g^2 m_b^4 \\over 16 \\pi^2 m_W^2 \\cos^2\\beta}\n\\log\\frac{m_{\\tilde{b}_1}^2 m_{\\tilde{b}_2}^2}{m_b^4},\n\\end{equation}\n\\begin{equation}\n\\label{delta2}\n\\Delta_2^2= {3 g^2 m_t^4 \\over 16 \\pi^2 m_W^2 \\sin^2\\beta}\n\\log\\frac{m_{\\tilde{t}_1}^2 m_{\\tilde{t}_2}^2}{m_t^4}.\n\\end{equation}\n{}From the above expressions one can easily derive the one-loop-corrected\neigenvalues $m_h$ and $m_H$, as well as the mixing angle $\\alpha$ associated\nwith the one-loop-corrected mass matrix (\\ref{cpeven1}).\nThe one-loop-corrected charged Higgs mass is given instead by\n\\begin{equation}\n\\label{sumrule}\nm_{H^{\\pm}}^2=\nm_W^2 + m_A^2 +\n\\Delta^2,\n\\end{equation}\nwhere, including $D$-term contributions to stop and sbottom masses,\n\\begin{eqnarray}\n\\label{approx1}\n\\Delta^2 & = &\n\\frac{3 g^2}{64 \\pi^2 \\sin^2 \\beta \\cos^2 \\beta m_W^2}\n\\nonumber \\\\\n& & \\nonumber \\\\\n& & \\times\n\\left\\{ \\frac{(m_b^2-m_W^2 \\cos^2 \\beta)(m_t^2-m_W^2 \\sin^2 \\beta)}\n{m_{\\tilde{t}_1}^2-m_{\\tilde{b}_1}^2} \\left[ f(m_{\\tilde{t}_1}^2) - f(m_{\\tilde{b}_1}^2) \\right]\n\\right. \\nonumber \\\\\n& & + \\left. \\frac{m^2_t m^2_b}{m_{\\tilde{t}_2}^2 - m_{\\tilde{b}_2}^2}\n\\left[ f(m_{\\tilde{t}_2}^2) - f(m_{\\tilde{b}_2}^2) \\right]\n- \\frac{2 m^2_t m^2_b}{m_t^2 - m_b^2}\n\\left[ f(m_t^2) - f(m_b^2) \\right] \\right\\}\n\\end{eqnarray}\nand\n\\begin{equation}\n\\label{effe}\nf (m^2) = 2 m^2 \\left( \\log \\frac {m^2}{Q^2} - 1 \\right).\n\\end{equation}\nThe most striking fact in eqs. (\\ref{cpeven1})--(\\ref{effe}) is that the\ncorrection $\\Delta_2^2$ is proportional to $(m_t^4\/m_W^2)$. This implies\nthat, for $m_t$ in the range of eq. (\\ref{mtop}), the tree-level predictions\nfor $m_h$ and $m_H$ can be badly violated, and so for the related\ninequalities. The other free parameter is $m_{\\tilde{q}}$, but the dependence\non it is much milder. To illustrate the impact of these results, we display\nin fig.~1 contours of the maximum allowed value of $m_h$ (reached for $m_A\n\\rightarrow \\infty$), in the $(m_t,\\tan \\beta)$ and $(m_t,m_{\\tilde{q}})$ planes, fixing $m_{\\tilde{q}} =\n1 \\; {\\rm TeV}$ and $\\tan \\beta=m_t\/m_b$, respectively. In the following, when making\nnumerical examples we shall always choose the representative value $m_{\\tilde{q}}\n= 1 \\; {\\rm TeV}$. To plot different quantities of physical interest\nin the $(m_A,\\tan \\beta)$ plane, which is going to be the stage of the following\nphenomenological discussion, one needs to fix also the value of $m_t$.\nIn this paper, whenever an illustration of the $m_t$ dependence is needed,\nwe work with the two representative values $m_t=120,160 \\; {\\rm GeV}$, which are\nsignificantly different but well within the range of eq. (\\ref{mtop}).\nOtherwise, we work with the single representative value $m_t=140 \\; {\\rm GeV}$.\nAs an example, we show in figs.~2--4 contours of constant $m_h$, $m_H$, and\n$m_{H^\\pm}$ in the $(m_A,\\tan \\beta)$ plane. Here and in the following we vary\n$m_A$ and $\\tan \\beta$ in the ranges\n\\begin{equation}\n\\label{bounds}\n0 \\le m_A \\le 500 \\; {\\rm GeV},\n\\;\\;\\;\\;\n1 \\le \\tan \\beta \\le 50 \\, .\n\\end{equation}\n\nThe effective-potential method allows us to compute also the leading\ncorrections\nto the trilinear and quadrilinear Higgs self-couplings. A detailed discussion\nand the full diagrammatic calculation will be given elsewhere. Here we\njust give the form of the leading radiative corrections to the\ntrilinear $hAA$, $HAA$, and $Hhh$ couplings, which will play an important\nrole in the subsequent discussion of Higgs-boson branching ratios.\nOne finds [\\ref{berz},\\ref{brignole2}]\n\\begin{equation}\n\\label{hhh1}\n\\lambda_{hAA}\n=\n\\lambda_{hAA}^0\n+\n\\Delta \\lambda_{hAA} \\, ,\n\\;\\;\\;\\;\n\\lambda_{HAA}\n=\n\\lambda_{HAA}^0\n+\n\\Delta \\lambda_{HAA} \\, ,\n\\;\\;\\;\\;\n\\lambda_{Hhh}\n=\n\\lambda_{Hhh}^0\n+\n\\Delta \\lambda_{Hhh} \\, ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{lhaa0}\n\\lambda_{hAA}^0\n=\n- {i g m_Z \\over {2 \\cos \\theta_W}}\n\\cos 2 \\beta \\sin (\\beta + \\alpha) \\, ,\n\\end{equation}\n\\begin{equation}\n\\label{lhhaa0}\n\\lambda_{HAA}^0\n=\n\\phantom{-} {i g m_Z \\over {2 \\cos \\theta_W}}\n\\cos 2 \\beta \\cos (\\beta + \\alpha) \\, ,\n\\end{equation}\n\\begin{equation}\n\\label{lhhh0}\n\\lambda_{Hhh}^0\n=\n- {i g m_Z \\over {2 \\cos \\theta_W}}\n[ 2 \\sin (\\beta + \\alpha) \\sin 2 \\alpha\n- \\cos (\\beta + \\alpha) \\cos 2 \\alpha] \\, ,\n\\end{equation}\nand, neglecting the bottom Yukawa coupling and the $D$-term contributions\nto squark masses\n\\begin{equation}\n\\label{deltahaa}\n\\Delta \\lambda_{hAA}\n=\n- {i g m_Z \\over {2 \\cos \\theta_W}}\n  {3 g^2 \\cos^2 \\theta_W \\over {8 \\pi^2}}\n  {\\cos  \\alpha \\cos^2 \\beta \\over  {\\sin^3 \\beta}}\n  {m_t^4 \\over m_W^4}\n  \\log\n  {m_{\\tilde{q}}^2 + m_t^2 \\over m_t^2},\n\\end{equation}\n\\begin{equation}\n\\label{deltahhaa}\n\\Delta \\lambda_{HAA}\n=\n- {i g m_Z \\over {2 \\cos \\theta_W}}\n  {3 g^2 \\cos^2 \\theta_W \\over {8 \\pi^2}}\n  {\\sin  \\alpha \\cos^2 \\beta \\over  {\\sin^3 \\beta}}\n  {m_t^4 \\over m_W^4}\n  \\log\n  {m_{\\tilde{q}}^2 + m_t^2 \\over m_t^2},\n\\end{equation}\n\\begin{equation}\n\\label{deltahhh}\n\\Delta \\lambda_{Hhh}\n=\n- {i g m_Z \\over {2 \\cos \\theta_W}}\n  {3 g^2 \\cos^2 \\theta_W \\over {8 \\pi^2}}\n  {\\cos^2  \\alpha \\sin \\alpha \\over  {\\sin^3 \\beta}}\n  {m_t^4 \\over m_W^4}\n  \\left( 3 \\log   {m_{\\tilde{q}}^2 + m_t^2 \\over m_t^2}\n       - 2 {m_{\\tilde{q}}^2 \\over m_{\\tilde{q}}^2 + m_t^2} \\right).\n\\end{equation}\nNotice that, besides the obvious explicit dependence, in\neqs.~(\\ref{hhh1})--(\\ref{deltahhh}) there is also an important implicit\ndependence on $m_t$ and $m_{\\tilde{q}}$, via the angle $\\alpha$, which is determined\nfrom the mass matrix of eqs.~(\\ref{cpeven1})--(\\ref{delta2}).\nWe also emphasize that neglecting the $D$-terms in the stop and sbottom\nmass matrices is guaranteed to give accurate results only for $m_t\n\\gamma\\gamma m_Z$. For $m_t \\sim m_Z$, one should make sure that the inclusion\nof $D$-terms does not produce significant modifications of our results.\nIn the case of the $h$ and $H$ masses, and of the mixing angle $\\alpha$,\ncomplete formulae are available, and this check can be easily performed.\nIn the case of the $hAA$, $HAA$ and $Hhh$ couplings, complete formulae are\nnot yet available. For the phenomenologically most important coupling\nat the LHC and SSC, $\\lambda_{Hhh}$, we have explicitly checked that\nthe inclusion of $D$-terms does not produce important modifications\nof our results.\n\nFinally, one should consider Higgs couplings to vector bosons and fermions.\nTree-level couplings to vector bosons are expressed in terms of gauge\ncouplings and of the angles $\\beta$ and $\\alpha$. The most important part\nof the radiative corrections is taken into account by using\none-loop-corrected formulae to determine $\\alpha$ from the input parameters.\nOther corrections are at most of order $g^2 m_t^2 \/ m_W^2$ and can be\nsafely neglected for our purposes. Tree-level couplings to fermions are\nexpressed in terms of the fermion masses and of the angles $\\beta$ and\n$\\alpha$. In this case, the leading radiative corrections can be taken\ninto account by using the one-loop-corrected expression for $\\alpha$ and\nrunning fermion masses, evaluated at the scale $Q$ which characterizes\nthe process under consideration. This brings us to the discussion of the\nrenormalization group evolution of the top and bottom Yukawa couplings\nin the MSSM. As boundary conditions, we assume as usual that $m_t(m_t)\n=m_t$ and $m_b(m_b)=m_b$, with $m_b=4.8 \\; {\\rm GeV}$ and $m_t$ numerical\ninput parameters. As stated in the Introduction, we assume in this\npaper that all supersymmetric particles are heavy.\nThen, since we want to compute Higgs-boson production cross-sections and\nbranching ratios, we are interested in the standard renormalization group\nevolution of $h_t(Q)$ [$h_b(Q)$] from $Q=m_t$ [$Q=m_b$] to $Q \\simeq\nm_{H^\\pm},m_H$, which is dominated by gluon loops.\n\nTo illustrate the behaviour of the Higgs couplings to vector bosons and\nfermions, as functions of the input parameters, we show in figs.~5--7\ncontours in the $(m_A,\\tan \\beta)$ plane of some of the correction factors\nappearing in table~\\ref{couplings}.\n\\section{LEP limits and implications}\nIn this section, we briefly summarize the implications of the previous results\non MSSM Higgs boson searches at LEP~I and LEP~II.\nPartial results were already\npresented in refs. [\\ref{bf},\\ref{erz3}].\n\nAs already clear from tree-level analyses, the relevant processes\nfor MSSM Higgs boson searches at LEP I are $Z \\rightarrow h Z^*$ and $Z \\rightarrow h A$,\nwhich play a complementary role, since their rates are proportional to\n$\\sin^2 (\\beta - \\alpha)$ and $\\cos^2 (\\beta - \\alpha)$, respectively. An important effect of radiative\ncorrections [\\ref{berz}] is to allow, for some values of the\nparameters, the decay $h \\rightarrow AA$, which would be kinematically forbidden\naccording to tree-level formulae. Experimental limits which take radiative\ncorrections into account have by now been obtained by the four LEP\ncollaborations [\\ref{lep}], using different methods to present and\nanalyse the data, and different ranges of parameters in the evaluation of\nradiative corrections. A schematic representation of the presently\nexcluded region of the $(m_A,\\tan \\beta)$ plane, for the standard parameter\nchoices discussed in sect.~2, is given in fig.~8, where the solid lines\ncorrespond to our na\\\"{\\i}ve\\footnote{\nWe fitted the experimental exclusion contours, corresponding to $m_t = 140\n\\; {\\rm GeV}$ and the other parameters as chosen here, with two numerical values\nfor $\\Gamma(Z \\rightarrow h Z^*)$ and $\\Gamma(Z \\rightarrow h A)$. We have then computed\nradiative corrections for the two values of $m_t$ considered here,\nassuming that the variations in experimental efficiencies are small\nenough not to affect our results significantly.} extrapolation of the\nexclusion contour given in the first of refs. [\\ref{lep}]. For a discussion of\nthe precise experimental bounds, we refer the reader to the above-mentioned\nexperimental publications.\n\nThe situation in which the impact of radiative corrections is most\ndramatic is the search for MSSM Higgs bosons at LEP II. At the time\nwhen only tree-level formulae were available, there was hope that LEP\ncould completely test the MSSM Higgs sector. According to tree-level formulae,\nin fact, there should always be a CP-even Higgs boson with mass smaller than\n($h$) or very close to ($H$) $m_Z$, and significantly coupled to the $Z$\nboson.\nHowever, as should be clear from the previous section, this result can be\ncompletely upset by radiative corrections. A detailed evaluation of the\nLEP II discovery potential can be made only if crucial theoretical parameters,\nsuch as the top-quark mass and the various soft supersymmetry-breaking masses,\nand experimental parameters, such as the centre-of-mass energy, the luminosity\nand the $b$-tagging efficiency, are specified. Taking for example $\\sqrt{s} =\n190 \\; {\\rm GeV}$, $m_t \\simgt 110 \\; {\\rm GeV}$, and our standard values for the soft\nsupersymmetry-breaking parameters, in the region of $\\tan \\beta$ significantly\ngreater than 1, the associated production of a $Z$ and a CP-even Higgs can be\npushed beyond the kinematical limit. Associated $hA$ production could be a\nuseful complementary signal, but obviously only for $m_h+m_A< \\sqrt{s}$.\nAssociated $HA$ production is typically negligible at these energies.\nTo give a measure of the LEP II sensitivity, we plot in fig.~8 contours\nassociated with two benchmark values of the total cross-section $\\sigma(\ne^+ e^- \\rightarrow hZ^*, HZ^*, hA, HA)$. The dashed lines correspond to $\\sigma =\n0.2 \\, {\\rm pb}$ at $\\sqrt{s} = 175 \\; {\\rm GeV}$, which could be seen as a rather\nconservative estimate of the LEP II sensitivity. The dash-dotted lines\ncorrespond to $\\sigma = 0.05 \\, {\\rm pb}$ at $\\sqrt{s} = 190 \\; {\\rm GeV}$, which\ncould be seen as a rather optimistic estimate of the LEP II sensitivity.\nIn computing these cross-sections, we have taken into account the finite\n$Z$ width, but we have neglected initial state radiation, which leads to\nsuppression near threshold. A more accurate estimate of the LEP II\nsensitivity can be found in ref. [\\ref{janot}].\nOf course, one should keep in mind that there is, at least in principle,\nthe possibility of further extending the maximum LEP energy up to values\nas high as $\\sqrt{s} \\simeq$ 230--240 GeV, at the price of introducing more\n(and more performing) superconducting cavities into the LEP tunnel\n[\\ref{treille}].\n\nIn summary, a significant region of the parameter space for MSSM Higgses\ncould be beyond the reach of LEP II, at least if one sticks to the reference\ncentre-of-mass energy $\\sqrt{s} \\simlt 190 \\; {\\rm GeV}$. The precise knowledge of\nthis region is certainly important for assessing the combined discovery\npotential of LEP and LHC\/SSC, but it does not affect the motivations and\nthe techniques of our study, devoted to LHC and SSC searches. Whether\nor not a Higgs boson will be found at LEP in the future, we want to\ninvestigate the possibilities of searching for all the Higgs states of the\nMSSM at large hadron colliders, in the whole region of parameter\nspace which is not already excluded at present. Even if a neutral Higgs\nboson is found at LEP, with properties compatible with the SM Higgs boson\nwithin the experimental errors, it will be impossible to exclude that it\nbelongs to the MSSM sector. The LHC and SSC could then play a role in\ninvestigating its properties and in looking for the remaining states of\nthe MSSM.\n\nSimilar considerations can be made for charged-Higgs searches at LEP~II\nwith $\\sqrt{s} \\simlt 190 \\; {\\rm GeV}$. In view of the $\\beta^3$ threshold\nfactor in $\\sigma ( e^+ e^- \\rightarrow H^+ H^-)$, and of the large background from\n$e^+ e^- \\rightarrow W^+ W^-$, it will be difficult to find the $H^{\\pm}$ at LEP II\nunless $m_{H^{\\pm}} \\simlt m_W$, and certainly impossible unless $m_{H^{\\pm}} < \\sqrt{s}\/2$.\nWe also know [\\ref{berz},\\ref{higgscorr}] that for\ngeneric values of the parameters there are no large negative radiative\ncorrections to the charged-Higgs mass formula, eq. (\\ref{mch0}). A comparison\nof figs.~4 and 8 indicates that there is very little hope of finding the\ncharged Higgs boson of the MSSM at LEP II (or, stated differently, the\ndiscovery of a charged Higgs boson at LEP II would most probably rule out\nthe MSSM).\n\\section{Branching ratios}\n\\subsection{Neutral Higgs bosons}\nThe branching ratios of the neutral Higgs bosons of the MSSM were\nsystematically studied in ref. [\\ref{kz}], using the tree-level\nformulae for masses and couplings available at that\ntime\\footnote{Also, the partial widths for the decays $h,H,A\n\\rightarrow\\gamma\\gamma$ were affected by numerical errors.}\n(previous work on the subject is summarized in ref. [\\ref{hunter}]).\nHere we present a systematic study which includes the radiative corrections\ndescribed in sect.~2. As usually done for the SM Higgs boson, we\nconsider the two-body decay channels\n\\begin{equation}\n\\label{smdecays}\nh,H,A \\longrightarrow\nc\\overline{c}, \\\nb\\overline{b}, \\\nt\\overline{t}, \\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi, gg, \\gamma\\gamma,\\\nW^*W^*,\\  Z^*Z^*,\\  Z\\gamma \\, .\n\\end{equation}\nFor consistency, we must also consider decays with one or two Higgs bosons\nin the final state\n\\begin{equation}\n\\label{newdecays}\nh \\rightarrow AA,\n\\ \\\nH \\rightarrow hh, \\, AA, \\, ZA,\n\\ \\\nA\\rightarrow Zh.\n\\end{equation}\nOn the other hand, we neglect here possible decays of MSSM Higgs bosons\ninto supersymmetric particles: as previously stated, we consistently assume\na heavy spectrum of $R$-odd particles, so that only $R$-even ones can be\nkinematically accessible in the decays of $h,H,$ and $A$. We perform our\nstudy in the framework of MSSM parameter space, with the representative\nparameter choices illustrated in sect.~2. The effects of changing the mass\nof the top quark, and the sensitivity to squark masses in the high-mass\nregion, will also be briefly discussed.\n\nThe partial widths for the decays of eq. (\\ref{smdecays}) that correspond\nto tree-level diagrams can be obtained from the corresponding formulae\nfor the SM Higgs boson (for a summary, see ref.~[\\ref{hunter}]), by simply\nmultiplying the various amplitudes by the supersymmetric correction\nfactors listed in table~1.\nFor decays that are described by loop diagrams, however, in the MSSM\none has to include some contributions that are absent in the SM.\nDiagrams corresponding to the exchange of $R$-odd supersymmetric particles\ngive negligible contributions to the\ncorresponding partial widths, in the limit of heavy supersymmetric-particle\nmasses that we have chosen for our analysis (in accordance with intuitive\nideas about decoupling). One must also include\nthe charged-Higgs loop contributions to the $\\gamma \\gamma$ and $Z \\gamma$\nfinal states. When considering instead the processes of eq. (\\ref{newdecays}),\nwe improve the tree-level formulae of ref. [\\ref{hunter}]\nnot only with the self-energy corrections to the mixing angle $\\alpha$,\nbut also with the vertex corrections of eqs. (\\ref{hhh1})--(\\ref{deltahhh}).\n\nQCD [\\ref{gorishnyh}] and electroweak [\\ref{kniehlbardin}]  ra\\-diative\ncor\\-rections  to the fermion-anti\\-fermion and the $WW$,\\ $ZZ$ channels\nhave been recently computed for the SM Higgs boson, $\\varphi$. They have\nbeen found to be small (less than $\\sim 20\\%$), with the exception of\nthe QCD corrections to the decays into charm and bottom quark pairs, which\nare large because of running-quark-mass effects.\nWe then included the QCD corrections as described in ref. [\\ref{zkwjsaa}].\nOne may also wonder\nwhether running-mass effects induced by the large top Yukawa coupling\ncould give further important effects. However, one can easily see that\nthese effects give corrections which are certainly less than 20\\%.\n\nThe QCD correction to $\\varphi \\rightarrow \\gamma\\gamma$ is also available, and\nknown to be negligibly small [\\ref{djouadigamgam}].\nSizeable QCD corrections are found, however, for the decay $\\varphi\n\\rightarrow gg$ [\\ref{djouadigg}].\nAlthough this effect is not important for the branching ratio study, since\n$\\varphi \\rightarrow gg$ is neither the dominant decay mode nor a useful channel for\ndetection, it still has to be included in the production cross-section of $h$\nvia the two-gluon fusion mechanism.\n\n\nAnother general and well-known property of the MSSM is that the\nself-interactions of the Higgs bosons are controlled, modulo the\nlogarithmic corrections discussed in sect.~2, by the $SU(2)$ and\n$U(1)$ gauge couplings. Therefore, the total widths of {\\it all}\nMSSM Higgs bosons, displayed in fig.~9,\nstay below 10 GeV in the whole parameter space we have considered.\n\nThe most important branching ratios for the neutral MSSM Higgs bosons are\nshown, as a function of the mass of the decaying particle, in figs.~10--12.\nTo avoid excessive proliferation of figures, we consider the two\nrepresentative values\n\\begin{equation}\n\\label{tbvalues}\n\\tan \\beta= 1.5, \\, 30,\n\\end{equation}\nand for each of these we vary $m_A$ between the experimental lower bound\nof fig.~8 ($m_A \\simeq 59 \\; {\\rm GeV}$ for $\\tan \\beta=1.5$, $m_A \\simeq 44 \\; {\\rm GeV}$ for\n$\\tan \\beta=30$) and the upper bound of eq. (\\ref{bounds}), assuming $m_t=140\\; {\\rm GeV}$\nand  $m_{\\tilde q}=1\\; {\\rm TeV}$.\n\nWe consider first the branching ratios of $h$ (fig.~10). We can clearly see\nthe effect of radiative corrections on the allowed range of $m_h$ for the\ngiven values of $\\tan \\beta$. For $m_A \\simlt 25 \\; {\\rm GeV}$, the decay $h \\rightarrow AA$ can\nbe kinematically allowed and even become the dominant mode. This decay channel\nwas important at LEP I, but since the corresponding region of parameter space\nis already excluded by experiment, this decay mode does not appear in fig.~10.\nThe dominant decay mode is then $h\\rightarrow b\\bar{b}$, whereas the $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ mode\nhas a branching ratio of about $8\\%$ throughout the relevant part of the\nparameter space. In fig.~10, one immediately notices the rather steep slopes\nfor the $c\\bar{c}$ and $\\gamma\\gamma$ branching ratios plotted versus $m_h$,\nwith larger effects for larger values of $\\tan \\beta$: their origin can be understood\nby looking at figs.~2 and 5--7, which show how $m_h$ and the $h$ couplings\nto heavy fermions and vector bosons vary in the $(m_A,\\tan \\beta)$ plane.\n\nIf the SM Higgs boson is in the intermediate mass region, $m_{\\varphi} =$\n70--140 GeV, at large hadron colliders a measurable signal can be obtained\nvia the $\\gamma\\gamma$  mode. Since the mass of the light Higgs $h$ is\nindeed below or inside this region, the $\\gamma\\gamma$ mode is also crucial\nfor the MSSM Higgs search. Furthermore, the $\\gamma \\gamma$ branching ratio\nas a function of the Higgs mass exhibits a rather peculiar behaviour, not only\nfor $h$ but also for $H$ and $A$, so a more detailed discussion is in order.\nThe partial width is given by\n\\begin{equation}\n\\label{gammaw}\n\\Gamma(\\phi\\rightarrow\\gamma\\gamma)\n={\\alpha^2 g^2\\over 1024\\pi^3}{m_{\\phi}^3\\over m_W^2}\n\\abs{\\sum_i I_i^{\\phi}(\\tau^\\phi_i)}^2 , \\qquad\n\\tau^{\\phi}_i={4m_i^2\\over m_{\\phi}^2},\n\\end{equation}\nwhere $\\phi=h,H,A$ and $i=f,W,H,\\tilde{f},\\tilde{\\chi}$ indicates\nthe contributions from ordinary fermions, charged gauge bosons, char\\-ged\nHiggs bosons, sfermions and charginos, respectively. The functions\n$I_\\phi^i(\\tau^i_{\\phi})$ are given by\n\\begin{eqnarray}\n\\label{ifunct}\nI_f^{\\phi} & = & F_{1\/2}^{\\phi}(\\tau^{\\phi}_f)N_{cf}e^2_f\\ R_f^{\\phi} \\, ,\n\\nonumber \\\\\nI_W^{\\phi} & = & F_{1}(\\tau^{\\phi}_W) R_W^{\\phi} \\, ,\n\\nonumber \\\\\nI_H^{\\phi} & = & F_{0}(\\tau^{\\phi}_H) R_H^{\\phi} {m_W^2\\over m_{\\phi}^2} \\, ,\n\\nonumber \\\\\nI_{\\tilde{f}}^{\\phi} & = & F_{0}(\\tau^{\\phi}_{\\tilde{f}})\nN_{cf}e^2_f  R_{\\tilde{f}}^{\\phi}{m_Z^2\\over m_{\\tilde{f}}^2},\n\\nonumber \\\\\nI_{\\tilde{\\chi}}^{\\phi} & = & F_{1\/2}^{\\phi}(\\tau^{\\phi}_{\\chi})\nR_{\\tilde{\\chi}}^{\\phi}{m_W\\over m_{\\tilde{\\chi}}},\n\\end{eqnarray}\nwhere $N_{cf}$ is $1$ for (s)leptons and $3$ for (s)quarks, and the\nsubscripts of the complex functions $F_{1\/2}^S(\\tau)$,\n$F_{1\/2}^P(\\tau)$, $F_{0}(\\tau)$, and $F_1(\\tau)$, which were\ncalculated in ref. [\\ref{vvzs}], indicate the spin of the\nparticles running in the loop. In the case of spin-$1\/2$\nparticles, the contribution is different for CP-even and CP-odd\nneutral Higgses. The symbols $R^{\\phi}_i$ denote the appropriate\ncorrection factors for the MSSM Higgs couplings: for $i=f,W$ they are\ngiven in table 1, for $i=H,\\tilde{f},\\tilde{\\chi}$ they can be found,\nfor example, in Appendix C of ref. [\\ref{hunter}]. The $W$ contribution\ndominates the $h \\rightarrow \\gamma \\gamma$ decay rate. The function $F_1$ is\nlarge at and above $\\tau=1$. For the $W$ contribution $\\tau =4m_W^2\/m_h^2\n> 1$, and increasing $m_h$ gives increasing values of $F_1$. The steep\ndependence of the branching ratio on $m_h$ is a consequence of the\nfast change of $\\sin^2 (\\beta - \\alpha)$ as $m_A$ is increased for fixed $\\tan \\beta$. This is\nfurther enhanced by the fact that the large interval $100 \\; {\\rm GeV} \\simlt\nm_A \\le 500\\; {\\rm GeV}$ is mapped into a very small interval (a few GeV)\nin $m_h$. We elucidate this effect by plotting in fig.~13 the\nbranching ratios  of $h$ as a function of $m_A$, for the same values of\nthe parameters as in fig.~10. We can see that the tip of the $gg, c \\overline{c}$\nand $\\gamma \\gamma$ curves in fig.~10 is mapped into a long\nplateau in fig.~13. We can also observe that in a large\nregion of the parameter space the $h \\rightarrow \\gamma \\gamma$ branching ratio\nhas a value somewhat smaller than (but comparable to) the corresponding\nbranching ratio for a SM Higgs of mass $m_h$. This is due to the\nfact that  all the $h$ couplings tend to the SM Higgs couplings for\n$m_A \\gamma\\gamma m_Z$; however, for the $h$ couplings to fermions the approach to\nthe asymptotic value is much slower than for the $h$ couplings to\nvector bosons, as can be seen from figs.~5 to 7.\nIn fig.~13, the branching ratios for the $W^*W^*$\nand $Z^*Z^*$ decays are also plotted, whereas they were omitted in\nfig.~10 in order to avoid excessive crowding of curves. However,\nfor our parameter choice they have little interest at large hadron\ncolliders, because of the small production rates and the large backgrounds.\n\nThe branching ratios of the heavy Higgs boson $H$, depicted in fig.~11,\nhave a rather complicated structure. We make here four remarks.\n\ni)\nThe $\\gamma\\gamma$\nmode has a steeply decreasing branching ratio with increasing $m_H$,\nexcept at small values of $\\tan \\beta$ and at the lower kinematical limit of $m_H$,\nwhere one or more of the $AA$, $ZA$ and $hh$ decay channels are open.\nThe steep fall of the $\\gamma\\gamma$ branching ratio at large values of $\\tan \\beta$\ncan be easily understood. The partial width $\\Gamma (H \\rightarrow\\gamma\\gamma)$\nis dominated by the $W$ contribution, proportional to $\\cos^2(\\beta-\\alpha)$.\nAs we can see in fig.~7, $\\cos^2(\\beta-\\alpha)$ decreases very fast, for\nincreasing $m_H$, at fixed values of $\\tan \\beta$. This steep decrease is slightly\ncompensated by the increase of $F_1(\\tau_W)$ at $\\tau_W \\leq 1$,\nwhich has a peak at the $W$ threshold $m_H=2 m_W$. Another peak in the\n$\\gamma \\gamma$ branching ratio is obtained, for small values of $\\tan \\beta$,\nat $m_H = 2 m_t$, where the top-quark loop gives the dominant contribution.\n\n\nii)\nThe complicated structure in the $H$ branching ratio curves is mainly due to\nthe $H\\rightarrow hh$ channel. For $m_H<2 m_t$, and not too high values of $\\tan \\beta$, this\ndecay mode is dominant whenever kinematics allows.  This channel is always\nopen at the lower kinematical limit of $M_H$. Increasing $M_H$ a little bit,\nhowever, it may become strongly suppressed, because for small\nincreasing values of $m_A$ the value of $m_h$ rises faster than that of $m_H$,\nso that the channel can become kinematically closed. Obviously, for\nsufficiently high values of $m_H$ the channel is always open.\nAt high values of $\\tan \\beta$, the mass region at the lower kinematical limit\nwhere $H\\rightarrow hh$ is open becomes smaller and smaller, explaining the presence\nof the almost vertical line in fig.~11.\nA further structure is present in this decay channel due to the coupling\nfactor $\\lambda_{Hhh}$ [see eqs. (\\ref{lhhh0}) and (\\ref{deltahhh})]\\@.\nThere are relatively small values of $m_H$ at which $\\lambda_{Hhh}$\naccidentally vanishes. Furthermore, for very large values of $m_H$ and $\\tan \\beta$\none has $\\alpha \\simeq 0$, $\\beta \\simeq \\pi\/2$, and therefore $\\lambda_{Hhh}\n\\simeq 0$. Unfortunately, even when it is dominant, this mode has very large\nbackgrounds, so it seems unlikely to give a measurable signal at large\nhadron colliders. The $H \\rightarrow AA$ mode is kinematically allowed only for\nvalues of $m_A$ below 50--60 GeV, in which case it can have a large branching\nratio, competing with the one for $H \\rightarrow hh$. The $H \\rightarrow Z A$ mode is\nkinematically allowed only in the region of parameter space which is already\nexcluded by the LEP I data.\n\niii)\n$H$ can decay at tree level into $ZZ\\rightarrow l^+l^-l^+l^-$, which is the\n`gold-plated' signature for the SM Higgs boson. Unfortunately, in the case\nof $H$ the branching ratio is smaller, and it decreases fast with increasing\n$\\tan \\beta$ and\/or $m_A$. For small $\\tan \\beta$ and $2 m_h < m_H < 2 m_t$, this mode is\nsuppressed by the competition with $H \\rightarrow hh$, and this effect is further\nenhanced by the inclusion of the radiative correction of eq. (\\ref{deltahhh}),\nwhich typically gives an additional $50 \\%$ suppression.\nNevertheless, as we shall see in the next section, the four-lepton channel\ncan give a measurable signal in some small region of the parameter space.\n\niv) The decay into $t\\bar{t}$ is dominant above threshold at moderate values\nof $\\tan \\beta$. But above $\\tan \\beta\\sim 8$ or so the $b\\bar{b}$ mode remains dominant\nand $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ has the typical $\\sim 10\\%$ branching ratio.\n\nFinally, we discuss the branching ratios of $A$, shown in fig.~12.\nThe $\\gamma\\gamma$ one is always small, although at small $\\tan \\beta$\nand slightly below the top threshold, $m_A\\sim 2 m_t$, it reaches a value\n$\\sim 8 \\times 10^{-4}$, which may give a measurable signal in a small\nisland of the parameter space. The behaviour of the $A \\rightarrow \\gamma\\gamma$\nbranching ratio can be easily understood by taking into account that the\npartial width is dominated by the top loop contribution.\nTwo features are important here. First, the function $F_{1\/2}(\\tau)$\nappearing in eq. (\\ref{ifunct}) has a strong enhancement at $\\tau\\sim 1$.\nFurthermore, the $t\\bar{t}A$ coupling gives a suppression factor $1\/(\\tan \\beta)^{2}$\nfor increasing values of $\\tan \\beta$.\n\nAt smaller values of $m_A$ and $\\tan \\beta$, there is a substantial branching\nratio to $Zh$, which however does not look particularly promising for\ndetection at large hadron colliders, because of the very large $Zb\\bar{b}$\nbackground. We can see that all the dominant decay modes of the $A$ boson\ncorrespond to channels which are overwhelmed by very large background,\nexcept perhaps the $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ mode, which, as we shall see in the next\nsection, may give a detectable signal for very high values of $\\tan \\beta$.\n\nWe have studied the neutral Higgs branching ratios also at $m_t=120,160,180\n\\; {\\rm GeV}$. Increasing the top mass has two major effects. First, the maximum\nvalue of $m_h$ increases (see fig.~1). Next, owing to the increased value of\nthe top threshold, the structure generated by the opening of the top decay\nchannel is shifted to higher mass values.\nWe also note that varying $m_{\\tilde{q}}$ in the range 0.5--2 TeV has\nnegligible effects on the branching ratio curves of figs.~10--12.\nFinally, if one chooses $m_t$ and $m_{\\tilde{q}}$ so large that $m_h > 130\n\\; {\\rm GeV}$, the $W^*W^*$ and $Z^*Z^*$ branching ratios can become relevant\nalso for $h$.\n\n\\subsection{Charged Higgs boson}\n\nIn the case of the charged Higgs boson, we considered only the two-body\ndecay channels\n\\begin{equation}\n\\label{chchan}\nH^+ \\rightarrow c \\overline{s}\\, , \\; \\tau^+ \\nu_{\\tau} \\, , \\; t \\overline{b} \\, , \\;\nW^+ h \\, .\n\\end{equation}\nTree-level formulae for the corresponding decay rates can be found, for\ninstance,\nin ref. [\\ref{hunter}]. Loop-induced decays such as $H^+ \\rightarrow W^+ \\gamma,\nW^+ Z$ have very small branching ratios [\\ref{loopch}] and are not relevant\nfor experimental searches at the LHC and SSC. Radiative corrections to\nthe charged-Higgs-boson mass formula were included according to eqs.\n(\\ref{sumrule}) and (\\ref{approx1}). The $H^+ W^- h$ coupling, proportional\nto $\\cos (\\beta - \\alpha)$, was evaluated with the one-loop corrected value\nof $\\alpha$. The leading QCD corrections to the $H^+ b \\overline{t}$ and\n$H^+ s \\overline{c}$ vertices  were parametrized, following refs. [\\ref{qcdch}],\nby running quark masses evaluated at a scale $Q \\sim m_{H^{\\pm}}$. The resulting\nbranching ratios for the charged Higgs boson are displayed in fig.~14, for\n$\\tan \\beta = 1.5, \\, 30$ and the standard parameter choice $m_t = 140 \\; {\\rm GeV}$, $m_{\\tilde{q}}\n= 1 \\; {\\rm TeV}$. One can see that the dominant factor affecting the branching ratios\nis the $m_{H^{\\pm}} = m_t+m_b$ threshold.\nAbove threshold, the $t \\overline{b}$ mode is dominant\nfor any value of $\\tan \\beta$ within the bounds of eq. (\\ref{bounds}).  Below\nthreshold, the dominant mode is $\\tau^+ \\nu_{\\tau}$, with the competing mode\n$c \\overline{s}$ becoming more suppressed for higher values of $\\tan \\beta$. For small\nvalues of $\\tan \\beta$, the $W^+ h$ decay mode can also be important, and even\ndominate, in a limited $m_{H^{\\pm}}$ interval, if the $W^+ h$ threshold opens up\nbefore the $t \\overline{b}$ one. The exact position of the two thresholds on\nthe $m_{H^{\\pm}}$ axis depends of course on $\\tan \\beta$, $m_t$, and $m_{\\tilde{q}}$. It is just a\nnumerical coincidence that in fig.~14a the two thresholds correspond\nalmost exactly.\nFor increasing values of $m_{H^{\\pm}}$ and $\\tan \\beta$, the numerical relevance of the\n$W^+ h$ branching ratio rapidly disappears, because of the $\\cos^2 (\\beta\n- \\alpha)$ suppression factor in the corresponding partial width.\n\nThe total  charged Higgs boson width is shown, as a function of $m_{H^{\\pm}}$ and for\n$\\tan \\beta = 1.5, \\, 3, \\, 10, \\, 30$, in fig.~9d. Again one can see the effects of\nthe $t \\overline{b}$ threshold, and also the $\\tan \\beta$-dependence of the couplings to\nfermions. In any case, the charged Higgs width remains smaller than 1 GeV for\n$m_{H^{\\pm}} < m_t + m_b$, and smaller than $10 \\; {\\rm GeV}$ for $m_{H^{\\pm}} < 500 \\; {\\rm GeV}$.\n\\section{Neutral-Higgs production cross-sections}\n\nThere is only a limited number of parton-level processes which can give\ninteresting rates for the production of the MSSM neutral Higgs bosons\n($\\phi=h,H,A$) at proton-proton supercolliders:\n\\begin{eqnarray}\n\\label{prodmech}\ng+g \\rightarrow \\phi \\, ,\\\\\nq + q \\rightarrow q + q+  W^* + W^* \\rightarrow q + q + \\phi \\, , \\\\\ng+g \\ \\ {\\rm or\\ \\ } q + \\overline{q}\\rightarrow b+\\overline{b}+\\phi \\, ,\\\\\ng+g \\ \\ {\\rm or\\ \\  } q + \\overline{q}\\rightarrow t+\\overline{t}+\\phi \\, , \\\\\nq+ \\overline{q} \\rightarrow W(Z) + \\phi \\, ,\n\\end{eqnarray}\nwhere $q$ denotes any quark flavour.\nThese processes are controlled by the Higgs couplings to heavy quarks and\ngauge bosons, whose essential features were summarized in table~1. We\nbriefly discuss here the corresponding cross-sections and the status of\ntheir theoretical description, emphasizing the features which are\ndifferent from the SM case. We shall always adopt the HMRSB structure\nfunctions [\\ref{hmrs}] with $\\Lambda^{(4)} = 190 \\; {\\rm MeV}$.\n\\vskip 0.5cm\n{\\bf Gluon fusion. \\ }\nIn the SM, $g g \\rightarrow \\phi$ [\\ref{georgi}] is the dominant production\nmechanism, the most\nimportant diagram being the one associated with the top-quark loop. In the\nMSSM, this is not always the case, since the correction factors of table~1\ngive in general suppression for the top contribution and enhancement for\nthe bottom one, and stop and sbottom loops could also play a role.\n\nThe leading-order amplitudes for the gluon-fusion processes\nare determined by the functions of eqs. (\\ref{ifunct}), with top,\nbottom, stop and sbottom intermediate states. For $m_{\\tilde{q}}=1\\; {\\rm TeV}$, the squark\ncontributions are very small, owing to the suppression factor $m_Z^2\n\/ m_{\\tilde{q}}^2$ in the corresponding $I_{\\tilde{q}}$ functions. For large values\nof $\\tan \\beta$, the bottom contribution can compete with the top one and even\nbecome dominant.\n\nQCD corrections to the gluon-fusion cross-section were recently evaluated\nin ref. [\\ref{djouadigg}], for a SM Higgs in the mass region below the\nheavy-quark threshold. In this region, QCD corrections increase the top\ncontribution by about $50\\%$. To a good approximation, the bulk of QCD\ncorrections can be taken into account by performing the replacement\n\\begin{equation}\n\\label{qcdgg}\n\\sigma_0 (g g \\rightarrow \\phi)\n\\longrightarrow\n\\sigma_0 (g g \\rightarrow \\phi)\n\\left[\n1 + \\left( \\frac{11}{2} + \\pi^2 \\right) \\frac{\\alpha_S}{\\pi}\n\\right],\n\\end{equation}\nat the renormalization scale $Q = m_{\\phi}$. This calculation, unfortunately,\nis not valid above the heavy-quark threshold, a region which is relevant for\nthe bottom contribution and for the top contribution to $H,A$ production, when\n$m_H,m_A>2 m_t$. Even below the heavy-quark threshold, the SM QCD corrections\nare applicable to $h$ and $H$ production, but not to $A$ production, because\nof the additional $\\gamma_5$ factor appearing at the $A q \\overline{q}$ vertex.\nIn view of this not completely satisfactory status of QCD corrections, we\ncalculate, conservatively, the top contribution without QCD corrections.\nHowever, when discussing the detectability of the different physics signals,\nwe shall take into account the results of ref. [\\ref{djouadigg}], when\napplicable. In the case of the bottom contribution, we use the running\n$m_b$, which leads to suppression.\n\nIn fig.~15 we display cross-sections for $gg \\rightarrow \\phi$ ($\\phi=h,H,A$),\nas functions of $m_{\\phi}$, for $\\tan \\beta=1.5,3,10,30$ and for LHC and SSC\nenergies. The SM Higgs cross-section is also shown for comparison. For\nlarge values of $\\tan \\beta$ and not too high values of $m_{\\phi}$, the\ncross-sections can be enhanced  with respect to the SM value.\nThis effect is due to the enhanced bottom-quark contribution,\nas apparent from table~1 and fig.~5. The fast disappearance of\nthis effect for increasing Higgs masses is due to the fast decrease\nof the function $F_{1\/2}^{\\phi} ( \\tau_b^{\\phi})$ as $\\tau_b^{\\phi}\n\\rightarrow 0$. When the neutral Higgs couplings to fermions are SM-like,\nthe gluon-fusion cross-sections approach the SM value, and are always\ndominated by the top contribution. The changes in the slopes of the\ncurves in fig.~15 are due to the competing top and bottom contributions.\nIn particular, one can notice an important threshold effect, for $m_A\n\\sim 2 m_t$, in the process $g g \\rightarrow A$, which can bring the corresponding\ncross-section above the SM value for low $\\tan \\beta$.\n\nAs a final remark, we notice that the LHC and SSC curves in fig.~15\nhave very similar shapes, with a scaling factor which is determined\nby the gluon luminosity and uniformly increases from $\\sim 2.5$ at\n$m_{\\phi} \\sim 100 \\; {\\rm GeV}$ to $\\sim 5$ at $m_{\\phi} \\sim 500 \\; {\\rm GeV}$.\n\n\\vskip 0.5cm\n{\\bf $W$ fusion.\\ }\nIn the SM case, the $W$-fusion mechanism [\\ref{wfusion}]\ncan compete with the gluon-fusion\nmechanism only for a very heavy ($m_{\\varphi} \\simgt 500 \\; {\\rm GeV}$) Higgs boson,\nowing to the enhanced $W_LW_L\\varphi$ coupling and to the relative increase of\nthe quark number-densities. In the MSSM, the correction factors for the\ncouplings to vector boson pairs (see table~1 and fig.~7) are always smaller\nthan 1, so that the MSSM $W$-fusion cross-sections are always smaller than\nthe SM one.\n\nWe illustrate this in fig.~16, where $W$-fusion cross-sections for $h$ and $H$\nare displayed, for the same $\\tan \\beta$  and $\\sqrt{s}$ values as in fig.~15.\nFor both $h$ and $H$, the SM cross-section is approached from below in\nthe regions of parameter space where $\\sin^2 (\\beta - \\alpha) \\rightarrow 1$\nand $\\sin^2 (\\beta - \\alpha) \\rightarrow 0$, respectively. In figs.~16b and 16d,\nfor $m_A \\rightarrow 0$ there is a positive lower bound on $\\sin^2 (\\beta\n- \\alpha)$ (see fig.~7), reflecting the fact that at the tree level\n$\\alpha \\rightarrow - \\beta$ in this limit, so the SM value is actually never\nreached. For increasing\n$m_H$, one can notice the fast decoupling of $H$ from $W$-pairs, as already\nobserved when discussing the total width. In leading order, $A$ does not\ncouple to $W$-pairs. A non-vanishing cross-section could be generated at\none loop, but such a contribution is completely negligible, since even\nfor $h$ and $H$ the $W$-fusion cross-section is small ($\\simlt 20 \\%$)\ncompared with the gluon-fusion cross-section. Finally, we observe that\nthe LHC and SSC cross-sections of fig.~16 differ by an overall factor\n$\\sim 3$ in the phenomenologically relevant region, $m_{\\phi} =$\n70--140 GeV.\n\n\\vskip 0.5 cm\n{\\bf Associated $\\ifmmode b\\bar{b} \\else $b\\bar{b}$ \\fi \\phi$ production. \\ }\nThis mechanism is unimportant in the SM, since its  cross-section is\ntoo small to give detectable signals [\\ref{zktth}]. In the MSSM model,\nhowever, for large values of $\\tan \\beta$ the\n$\\ifmmode b\\bar{b} \\else $b\\bar{b}$ \\fi \\phi$ couplings can be strongly enhanced. Then for not too high\nvalues of the Higgs masses, a significant fraction of the total cross-section\nfor neutral Higgs bosons can be due to this mechanism.\n\nThe associated $b \\overline{b} \\phi$ production involves two rather different mass\nscales, $m_{\\phi} >> m_b$, therefore at higher orders large logarithmic\ncorrections of order\n$$\n\\alpha_s^n {\\rm ln}^n \\left( {m_{\\phi}\\over m_b} \\right) \\,\n$$\nmay destroy the validity of the Born approximation, depending on the value\nof $m_{\\phi}$. One needs an\nimproved treatment where these logarithms are resummed to all orders.\nThe origin of these logarithms is well understood. Part of them\ncome from configurations where the gluons are radiated collinearly\nby nearly on-shell bottom quarks, which are obtained by splitting\nthe initial gluons into a $\\ifmmode b\\bar{b} \\else $b\\bar{b}$ \\fi$ pair.\nThis type of logarithms are responsible for\nthe QCD evolution of the effective $b$-quark density within the proton:\nthey were carefully analysed and resummed to all orders,\nand it was found [\\ref{sophabgun}] that the corrections are positive\nand increase with the Higgs-boson mass.\nA second subset of logarithms lead to running quark mass effects.\nAn analysis where both effects are  treated simultaneously is still\nmissing. In view of this ambiguity, we interpolated the existing\nresults by using the Born approximation with the bottom quark mass\nadjusted to the fixed value $m_b=4\\; {\\rm GeV}$. However, one should keep in\nmind that the theoretical estimate in this case has a large (factor of 2)\nuncertainty.\n\nIn fig.~17 we display cross-sections for associated $\\ifmmode b\\bar{b} \\else $b\\bar{b}$ \\fi \\phi$ production,\nfor the same $\\tan \\beta$  and $\\sqrt{s}$ values as in fig.~15.\nComparing the cross-sections of figs.~15 and 17, we can see that\nthe $h \\ifmmode b\\bar{b} \\else $b\\bar{b}$ \\fi$ cross-section can give at most a 20\\% correction\nto the total $h$ cross-section. The $H\\ifmmode b\\bar{b} \\else $b\\bar{b}$ \\fi$ and $A \\ifmmode b\\bar{b} \\else $b\\bar{b}$ \\fi$\ncross-sections, however, can be even larger than the corresponding\ngluon-fusion cross-sections for $\\tan \\beta \\simgt 10$.\nComparing the LHC and SSC curves of fig.~17, one can notice\na rescaling factor varying from $\\sim 3$ to $\\sim 8$ in the $m_{\\phi}$\nregion from 60 to 500 GeV.\n\n\\vskip 0.5cm\n{\\bf Associated $W \\phi$ ($Z \\phi$) production. \\ }\nThis mechanism [\\ref{bjorken}] is the hadron collider analogue of the SM Higgs\nproduction mechanism at LEP, with the difference that at hadron colliders\n$W\\phi$ production is more important than $Z\\phi$ production. In the $Z\\phi$\ncase,\nthe event rate at the LHC and SSC is too low to give a detectable signal,\nboth in the SM and (consequently) in the MSSM. The $W \\phi$ mechanism has\nconsiderable importance at the LHC for $\\phi=h,H$ and in the Higgs mass\nrange $m_\\phi=$ 70--140 GeV, where a measurable signal may be obtained\nfrom final states consisting of two isolated photons and one isolated lepton.\nThe calculation of the cross-section is well understood, including\nthe QCD corrections, since it has a structure similar to the Drell-Yan\nprocess, with the same next-to-leading-order corrections (for a recent\nstudy concerning the numerical importance of the QCD corrections see\nref.~[\\ref{willen}]). The QCD corrections are positive, and amount to\nabout 12\\% if one chooses $Q^2=\\hat{s}$ as the scale of $Q^2$ evolution.\nThe production cross-sections of $h$ and $H$ are obtained by rescaling\nthe SM model cross-section by the appropriate correction factors\ngiven in table~1.\n\nIn fig.~18, cross-sections for $Wh$ and $WH$ are displayed,\nas functions of corresponding Higgs masses,\nfor the same $\\tan \\beta$  and $\\sqrt{s}$ values as in fig.~15.\nSince the SUSY correction factors are the same, the approach to the\nSM case and the irrelevance of $WA$ production can be described in the same\nway as for the $W$-fusion mechanism.\n\nIn the phenomenologically relevant region, $m_{\\phi} =$ 70--140 GeV,\nthe scaling factor between the LHC and SSC curves is $\\sim 2.5$.\n\n\\vskip 0.5cm\n{\\bf Associated $\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi \\phi$ production. \\ }\nIn the SM, the Born cross-section formula for this process is the same\nas for the $b \\overline{b} \\varphi$ case [\\ref{zktth}]. In the MSSM case,\none just needs to insert the appropriate SUSY correction factors, as\nfrom table~1. Note, however, that the leading-order QCD calculation is\nmore reliable in this case, since in the $t \\overline{t} \\phi$ case\none does not have two very different physical scales when $m_{\\phi}$\nis in the intermediate mass region. The next-to-leading QCD corrections are\nnot known, therefore the Born cross-section still suffers from a relatively\nlarge ($\\sim 50\\%$) scale ambiguity.\n\nIn fig.~19, the production cross-sections for $\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi \\phi$ ($\\phi=h,H,A$) are\nplotted, as functions of the corresponding Higgs mass,\nfor the same $\\tan \\beta$  and $\\sqrt{s}$ values as in fig.~15.\nIn general, the MSSM cross-sections are\nsmaller than the SM one, which is approached in the limit\nin which the $\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi \\phi$ coupling becomes SM-like.\nA possible exception is the $t \\overline{t} H$ cross-section for small values\nof $m_A$ and $\\tan \\beta$, since in this case the corresponding coupling can\nbe slightly enhanced with respect to the SM one.\n\nIn the phenomenologically relevant range, $m_{\\phi} =$ 70--140 GeV,\nthe rescaling factor between the LHC and SSC curves in fig.~19 varies\nfrom $\\sim 6$ to $\\sim 7$.\n\n\\vskip 0.5 cm\nIn the phenomenologically allowed range of eq. (\\ref{mtop}), the top-mass\ndependence of the cross-sections of figs.~15--19 is not negligible, but it\ndoes not change qualitatively the previous considerations. The largest\neffect comes from the increase of the upper limit on $m_h$  for increasing\ntop mass (see fig.~1). This induces a shift in the limiting values for the\n$h$ and $H$ production cross-sections. There are also obvious\nkinematical top-mass\neffects in the gluon-fusion mechanism and in the $\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi \\phi$ mechanism, which\nare well understood from SM studies [\\ref{zkwjsaa}].\nIn the MSSM, additional effects are\ngiven by the radiative corrections to the relevant Higgs couplings,\nwhich were discussed in sect.~2.\n\n\\section{Physics signals}\n\\subsection{Neutral Higgs bosons}\nWe now calculate the rates for a number of processes that could\nprovide evidence for one or more of the neutral MSSM Higgs\nbosons at the LHC and SSC, and we summarize our results with\nthe help of contour plots in the $(m_A,\\tan \\beta)$ plane. We  consider\nproduction cross-sections, folded with branching ratios, for the\nfollowing signals:\n\\begin{itemize}\n\\item\ntwo isolated photons;\n\\item\none isolated lepton and two isolated photons;\n\\item\nfour isolated charged leptons;\n\\item\na pair of tau leptons.\n\\end{itemize}\nIn the SM case, the first two signals are relevant for the region of\nintermediate Higgs mass, $70\\; {\\rm GeV} < m_{\\phi} < 140 \\; {\\rm GeV}$, the third one is the\nso-called `gold-plated' signal in the high-mass region $130\\; {\\rm GeV} < m_{\\phi}\n< 800 \\; {\\rm GeV}$, and the $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ signal appears to be hopelessly difficult.\n\nIn a complete phenomenological study, one would like to determine precisely\nthe statistical significance of the different physics signals. This would\nrequire, besides the computation of total signal rates, the calculation of\nthe backgrounds, the determination of the\nefficiencies (for both signals and backgrounds) due to kinematical cuts and\ndetector effects,  the optimization of the kinematical cuts to achieve the\nbest signal$\/$background ratio, etc.\nSuch a complete analysis would require the specification of several detector\nand machine parameters, and goes beyond the aim\nof the present paper. Instead, we try here to present total rates for\nwell-defined physics signals, in a form which should be useful as a\nstarting point for dedicated experimental studies.\n\nAs the only exception, to illustrate with an example how our results can be\nused to establish the statistical significance of a given physics signal in\na given detector, we shall describe the case of the `two-isolated-photons'\nsignal, using the results of recent simulation works.\nA similar procedure should be adopted for\nany other physics signal, detector, and collider, once complete results of\nsimulation works are available. In many cases, the existing results\nfrom previous background and simulation studies, carried out for the SM,\ncan also be used to draw conclusions concerning the MSSM case. We mention,\nhowever, two important differences: 1) the total widths of $H$ and $A$\nremain small even in the high-mass region; 2) for large $\\tan \\beta$, the number\nof signal events in the $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ final state is significantly higher than\nin the SM case.\n\n\\vskip 0.5cm\n{\\bf Inclusive two-photon channel.\\ }\nIn fig.~20 we display cross-sections times branching ratios for the\ninclusive production of $\\phi=h,H,A$, followed by the decay $\\phi \\rightarrow\n\\gamma \\gamma$, as functions of $m_{\\phi}$, and for\nthe same parameter choices and energies as in fig.~15. We sum the\ncontributions of the gluon-fusion, $W$-fusion, and $\\ifmmode b\\bar{b} \\else $b\\bar{b}$ \\fi\\phi$ mechanisms.\nFor comparison, the SM value is also indicated. The QCD corrections of\nref. [\\ref{djouadigg}] are not included, for the reasons explained in\nthe previous section. In the case of $h$ and $H$,\nthe signal rates are always smaller than in the SM, and approach the SM\nvalues at the upper and lower edge of the allowed $m_h$ amd $m_H$ ranges,\nrespectively. The rather steep slope characterizing the approach to the SM\nlimit, for varying Higgs mass and fixed $\\tan \\beta$, is a reflection of the property\nof the branching ratios discussed in sect.~4. Also the structure in figs.~20b\nand 20e can be attributed to the threshold behaviour of the $H \\rightarrow hh$\nchannel. The signal rate for the CP-odd $A$ boson is extremely small for\n$\\tan \\beta \\simgt 3$. However, we observe that in a small\nregion of the parameter space, for $m_A$ just below $2 \\, m_t$ and\n$\\tan \\beta \\simlt 3$, the rate can become larger than the SM one: nevertheless,\nin general it is still too low to produce a detectable signal, unless\none chooses $\\tan \\beta \\sim 1$ and $m_A \\sim 2 m_t$.\n\nFor the inclusive two-photon channel, the results of detailed simulations\nof signal and SM background are now available, for some of the LHC detector\nconcepts [\\ref{cseez},\\ref{evian},\\ref{cseezcap},\\ref{unal}]. For\nillustrative purposes, in the following example we shall follow the\ntreatment of ref.~[\\ref{cseezcap}]. In the mass range $m_\\phi=$ 80--150 GeV,\nand assuming $10^5 \\, {\\rm pb}^{-1}$ integrated luminosity, this LHC\nsimulation considers a fairly wide range of detector performances,\nwhich affect the significance of the signal. For an energy resolution\n$\\Delta E \/ E = [2 \\% \/ \\sqrt{E(\\!\\!\\; {\\rm GeV})}] +0.5 \\%$, ref.~[\\ref{cseezcap}]\nobtains a $10^4$ efficiency for rejecting jets faking an isolated photon in\nthe relevant $p_T$ region. Applying standard kinematical cuts, this simulation\nfinds $\\sim$ 40--50$\\%$ kinematical acceptance, with an additional $\\sim$\n30--40$\\%$ loss due to isolation cuts and reconstruction efficiency for\nthe isolated photons. Typically, for a SM Higgs with $m_{\\varphi} \\sim\n100 \\; {\\rm GeV}$, one obtains $\\sim 10^3$ signal events over $\\sim 10^4$ background\nevents, corresponding to a statistical significance $S\/\\sqrt{B} \\sim 10$.\nMore generally, ref. [\\ref{cseezcap}] determined the statistical\nsignificance of the signal for given values of the generic Higgs mass\n$m_{\\phi}$ and of the signal rate $\\sigma \\cdot BR (\\phi \\rightarrow \\gamma \\gamma)$\n(see fig.~21). In our opinion, this is an excellent way of summarizing the\nsimulation work, since it gives the possibility of studying alternatives\nto the SM case, and in particular the MSSM. The dashed line in fig.~21\ncorresponds to the signal for the SM Higgs, which includes both the\ngluon-fusion and the $W$-fusion production mechanisms, and also\nthe QCD corrections of ref. [\\ref{djouadigg}]. One can see from fig.~21\nthat for such optimistic detector parameters there is some margin for\ndetecting smaller rates than in the SM. Clearly the SUSY Higgs search\nfurther enhances the need for the best possible  $m_{\\gamma\\gamma}$\nresolution and $\\gamma$-jet rejection.\n\nIn extending the SM analysis to the MSSM, one should pay attention to the\napplicability of the QCD corrections of ref. [\\ref{djouadigg}] to the\ngluon-fusion cross-section.\nWe have checked that in the phenomenologically relevant region, which\ncorresponds to $h$ or $H$ in the intermediate mass range, and to signal rates\nwithin an order of magnitude from the SM one, the gluon-fusion mechanism is\ndominated by the top-quark loop. Since in this region the gluon-fusion\nmechanism accounts for $\\sim 80 \\%$ of the total cross-section, and the\ncorrection is roughly a multiplicative factor 1.5, as a rule of thumb we\ncan take it into account by multiplying the total cross-section by a factor\n$\\sim 1.4$.\n\nIn fig.~22 we show contour plots in the $(m_A,\\tan \\beta)$ plane, corresponding\nto fixed values of $\\sigma \\cdot BR (\\phi\\rightarrow\\gamma\\gamma)$ ($\\phi=h,H$).\nQCD corrections have been included according to ref. [\\ref{djouadigg}].\nThe region where the rate is large enough to promise a measurable\nsignal is rather large for $h$, is concentrated in a small strip for $H$,\nand is possibly a very small area, just below $m_A = 2 m_t$ and just above\n$\\tan \\beta = 1$, for $A$. In our representative\nexample [\\ref{cseezcap}], we can now evaluate the statistical\nsignificance of the `two-isolated-photons' signal at any point of the\n$(m_A,\\tan \\beta)$ plane, by just combining the information contained in figs.~22,\n2, 3, and 21. In the case of $h$ searches, and for $m_h \\simgt\n90 \\; {\\rm GeV}$, a signal rate beyond 40 fb should give detectable signals.\nA signal rate of 30 fb is the borderline of detectability for one year\nof running, and signal rates below 20 fb appear extremely difficult to\ndetect. In the case of $H$, which has higher mass, a signal rate of 20 fb\nappears to be the borderline of what can be achieved in one year of running.\nIn the case of $A$, the interesting mass region is $m_A \\sim 2 m_t$:\nfor $m_A = 250 \\; {\\rm GeV}$, and taking $\\sigma \\cdot BR (A \\rightarrow \\gamma \\gamma)\n= 3 \\; {\\rm fb}$ as a plausible discovery limit at the LHC [\\ref{cseezcap}],\na signal for $A \\rightarrow \\gamma \\gamma$ will be found only if $\\tan \\beta \\simlt 1.5$.\n\n\\vskip 0.5cm\n{\\bf One isolated lepton and two isolated photons. \\ }\nThis signal can come from either $W \\phi$ or $\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi \\phi$ production.\nIn the latter case, two or more isolated jets are also produced.\nThe physics signals from $W \\phi$ production are particularly important\nat the LHC, and were studied in ref. [\\ref{rkzkwjs}]. The\nimportance of the physics signals from $t \\overline{t} \\phi$ production\nwas recently emphasized in ref. [\\ref{marcianopaige}].\nThe production rates, multiplied by the $\\phi \\rightarrow \\gamma \\gamma$\nbranching ratio, are shown in fig.~23. We can see that,\nsimilarly to the inclusive $\\gamma\\gamma$ channel, the rates for\n$Wh$, $WH$, $\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi h$, $\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi H$ are always smaller than the  SM value,\nwhich represents the boundary curve in the limit of large $\\tan \\beta$ and\nlarge (small) $m_A$ for $h$ ($H$).\n{}From figs.~23g and 23j one can see that $t \\overline{t} A$ production can\ngive a $l l \\gamma$ signal larger than in the SM for small $\\tan \\beta$ and\nnear the $m_A = 2 m_t$ threshold, but even in this case the rate appears\nto be too small for detection.\nWe emphasize that the production rates shown in fig.~23\ndo not include the branching ratios of leptonic $W$ and\nsemileptonic $t$ decays. If top decays\nare as in the SM, one should still include a combinatorial factor of\n2, coming from the fact that both top and antitop can decay semileptonically.\nOn the other hand, in the MSSM there is the possibility of $t \\rightarrow b H^+$\ndecays, where the subsequent $H^+$ decay cannot produce a direct lepton\n$l=e,\\mu$. We shall take this possibility into account in the following,\nbut its impact on the detectability of the $l \\gamma \\gamma$ signal is rather\nsmall. The only case in which this effect is not completely negligible is\nfor $H$, when $m_A \\simlt 100 \\; {\\rm GeV}$ and $\\tan \\beta \\simlt 4$ or $\\tan \\beta \\simgt 10$,\nin which case the $t \\rightarrow b H^+$ branching ratio can play a role.\n\n{}From parton-level simulations [\\ref{cseez},\\ref{kuntrocs},\\ref{mangano}],\nfor a SM Higgs of about 100 GeV, one typically obtains $\\sim (12+15)$ and\n$\\sim (3+11)$ $l \\gamma \\gamma$ signal events at the LHC and SSC,\nrespectively. Here we assumed $10^5$ ${\\rm pb}^{-1}$ of integrated luminosity\nfor the LHC and $10^4$ ${\\rm pb}^{-1}$ for the SSC. The quoted numbers\nseparately show the contributions from $W \\varphi$ and $t \\overline{t} \\varphi$\nproduction. Furthermore, they include losses due to acceptances ($\\sim\n30 \\%$), and lepton and photon detection efficiencies [$\\epsilon \\sim\n(0.9)^3$]. The total background is roughly 20--30$\\%$ of the signal and\nis dominated by the irreducible $W\\gamma\\gamma$ and $\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi\\gamma\\gamma$\ncontributions. There are many different contributions to the reducible\nbackground ($b \\overline{b} g, b \\overline{b} \\gamma, b \\overline{b} \\gamma \\gamma,\nW j \\gamma, \\ldots$). Parton-level simulations indicate that they can\nbe suppressed well below the irreducible background, provided one assumes,\nas for the inclusive $\\gamma \\gamma$ case, excellent detector performances:\na $\\gamma$-jet rejection factor $\\simgt 3 \\times 10^3$ and a suppression\nfactor $\\simgt 7$ for the leptons from $b$-decays after isolation cuts.\n\nClearly, there is very little margin (a factor of 2?) to be sensitive to\nsignal rates smaller than in the SM. In fig.~24, we show contour plots\ncorresponding to fixed values for the quantity\n\\begin{equation}\n\\label{ldef}\n\\begin{array}{l}\nL_{\\phi} \\equiv \\left[ \\sigma \\cdot BR \\left( l \\gamma \\gamma \\right)\n\\right]_{\\phi}\n\\\\\n\\\\\n\\phantom{L_{\\phi}}\n= \\bigl[ 2 \\sigma (\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi \\phi) \\cdot BR ( t \\rightarrow W b)  + \\sigma ( W \\phi) \\bigr]\n\\cdot BR(\\phi\\rightarrow\\gamma\\gamma) \\cdot BR (W \\rightarrow l \\nu) \\, ,\n\\end{array}\n\\end{equation}\nfor the same choice of $m_t$ and $m_{\\tilde{q}}$ as in fig.~15, and for LHC and SSC\nenergies. In eq. (\\ref{ldef}), $l=e,\\mu$ and we have not considered the\nstrongly suppressed possibility\nof getting a light charged lepton from both top and antitop.\n\n\\vskip 0.5cm\n{\\bf The four-lepton channel. \\ }\nThe channel $\\varphi \\rightarrow Z^* Z^* \\rightarrow l^+ l^- l^+ l^-$ ($l=e,\\mu$) gives the\nso-called `gold-plated' signal for the SM Higgs in the mass range\n$m_{\\varphi} =$ 130--800 GeV. Below\n$m_{\\varphi} \\sim 130 \\; {\\rm GeV}$, both the total rate and the acceptance decrease\nvery rapidly, leading to too small a signal for detection. For all three\nneutral Higgs bosons of the MSSM, the rates in this channnel are always smaller\nthan in the SM. In the case of $A$, there is no $AZZ$ coupling at tree level,\nand loop corrections cannot generate measurable rates in the four-lepton\nchannel. As for $h$, if $m_t \\simlt 180 \\; {\\rm GeV}$ and $m_{\\tilde{q}} \\simlt 1 \\; {\\rm TeV}$,\none can see from fig.~1 that $m_h \\simlt 130 \\; {\\rm GeV}$. Therefore, the\n$h \\rightarrow Z^* Z^* \\rightarrow 4 l$ signal does not have chances of detection at the\nLHC and SSC, unless one chooses extremely high values for $m_t$ and\n$m_{\\tilde{q}}$ or one has superb resolution and acceptance for leptons.\nThe situation is somewhat better in the case of $H$, despite the strong\nsuppression with respect to the SM, due to the competition with the\n$hh, b \\overline{b}, t \\overline{t}$ channels, as discussed in sect.~3.\n\nIn fig.~25, we show signal rates for the SM Higgs boson and for $H$, for\nthe same choices of parameters as in fig.~15. The threshold effects and\nthe suppression for large values of $\\tan \\beta$ are clearly visible.\n\nThe LHC and SSC discovery potential can be estimated by using the\nresults of simulations carried out for the SM\n[\\ref{froid},\\ref{nisati},\\ref{dellanegra},\\ref{evian},\\ref{sdcgem}],\ntaking also into account that $\\Gamma_H <\n2 \\; {\\rm GeV}$ all over the mass region of interest, $m_H \\simlt 2 m_t$.\nAssuming excellent lepton momentum resolution, in the\nmass range $2 m_Z < m_H < 2 m_t$ a signal rate $\\sim 20$ smaller than in\nthe SM could still lead to a detectable signal. In fig.~26, we show\ncontour plots in the $(m_A,\\tan \\beta)$ plane, corresponding to fixed values of\n$\\sigma \\cdot BR ( H \\rightarrow 4  l)$. QCD corrections have been included according\nto ref. [\\ref{djouadigg}]. In view of the strong sensitivity to the value\nof $m_t$, we show contours for $m_t=120,140,160 \\; {\\rm GeV}$, for LHC and SSC\nenergies, and for $m_{\\tilde{q}}=1 \\; {\\rm TeV}$. The two almost vertical dashed lines\ncorrespond to $m_H = 2 m_Z$ and to $m_H = 2 m_t$. For $m_H > 2 m_Z$,\na detectable signal could be obtained up to $\\tan \\beta \\sim 5$. Notice that\nthe experimental acceptances change with $m_H$; in particular, in the region\n$m_H < 2 m_Z$ they fastly decrease with decreasing $m_H$: for a realistic\nassessment of the discovery limits in this mass region, one should take this\nand other effects into account. Anyway, the prospects for detection for\n$m_H < 2 m_Z$ do not look good if $m_t \\simlt 150 \\; {\\rm GeV}$ and $m_{\\tilde{q}} \\simlt\n1 \\; {\\rm TeV}$.\n\n\\vskip 0.5cm\n{\\bf The $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ channel. \\ }\nFor the SM Higgs boson, the $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ decay channel has been\nfound hopelessly  difficult for discovery [\\ref{dilella},\\ref{kbos}],\nsince this channel has bad mass resolution and overwhelmingly large\nbackground coming from the production of $\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi$, $WW +$ jets,\nDrell-Yan pairs, $Z + $ jets, $\\ifmmode b\\bar{b} \\else $b\\bar{b}$ \\fi + $ jets, \\ldots .\nThe bad resolution is due to the fact that the tau-decay products\nalways include one or more neutrinos, which carry away energy;\ntherefore one cannot reconstruct the signal as a resonance peak.\nThe situation is improved if the Higgs is produced with large transverse\nmomentum that is balanced by a jet [\\ref{kellis}].\nIn this case one can use the approximation\n\\begin{equation}\n\\label{taumiss}\n p_{\\nu}^{(1)}{\\vec{p}_{l,T}^{\\; (1)}\\over E_{l}^{(1)}}  +\n p_{\\nu}^{(2)}{\\vec{p}_{l,T}^{\\; (2)}\\over E_{l}^{(2)}}  =\n \\vec{p}^{\\rm \\;  miss}_{\\rm T}\n\\end{equation}\nto reconstruct the transverse momenta of the neutrinos and hence\nthe invariant mass of the tau pair. In the above equation,\n$ p_{\\nu}^{(i)}$ denotes the total transverse momentum of\nthe neutrinos coming from the decay of $\\tau^{(i)}$, $i=1,2$,\nwhile $\\vec{p}_{l,T}^{\\; (i)}$ and $E_{l}^{(i)}$ denote the lepton momenta\nand energies, respectively. It was shown in ref. [\\ref{dilella}]\nthat, in the mass range $m_{\\phi}=$ 70--140 GeV,\na mass resolution of $\\sim$13--17$\\%$ can be achieved.\nThis method can also be used for the hadronic decay modes,\ntaking advantage of the fact that the rate is higher by a factor\nof $\\sim 3.5$. When a tau decays hadronically,\nthe hadrons have very low multiplicity and invariant mass,\nand these properties might be used to recognize the `$\\tau$-jet'\n[\\ref{ua1}].\nThere is a price for the better mass resolution.\nTagging on a large-$p_{\\rm T}$ jet can reduce the rate\nby an order of magnitude. Furthermore, at $10^5 \\, {\\rm pb}^{-1} \/ {\\rm year}$\nluminosity, the presence of pile-up deteriorates significantly the\nmeasurement of $\\vec{p}_T^{\\rm \\;\\, miss}$, and\ntherefore the $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ signal can only be studied\nwith this method at lower luminosities, $\\sim 10^4 \\, {\\rm pb}^{-1} \/ {\\rm\nyear}$~\\footnote{Alternatively, at high luminosity one may try to just search\nfor an excess of events in the $e^{\\pm} \\mu^{\\mp}$ or $l^{\\pm} +$\n`$\\tau$-jet' channels.}. While these difficulties appear prohibitive\nin the case of the SM, the\nsituation is not entirely negative in the MSSM.\n\nIn fig.~27, we display signal rates for $\\phi \\rightarrow \\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ production\n($\\phi=h,H,A$), for the same parameter choices as in fig.~15, together\nwith the SM values. We can see that for large values of $\\tan \\beta$  the\nproduction rates can become much larger than in the SM. In the case of\n$h$ production, for $\\tan \\beta=30$ the enhancement can be more than one order\nof magnitude, and increases with decreasing values of $m_h$ (see figs.~27a\nand 27d). Huge enhancements can be obtained also for $H$ and $A$, thanks to\nthe properties of the $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ branching ratios discussed in sect.~4.\nNote for example that, at the LHC, for $m_H,m_A \\sim 500 \\; {\\rm GeV}$ and $\\tan \\beta \\simgt\n10$, we get  $\\sigma \\cdot BR(H,A \\rightarrow \\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi) \\geq 20 \\; {\\rm pb}$, while for\n$m_A,m_H \\sim 120 \\; {\\rm GeV}$ and $\\tan \\beta \\simgt 30$, we obtain $\\sigma \\cdot BR(H,A\n\\rightarrow \\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi) \\geq 30 \\; {\\rm pb}$. The rates for the SSC are rescaled by the\nfactor already discussed in the previous section.\n\nIn order to assess, for any given mass, the cross-section values above\nwhich one obtains a measurable signal over the large background, detailed\nsimulations are needed. Preliminary studies have been reported for the\nlepton channel $e^{\\pm} \\mu^{\\mp}$ in ref. [\\ref{dilella}] and for the\nmixed channel $l^{\\pm} +$ `$\\tau$-jet' in ref. [\\ref{pausstau}]. In the second\ncase, the difficulty of recognizing a `$\\tau$-jet' may be compensated by the\nhigher rate of this channel. The preliminary analysis of ref. [\\ref{pausstau}]\nfinds for the LHC sensitivity to values of $\\sigma \\cdot BR(\\phi\\rightarrow\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi)$\ndown to $\\sim 10 \\; {\\rm pb}$ in the low-mass region $m_{\\phi} \\sim 100 \\; {\\rm GeV}$\nand $\\sim 1 \\; {\\rm pb}$ in the high-mass region $m_{\\phi} \\sim 400 \\; {\\rm GeV}$.\nThis result cannot be easily rescaled to the SSC case, since a large\nmass interval is involved and the SSC luminosity gives more favourable\nexperimental conditions for the srudy of this channel.\n\nIn fig.~28, we show contour plots corresponding to\nfixed values of $\\sigma \\cdot BR(\\phi\\rightarrow\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi)$,\nfor the same values of $m_t$ and $m_{\\tilde{q}}$ as in fig.~15.\n\n\\subsection{ Charged Higgs boson }\nWe now move to the discussion of possible physics signals associated\nwith the charged Higgs boson. The phenomenology of the charged Higgs\nboson at hadron colliders was previously discussed in refs. [\\ref{lhcch}].\nThe benchmark mass value for charged-Higgs-boson searches at the LHC and\nSSC is $m_{H^{\\pm}} = m_t - m_b$. For lower values of $m_{H^{\\pm}}$, the dominant\nproduction mechanism at large hadron colliders is $gg \\rightarrow t \\overline{t}$, followed\nby $t \\rightarrow H^+ b$. For higher values of $m_{H^{\\pm}}$, the dominant production\nmechanism is $g b \\rightarrow t H^+$. As far as detectable signals are concerned,\nthis last case appears hopeless, in view of the suppressed cross-section\nand of the large backgrounds coming from QCD subprocesses. The first\ncase appears instead to be experimentally viable over a non-negligible\nregion of parameter space. Given the known $t \\overline{t}$ production cross-section,\none can compute the $t \\rightarrow b H^+$ branching ratio according to well-known\nformulae, parametrizing again the leading QCD corrections by running\nmasses evaluated at a scale $Q \\sim m_{H^{\\pm}}$. The charged Higgs branching ratios\nwere discussed in the previous section, where it was found that the\n$\\tau^+ \\nu_{\\tau}$ mode dominates in the mass range under consideration.\nThe experimental signal of a charged Higgs would then be a violation\nof lepton universality in semileptonic top decays. As a convenient indicator,\none can consider the ratio\n\\begin{equation}\n\\label{rfelc}\nR =\n{\nBR(t \\rightarrow \\tau^+ \\nu_{\\tau} b)\n\\over\nBR(t \\rightarrow \\mu^+ \\nu_{\\mu} b)\n}\n\\equiv\n1 + \\Delta R \\, ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{deltar}\n\\Delta R =\n{\nBR(t \\rightarrow H^+ b) \\cdot BR ( H^+ \\rightarrow \\tau^+ \\nu_{\\tau})\n\\over\nBR(t \\rightarrow W^+ b) \\cdot BR ( W^+ \\rightarrow \\mu^+ \\nu_{\\mu})\n}\n{}.\n\\end{equation}\nPreliminary investigations [\\ref{lhcch}] show that the experimental\nsensitivity could reach $\\Delta R \\simgt 0.15$ at the LHC. At the\nSSC the increased $t \\overline{t}$ production cross-section is likely to\ngive better sensitivity. In fig.~29, we display\ncontour lines of $\\Delta R$ in the $(m_A,\\tan \\beta)$ plane, for the three\nrepresentative values $m_t = 120, \\, 140, \\,\n160 \\; {\\rm GeV}$. The dashed lines denote\nthe kinematical limit $m_{H^{\\pm}} = m_t - m_b$. One can see that the most\ndifficult values of $\\tan \\beta$ are those between 3 and 10, and that the process\nunder consideration could give access to values of $m_A$ as high as 80--120\nGeV for top-quark masses in the range 120--160 GeV.\n\n\n\n\\section{Conclusions and outlook}\n\nIn this paper we carried out a systematic analysis of the possible\nphysics signals of the MSSM Higgs sector at the LHC and SSC,\nassuming that the supersymmetric (R-odd) particles are heavy enough\nnot to affect significantly the production cross-sections and the\nbranching ratios of the MSSM Higgs particles. As independent\nparameters in the Higgs sector, we chose $m_A$ and $\\tan \\beta$, and we\nconsidered the theoretically motivated region of the parameter space\n$$\n0 \\le m_A \\le 500 \\; {\\rm GeV},\n\\;\\;\\;\\;\n1 \\le \\tan \\beta \\le 50 \\, .\n$$\nWe assumed $m_{\\tilde{q}}=1 \\; {\\rm TeV}$ and negligible mixing in the squark sector.\nWe included\nthe most important radiative corrections to the Higgs masses $m_h$, $m_H$,\n$m_{H^{\\pm}}$, and to the Higgs couplings to fermions and vector bosons.\nWe also included the most important radiative corrections to the three-point\ncouplings of the neutral Higgs bosons.\n\nWe estimated the discovery potential of LEP I and LEP II, and we carried\nout detailed cross-section calculations for the LHC and SSC. We singled\nout four classes of final states ($\\gamma\\gamma$, $l^{\\pm}\\gamma\\gamma$,\n$l^+l^-l^+l^-$, $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$) which could provide significant signals for\nneutral Higgs bosons at the LHC and SSC, and we also examined possible\nsignals of charged Higgs bosons in top decays.\n\nWe calculated all the relevant branching ratios, and the cross-sections\nfor all the relevant production mechanisms.\nWe presented our results with the help of branching-ratio curves,\ncross-section curves, signal-rate curves and contour plots in the\n$(m_A,\\tan \\beta)$ plane. We did not perform new background\nstudies, but we pointed out that, using the results of our calculations\nand of the existing simulations carried out for the SM Higgs, supplemented\nby estimates of the acceptances and efficiencies of typical experiments,\nin many cases one can draw conclusions concerning the discovery range. In\nsome cases, as for the $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ channel, further simulation work appears\nto be needed in order to reach firm conclusions. Nevertheless, some\npreliminary conclusions can already be drawn and will now be summarized.\n\nAt large hadron colliders, the MSSM Higgs search is, in general, more\ndifficult than the SM Higgs search.\nThis is due to the fact that, in a large region of the parameter\nspace, at least one of the MSSM neutral Higgs bosons\nis in the intermediate mass region, $80 \\; {\\rm GeV} \\simlt m_{\\phi} \\simlt 140 \\; {\\rm GeV}$,\nbut with rates in the $\\gamma\\gamma$ and $l\\gamma\\gamma$ channels which can\nbe significantly suppressed with respect to the SM case. Similarly,\nneutral Higgs bosons with $m_{\\phi} \\simgt 130 \\; {\\rm GeV}$ have typically\nstrongly suppressed rates in the $l^+l^-l^+l^-$ channel. On the contrary,\nin the MSSM, for rather large values of $\\tan \\beta$, one can obtain a much\nlarger signal rate in the $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ channel than in the SM.\nFinally, $t \\rightarrow b H^+$ decays,\nfollowed by $H^+ \\rightarrow \\tau^+ \\nu_{\\tau}$, can give detectable signals\nonly in a rather restricted region of the parameter space.\n\nAs an example, we now try to assess the discovery potential of the\ndifferent channels  for the representative parameter choice $m_t =\n140 \\; {\\rm GeV}$, $m_{\\tilde{q}} = 1 \\; {\\rm TeV}$, working as usual in the $(m_A,\\tan \\beta)$ plane.\n\nTo begin with, we recall the expectations for LEP II.\nThe size of the LEP II discovery region depends rather strongly on\n$m_t$ and $m_{\\tilde{q}}$, and on the assumed energy and luminosity.\nFor standard machine parameters, LEP II cannot test the\nwhole parameter space allowed by the present data.\nLooking back at fig.~8, one may tentatively say that LEP II will give us\n(if no Higgs boson is discovered) lower limits of about $m_A \\simgt$\n70--100 GeV and $\\tan \\beta \\simgt$ 3--8 for $m_t = 120 \\; {\\rm GeV}$, $\\tan \\beta \\simgt$\n1.5--3 for $m_t = 160 \\; {\\rm GeV}$.\n\nWe observe that the LHC and SSC will test the MSSM\nHiggs sector in a largely complementary region of the ($m_A,\\tan \\beta$) plane.\nA pictorial summary of the discovery potential of the different channels\nis presented in fig.~30. We emphasize once again that the final discovery\nlimits will depend on the machine and detector properties, as well as on\nthe actual values of the top and the soft supersymmetry-breaking masses.\nWe therefore drew fig.~30 just for illustrative purposes, to exemplify\na particularly convenient way of considering all the discovery channels\nat once.\n\nThe $\\gamma \\gamma$ and $l \\gamma \\gamma$ channels are important in\napproximately the same region of the parameter space, $m_A \\gamma\\gamma m_Z$ for\n$h$ and $50 \\; {\\rm GeV} \\simlt m_A \\simlt 100 \\; {\\rm GeV}$ for $H$. Therefore, these\ntwo channels can be experimentally cross-checked one against the other,\nreinforcing the significance of a possible signal.\nAs an optimistic discovery limit for $h$, we show in fig.~30 the contour line\n$\\sigma \\cdot\nBR (h \\rightarrow \\gamma \\gamma) = 30 \\; {\\rm fb}$ at the LHC,  corresponding to\n$m_A \\simgt 200 \\; {\\rm GeV}$. This contour line is shown only for $m_h \\simgt\n80 \\; {\\rm GeV}$. In the region of the parameter space to the right of this line\n[indicated  by the labels $h\\rightarrow\\gamma\\gamma$ and $l + (h\\rightarrow\\gamma\\gamma)$],\nit is expected that measurable signals will be found, assuming detector and\nmachine parameters as discussed in refs. [\\ref{evian}]. Approximately the same\ncontour line is obtained by taking $\\sigma \\cdot\nBR (h \\rightarrow \\gamma \\gamma) = 85 \\;\n{\\rm fb}$ at the SSC. This indicates that, in the inclusive $\\gamma \\gamma$ channel,\nthe discovery range of the LHC and SSC will be the same if the luminosity\nat the LHC will be  $\\sim 3$ times higher than at the SSC and if the\ndetectors used at the two machines will have similar efficiencies in\nsuppressing the backgrounds.\nVery similar discovery lines can be drawn by considering  the $l \\gamma\n\\gamma$ channel and taking $\\sigma \\cdot\nBR [l + (h\\rightarrow\\gamma\\gamma)] \\sim 0.8\n\\; {\\rm fb}$ for the LHC and $\\sim 4 \\; {\\rm fb}$ for the SSC.\n\nIn fig.~30 we also show the contour line for $\\sigma\n\\cdot BR (H \\rightarrow\\gamma\\gamma)\n= 20 \\; {\\rm fb}$ at the LHC, corresponding to $\\sim 55 \\; {\\rm fb}$ at the SSC.\nThe slightly smaller signal rate was chosen to account for the improved\nefficiencies at higher Higgs-mass values. The contour line defines a\nnarrow strip around $m_A\\sim 75\\; {\\rm GeV}$ (shaded in fig.~30), where\nthe discovery of $H$ is expected to be possible both in the\n$\\gamma \\gamma$ and in the $l \\gamma \\gamma$ channels [for lack\nof space the label $l + (H \\rightarrow \\gamma \\gamma)$ has been omitted].\n\nThe four-lepton channel is important mainly for $H$, in the mass region\n$2 m_Z \\simlt m_H \\simlt 2 m_t$, which translates into $150 \\; {\\rm GeV} \\simlt\nm_A \\simlt 2 m_t$, and for relatively small $\\tan \\beta$. As a reference\nvalue for discovery in this mass region, we take $\\sigma \\cdot\nBR (H \\rightarrow 4 l)\n= 1 \\; {\\rm fb}$ for the LHC, which corresponds to $\\sigma \\cdot\nBR (H \\rightarrow 4 l)\n\\sim 3 \\; {\\rm fb}$ for the SSC. This contour defines the area in fig.~30\nindicated  by the label $H\\rightarrow 4l$.\nIn a small part of this area, corresponding to $m_A \\sim m_t$ and\n$\\tan \\beta \\sim 1$, $A \\rightarrow \\gamma \\gamma$ could also give a detectable signal.\n\nIn the region of very large $\\tan \\beta$, and moderately large $m_A$, one\ncould take advantage of the enhanced production cross-sections and of\nthe unsuppressed decays into $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ to obtain a visible signal for\none or more of the MSSM neutral Higgs bosons, and in particular for\n$H$ and $A$, whose masses can be significantly larger than 100 GeV.\nThe simulation work for this process is still at a rather early stage\n[\\ref{pausstau}], so that no definite conclusion can be drawn yet.\nFor reference, the dotted line in fig.~30 corresponds to a (somewhat\narbitrary) interpolation of $\\sigma\n\\cdot BR (\\phi \\rightarrow \\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi) \\sim 10 \\, {\\rm pb}$ at\n$m_{\\phi} = 100 \\; {\\rm GeV}$ and $\\sigma \\cdot\nBR (\\phi \\rightarrow \\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi) \\sim 1\\, {\\rm pb}$ at\n$m_{\\phi} = 400 \\; {\\rm GeV}$, for LHC energy and summing over the $\\phi=H,A$\nchannels.\n\nFinally, in the region of parameter space corresponding to $m_A\n\\simlt 100 \\; {\\rm GeV}$, a violation of lepton universality due to the decay\nchain $t \\rightarrow b H^+ \\rightarrow b \\tau^+ \\nu_{\\tau}$ could indicate the existence of\nthe charged Higgs boson. This region is indicated by the label\n$H^{\\pm}\\rightarrow \\tau\\nu$ in fig.~30. Its right border is defined\nby the contour line of $R = 1.15$, where $R$ was defined in eq.\n(\\ref{rfelc}).\n\nBy definition, our contour lines do not take into account changes in the\nacceptances and efficiencies, which are expected in realistic experimental\nconditions, and depend on the Higgs mass and on the detector. We therefore\nexpect some deformations of our contours once discovery lines are extracted\nfrom realistic experimental simulations [\\ref{cseezcap},\\ref{pausstau}].\n\nAs a last piece of information we also display in fig.~30 the border of the\nexpected discovery region at LEP II, which depends rather sensitively,\nas already discussed, on the assumed values of the machine energy and\nluminosity.  We then show two representative lines: the lower dashed\nline corresponds to $\\sigma (e^+ e^- \\rightarrow\nhZ, HZ, hA, HA) = 0.2 \\, {\\rm pb}$ at $\\sqrt{s} = 175 \\; {\\rm GeV}$,\nwhile the upper dashed line corresponds to $\\sigma (e^+ e^- \\rightarrow\nhZ, HZ, hA, HA) = 0.05 \\, {\\rm pb}$ at $\\sqrt{s} = 190 \\; {\\rm GeV}$.\n\nUsing the result summarized in fig.~30,\nwe can draw several important qualitative\nconclusions:\n\n\\begin{itemize}\n\\item\nThe discovery potentials of LEP and the LHC\/SSC show a certain\ncomplementarity.\nThe discovery region at LEP covers all $\\tan \\beta$ values at small values\nof $m_A$, and all $m_A$ values at small values of $\\tan \\beta$, while at\nthe LHC\/SSC one should be sensitive to the large $\\tan \\beta$, large\n$m_A$ region.\n\\item\nOne may ask whether the LHC and SSC, combined with LEP II,\ncan explore the full parameter space of the MSSM Higgs sector, being\nsensitive to at least one signal in each point of the $(m_A,\\tan \\beta$)\nplane, for all plausible values of $m_t$ and $m_{\\tilde{q}}$. At present,\nthis question cannot  be answered positively.\nThe union of the regions where one should find signals\nat least for one Higgs boson does not cover the whole parameter\nspace: the discovery region has a hole in the middle of the parameter space.\nFor our parameter choice, the most difficult region appears\nto be the cross-hatched area around $m_A=150 \\; {\\rm GeV}$ and $\\tan \\beta= 10$.\nTherefore we cannot claim yet the existence of\na `no-lose' theorem for the MSSM Higgs search.\n\\item\nOne may also ask if there are regions of parameter space where one\ncan find more than one signal from the MSSM Higgs sector. The\nanswer is that around $m_A=200 \\; {\\rm GeV}$ and $\\tan \\beta < 5$ one can\ndiscover $h$ at LEP II and $H$ at the LHC\/SSC in the four-lepton\nchannel. There is a somewhat smaller region above $m_A=200\\; {\\rm GeV}$\nwhere one can also find $h$ in the $\\gamma\\gamma$ and $l \\gamma\\gamma$\nchannels. Furthermore, at high values of $\\tan \\beta $ ($\\simgt 20$) and\nat $m_A > 200 \\; {\\rm GeV}$ one may discover $A$ and $H$ in the $\\tau\\tau$ channel\nand $h$ in the $\\gamma\\gamma$ and $l \\gamma\\gamma$ channels, although\nthe separation of $H$ and $A$ appears to be impossible, due to their\nalmost perfect degeneracy in mass. This part of the parameter space is\ninaccessible at LEP II. The discovery region for $H$ in the $\\gamma\\gamma$\nand $l, \\gamma\\gamma$ channels, corresponding to low values of $m_A$,\nlargely overlaps with the LEP II discovery region and with the discovery\nregion related to charged-Higgs production in top decays. In the low $\\tan \\beta$,\nfor $80 \\; {\\rm GeV} \\simlt m_A \\simlt 180 \\; {\\rm GeV}$ and $m_A \\simgt 2 m_t$ one has\na signal at LEP II and no signal at the LHC and SSC, since $m_h$\nis too small for detection.\n\\end{itemize}\n\nFinally, we would like to make some comments on the theoretical\nuncertainties and on possible future studies.\n\nOur values for the signal rates depend on several phenomenological\ninput parameters, as the value of the bottom mass, the parton-number\ndensities and the value of $\\alpha_S$. The given cross-sections and\nbranching ratios will change if the input parameters are varied in\ntheir allowed range. Also, for some\nproduction mechanisms, only the Born cross-sections are known.\nWe estimate that the theoretical errors of the calculated rates\nvary from about 30\\%, in the case of the $\\gamma\\gamma$ channel,\nup to about a factor of 2 when the $b \\overline{b} \\phi$ or $t \\overline{t} \\phi$\nproduction mechanisms are important.\n\nWe did not study all effects associated to variations of the parameters\nin the SUSY ($R$-odd) sector.\nIt would be interesting to examine the case when some of the\nHiggs bosons are allowed to decay into $R$-odd SUSY particles,\nor the effects of squark mixing.\nMore importantly, serious simulation work is still needed, in particular\nfor the $\\ifmmode \\tau^+\\tau^-\\else $\\tau^+\\tau^-$ \\fi$ and the $l\\gamma\\gamma$ channel.\n\n\\section*{Note added}\nAfter the completion of most of the work presented in this paper, which\nwas anticipated in many talks [\\ref{talks}], we received a number of papers\n[\\ref{copies},\\ref{barger}] dealing with different subsets of the material\npresented here, and reaching similar conclusions. Reference [\\ref{barger}]\nalso contains the generalization of eqs. (\\ref{hhh1}--\\ref{deltahhh})\nto the case of arbitrary mixing in the stop and sbottom mass matrices,\nbut still neglecting the $D$-term contributions.\n\\section*{Acknowledgements}\nWe are grateful to G.~Altarelli for discussions, and for insisting\nthat we should carry out this study. We also thank A.~Brignole, D.~Denegri,\nJ.~Ellis, L.~Fayard, D.~Froidevaux, J.-F.~Grivaz, P.~Janot, F.~Pauss,\nG.~Ridolfi, C.~Seez, T. Sj\\\"ostrand, D.~Treille, J.~Virdee and P.~Zerwas for\nuseful discussions and suggestions.\n\\newpage\n\\section*{References}\n\\begin{enumerate}\n\\item\n\\label{LPHEP91}\nJ. Carter,  M. Davier and  J. Ellis, Rapporteur's talks given at the\nLP-HEP '91 Conference, Geneva, 1991, to appear in the Proceedings, and\nreferences therein.\n\\item\n\\label{higgs}\nM.S. Chanowitz, Ann. Rev. Nucl. Part. Phys. 38 (1988) 323;\n\\\\\nM. Sher, Phys. Rep. 179 (1989) 273;\n\\\\\nR.N. 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Wisconsin preprint MAD-PH-680 (1991) (revised).\n\\end{enumerate}\n\\newpage\n\\section*{Figure captions}\n\\begin{itemize}\n\\item[Fig.1:]\nContours of $m_h^{max}$ (the maximum value of $m_h$, reached for $m_A \\rightarrow\n\\infty$): a)~in the $(m_t,\\tan \\beta)$ plane, for $m_{\\tilde{q}} = 1 \\; {\\rm TeV}$; b)~in\nthe $(m_t,m_{\\tilde{q}})$ plane, for $\\tan \\beta = m_t \/ m_b$.\n\\item[Fig.2:]\nContours of $m_h$ in the $(m_A,\\tan \\beta)$ plane, for $m_{\\tilde{q}} = 1 \\; {\\rm TeV}$\nand a)~$m_t = 120 \\; {\\rm GeV}$, b)~$m_t = 160 \\; {\\rm GeV}$.\n\\item[Fig.3:]\nContours of $m_H$ in the $(m_A,\\tan \\beta)$ plane, for $m_{\\tilde{q}} = 1 \\; {\\rm TeV}$\nand a)~$m_t = 120 \\; {\\rm GeV}$, b)~$m_t = 160 \\; {\\rm GeV}$.\n\\item[Fig.4:]\nContours of $m_{H^\\pm}$ in the $(m_A,\\tan \\beta)$ plane, for $m_{\\tilde{q}} =\n1 \\; {\\rm TeV}$.\nThe solid lines correspond to $m_t = 120 \\; {\\rm GeV}$,\nthe dashed ones to $m_t=160 \\; {\\rm GeV}$.\n\\item[Fig.5:]\nContours of $\\sin^2 \\alpha \/ \\cos^2 \\beta$ in the $(m_A,\\tan \\beta)$ plane,\nfor $m_{\\tilde{q}} = 1 \\; {\\rm TeV}$.\nThe solid lines correspond to $m_t = 120 \\; {\\rm GeV}$,\nthe dashed ones to $m_t=160 \\; {\\rm GeV}$.\n\\item[Fig.6:]\nContours of $\\cos^2 \\alpha \/ \\sin^2 \\beta$ in the $(m_A,\\tan \\beta)$ plane,\nfor $m_{\\tilde{q}} = 1 \\; {\\rm TeV}$.\nThe solid lines correspond to $m_t = 120 \\; {\\rm GeV}$,\nthe dashed ones to $m_t=160 \\; {\\rm GeV}$.\n\\item[Fig.7:]\nContours of $\\sin^2 (\\beta - \\alpha)$ in the $(m_A,\\tan \\beta)$ plane,\nfor $m_{\\tilde{q}} = 1 \\; {\\rm TeV}$.\nThe solid lines correspond to $m_t = 120 \\; {\\rm GeV}$,\nthe dashed ones to $m_t=160 \\; {\\rm GeV}$.\n\\item[Fig.8:]\nSchematic representation of the present LEP I limits and of the future\nLEP II sensitivity in the $(m_A,\\tan \\beta)$ plane, for $m_{\\tilde{q}} = 1 \\; {\\rm TeV}$\nand a)~$m_t = 120 \\; {\\rm GeV}$, b)~$m_t = 160 \\; {\\rm GeV}$.\nThe solid lines correspond to the present LEP I limits.\nThe dashed lines correspond to $\\sigma (e^+ e^- \\rightarrow\nhZ, HZ, hA, HA) = 0.2 \\, {\\rm pb}$ at $\\sqrt{s} = 175 \\; {\\rm GeV}$, which\ncould be seen as a rather conservative estimate of the LEP II\nsensitivity.\nThe dash-dotted lines correspond to $\\sigma (e^+ e^- \\rightarrow\nhZ, HZ, hA, HA) = 0.05 \\, {\\rm pb}$ at $\\sqrt{s} = 190 \\; {\\rm GeV}$, which\ncould be seen as a rather optimistic estimate of the LEP II\nsensitivity.\n\\item[Fig.9:]\nTotal widths of the MSSM Higgs bosons, as functions of their respective\nmasses, for $m_t=140 \\; {\\rm GeV}$, $m_{\\tilde{q}} = 1 \\; {\\rm TeV}$ and $\\tan \\beta=1.5,3,10,30$: a)~$h$;\nb)~$H$; c)~$A$; d)~$H^\\pm$.\n\\item[Fig.10:]\nBranching ratios for $h$, as functions of $m_h$, for $m_t=140 \\; {\\rm GeV}$,\n$m_{\\tilde{q}} = 1 \\; {\\rm TeV}$ and: a)~$\\tan \\beta=1.5$; b)~$\\tan \\beta=30$.\n\\item[Fig.11:]\nBranching ratios for $H$, as functions of $m_H$, for $m_t=140 \\; {\\rm GeV}$,\n$m_{\\tilde{q}} = 1 \\; {\\rm TeV}$ and: a)~$\\tan \\beta=1.5$; b)~$\\tan \\beta=30$.\n\\item[Fig.12:]\nBranching ratios for $A$, as functions of $m_A$, for $m_t=140 \\; {\\rm GeV}$,\n$m_{\\tilde{q}} = 1 \\; {\\rm TeV}$ and: a)~$\\tan \\beta=1.5$; b)~$\\tan \\beta=30$.\n\\item[Fig.13:]\nBranching ratios for $h$, as a function of $m_A$, for $m_t=140 \\; {\\rm GeV}$,\n$m_{\\tilde{q}} = 1 \\; {\\rm TeV}$ and: a)~$\\tan \\beta=1.5$; b)~$\\tan \\beta=30$.\n\\item[Fig.14:]\nBranching ratios for $H^\\pm$, as functions of $m_{H^\\pm}$, for\n$m_t=140 \\; {\\rm GeV}$, $m_{\\tilde{q}} = 1 \\; {\\rm TeV}$ and: a)~$\\tan \\beta=1.5$; b)~$\\tan \\beta=30$.\n\\item[Fig.15:]\nCross-sections for neutral Higgs production, via the gluon-fusion\nmechanism, as functions of the corresponding masses and for $m_t=140 \\; {\\rm GeV}$,\n$m_{\\tilde{q}} = 1 \\; {\\rm TeV}$, $\\tan \\beta=1.5,3,10,30$: a)~$h$, LHC; b)~$H$, LHC; c)~$A$,\nLHC; d)~$h$, SSC; e)~$H$, SSC; f)~$A$, SSC. QCD corrections are not\nincluded.\n\\item[Fig.16:]\nCross-sections for $h$ and $H$ production, via the $W$-fusion mechanism, as\nfunctions of the corresponding masses,\nfor the same parameter choices as in fig.~15:\na)~$h$, LHC; b)~$H$, LHC; c)~$h$, SSC; d)~$H$, SSC.\n\\item[Fig.17:]\nCross-sections for associated $\\ifmmode b\\bar{b} \\else $b\\bar{b}$ \\fi \\phi$ production, as functions of the\ncorresponding Higgs masses, for\nthe same parameter choices as in fig.~15:\na)~$h$, LHC; b)~$H$, LHC; c)~$A$,\nLHC; d)~$h$, SSC; e)~$H$, SSC; f)~$A$, SSC.\n\\item[Fig.18:]\nCross-sections for associated $W \\phi$ production, as functions of\nthe corresponding Higgs masses and for\nthe same parameter choices as in fig.~15:\na)~$h$, LHC; b)~$H$, LHC; c)~$h$, SSC; d)~$H$, SSC.\n\\item[Fig.19:]\nCross-sections for associated $\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi \\phi$ production, as functions of the\ncorresponding Higgs masses and\nfor the same parameter choices as in fig.~15:\na)~$h$, LHC; b)~$H$, LHC; c)~$A$,\nLHC; d)~$h$, SSC; e)~$H$, SSC; f)~$A$, SSC.\n\\item[Fig.20:]\nCross-sections times branching ratios for inclusive Higgs production (the\ngluon-fusion, $W$-fusion, and $\\ifmmode b\\bar{b} \\else $b\\bar{b}$ \\fi \\phi$ contributions are summed) and decay\nin the $\\gamma\\gamma$ channel, for the same parameter choices as in fig.~15:\na)~$h$, LHC; b)~$H$, LHC; c)~$A$, LHC; d)~$h$, SSC; e)~$H$, SSC;\nf)~$A$, SSC. For the sake of comparison, the SM values are also indicated.\nQCD corrections to the gluon-fusion mechanism are not included.\n\\item[Fig.21:]\nSignificance of the inclusive $\\phi \\rightarrow \\gamma \\gamma$ signal, in the\nplane defined by $m_{\\phi}$ and $\\sigma \\cdot BR (\\phi \\rightarrow \\gamma \\gamma)$,\nfor the CMS detector proposal at the LHC, with an energy resolution\n$\\Delta E \/ E = [2 \\% \/ \\sqrt{E(\\!\\!\\; {\\rm GeV})}] +0.5 \\%$.\nThe solid lines are contours of\nconstant $S\/\\sqrt{B}$, where $S$ is the signal and $B$ is the background.\nThe dashed line corresponds to the SM Higgs, including QCD corrections\nto the gluon-fusion mechanism. Courtesy of C. Seez [\\ref{cseezcap}].\n\\item[Fig.22:]\nContours of constant cross-sections times branching ratios,\nin the $(m_A,\\tan \\beta)$ plane, for the inclusive $\\phi\\rightarrow\\gamma\\gamma$\nchannel: a)~$h$, LHC; b)~$H$, LHC; c)~$h$, SSC; d)~$H$, SSC.\nThe choice of $m_t$ and $m_{\\tilde{q}}$ is the same as in fig.~15, and\nQCD corrections to the gluon-fusion mechanism are included.\n\\item[Fig.23:]\nCross-sections for associated $W\\phi$ and $\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi \\phi$ production,\ntimes branching ratios for the $\\phi\\rightarrow\\gamma\\gamma$ channel, for\nthe same parameter choices as in fig.~15:\na)~$Wh$, LHC; b)~$WH$, LHC; c)~$Wh$, SSC; d)~$WH$, SSC;\ne)~$\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi h$, LHC; f)~$\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi H$, LHC; g)~$\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi A$, LHC;\nh)~$\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi h$, SSC; i)~$\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi H$, SSC; j)~$\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi A$, SSC.\nFor the sake of comparison, the SM values are also indicated.\n\\item[Fig.24:]\nContours of constant  $L_{\\phi} =\n[ 2 \\sigma (\\ifmmode t\\bar{t} \\else $t\\bar{t}$ \\fi \\phi) \\cdot BR ( t \\rightarrow W b)  + \\sigma ( W \\phi) ]\n\\cdot BR(\\phi\\rightarrow\\gamma\\gamma) \\cdot BR (W \\rightarrow l \\nu)$,\nfor the same choice of $m_t$ and $m_{\\tilde{q}}$ as in fig.~15:\na)~$h$, LHC; b)~$H$, LHC; c)~$h$, SSC; d)~$H$, SSC.\n\\item[Fig.25:]\nCross-sections for inclusive $H$ production (the\ngluon-fusion, $W$-fusion and $\\ifmmode b\\bar{b} \\else $b\\bar{b}$ \\fi \\phi$ contributions are summed) and decay\nin the $Z^* Z^* \\rightarrow 4 l^{\\pm}$ channel ($l=e,\\mu$),\nfor the same parameter choices as in fig.~15: a)~LHC; b)~SSC.\nFor the sake of comparison, the SM values are also indicated.\nQCD corrections to the gluon-fusion mechanism are not included.\n\\item[Fig.26:]\nContours of constant cross-sections times branching ratios for $H \\rightarrow Z^* Z^*\n\\rightarrow 4 l^{\\pm}$, for the same choice of $m_{\\tilde{q}}$ as in fig.~15 and:\na)~$m_t = 140 \\; {\\rm GeV}$, LHC;\nb)~$m_t = 140 \\; {\\rm GeV}$, SSC;\nc)~$m_t = 120 \\; {\\rm GeV}$, LHC;\nd)~$m_t = 120 \\; {\\rm GeV}$, SSC;\ne)~$m_t = 160 \\; {\\rm GeV}$, LHC;\nf)~$m_t = 160 \\; {\\rm GeV}$, SSC.\nQCD corrections to the gluon-fusion mechanism are included.\n\\item[Fig.27:]\nCross-sections times branching ratios for $\\phi \\rightarrow \\tau^+ \\tau^-$,\nfor the same parameter choices as in fig.~15: a)~$h$, LHC; b)~$H$, LHC;\nc)~$A$, LHC; d)~$h$, SSC; e)~$H$, SSC; f)~$A$, SSC.\nFor the sake of comparison, the SM values are also indicated.\n\\item[Fig.28:]\nContours of constant cross-sections times branching ratios for $\\phi \\rightarrow\n\\tau^+ \\tau^-$, for the same choice of $m_t$ and $m_{\\tilde{q}}$ as in fig.~15:\na)~$h$, LHC; b)~$H$, LHC; c)~$A$, LHC; d)~$h$, SSC; e)~$H$, SSC;\nf)~$A$, SSC. The vertical lines in c) and f) correspond to $m_A =\n60 \\; {\\rm GeV}$.\n\\item[Fig.29:]\nContours of constant $\\Delta R =[ BR(t \\rightarrow H^+ b) \\cdot BR ( H^+ \\rightarrow\n\\tau^+ \\nu_{\\tau})] \/ [ BR(t \\rightarrow W^+ b) \\cdot BR ( W^+ \\rightarrow \\mu^+ \\nu_{\\mu})]$,\nfor $m_{\\tilde{q}} = 1 \\; {\\rm TeV}$ and: a)~$m_t = 140 \\; {\\rm GeV}$; b)~$m_t = 120 \\; {\\rm GeV}$;\nc)~$m_t = 160 \\; {\\rm GeV}$.\n\\item[Fig.30:]\nPictorial summary of the discovery potential of large hadron colliders\nfor $m_{\\tilde{q}} = 1 \\; {\\rm TeV}$ and $m_t=140 \\; {\\rm GeV}$.\n\\end{itemize}\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn classical analysis, Peano curves are usually thought of as curiosities. It is exciting and counterintuitive that they exist, but they are not useful for solving problems. The purpose of this paper is to show that in fractal analysis, in contrast, Peano curves may play an important role in constructing Laplacians and studying their properties. In fact, this approach has already been used to construct Laplacians on Julia sets of complex polynomials \\cite{julia},\\cite{famofjulia},\\cite{JuliaSetsIII}.\n\nIf $X$ is a compact topological space, we will use the term Peano curve for any continuous mapping $\\gamma$ from the circle (parameterized by $t\\in [0,1]$ with $0\\equiv 1$) onto $X$. It is well known that $\\gamma$ cannot be one-to-one (except in the trivial case in which $X$ is homeomorphic to a circle), so there must be values in $[0,1]$ that are mapped to the same point in $X$, say $\\gamma (t_1)=\\gamma (t_2)$. We will say that such $t_1$ and $t_2$ are identified and write $t_1\\equiv t_2$. If we consider all possible identifications, then we obtain a model of $X$ as a circle with identifications, exactly the point of view adopted in \\cite{julia},\\cite{famofjulia},\\cite{JuliaSetsIII}.. Typically the number of identifications will be uncountable. Our goal is to obtain a countable sequence of identifications that is appropriately dense in the set of all identifications, so that if we take an initial segment of identifications, we will obtain a useful approximation for $X$.\n\nIn the case of Julia sets, the Peano curve is defined by means of applying the Riemann mapping theorem to the component of infinity of the complement of the Julia set. Despite this rather abstract construction, it turned out that it was possible to find a useful sequence of identified points. In this paper, Peano curves will be given as a limit of a sequence of curves defined by a self-similar type iteration scheme, and the self-intersections of the approximating curves will give us the sequence of identifications of the limiting Peano curve. It is interesting to note that many of the examples of Peano curves to planar domains that are found in textbooks are also constructed as limits of simpler curves obtained by iterating some self-similar scheme, but the approximating curves have no self-intersections. From our point of view this is a rather silly choice.\n\nGiven a finite set of identifications on the circle, we have a natural graph structure where the identified points are the vertices, and the edges join consecutive points around the circle. If we assign positive weights $\\mu(t_j)$ to the points, thought of as a discrete measure on the set of vertices, and non-negative weights $c(t_j,t_{j+1})$ to the edges, thought of as conductances on an electrical network associated to the graph, then we may define a graph Laplacian\n\n\\begin{equation}\n-\\Delta u(x)=\\frac{1}{\\mu (x)}\\sum_{x \\sim y} c(x,y)(u(x)-u(y)),\n\\end{equation}\n\nwhere the sum is taken over all $y$ neighboring $x$ \\cite{str}. For simplicity we may take $c(x,y)=0$ if $x$ and $y$ are not connected by an edge. Notice that self-edges are possible if $x$ and $y$ are identified neighboring points, but they do not contribute to the Laplacian. We want the discrete measures on the approximating graphs to converge to some natural measure on $X$. In all the examples considered here, we will use the simple choice of taking $\\mu(x)=\\sum\\frac{1}{2}\\left(t_{j+1}-t_{j-1}\\right)$ where the sum is taken over all points in the set of identified points denoted by $x$. For the conductances, the simplest choice is $c\\left(t_j,t_{j+1}\\right)=\\frac{1}{t_{j+1}-t_j}$, so the length of the interval $[t_j,t_{j+1}]$ is taken to be the resistance in the electrical network. This is not always the optimal choice, however, as is apparent in the previous work on Julia sets. In particular we will make a more sophisticated choice for one of our examples, the pentagasket. In order to obtain a Laplacian on $X$ in the limit it will be necessary to renormalize the sequence of Laplacians on the approximating graphs, and this becomes a highly nontrivial problem.\n\nIn this paper we will look at five examples of self-similar sets $X$ with carefully chosen Peano curves. Only three of them are fractal. The nonfractal sets are the equilateral triangle $T$ with Neumann boundary conditions, and the square torus $T_o$. These may be regarded as \"controls,\" since we know exactly what the eigenvalues and eigenfunctions of the Laplacian are. We can then see how well the Peano curves reproduce the known results. It would of course be silly to claim that this is an optimal way to develop properties of the usual planar Laplacian; however we will show that it reveals some insights into the eigenfunctions to view them as functions on the circle pulled back via the Peano curves. \n\nThe first fractal example we consider is the pentagasket, PG (Figure 1.1). It is a highly symmetric example of Kigami's class of postcritically finite (PCF) self-similar sets \\cite{ki} \\cite{str}. It has a fully symmetric self-similar Laplacian, whose properties were studied in detail in \\cite{Spec}. Since it does not satisfy spectral decimation, many of these properties have only been observed experimentally. We construct a Peano curve that yields a slightly different sequence of graph approximations than was considered in \\cite{Spec}. By following the outline from the work on Julia sets, we are able to find the correct renormalization constant and give an independent construction of the Laplacian, together with numerical results fully consistent with those in \\cite{Spec}. In this sense, the case of PG may also be considered a \"control.\"\n\n\\begin{figure}\n\\includegraphics[scale=0.5, trim=0mm 0mm 0mm 0mm,clip]{PG_Fig11}\n\\caption{Pentagasket}\n\\end{figure}\n\nThe second fractal example is the octagasket, OG (Figure 1.2). This fractal is not PCF, and in fact there is no proof yet of the existence of a symmetric self-similar Laplacian on OG. Nevertheless, experimental evidence of the existence of a Laplacian and properties of the spectrum using the method of \"outer approximation\" was presented in \\cite{BHS}. Our Peano curve approach gives independent experimental evidence for the existence of the Laplacian with data that is consistent with \\cite{BHS}. What is more, we are able to give concrete conjectures concerning the structure of the spectrum; for example, we find precise locations in the spectrum for spectral gaps that were noticed in \\cite{BHS}. The existence of spectral gaps is still quite a mystery. It is possible to prove existence for PCF fractals that enjoy spectral decimation, but the proof is just technical and yields no ``ideological\" explanation for them. The significance of spectral gaps is pointed out forcefully in \\cite{FourierSeries}.\n\n\\begin{figure}[h!]\n\\includegraphics[scale=0.5, trim=0mm 0mm 0mm 0mm,clip]{OG_Fig12}\n\\caption{Octagasket}\n\\end{figure}\n\nThe third fractal example we consider is the Magic carpet MC, recently introduced in \\cite{kform}. This fractal is obtained by modifying the construction of the Sierpinski carpet SC (Figure 1.3) to immediately sew up all cuts that are made, as illustrated in Figure 1.4. Thus MC is a limit of closed surfaces, and geometrically these surfaces are flat everywhere with the exception of a finite number of point singularities carrying negative curvature. Again, there is no proof of the existence of a symmetric self-similar Laplacian on MC, but the results in this paper together with \\cite{kform} give strong experimental evidence. The evidence suggests that analysis on MC should be similar to Euclidean spaces of dimension exceeding two, with points having zero capacity.\n\n\\begin{figure}\n\\includegraphics[scale=0.5, trim=0mm 0mm 0mm 0mm,clip]{SC_Fig13}\n\\caption{Sierpinski Carpet}\n\\end{figure}\n\nThe paper is organized as follows. In section 2 we give descriptions of all the Peano curves, as limits of piecewise linear maps obtained recursively using specified substitution rules. We are not claiming that these are the unique curves, or even that they are optimal in any sense. \n\nIn section 3 we discuss the example of PG. We describe the method of creating graph approximations and associated graph Laplacians from the Peano curve, and relate this to the Laplacian constructed by Kigami \\cite{ki}(see also \\cite{FourierSeries} and \\cite{Spec}). We use the same method for the other examples, the octagasket (section 4), the magic carpet (section 5), and the torus and triangle (section 6). The existence of the Laplacian in the limit for OG and MC has not yet been established, and we do no see any way to use the Peano curve construction to resolve this problem. For the torus and the triangle the Laplacian and its spectrum are well known. The graphs of the eigenfunctions pulled back to the circle via the Peano curve for the triangle are new and appealing.\n\n\\begin{figure}\n\\begin{minipage}[h!]{0.40\\linewidth}\n\\includegraphics[scale=.25,trim = 0mm 0mm 0mm 0mm, clip]{Figure14a}\n\\end{minipage}\n\\begin{minipage}[h!]{0.40\\linewidth}\n\\includegraphics[scale=.25,trim = 0mm 0mm 0mm 0mm, clip]{Figure14b}\n\\end{minipage}\n\\caption*{FIGURE 1.4. To transform SC into MC, \"sew up\" torus-type edge identifications, as shown here on level 0 and level 1. Point singularities of infinite negative curvature occur at the identified interior points markes at level 1}\n\\end{figure}\n\nWe have not attempted to obtain higher order accuracy in approximating eigenvalues by using more sophisticated numerical methods, such as the finite element method. It should not be difficult to adopt such methods to our examples, if so desired.\n\nWe include many numerical results, such as tables of eigenvalues and graphs of eigenfunctions. The reader should see the website \\emph{www.math.cornell.edu\/ $\\sim$ dmolitor} for more data.\n\n\nThis paper should be regarded as a \"proof of concept\" paper, showing by example that the method of Peano curves can be an effective tool in studying analysis on fractals. We hope it will prove useful for other fractals. \n\n\n\\section{Constructing the Peano Curves}\nAll the Peano curves we consider are limits of piecewise linear maps $\\gamma_m$, where the passage from $\\gamma_m$ to $\\gamma_{m+1}$ is given by a set of substitution rules for replacing each linear segment of $\\gamma_m$ by a union of consecutive segments of $\\gamma_{m+1}$ with the same endpoints. To make this clear we start by describing a Peano curve whose image is the Sierpinski gasket (SG, Figure 2.1).\n\\begin{figure}\n\\includegraphics[scale=.5,trim = 15mm 45mm 75mm 65mm, clip]{fig2_1}\n\\caption{Sierpinski Gasket.}\n\\end{figure}\n We will not discuss this example in detail below because it gives rise to exactly the same approximations to the Laplacian on SG as initially constructed by Kigami \\cite{ki} \\cite{str}. The first approximation $\\gamma_0$ simply traces out an equilateral triangle at constant speed. The substitution rules are shown in Figure 2.2. Each linear segment traces out an interval in one of the three orientations of the sides of the triangle. Each segment of $\\gamma_m$ is parameterized by an interval $\\left[\\frac{k}{3^{m+1}},\\frac{k+1}{3^{m+1}}\\right]$ and is shown as a dotted line in the figure. It's replacement, three intervals parameterized by $\\left[\\frac{3k}{3^{m+2}},\\frac{3k+1}{3^{m+2}}\\right],\\ \\left[\\frac{3k+1}{3^{m+2}},\\frac{3k+2}{3^{m+2}}\\right],\\ \\left[\\frac{3k+2}{3^{m+2}},\\frac{3k+3}{3^{m+2}}\\right]$ traces out the solid lines, with the same direction as the dotted line.\n\\begin{figure}\n\\begin{minipage}[h!]{0.28\\linewidth}\n\\includegraphics[scale=.28,trim = 0mm 0mm 0mm 0mm, clip]{fig2_2av4}\\\\a.\n\\end{minipage}\n\\begin{minipage}[h!]{0.28\\linewidth}\n\\includegraphics[scale=.28,trim = 0mm 0mm 0mm 0mm, clip]{Fig2_2bv4}\\\\b.\n\\end{minipage}\n\\begin{minipage}[h!]{0.28\\linewidth}\n\\includegraphics[scale=.28,trim = 0mm 0mm 0mm 0mm, clip]{fig2_2v2}\\\\c.\n\\end{minipage}\n\\caption{Shows the substitution rule for the three types of line segments encountered in SG. The dotted line represents the line from the previous level that is to be replaced. The replacement curve has the same beginning and end point as the line in the previous level.}\n\\end{figure}\nFigure 2.3 shows the image of $\\gamma_0, \\gamma_1$ and $\\gamma_2$ with arrows to show the direction and vertices labeled $k$ to indicate $\\gamma_m\\left(\\frac{k}{3^{m+1}}\\right)$.\n\\begin{figure}\n\\begin{minipage}[h!]{0.32\\linewidth}\n\\includegraphics[scale=.25,trim = 0mm 0mm 0mm 0mm, clip]{fig2_3av3}\\\\a.\n\\end{minipage}\n\\begin{minipage}[h!]{0.32\\linewidth}\n\\includegraphics[scale=.25,trim = 0mm 0mm 0mm 0mm, clip]{Fig2_3bv2}\\\\b.\n\\end{minipage}\n\\begin{minipage}[h!]{0.32\\linewidth}\n\\includegraphics[scale=.25,trim = 0mm 0mm 0mm 0mm, clip]{fig2_3cv2}\\\\c.\n\\end{minipage}\n\\caption*{FIGURE 2.3. The image of $\\gamma_m$ for $m=0,1,2$ in units of $(\\frac{1}{3})^{m+1}$}\n\\end{figure}\nNote that $\\gamma_m\\left(\\frac{k}{3^{m+1}}\\right)=\\gamma_{m+1}\\left(\\frac{3k}{3^{m+2}}\\right)$ etc., so the value of the limiting curve $\\gamma$ at a value $\\frac{k}{3^{m+1}}$ is the same as for all $\\gamma_{m'}$ with $m'\\geq m$. For a generic value of $t$, however, it is not so obvious what the point $\\gamma (t)$ on SG is exactly. In Figure 2.4,\n\\begin{figure}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.6,trim = 0mm 0mm 0mm 0mm, clip]{Fig2_4a}\\\\a.\n\\end{minipage}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.45,trim = 0mm 0mm 0mm 0mm, clip]{fig2_4b}\\\\b.\n\\end{minipage}\n\\caption*{FIGURE 2.4. Shows the identifications on the Peano curves corresponding to the first and second level graph approximations of the Sierpinski Gasket. Figure 2.4.a corresponds to the first level graph approximation  in units of $\\frac{1}{9}$ and figure 2.4b corresponds to the second in units of $\\frac{1}{27}$. Note that for each level $m$, the points $0, 3^m$ and $2\\cdot 3^m$ remain unidentified. Also, identifications from previous levels carry on as permanent fixtures in each of the higher levels.}\n\\end{figure}\n we show the identifications on the circle for the curves $\\gamma_1$ and $\\gamma_2$. These identifications persist for the limit curve $\\gamma$ and all identifications arising from $\\gamma$ are limits of those produced by $\\gamma_m$ as $m\\to\\infty$, although it is not at all obvious what these are.\n\nTo indicate the complexity of this simple example, we ask the question: what is the image of $\\gamma \\left(\\left[0,\\frac{1}{3}\\right]\\right)$, the first third of the circle? Figure 2.5\n\\begin{figure}\n\\includegraphics[scale=.5,trim = 15mm 50mm 75mm 60mm, clip]{fig2_5}\n\\caption*{FIGURE 2.5. The image of $\\gamma_5$ $[0,\\frac{1}{3}]$.}\n\\end{figure}\n shows $\\gamma_7 \\left(\\left[0,\\frac{1}{3}\\right]\\right)$, which suggests $\\left(\\left[0,\\frac{1}{3}\\right]\\right)=SG$. It is in fact easy to prove this, since $\\gamma_m\\left(\\left[0,\\frac{1}{3}\\right]\\right)$ satisfies a self-similar identity which uniquely characterizes SG. Thus the restriction of $\\gamma$ to $\\left(\\left[0,\\frac{1}{3}\\right]\\right)\\ \\left(\\text{or }\\left[\\frac{1}{3},\\frac{2}{3}\\right] \\text{ or }\\left[\\frac{2}{3},1\\right]\\right)$ is another Peano curve describing SG. Why did we not use this Peano curve to start with?\nThere are two reasons: \n1) the Peano curve destroys the dihedral-3 symmetry,\n2) the approximations have no self-intersections.\n(If we were interested in creating a Peano curve from $\\mathbb{R}$ to an infinite blow up of SG we would use these curves as the building blocks.) But this observation has the simple consequence that the original Peano curve $\\gamma$ must pass through each point (with the exception of the three vertices of the original triangle) at least three times, once for each third of the circle. Of course the smaller curves must have self-intersections as well, so there must certainly be some equivalence classes of identified points with four elements. Nevertheless, the approximating curves only produce equivalence classes with two elements. There are many interesting questions about the size of equivalence classes that we are not able to answer; are there infinite equivalence classes? If not, is there a maximal size? What is the size of a \"typical\" equivalence class? What is the complete equivalence class of the junction points $\\frac{k}{3^{m+1}}$? Despite the vexing nature of these unanswered questions, the identifications from the approximating curves $\\gamma_m$ give a perfect picture of the Laplacian on SG. In this case, what you do not know does not hurt you.\n\nThe next Peano curve we describe maps to the Pentagasket (PG). The first level approximation $\\gamma_1$ traces out a five-sided star. We think of this as consisting of five line segments, but because of the intersections, each segment consists of three pieces. We choose to traverse the pieces at different speeds, faster on the longer end pieces and slower on the short middle piece, so that the arrival times will be consistent from level to level. In general, the identified points for $\\gamma_m$ will be of the form $\\frac{k}{5^{m+1}}$, where $k\\equiv 1$ or $4\\mod 5$. Figure 2.6 shows $\\gamma_1$ and $\\gamma_2$ $\\left(\\text{we write }k\\text{ for }\\frac{k}{5^{m+1}}\\right)$.\n\\begin{figure}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.4,trim = 0mm 0mm 0mm 0mm, clip]{Fig2_6a}\n\\end{minipage}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.4,trim = 0mm 0mm 0mm 0mm, clip]{fig2_6b}\n\\end{minipage}\n\\caption*{FIGURE 2.6. Shows the first and second level graph approximations of PG with the locations corresponding to locations on the Peano curve labeled on each of the vertices. In units of $\\frac{1}{25}$ for the first level and $\\frac{1}{125}$ for the second level.}\n\\end{figure}\nThe substitution rule is illustrated in Figure 2.7 and is the same for all five rotations of the original dotted line.\n\n\\begin{figure}\n\\includegraphics[scale=0.5,trim = 25mm 25mm 25mm 25mm, clip]{Fig271}\n\\caption*{FIGURE 2.7. The substitution rule for one of the five directions.}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.4,trim = 0mm 0mm 0mm 0mm, clip]{Fig2_8a}\\\\a.\n\\end{minipage}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.4,trim = 0mm 0mm 0mm 0mm, clip]{fig2_8bv2}\\\\b.\n\\end{minipage}\n\\caption*{FIGURE 2.8. Shows the identifications on the Peano curve corresponding to the first and second level graph approximations of PG in units of $\\frac{1}{25}$ and $\\frac{1}{125}$, respectively. Note that the points of the form $5^{m-1}$, where $m$ is the level of the graph approximation remain unidentified throughout all levels of the graph approximations.}\n\\end{figure}\nIn Figure 2.8\nwe show the identifications on the circle arising from $\\gamma_1$ and $\\gamma_2$. We include the unidentified points of the form $\\frac{k}{5^{m+1}}$, with $k\\equiv 0\\mod 5$ as these are turning points for the curves and we will use these in forming the approximating graphs and associated Laplacians. We see that there are two distinct types of edges in the graph of length $\\frac{1}{5^{m+1}}$ and $\\frac{3}{5^{m+1}}$ and we will assign different conductances to the two types. All equivalence classes of identified points consist of pairs and identifications persist from level to level. In this case we note that the image of one fifth of the circle is not all of PG, but rather a smaller self-similar set defined by an iterated function system of five similarities with a smaller contraction ration and a different set of fixed points (See Figure 2.9).\n\\begin{figure}\n\\includegraphics[scale=.3,trim = 0mm 0mm 0mm 0mm, clip]{1_5imagepent}\n\\caption*{FIGURE 2.9. Shows the image of the first fifth of the Peano curve on PG. Each of the subsequent one-fifth sections of the Peano curve are reflections or rotations of this image.}\n\\end{figure}\n\nNext we describe the Peano curve for OG. We begin with $\\gamma_0$ that traces around an octagon clockwise and then turns around and traces around counterclockwise. The substitution rule is shown in Figure 2.10. \n\\begin{figure}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.45,trim = 10mm 15mm 30mm 10mm, clip]{fig2_10av3}\n\\end{minipage}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.45,trim = 10mm 15mm 10mm 10mm, clip]{fig2_10bv3}\n\\end{minipage}\n\\caption*{FIGURE 2.10. Shows the substitution rule for OG. The dotted arrow shows the line from the previous level of the graph approximation that is to be replaced. Note that the two substitutions shown differ in their direction and are reflections of each other. The eight rotations of these dotted lines corresponding to different possible edges of OG graph approximations use the same substitutions.}\n\\end{figure}\nNote that the direction of the edge is significant. For $\\gamma_m$ the identified points are of the form $\\frac{k}{2\\cdot 8^{m+1}}$. Some equivalence classes have two points, and we call these \\emph{outer points}, while others have four points and we call these \\emph{inner points}. Figure 2.11\n\\begin{figure}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.45,trim = 10mm 15mm 30mm 10mm, clip]{fig2_11a}\n\\end{minipage}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.35,trim = 10mm 15mm 10mm 10mm, clip]{fig2_11b}\n\\end{minipage}\n\\caption*{FIGURE 2.11. The paths $\\gamma_1$ and $\\gamma_2$ on OG}\n\\end{figure}\nshows $\\gamma_0$ and $\\gamma_1$. Figure 2.12\n\\begin{figure}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.40,trim = 0mm 0mm 0mm 0mm, clip]{fig2_12_b_two}\\\\a.\n\\end{minipage}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.40,trim = 0mm 0mm 0mm 0mm, clip]{fig2_12_four}\\\\b.\n\\end{minipage}\n\\caption*{FIGURE 2.12. Shows the identifications of outer points(a) and inner points(b) from $\\gamma_2$} \n\\end{figure}\nshows the identifications of outer and inner points from $\\gamma_2$.\n\nWe note that the restriction of $\\gamma$ to the interval $\\left[ 0,\\frac{1}{2}\\right]$ maps onto OG, but it destroys the symmetry and fails to give the full set of identifications.\n\nNext we describe the Peano for the Magic Carpet (MC). This construction requires a bit of imagination, since MC does not embed in the plane. The curves $\\gamma_m$ approximating $\\gamma$ may be visualized as mappings to the Sierpinski Carpet (SC) with jump discontinuities, where the jumps connect points of SC that are identified to create MC. We will take $\\gamma_m$ to be the union of $2\\cdot 8^{m}$ line segments from $\\left[\\frac{k}{2\\cdot 8^{m+1}},\\frac{k+1}{2\\cdot 8^{m+1}}\\right]$ to each of two boundary edges of each $m$-cell in MC.\nIn Figure 2.13\n\\begin{figure}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.35,trim = 0mm 0mm 0mm 0mm, clip]{fig2_13a}\\\\a.\n\\end{minipage}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.35,trim = 0mm 0mm 0mm 0mm, clip]{fig2_13b}\\\\b.\n\\end{minipage}\n\\caption*{FIGURE 2.13: The path $\\gamma_1$ for MC. In a. we show the actual path with the points 2,3,7,8,11 and 14 identified. In b., the dashed arrows show the symbolic description, while the solid arrow illustrates a jump between identified points.}\n\\end{figure}\nwe show both the symbolic description of $\\gamma_1$ and the actual path, which has a single jump discontinuity at $\\frac{1}{2}$. We note that this gives rise to a set of six identified points $\\{2,3,7,8,11,14\\}$ and the rest identified in pairs $\\{0,5\\},\\{1,12\\},\\{4,13\\},\\{6,9\\}$ and $\\{10,15\\}$, mainly because of the MC identifications. The substitution rule in symbolic form is shown in Figure 2.14.\n\\begin{figure}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.6,trim = 0mm 0mm 0mm 0mm, clip]{fig2_14a}\\\\a.\n\\end{minipage}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.35,trim = 0mm 0mm 0mm 0mm, clip]{fig2_14b}\\\\b.\n\\end{minipage}\n\\caption*{FIGURE 2.14. Shows one of  the symbolic substitution rule for MC.The others are simply rotations of this one.}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[scale=.45,trim = 0mm 0mm 0mm 0mm, clip]{fig2_15}\n\\caption*{FIGURE 2.15. The path $\\gamma_2$ on MC}\n\\end{figure}\nFigure 2.15 shows the path $\\gamma_2$. We note that every edge at level $m$ is traversed exactly once by $\\gamma_m$.\n\nThis Peano curve differs from the others in that the identifications at level $m$ do not persist exactly at level $m+1$. In fact if $\\frac{k}{2\\cdot 8^{m+1}}\\sim\\frac{j}{2\\cdot8^{m+1}}$ at level $m$ where $k$ is even and $j$ is odd, then $\\frac{8k}{2\\cdot8^{m+2}}\\sim \\frac{8j\\pm 3}{2\\cdot 8^{m+2}}$ at level $m+1$.\n\nA slight modification of the MC construction gives rise to a Peano curve to the square torus T$_0$. Figure 2.16\nshows how to modify the substitution rule from Figure 2.14. Figure 2.17 shows the actual path of $\\gamma_1$.\n\n\\begin{figure}\n\\begin{minipage}{0.49\\linewidth}\n\\includegraphics[scale=.6,trim = 0mm 0mm 0mm 0mm, clip]{Fig216Part1}\n\\end{minipage}\n\\begin{minipage}{0.49\\linewidth}\n\\includegraphics[scale=.6,trim = 0mm 0mm 0mm 0mm, clip]{Fig216Part2}\n\\end{minipage}\n\\caption*{FIGURE 2.16. Shows one of the symbolic substitution rules for $T_0$. The others are simply rotations of this one.}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[scale=.65,trim = 0mm 0mm 0mm 0mm, clip]{Fig217CorrectedV1}\n\\caption*{FIGURE 2.17. Path of $\\gamma_1$ on $T_0$}\n\\end{figure}\n\nNow $\\gamma_m$ identifies points of the form $\\frac{k}{2\\cdot 9^{m}}$ in pairs. Again in this example we note that identifications do not persist from level to level. \n\nThe final example gives a Peano curve to the equilateral triangle $T$. It may be thought of as a modification of the first example (SG). The substitution rule is shown in Figure 2.18 (to be compared with Figure 2.2), and the paths $\\gamma_0,\\gamma_1$ and $\\gamma_2$ are shown in Figure 2.19 (to be compared with Figure 2.3). We note that there are now three types of points:\n\n\\begin{figure}\n\\centering\n\\begin{adjustwidth}{-.0in}{-0in}\n\\begin{minipage}[h!]{0.49\\linewidth}\n\\includegraphics[scale=.4,trim = 5mm 40mm 5mm 5mm, clip]{Sec2Fig218}\n\\end{minipage}\n\\end{adjustwidth}\n\\caption*{Figure 2.18. Shows the substitution rule for the triangle}\n\\end{figure}\n\n\n\\begin{table}\n\n\\centering\n\\begin{adjustwidth}{-1.3in}{-1.0in}\n\\begin{tabular}{c c c}\n\n\\includegraphics[scale=.5,trim = 0mm 55mm 0mm 0mm, clip]{Figure219_1}\n&\n\\includegraphics[scale=.45,trim = 0mm 0mm 0mm 0mm, clip]{Figure219_2}\n&\n\\includegraphics[scale=.45,trim = 0mm 0mm 0mm 0mm, clip]{TrianglePic219_C}\\\\\n\n\n\\end{tabular}\n\\caption*{Figure 2.19. The paths $\\gamma_0$, $\\gamma_1$, and $\\gamma_2$ for the triangle}\n\\end{adjustwidth}\n\\end{table}\n\n\n\n(i) the three corner points that are not identified;\n(ii) the other points along the boundary that are identified in groups of three;\n(iii) interior points that are identified in groups of six. \nSimilarly, there are two types of edges: (i) boundary edges that are traversed once in the counterclockwise direction;\n(ii) interior edges that are traversed twice, both times in the same direction. For the boundary edges there is no choice of which of the substitution rules to use in order for the curve to stay inside the triangle. Since the interior edges bound two distinct cells, we need to make use of both alternatives, and our convention is to go into the central cell first and the peripheral cell the second time we traverse the edge. At level $m$, the identified points are of the form $\\frac{k}{3\\cdot 4^{m}}$, and in this example the identifications persist from level to level. The boundary of the triangle is the image of a Cantor set in the circle, as shown in Figure 2.20.\n\n\\begin{figure}\n\\includegraphics[scale=.4,trim = 0mm 0mm 0mm 0mm, clip]{CantorApproxLev5}\n\\caption*{FIGURE 2.20. The Cantor set on the circle whose image under $\\gamma$ corresponds to the boundary of $T$.}\n\\end{figure}\n\n\n\\section {The Pentagasket}\nIn this section we use the Peano curve $\\gamma$ for the Pentagasket (PG) and its approximations $\\gamma_m$ to define both an energy and a Laplacian that are self-similar and symmetric with respect to the dihedral-5 symmetry group that acts on PG. Since PG belongs to Kigami's PCF class, the energy and Laplacian are unique up to constant multiples, so our construction may be compared with previous constructions, in particular \\cite{Spec}.\n\nLet $\\Gamma_m$ denote the graph determined by $\\gamma_m$ as described in section two. We initially assign conductances to edges as follows:\n\\begin{equation}\nc(x,y)=\\begin{cases}\n1\\text{ if }[x,y]\\text{ has length } \\frac{1}{5^{m+1}}\\\\\nb\\text{ if }[x,y]\\text{ has length } \\frac{3}{5^{m+1}}\n\\end{cases},\\end{equation}\nwhere $b$ is a constant to be determined. Then set \n\\begin{equation}\nE_m(u)=\\sum_{x\\sim y} c(x,y)(u(x)-u(y))^2.\n\\end{equation}\nGiven a function $u$ on the vertices of $\\Gamma_m$, we let $\\bar u$ denote the extension of $u$ to the vertices of $\\Gamma_{m+1}$ that minimizes the value of $E_{m+1}$, and then consider the renormalization equation\n\\begin{equation}\nE_{m+1}(\\bar u)=r E_m(u)\n\\end{equation}\nfor a renormalization factor $r$ to be determined. There is a unique choice of values for $b$ and $r$ for which (3.3) holds, namely\n\\begin{equation}\n\\begin{cases}\nb=\\frac{1+\\sqrt{161}}{10} \\\\\nr= \\frac{\\sqrt{161}-9}{8}\n\\end{cases}.\n\\end{equation}\nNote that the value of $r$ agrees with the value given in \\cite{Spec}.\nThen we define the renormalized energy\n\\begin{equation}\n\\mathcal{E}_m (u)=r^{-m}E_m (u),\n\\end{equation}\nmaking $\\mathcal{E}_m (u)$ an increasing sequence, so \n\\begin{equation}\n\\mathcal{E} (u)=\\lim_{m\\to\\infty}\\mathcal{E}_m (u).\n\\end{equation}\nHere $u$ is any continuous function on the circle that respects all identifications made by $\\gamma_m$ for all $m$, hence $u(t)=u(s)$ if $\\gamma(t)=\\gamma (s)$. We define $dom\\, \\mathcal{E}$ to be those functions with $\\mathcal{E}(u)<\\infty$. If $u,v\\in dom\\,\\mathcal{E}$, then \n\\begin{equation}\n\\mathcal{E}(u,v)=\\lim_{m\\to\\infty}\\mathcal{E}_m(u,v),\n\\end{equation} \nand $dom\\, \\mathcal{E}\\mod $ constants becomes a Hilbert space with this inner product.\n\nThe standard Lebesgue measure on the circle is pushed forward by $\\gamma$ to the standard self-similar probability measure $\\mu$ on the pentagasket. We may define the Laplacian via the weak formulation\n\\begin{equation}\n\\mathcal{E}(u,v)=-\\int (\\Delta u)vd\\mu,\n\\end{equation}\nfor all $v\\in dom\\,\\mathcal{E}$. Note that this is the Neumann Laplacian if we choose a boundary for the pentagasket. Typically one takes the boundary to be either the five points of the initial star, or just three of the five. It is important to observe that for PG, in contrast to the SG, there is nothing special about the local geometry in a neighborhood of a boundary point: there are infinitely many points with the same local geometry. Because Neumann boundary conditions are \"natural,\" the Laplacian behaves the same way at all these locally isometric points regardless of which ones we designate as the boundary.\n\nOf course we want a simpler method to give $\\Delta u$ as a limit of discrete graph Laplacians $\\Delta_m$ on $\\Gamma_m$, and for this we need to approximate the measure $\\mu$ by a discrete measure on the vertices of $\\Gamma_m$ to get a formula of the form (1.1). For each point of the form $\\frac{k}{5^{m+1}}$ we assign the weight $\\frac{1}{5^{m+1}}$ if $k\\equiv 0\\mod 5$, and $\\frac{2}{5^{m+1}}$ if $k\\equiv 1$ or $4\\mod 5$, and $\\mu(x)$ is the sum of the weights of all the points in the equivalence class $x$. Of course the equivalence class just consists of the singleton $\\frac{k}{5^m+1}$ if $k\\equiv 0\\mod 5$, so the $\\mu(x)=\\frac{1}{5^{m+1}}$, while if $k\\equiv 1$ or $4\\mod 5$, then the equivalence class consists of two points of the same type, so $\\mu(x)=\\frac{4}{5^{m+1}}$.\n\n\\begin{table}[htb!]\n\\begin{adjustwidth}{-1.2in}{-1in}\n\\begin{center}\n$\\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c} \n\\hline\n\\multicolumn{3}{|c|}{\\text {Level 1}}\n& \\multicolumn{3}{|c|}{\\text{Level 2}}\n& \\multicolumn{3}{|c|}{\\text{Level 3}}\n& \\multicolumn{3}{|c|}{\\text{Level 4}}\\\\\n\\hline\n\\#  & \\text{Mult} & \\text{Eigenvalue} &  \\# & \\text{Mult} & \\text{Eigenvalue}& \\# & \\text{Mult} & \\text{Eigenvalue}& \\# & \\text{Mult} & \\text{Eigenvalue}\\\\\n\\hline 1 & 1 & 0 & 1 & 1 & 0 & 1&1& 0 & 1 & 1 & 0\\\\ \\hline \n\\hline 2 & 2 & 28.6410 & 2 & 2 & 12.5186 &2&2& 12.6700&2& 2 &12.6832\\\\ \\hline \n\\hline 4&2&28.9251&4&2&30.6109 &4&\t2&31.3706 &4&2 &31.4492 \\\\ \\hline\n\\hline  6&2&119.5409&6 &5& 143.2049&6& 5& 135.7523&6& 5& 137.4025  \\\\  \\hline \n\\hline 8&2&132.5555&11 &1& 168.8936&11& 1& 164.5714&11& 1& 166.9378\\\\ \\hline\n\\hline 10&1&135.5536&12&2&182.4264&12& 2& 182.3916&12& 2 &185.2678\\\\ \\hline\n\\hline &&& 14&2&215.2990&14&2& 239.2249&14& 2& \t244.1480\\\\ \\hline\n\\hline  &&& 16&5&415.7326&16&5& 331.9515&16& 5& 340.1929\\\\ \\hline\n\\hline &&& 21&2&430.6319&21& 2& 435.5986&21& 2& 453.4902\\\\ \\hline\n\\hline  &&& 23&2&454.5580&23& 2& 562.4423&23& \t2 \t&596.8892 \\\\ \\hline\n\\hline &&&\t25&1&463.5525&25& 1 &629.634&25& 1 &677.4916\\\\ \\hline\n\\hline  & && 26&5&597.7066 &26&20& 1552.9561&26&20&1472.1417\\\\ \\hline\n\\end{array}$\n\\vspace{5mm}\n\\caption{ Eigenvalues of the Pentagasket}\n\\end{center}\n\\end{adjustwidth}\n\\end{table}\n\nIn defining the discrete Laplacian via (1.1), we renormalize the conductances by $r^{-m}$ to make $\\Delta=\\lim_{m\\to\\infty}\\Delta_m$ without further renormalization factors. In Table 3.1 we show the eigenvalues with multiplicities for $m=$1, 2, 3, and $4$. It is clear that the eigenvalues are converging to a limit as $m\\to\\infty$, and the multiplicities and values up to a constant multiple agree with those computed in \\cite{Spec}.\n\nIn Figures 3.1 and 3.2 we show graphs of eigenfunctions for $m=$2, and 3 as functions on the circle (here [0,1] with $0\\equiv 1$) that respect identifications. We emphasize that this gives a new way to visualize eigenfunctions. The requirement that the functions respect identifications is a very stringent requirement. There are essentially no \"nice\" functions (for example, differentiable functions) with this property except for constants, and it is possible to visually recognize the patterns characteristic of these graphs.\n\n\\begin{table}[htb!]\n\\centering\n\\begin{adjustwidth}{-1.2in}{-1.0in}\n\n\\begin{tabular}{  c c c c }\n$ \\text{Eigenfunction \\# 2 \\& 3} $ &   $\\text{Eigenfunction \\# 4 \\& 5}$ \\\\\n\\includegraphics[height=1.25in,width=1.75in]{Plseigenval2_Iteration3}  \\includegraphics[height=1.25in,width=1.75in]{Mineigenval2_Iteration3}&\n \\includegraphics[height=1.25in,width=1.75in]{Pluseigenval4_Iteration3}  \\includegraphics[height=1.25in,width=1.75in]{Mineigenval4_Iteration3}\\\\ \n\\\\\n $\\text{Eigenfunction \\# 11} $ \\\\\n \\includegraphics[height=1.25in,width=1.75in]{eigenval11_Iteration3}& \\\\\n\\end{tabular}\n\\end{adjustwidth}\n\\caption*{FIGURE 3.1. Eigenfunctions of the Pentagasket at Level 2}\n\\end{table}\n\n\\begin{table}[h!]\n\\centering\n\\begin{adjustwidth}{-1.2in}{-1.0in}\n\n\\begin{tabular}{  c c c c }\n$ \\text{Eigenfunction \\# 2 \\& 3} $ &   $\\text{Eigenfunction \\# 4 \\& 5}$ \\\\\n\\includegraphics[height=1.25in,width=1.75in]{EigVal2_Iteration4}  \\includegraphics[height=1.25in,width=1.75in]{MinEigVal2_Iteration4}&\n \\includegraphics[height=1.25in,width=1.75in]{PlusEigVal4_Iteration4}  \\includegraphics[height=1.25in,width=1.75in]{MinEigVal4_Iteration4}\\\\ \n$\\text{Eigenfunction \\#6}$ & $\\text{Eigenfunction \\#11}$\\\\\n\\includegraphics[height=1.25in,width=1.75in]{EigVal6_Iter4}&\\includegraphics[height=1.25in,width=1.75in]{PEigVall11_Iteration4}\n\\\\\n$\\text{Eigenfunction \\# 16}  $ & $\\text{Eigenfunction \\# 51} $ \\\\\n\\includegraphics[height=1.25in,width=1.75in]{EigVal16_Iteration4}&\n\\includegraphics[height=1.25in,width=2.25in]{EigVal51_Iteration4} \\\\\n\\\\\n\\end{tabular}\n\n\\caption*{FIGURE 3.2. Eigenfunctions of the Pentagasket at Level 3}\n\\end{adjustwidth}\n\\end{table}\n\n\nThe connection with the dihedral-5 group of symmetries is very straightforward in this realization. Rotations through the angle $\\frac{2\\pi j}{5}$ in PG correspond to translations $t\\to t+\\frac{j}{5}$, and reflections correspond to $t\\to \\frac{j}{5}-t$. For eigenspaces of multiplicity two, such as $\\#$2 $\\&$3 and $\\#$4 $\\&$5, we can find a basis of eigenfunctions satisfying the symmetry condition $u(1-t)=u(t)$ and the skew-symmetry condition $u(1-t)=-u(t)$, and we have automatically enforced this dichotomy in our choice of graphs. Eigenfunctions corresponding to multiplicity one, such as $\\#$11, will have periodicity $u\\left (t+\\frac{1}{5}\\right)=u(t)$, and will be skew-symmetric with respect to all reflections ( except in the trivial case of constants), as was shown in \\cite{Spec}. Within eigenspaces of multiplicity five it is possible to find eigenfunctions that are symmetric with respect to both translations and reflections, such as $\\#$6 (out of $\\#$ 6-10). All these symmetries are immediately visible from the graphs. \n\n\n\nWe can also see miniaturization of eigenfunctions. Consider an eigenspace of multiplicity two with eigenvalue $\\lambda$. Then $5r^{-1}\\lambda$ will be an eigenspace of multiplicity five, and if $u$ is the reflection symmetric $\\lambda$-eigenfunction then $u(5t)$ is the reflection symmetric periodic $5r^{-1}\\lambda$ eigenfunction. This fact is proven in \\cite{Spec}, but is visually obvious from the graphs of $\\#$ 2 and $\\#$ 4 on level 2 and $\\# 6$ and $\\# 16$ on level 3. Similarly, if $u$ is a $\\lambda$-eigenfunction for an eigenspace of multiplicity one with $\\lambda\\neq 0$, then it is shown in \\cite{Spec} that $u$ is reflection skew-symmetric and $u(5t)$ is a reflection skew-symmetric periodic $5r^{-1}\\lambda$-eigenfunction, and this eigenspace has multiplicity five.This is seen in $\\#11$ on level 2 and $\\#51$ on level 3. Here we find the periodic eigenfunctions in the multiplicity five eigenspaces via periodization. \nIn Figure 3.3 we show the log-log graphs of the eigenvalue counting function, $\\rho(x)=\\#\\{\\lambda_j \\le x \\}$ and the Weyl Ratio, $WR(x)=\\frac{\\rho(x)}{x^\\beta}$ for $\\beta=\\frac{\\text{log}5}{\\text{log}5-\\text{log}5}\\approx 0.675$ on levels 2,3,and 4. We can begin to see evidence of the asymptotic multiplicative periodicity of the Weyl ratio, $WR(5x)\\approx WR(x)$, in the level 4 graph\n\\vspace{.1in}\n\n\n\n\n\\begin{table}[h!]\n\\centering\n\\begin{adjustwidth}{-1.0in}{-1.0in}\n\\vspace{3mm}\n\\begin{tabular}{  c c c c }\n$\\text{Eigenvalue Counting Function}$\\\\\n\\\\\n$\\text{ Level 2}$ & $\\text{ Level 3}$ & $\\text{Level 4}$ \\\\\n\\\\\n \\includegraphics[height=1.25in,width=1.75in]{PentaLevel2EigCount}&  \\includegraphics[height=1.25in,width=1.75in]{NewPowerEigCountPent3} &  \\includegraphics[height=1.25in,width=1.75in]{NewPowerEigCountPent4}\\\\ \n\\\\\n$\\text{Weyl Ratios}$ \\\\\n\\\\\n$ \\text{Level 2}$ & $ \\text{Level 3}$ & $\\text{Level 4} $\\\\\n\\\\\n \\includegraphics[height=1.25in,width=1.75in]{PentaLevel2WeylRatio} &  \\includegraphics[height=1.25in,width=1.75in]{NewPowerWeylRatioPent3} &  \\includegraphics[height=1.25in,width=1.75in]{NewPowerWeylRatioPent4}\\\\\n\\end{tabular}\n\\vspace{1mm}\n\n\\caption*{FIGURE 3.3. We show the log-log graphs of the eigenvalue counting function, $\\rho(x)=\\#\\{\\lambda_j \\le x \\}$ and the Weyl Ratio, $WR(x)=\\frac{\\rho(x)}{x^\\beta}$ for $\\beta=\\frac{\\text{log}5}{\\text{log}5-\\text{log}5}\\approx 0.675$ on levels 2,3,and 4. We can begin to see evidence of the asymptotic multiplicative periodicity of the Weyl ratio, $WR(5x)\\approx WR(x)$, in the level 4 graph.}\n\\end{adjustwidth}\n\\end{table}\n\n\n\n\n\n\n\\section{The Octagasket}\n\nIt is believed that the Octagasket (OG) has a symmetric self-similar energy $\\mathcal{E}$ with $dom\\mathcal{E}$ containing only continuous functions (equivalently, points have positive capacity), and an associated Laplacian defined by $(3.8)$ with the standard self-similar measure $\\mu$. There is no proof for these conjectures at present. Experimental evidence for the existence of the Laplacian was provided in \\cite{BHS}, and this paper will provide independent evidence. The Peano curve $\\gamma$ and its approximations $\\gamma_m$ give us a sequence of graph approximations $\\Gamma _m$, and we can define a graph energy $E_m$ by giving single edges conductance 1 and double edges conductance $2$. We expect that there is an energy renormalization factor $r<1$ such that \n\\begin{equation}\n\\mathcal{E}(u,v)=\\lim_{m\\to\\infty} r^{-m} E_m (u,v)\\text{ for }u,v\\in dom\\mathcal{E},\n\\end{equation}\nand we will give a rough estimate for $r$ based on our experimental data. We note that OG is not PCF, and there is no clean formula relating $E_{m+1}(\\bar u)$ and $E_{m}( u)$ when $\\bar u$ is the $E_{m+1}$ minimizing extension of $u$ analogous to (3.3). It is natural to approximate $\\mu$ by the discrete measure on vertices of $\\Gamma_m$ that assigns weights $\\frac{1}{2\\cdot 8^{m+1}}$ to each point $\\frac{k}{2\\cdot 8^{m+1}}$, so outer vertices get weight $\\frac{1}{8^{m+1}}$ and inner vertices get weight $\\frac{2}{8^{m+1}}$. We define an unnormalized discrete Laplacian $\\Delta_m$ on $\\Gamma_m$ by \n\\begin{equation}\n-\\Delta_m u(x)=4\\left(u(x)-\\text{Ave}(u(y))\\right)\n\\end{equation}\nwhere $\\text{Ave}(u(y))$ denotes the average value of $u$ on the 2 or 4 neighboring points. (The constant 4 is just for convenience.) Note that this is a different convention than the one used in the case of PG. Now we expect\n\\begin{equation}\n-\\Delta u=lim_{m\\to\\infty} \\left (\\frac{8}{r} \\right )^m\\Delta_m u.\n\\end{equation}\nThe ratios of the corresponding eigenvalues for $\\Delta_m$ and $\\Delta_{m+1}$ give an estimate for the renormalization factor $\\frac{8}{r}\\approx 14.9$, which means\n $r\\approx 0.537$. In Table 4.1 we give the eigenvalues for $m=1,2,3$ and the ratios.\nIn Table 4.2 we give the same data for the renormalized eigenvalues, where we have multiplied by $(\\frac{8}{r})^m$.\nIt is clear that the multiplicities remain the same as $m$ increases (at least in the lower portion of the spectrum). The values are in reasonable agreement with those found in \\cite{BHS}.\n\n\n\\begin{table}\n\\begin{adjustwidth}{-.5in}{-.5in}\n\\begin{center}\n$\\begin{array}{|c|c|c|c|c|c|c|c|c|||c|c|c|} \n\\hline\n\\multicolumn{3}{|c|}{\\text {Level 1}}\n& \\multicolumn{3}{|c|}{\\text{Level 2}}\n& \\multicolumn{3}{|c|}{\\text{Level 3}}\n&\\multicolumn{2}{|c|}{\\text{Ratio}} \\\\\n\\hline\n\\#  & \\text{Mult} & \\text{Eigenvalue} &  \\# & \\text{Mult} & \\text{Eigenvalue}& \\# & \\text{Mult} & \\text{Eigenvalue}& \\frac{\\lambda_1}{\\lambda_2}&\\frac{\\lambda_2}{\\lambda_3}\\\\\n\\hline 1 & 1 & 0 & 1 & 1 & 0 & 1&1&0&&\\\\ \\hline \n\\hline 2 & 2 & 0.111 & 2 & 2 & 0.0074& 2 & 2 & 0.0005 &14.802 & 14.938\\\\ \\hline \n\\hline 4 & 2 & 0.396 & 4 & 2 & 0.0282 & 4 & 2 & 0.0018 & 14.027 & 14.908\\\\ \\hline\n\\hline 6 & 2 & 0.770& 6 & 2 &  0.0570  & 6 & 2 & 0.0038 & 13.495 & 14.897\\\\  \\hline \n\\hline 8 & 3 & 1.171 & 8 & 1 & 0.0784 & 8 & 1 &0.0052 &&14.960\\\\ \\hline\n\\hline 11 &2 & 1.276 & 9 & 2 & 0.1108 & 9 & 2 & 0.0074 &&14.794\\\\ \\hline\n\\hline 13 & 2 & 1.500 & 11 & 2 & 0.1157&11&2&0.0077&&14.852\\\\ \\hline\n\\hline 15 & 2 & 1.506& 13 & 2 & 0.1251&13&2&0.0083&&14.941\\\\ \\hline\n\\hline 17 & 2 & 3.109 & 15 & 2 & 0.1263&15&2&0.0084&&14.971\\\\ \\hline\n\\hline 19 & 2 & 3.299 & 17 & 2 & 0.2291&17&2&0.0154&&14.803\\\\ \\hline\n\\hline 21 & 2 &3.465 & 19 & 1& 0.2362&19&1&0.0157&&15.034\\\\ \\hline\n\\hline 23 & 4  & 4.000 & 20 & 2 & 0.2412&20&2&0.0165&& 14.590\\\\ \\hline\n\\hline 27& 2 &4.534 & 22 & 2 &  0.2771  &22&2&0.0189&&14.605 \\\\ \\hline\n\\hline 29 & 2 & 4.700 & 24 & 1 & 0.3021&24&1&0.0205&&14.691 \\\\ \\hline\n\\hline 31 & 2 &4.890 & 25 & 2 & 0.3961&25&2&0.0282&&14.027 \\\\ \\hline\n\\hline 33 & 2 & 6.493 & 27 & 2 & 0.4237 &27&2&0.0300&&14.120 \\\\ \\hline\n\\hline 35 & 2 & 6.499  &29&2&0.4261&29&2&0.0301&&14.136  \\\\ \\hline\n\\hline 37 & 2 & 6.723 & 31& 2 &0.4561& 31 &2&0.0321&&14.204 \\\\ \\hline\n\\hline 39 & 3 & 6.828 & 33&2& 0.5912&33&2&0.0425&&13.909\\\\ \\hline\n\\hline 42 & 2 & 7.229 & 35 & 2 & 0.5984&35&2&0.0428&&13.970\\\\ \\hline\n\\hline 44 & 2 &  7.603 &37& 1&  0.6249&37&1&0.0445&&14.035\\\\ \\hline\n\\hline 46 & 2 &  7.889 & 38 &2& 0.6650&38&2&0.0479&&13.871\\\\ \\hline\n\\hline 48           & 1 & 8.000  &40&1 & 0.7536&40&1&0.0542&&13.903\\\\ \\hline\n\\hline &&&41& 2 &0.7700 &41& 2 &0.0570&&13.495\\\\ \\hline\n\\hline &&&43&2&0.8100&43&2&0.0598&&13.526\\\\ \\hline\n\\hline &&&45&2&0.8525&45&2&0.0631&&13.504\\\\ \\hline\n\\hline &&&47&2&0.8866&47&2&0.0656&&13.507\\\\ \\hline\n\\hline &&&49&2&0.9328&49&2&0.0696&&13.401\\\\ \\hline\n\\hline &&&51&2&0.9772&51&2&0.0729&&13.389\\\\ \\hline\n\\hline &&&53&2&0.9891&53&2&0.0735&&13.446\\\\ \\hline\n\\hline &&&55&1&1.0151&55&1&0.0750&&13.532\\\\ \\hline\n\\hline &&&56&1&1.0314&56&1&0.0784&&13.146\\\\ \\hline\n\\hline &&&57&3&1.1715&57&1&0.0843&&\\\\ \\hline\n\\hline &&&60&2&1.1810&58&1&0.0901&&\\\\ \\hline\n\\hline &&&62&2&1.1933&60&2&0.0973&&\\\\ \\hline\n\\hline &&&64&2&1.2025&62&2&0.1024&&\\\\ \\hline\n\\hline &&&66&2&1.2201&64&1&0.1042&&\\\\  \\hline\n\\end{array}$\n\\vspace{5mm}\n\\caption{Eigenvalues of OG}\n\\end{center}\n\\end{adjustwidth}\n\n\\end{table}\n\n\n\n\\begin{table}\n\\centering\n$\\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{3}{|c|}{\\text {Level 1}}\n& \\multicolumn{3}{|c|}{\\text{Level 2}}\n& \\multicolumn{3}{|c|}{\\text{Level 3}} \\\\\n\\hline\n\\#  & \\text{Mult} & \\text{Eigenvalue} &  \\# & \\text{Mult} & \\text{Eigenvalue}& \\# & \\text{Mult} & \\text{Eigenvalue} \\\\\n\\hline 1 & 1 & 0 & 1 & 1 & 0 & 1&1&0\\\\ \n\\hline 2 & 2 &1.652 & 2 & 2& 1.662 & 2& 2 &  1.659 \\\\ \\hline \n\\hline 4 & 2 & 5.902 &4& 2 & 6.269 & 4 & 2 & 6.265\\\\ \\hline\n\\hline 6 & 2 &  11.473 & 6 & 2 & 12.667 & 6  & 2 &  12.669 \\\\  \\hline \n\\hline 8 & 3 & 17.456&8 & 1 & 17.418 & 8& 1 & 17.348 \\\\ \\hline\n\\hline 11 & 2 & 19.018 &9 & 2 & 24.614 & 9 & 2 & 24.789 \\\\ \\hline\n\\hline 13 & 2 & 22.355 & 11& 2 &25.693 & 11& 2&25.775  \\\\ \\hline\n\\hline 15 & 2 & 22.439 & 13 & 2 & 27.773 & 13 & 2 & 27.697 \\\\ \\hline\n\\hline 17 & 2 &  46.337 & 15& 2 & 28.039& 15 & 2 &  27.905\\\\ \\hline\n\\hline 19 & 2& 49.160 & 17 & 2 &  50.878 & 17 & 2 & 51.209 \\\\ \\hline\n\\hline 21 & 2 &  51.642 & 19 & 1 &  52.445 & 19& 1 & 51.974 \\\\ \\hline\n\\hline 23       & 4 & 59.600  & 20 & 2 &  53.551 & 20 & 2&  54.688 \\\\ \\hline\n\\hline 27    & 2 &  67.557 & 22 & 2 & 61.527 & 22& 2 & 62.769\\\\ \\hline\n\\hline 29 & 2 &  70.039 & 24& 1 &  67.071 & 24& 1 & 68.021\\\\ \\hline\n\\hline 31  & 2 & 72.862 & 25 & 2 &   87.942&25 & 2 & 93.410\\\\ \\hline\n\\hline 33 & 2 & 96.760 & 27 & 2 &  94.083  &27 & 2 & 99.274  \\\\ \\hline\n\\hline 35 & 2 & 96.844 &29 & 2 &  94.616 &29 & 2 & 99.727  \\\\ \\hline\n\\hline 37 & 2 & 100.181  & 31& 2 &101.269 & 31 & 2 & 106.231 \\\\ \\hline\n\\hline 39 & 3 & 101.743 &33 & 1 &  131.263 &33& 2 & 140.612\\\\ \\hline\n\\hline 42 & 2 & 107.726 & 35 & 2 & 132.857  & 35 & 2 & 141.695\\\\ \\hline\n\\hline 44 & 2 & 113.297 & 37& 1 & 138.734 & 37 & 1 & 147.282\\\\ \\hline\n\\hline 46 & 2 & 117.548 & 38& 2 &  147.636& 38 & 2 &158.585\\\\ \\hline\n\\hline 48 & 1 & 119.200 & 40  &  1 & 167.317 & 40 & 1 & 179.305\\\\ \\hline\n\\hline &&&41&2 & 170.961 & 41 & 2 & 188.748 \\\\ \\hline\n\\hline &&&43&2&179.830&43&2&198.093\\\\ \\hline\n\\hline &&&45&2&189.270&45&2&208.827\\\\ \\hline\n\\hline &&&47&2&196.836&47&2&217.132\\\\ \\hline\n\\hline &&&49&2&207.097&49&2&230.250\\\\ \\hline\n\\hline &&&51&2&216.965&51&2&241.441\\\\ \\hline\n\\hline &&&53&2&219.590&53&2&243.324\\\\ \\hline\n\\hline &&&55&1&225.377&55&1&248.161\\\\ \\hline\n\\hline &&&56&1&228.994&56&1&259.535\\\\ \\hline\n\\hline &&&57&3&260.100&57&1&278.937\\\\ \\hline\n\\hline &&&60&2&262.198&58&2&298.082\\\\ \\hline\n\\hline &&&62&2&264.926&60&2&322.073\\\\ \\hline\n\\hline &&&64&2&266.978&63&2&338.988\\\\ \\hline\n\\hline &&&66&2&270.885&64&1&344.883\\\\  \\hline\n\\end{array}$\n\\vspace{5mm}\n\\caption{Renormalized eigenvalues of OG}\n\\end{table}\n\n\nThere are certain patterns to the eigenvalues of $-\\Delta_m$ that are quite striking. The first is that, since the graph $\\Gamma_m$ is bipartite (the even and odd numbers of $k$ in $\\frac{k}{2\\cdot 8^m}$ alternate), eigenvalues come in pairs: if $-\\Delta_m u=\\lambda u$, then\n\n\\begin{equation}\n-\\Delta_m u^*=\\left(8-\\lambda \\right )u^*,\\text{ where}\n\\end{equation}\n\n\\begin{equation}\nu^*\\left (\\frac{k}{2\\cdot 8^m} \\right)=(-1)^k u \\left (\\frac{k}{2\\cdot 8^m} \\right),\n\\end{equation}\nIt does not seem that this observation has any consequences for the spectrum of $-\\Delta$, since only the lower portion of the spectrum of $-\\Delta_m$ is relevant. A more significant observation is the miniaturization of eigenfunctions: each eigenvalue of $-\\Delta_m$ is also an eigenvalue of $-\\Delta_{m+1}$ with the same multiplicity (with three exceptions to be explained below), and the corresponding eigenfunction of $-\\Delta_m$ is \"miniaturized\" to create the eigenfunctions of $-\\Delta_{m+1}.$ The rule for miniaturization depends on the representation of the dihedral-8 symmetry group that the eigenspace corresponds to, as explained in \\cite{BHS}. There are three 2-dimensional representations, labeled $2_1,\\,2_2,\\,2_3$ and four 1-dimensional representations labeled $1\\pm \\pm$ for symmetry or skew symmetry with respect to the reflections through the centers of edges of the octagon and the reflections through the vertices of the octagon. The 2-dimensional representations miniaturize to representations of the same type, while for the 1-dimensional representations, the miniaturization rule is \n\\begin{equation}\n\\begin{cases}\n1++\\to1++\\\\\n1+-\\to 1+-\\\\\n1-+\\to1++\\\\\n1--\\to1+-\n\\end{cases}\n\\end{equation}\n\nFor an example of the first rule, a constant miniaturizes to a constant. An example of the third rule is illustrated by Figure 4.1.\\\\\n\n\n\\begin{figure}[h!]\n\\begin{minipage}[h!]{0.48\\linewidth}\n\\begin{center}\n\\includegraphics[trim = 23mm 10mm 0mm 0mm, clip,height=2.5in]{Fig4_1m1}\n\\end{center}\nm=0\n\\end{minipage}\n\\hspace{0.01cm}\n\\begin{minipage}[h!]{0.48\\linewidth}\n\\begin{center}\n\\includegraphics[trim = 15mm 15mm 0mm 0mm, clip,height=2.5in]{Fig41m2}\n\\end{center}\nm=1\n\\end{minipage}\n\\caption{Shows the miniaturization rule for the third symmetry type, $1-+\\to 1+-$.}\n\\label{fig:figure4.1}\n\\end{figure}\n\n\n\n(Since the $1+-$ and $1-+$ representations do not occur for m=0, it is not practical to illustrate the other two cases.)\n\nThe exceptional miniaturizations correspond to the eigenvalues $4\\pm2\\sqrt{2}$ and $4$. In fact the multiplicity of $4\\pm2\\sqrt{2}$ is three for each $m\\geq 1$. Figure 4.2 shows the $m=1$ case with symmetry types $1++$, $1+-$ and $1-+$.\n\n\n\\begin{figure}[h!]\n\\includegraphics[trim = 35mm 35mm 35mm 35mm, clip,height=2.5in]{fig42Slide1}\n\\includegraphics[trim = 35mm 35mm 35mm 35mm, clip,height=2.5in]{fig42Slide2}\\\\\n\\includegraphics[trim = 35mm 35mm 35mm 35mm, clip,height=2.5in]{fig42Slide3}\n\\caption{Shows the $m=1$ case for symmetry types $1++$, $1+-$ and $1-+$ for eigenvalue $4\\pm 2\\sqrt{2}$.}\n\\label{fig:figure2}\n\\end{figure}\n\nThe $\\lambda=4$ eigenspace has multiplicity that grows with $m$ (4, 20, 164 for $m=1,\\,2,\\,3$). \nAgain, it does not appear that these exceptional eigenspaces contribute to the spectrum of $-\\Delta$, which appears to only have multiplicities one and two.\n\nWe call an eigenvalue (or eigenspace) of $-\\Delta_m$ \\emph{primitive} if it is not an eigenvalue of $-\\Delta_{m-1}$, otherwise we call it a \\emph{derived} eigenvalue. In the limit we expect that the primitive eigenspaces of $-\\Delta$ do not contain any miniaturized eigenfunctions. Just as in the case of PG, as proven in \\cite{Spec}, there are restrictions of the types of 1-dimensional representations that can appear in primitive eigenspaces. \n\n\\begin{table}\n\\centering\n\\begin{adjustwidth}{-1.0in}{-1.0in}\n\\begin{tabular}{  c c c c }\n$\\text{Eigenfunction \\# 2 \\& 3}$ & $\\text{Eigenfunction \\# 4 \\& 5}$ \\\\\n\\\\\n\\includegraphics[height=1.25in,width=1.75in]{PlusEigVal1_Lev1}  \\includegraphics[height=1.25in,width=1.75in]{MinEigVal1_Lev1}&\n \\includegraphics[height=1.25in,width=1.75in]{PlusEigVal4_Lev1}  \\includegraphics[height=1.25in,width=1.75in]{MinEigVal4_Lev1}\\\\ \n\\\\\n$ \\text{Eigenfunction \\# 6 \\& 7}$ \\\\\n\\\\\n\\includegraphics[height=1.25in,width=1.75in]{PlusEigVal6_Lev1} \\includegraphics[height=1.25in,width=1.75in]{MinEigVal6_Lev1}& \n\\end{tabular}\n\\vspace{1mm}\n\n\\caption*{FIGURE 4.3: Eigenfunctions of OG at Level 1}\n\\end{adjustwidth}\n\\end{table}\n\n\\begin{theorem}\nA primitive eigenfunction of $-\\Delta_m$, other than a constant, cannot have $1++$ or $1+-$ symmetry.\n\\end{theorem}\n\\begin{proof}\nSuppose $-\\Delta_mu=\\lambda u$ and $u$ is symmetric with respect to the reflections through the centers of the sides of the octagon. Then we could construct an eigenfunction $v$ of $-\\Delta_{m-1}$ by blowing up $u$ on a one-cell, so $v(t)=u\\left(\\frac{1}{8}t\\right)$ in the Peano curve parameterization, or $v(x)=u\\left (F_j x \\right)$, where $F_j$ is one of the contractions in the IFs that defines the octagasket. This would show that $u$ is not primitive. The only nontrivial fact to verify is that $v$ satisfies the eigenvalue equation along the \"boundary\" of $\\Gamma_{m-1}$, since for \"interior\" vertices the equations $-\\Delta_m u=\\lambda u$ and $-\\Delta_{m-1} v=\\lambda v$ are identical. But this is exactly where the symmetry hypothesis comes in: in $\\Gamma_m$ there will be four neighbors and in $\\Gamma_{m-1}$ there will be two neighbors, but the average over the neighbors will be the same.\n\\end{proof}\n\nThe Peano curve we are using does not respect all the symmetries of the dihedral-8 symmetry group. However, the reflection in the line through the initial point $\\gamma(0)$ is represented by the transformation $t\\to t+\\frac{1}{2}$, so we can sort eigenfunctions into symmetric and skew-symmetric ones with respect to this symmetry. Of course one-dimensional eigenspaces will automatically be one or the other. The two-dimensional eigenspaces will have a basis consisting of one symmetric and one skew-symmetric. Figures 4.3 and 4.4 \nshow the graphs of typical eigenfunctions at levels 2 and 3. We observe immediately that primitive $2_1$ and $2_3$ type eigenspaces contain a symmetric eigenfunction satisfying $u \\bigr( t+\\frac{1}{4}\\bigl) =-u(t)$, for example $\\# 2$ and $\\#6$, while the primitive $2_2$ type eigenspace contains symmetric eigenfunctions satisfying $u\\bigr(t+\\frac{1}{8}\\bigl)=-u(t)$, hence $u\\bigr(t+\\frac{1}{4}\\bigl)=u(t)$, for example $\\# 4$. A primitive $1-+$ eigenfunction will satisfy $u\\bigr(t+\\frac{1}{16}\\bigl)=-u(t)$, hence $u\\bigr(t+\\frac{1}{8}\\bigl)=u(t)$ while a primitive $1--$ eigenfunction will satisfy $\\tilde u \\bigr( t+\\frac{1}{16}\\bigl)=\\tilde u(t)$, where\n\\begin{equation}\n\\tilde{u}(t)=\\begin{cases} u(t)\\text{ if }0\\leq t\\leq\\frac{1}{2}\\\\\n-u(t)\\text{ if }\\frac{1}{2}\\leq t\\leq 1,\n\\end{cases}\n\\end{equation}\n\nfor example $\\smaller \\# \\hspace{0.8mm}   24$. \n\nMiniaturization will decrease the periods by a factor of $\\frac{1}{8}$. For example, $\\# \\hspace{0.8mm}9$ has period $\\frac{1}{16}$ (coming from the period $\\frac{1}{2}$ of $\\#\\hspace{0.8mm}2$) and $\\#\\hspace{0.8mm} 25$ has period $\\frac{1}{32}$ ( coming from the period $\\frac{1}{4}$ of $\\#\\hspace{0.8mm} 4$). In Figure 4.5 we show explicitly the miniaturization of  $ \\smaller\\#\\hspace{0.8mm} 2  \\hspace{0.8mm}\\&\\hspace{0.8mm} 3$  at level 1 to $\\#\\hspace{0.8mm} 8 \\hspace{0.8mm}\\& \\hspace{0.8mm}9$ at level 2. \n\nIn Figure 4.6 we show the eigenvalue counting function and the Weyl ratio at levels 1,2, and 3, using the values $\\beta=0.7213$ which was obtained experimentally. We do not observe any evidence of asymptotic multiplicative periodicity for the Weyl Ratio. \n\n\\begin{table}\n\\centering\n\\begin{adjustwidth}{-1.2in}{-1.0in}\n\n\\begin{tabular}{  c c c c }\n$ \\text{Eigenfunction \\# 2 \\& 3} $ &   $\\text{Eigenfunction \\# 4 \\& 5}$ \\\\\n\\includegraphics[height=1.25in,width=1.75in]{EigFunkOct3Plus2}\\includegraphics[height=1.25in,width=1.75in]{EigFunkOct3Min2}&\n \\includegraphics[height=1.25in,width=1.75in]{EigFunkkOct3Plus4}  \\includegraphics[height=1.25in,width=1.75in]{EigFunkkOct3Min4}\\\\ \n\\\\\n$\\text{Eigenfunction \\# 6 \\& 7}  $ & $\\text{Eigenfunction \\# 8} $ \\\\\n\\includegraphics[height=1.25in,width=1.75in]{EigFunkOct3Plus6} \\includegraphics[height=1.25in,width=1.75in]{EigFunkOct3Min6}&\n\\includegraphics[height=1.25in,width=2.25in]{EigFunkOct3Sol8} \n\\\\\n$ \\text{Eigenfunction \\# 9 \\& 10}  $ & $\\text{Eigenfunction \\# 24}$ \\\\\n\\includegraphics[height=1.25in,width=1.75in]{EigFunkOct3Plus9} \\includegraphics[height=1.25in,width=1.75in]{EigFunkOct3Min9}&\n\\includegraphics[height=1.25in,width=1.75in]{EigFunkOct3Sol24}\\\\\n\\\\\n $\\text{Eigenfunction \\# 25 \\& 26}$ \\\\\n\\includegraphics[height=1.25in,width=1.75in]{EigFunkOct3Plus25} \\includegraphics[height=1.25in,width=1.75in]{EigFunkOct3Min25}\\\\\n\\end{tabular}\n\\vspace{2mm}\n\\caption*{FIGURE 4.4: Eigenfunctions of the Octagasket at Level 3}\n\\end{adjustwidth}\n\\end{table}\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ c c c  }\n\\includegraphics[height=1.25in,width=1.75in]{PlusEigVal5_Lev2} && \\includegraphics[height=1.25in,width=1.75in]{MinEigVal5_Lev2}\\\\\n\\includegraphics[height=1.25in,width=1.75in]{ZOOMOctPlus9} && \\includegraphics[height=1.25in,width=1.75in]{ZOOMOct9Min}\\\\\n\\end{tabular}\n\\vspace{1mm}\n\n\n\\caption*{FIGURE 4.5. The top row shows $\\#$ 9 $\\&$10 at level 2, and the bottom row shows a zoom in by a factor of $\\frac{1}{8}$, to be compared with $\\#$ 2,3 on level 1 in Figure 4.3. }\n\\end{table}\n\n\n\n\nIt was observed in \\cite{BHS} that the spectrum of OG appears to have spectral gaps (ratios $\\frac{\\lambda_{k+1}}{\\lambda_k}$ considerably larger than 1 ). On the basis of our data we see gaps occurring for values of $k$ divisible by 16. In Table 4.3 we present this data.\n\n\n\\newpage \n\n\\begin{table}\n\\centering\n$\\begin{array}{|c|c|c|} \n\\hline k & 16k& \\frac{\\lambda_{16k+1}}{\\lambda_{16k}}\\\\ \\hline\n\\hline 1 & 16 & 1.8350\\\\ \\hline\n\\hline 2 & 32 & 1.3236\\\\ \\hline\n\\hline 7 & 112 & 1.554\\\\ \\hline\n\\hline 15 & 240 & 1.168\\\\ \\hline\n\\hline 54 & 864 & 1.768\\\\ \\hline\n\\end{array}$\n\\vspace{5mm}\n\\caption{Spectral Gaps of the Octagasket at Level 3}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{adjustwidth}{-1.0in}{-1.0in}\n\n\\vspace{3mm}\n\\begin{tabular}{  c c c c c}\n$\\text{Eigenvalue Counting Functions}$&& \\\\\n\\\\\n$\\text{ Level 1}$ & $\\text{Level 2}$ & $\\text{Level 3}$ \\\\\n\\\\\n  \\includegraphics[height=1.25in,width=1.75in]{NewPowOctEigCount1}&  \\includegraphics[height=1.25in,width=1.75in]{NewPowOctEigCount2}&  \\includegraphics[height=1.25in,width=1.75in]{NewPowOctEigCount3}\\\\ \n\\\\\n\\\\\n$\\text{Weyl Ratios}$ &&\\\\\n\\\\\n$ \\text{Level 1} $ & $\\text{Level 2}$ & $\\text{Level 3}$ \\\\\n\\\\\n \\includegraphics[height=1.25in,width=1.75in]{NewPowOctWeylRatio1} & \\includegraphics[height=1.25in,width=1.75in]{NewPowOctWeylRatio2} & \\includegraphics[height=1.25in,width=1.75in]{NewPowOctWeylRatio3}\\\\\n\\end{tabular}\n\\vspace{2mm}\n\n\\caption*{FIGURE 4.6. Eigenvalue counting function and Weyl Ratio of OG}\n\\end{adjustwidth}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The Magic Carpet}\n\nThe approximations to $\\gamma_m$ to the Peano Curve to MC give rise to a sequence of graphs $\\Gamma_m$, and the graph Laplacians $\\Delta_m$ are candidates for approximations to a Laplacian $\\Delta$ on MC, with appropriate renormalization. We choose to normalize $\\Delta_m$ by \n\\begin{equation} \n-\\Delta_m u(x)=12 (u(x) - \\text{Ave}(u(y)) \\end{equation} \nanalogous to (4.2). Note, however, that some points $x$ will have neighbors that are identified with it, so more explicitly\n\n\\begin{equation}\n -\\Delta_m (x) =\n\\begin{cases}\n12u(x) - 3 \\sum_{y\\sim x} u(y) & \\text{if $x$ has 2 identifications} \\\\\n12 u(x) -\\sum_{y\\sim x} u(y) & \\text{if $x$ has 6 identifications and 12 distinct neighbors}\\\\\n8u(x) -\\sum_{y\\sim x} u(y) & \\text{if $x$ has 6 identifications and 8 distinct neighbors}\\\\\n\\end{cases}\n \\end{equation}\n\nThe third case occurs exactly when $x$ is a singular point introduced at level $m$, and the second case occurs when $x$ is a singular point introduced at level $m'<m$. We note that these approximate Laplacians agree exactly (except for a different renormalization constant) with the approximate Laplacians for zero-forms studied in \\cite{kform}, and indeed the eigenvalues shown in Table 5.1 are equal to six times the eigenvalues computed in \\cite{kform}. We would like to believe that a limit as in (4.3) exists for an appropriate choice of $r$. Experimentally it appears that $r\\approx1.25$ and $\\frac{8}{r} \\approx 6.45$. This is very interesting because it means that the energy renormalization factor $r^m$ blows up. This would imply that the associated energy has points with zero capacity and functions of finite energy need not be continuous, as in Euclidean space of dimensions $ >2$. This would make it more challenging to define energy on MC as a limit of graph energies on $\\Gamma_m$. The spectrum of $-\\Delta_1$ is exactly $\\{ 0, 9, \\frac{29-\\sqrt{73}}{2}, 15, 15, \\frac{29 +\\sqrt{73}}{2} \\}$ and the associated non-constant eigenfunction are shown in Figures 5.1 -5.3.\n\n\\newpage\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=2.5in,width=2.5in]{figure1.png}\n\\caption*{FIGURE 5.1. $\\lambda=9$}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=2.5in,width=2.5in]{figure2.png}\n\\caption*{FIGURE 5.2. $\\lambda=15$. Note that the eigenfunction may also be rotated.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=3in,width=3in]{Figure53}\n\\caption*{FIGURE 5.3. $\\lambda=\\frac{29 \\pm \\sqrt{73}}{2}$,$a=\\frac{12}{\\lambda-12}$,$b=-1$, $c=\\frac{8}{\\lambda-8}$}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}[!ht]\n\\centering\n\\begin{minipage}[h!]{0.48\\linewidth}\n\\begin{center}\n\\includegraphics[height=2.5in,width=2.5in]{figure5.png}\n\\hspace{0.01cm}\n\\includegraphics[height=2.5in,width=2.5in]{figure6.png}\\\\\n\\end{center}\n\\end{minipage}\n\\vspace{0.2cm}\n\\begin{minipage}[h!]{0.45\\linewidth}\n\\begin{center}\n\\includegraphics[height=2.5in,width=2.5in]{figure7.png}\n\\end{center}\n\\end{minipage}\n\\caption*{FIGRUE 5.4: Basis for $\\lambda=9$ eigenspace for $m=2$ }\n\\label{fig:figure2}\n\\end{figure}\n\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=3in,width=3in]{figure8.png}\n\\caption*{FIGURE 5.5:$\\lambda=12$}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=3in,width=3in]{figure9.png}\n\\caption*{FIGURE 5.6:$\\lambda=5$}\n\\end{center}\n\\end{figure}\n\nWith respect to the dihedral-4 symmetry group of MC, we have a single two-dimensional representations denoted 2 and four one-dimensional representation $ 1\\pm \\pm$, where the first $\\pm$ denotes symmetry or skew-symmetry with respect to the diagonal reflections, and the second $\\pm$ denotes symmetry or skew-symmetry with respect to horizontal and vertical reflections. With this notation, the eigenspace corresponding to $\\lambda =15$ corresponds to representation 2, the eigenspace $\\lambda =9$ corresponds to representation $1 - +$, and the eigenspaces $\\lambda = \\frac{29\\pm\\sqrt{75}}{2}$ correspond to the representation $1 + +$.\n\nWe have the same miniaturization in passing from $-\\Delta_{m-1}$ to $-\\Delta_m$ as in the case of SC as described in \\cite{BHS}. The exceptions are $\\lambda =9$, where the multiplicity is $\\frac{2\\cdot8^{m-1}+5}{7}$, and $\\lambda=15$, where the multiplicity is $\\frac{9\\cdot8^{m-1}+5}{7}$. In Figure 5.4 we show a basis for the $\\lambda=9$ eigenspace with $m=2$, consisting of one $1 - -$ representation and two $1 - +$ representations. Note that the third basis element is the one given by miniaturization. There are two other simple eigenvalues, $\\lambda=5$ and $\\lambda=12$ that appear with multiplicity one for $m \\ge 2$. We show the eigenfunctions for $m=2$ in Figures 5.5 and 5.6, both $ 1 + -$ representations.\n\n\n\n\n\n\n\n\n\\begin{table}[h!]\n\\centering\n\n\\vspace{5mm}\n\n\\begin{tabular}{  c c c c }\n$\\text{Eigenfunction \\# 2} $ &   $\\text{Eigenfunction \\# 3 \\& 4}$ \\\\\n\\includegraphics[height=1.25in,width=1.75in]{Magic2Lev3} &\n \\includegraphics[height=1.25in,width=1.75in]{Magic3Lev3} \\includegraphics[height=1.25in,width=1.75in]{Magic4Lev3}\\\\\n\\\\\n$ \\text{Eigenfunction \\# 5}  $ & $ \\text{Eigenfunction \\# 6} $ \\\\\n\\includegraphics[height=1.25in,width=1.75in]{Magic5Lev3}&\n\\includegraphics[height=1.25in,width=1.75in]{Magic6Lev3}\\\\\n\\\\ \n\n$\\text{Eigenfunction \\# 18}$ & $\\text{Eigenfunction \\#27 \\& 28}$\\\\\n\\includegraphics[height=1.25in, width=1.75in]{Magic18Lev3}&  \\includegraphics[height=1.25in,width=1.75in]{Magic27Lev3} \\includegraphics[height=1.25in, width=1.75in]{Magic28Lev3}\\\\\n\\\\\n$\\text{Eigenfunction \\# 29}$ & $\\text{Eigenfunction \\# 54} $ \\\\\n \\includegraphics[height=1.25in,width=1.75in]{Magic29Lev3} & \\includegraphics[height=1.25in, width=1.75in]{Magic54Lev3}\\\\\n\\\\\n\n\\end{tabular}\n\\caption*{FIGURE 5.7. Eigenfunctions of the Magic Carpet at Level 3}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{adjustwidth}{-1.0in}{-1.0in}\n\\vspace{3mm}\n\\begin{tabular}{  c c c c }\n$\\text{Eigenvalue Counting Function}$\\\\\n\\\\\n$\\text{ Level 2}$ & $\\text{ Level 3}$ & $\\text{Level 4}$ \\\\\n\\\\\n \\includegraphics[height=1.25in,width=1.75in]{CarpetLevel2EIGENCOUNT}&  \\includegraphics[height=1.25in,width=1.75in]{CarpetLevel3EIGENCOUNT} &  \\includegraphics[height=1.25in,width=1.75in]{CarpetLevel4EIGENCOUNT}\\\\ \n\\\\\n$\\text{Weyl Ratios}$ \\\\\n\\\\\n$ \\text{Level 2}$ & $ \\text{Level 3}$ & $\\text{Level 4} $\\\\\n\\\\\n \\includegraphics[height=1.25in,width=1.75in]{PentWeyl2} &  \\includegraphics[height=1.25in,width=1.75in]{CarpetLevel3WEYLRATIO} &  \\includegraphics[height=1.25in,width=1.75in]{CarpetLevel4WEYLRATIO}\\\\\n\\end{tabular}\n\\vspace{1mm}\n\n\\caption*{FIGURE 5.8. Eigenvalue Counting Function and Weyl Ratio of the Magic Carpet}\n\\caption*{$\\beta=1.2$}\n\\end{adjustwidth}\n\\end{table}\n\n\n\n\\begin{table}[!ht]\n\\begin{adjustwidth}{-1.3in}{-1.3in}\n\\begin{center}\n$\\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|||c|c|c|}\n\\hline\n\\multicolumn{3}{|c|}{\\text {Level 1}}\n& \\multicolumn{3}{|c|}{\\text{Level 2}}\n& \\multicolumn{3}{|c|}{\\text{Level 3}}\n& \\multicolumn{3}{|c|}{\\text{Level 4}}\n&\\multicolumn{2}{|c|}{\\text{Ratio}} \\\\\n\\hline\n\\#  & \\text{Mult} & \\text{Eiv} &  \\# & \\text{Mult} & \\text{Eiv}& \\# & \\text{Mult} & \\text{Eiv}& \\# & \\text{Mult} & \\text{Eiv}& \\frac{\\lambda_2}{\\lambda_3}&\\frac{\\lambda_3}{\\lambda_4}\\\\\n\\hline 1 & 1 & 0 & 1 & 1 & 0 & 1&1&0&1&1&0&&\\\\  \\hline\n\\hline 2 &1&9.000& 2 & 1&1.726& 2&1& 0.274&2&1&0.0429&6.281&6.406\\\\ \\hline\n\\hline 3&1&10.228&3&2&2.674&3&2&0.441&3&2&0.068&6.069&6.442\\\\ \\hline\n\\hline 4&2& 15.000&5&1&2.697&5&1&0.458&5&1&0.072&5.885&6.365\\\\ \\hline\n\\hline 6&1&18.772&6&1&5.000&6&1&0.869&6&1&0.138&5.752&6.294\\\\ \\hline\n\\hline &&&7&2&5.515&7&2&0.923&7&2&0.146&5.586&6.296\\\\ \\hline\n\\hline &&&9&1&5.917&9&1&0.987&9&1&0.154&5.993&6.402\\\\ \\hline\n\\hline &&&10&1&6.580&10&1&1.112&10&1&0.173&5.915&6.423\\\\ \\hline\n\\hline &&&11&1&7.102&11&1&1.304&11&1&0.207&5.444&6.290\\\\ \\hline\n\\hline &&&12&2&7.808&12&2&1.431&12&2&0.223&5.455&6.412\\\\ \\hline\n\\hline &&&14&3&9.000&14&2&1.610&14&2&0.232&&6.386\\\\ \\hline\n\\hline &&&17&2&9.475&16&1&1.620&16&1&0.257&&6.305\\\\ \\hline\n\\hline &&&19&1&10.147&17&1&1.709&17&1&0.272&&6.277\\\\ \\hline\n\\hline &&&20&1&10.228&18&1&1.726&18&1&0.274&&6.281\\\\ \\hline\n\\hline &&&21&1&11.261&19&1&2.044&19&1&0.331&&6.168\\\\ \\hline\n\\hline &&&22&1&11.347&20&1& 2.321&20&1&0.375&&6.177\\\\ \\hline\n\\hline &&&23&2&11.796&21&2&2.354&21&2&0.379&&6.193\\\\ \\hline\n\\hline &&&25&1&12.000&23&1&2.501&23&1&0.411&&6.084\\\\ \\hline\n\\hline &&&26&1&13.893&24&2&2.594&24&2&0.423&&6.131\\\\ \\hline\n\\hline &&&27& 2&13.998&26&1&2.613&26&1&0.430&&6.068\\\\ \\hline\n\\hline &&&29&1&14.675&27&2&2.674&27&2&0.440&&6.070\\\\ \\hline\n\\hline &&&30&11&15.000&29&1&2.697&29&1&0.457&&5.896\\\\ \\hline\n\\hline &&&41&1&18.663&30&1&2.773&30&1&0.458&&6.051\\\\ \\hline\n\\hline &&&42&1&18.6701&31&2&2.846&31&2&0.472&&6.021\\\\ \\hline\n\\hline &&&43&1&18.772&33&1&3.108&33&1&0.518&&5.993\\\\ \\hline\n\\hline &&&44&2&19.087&34&2&3.314&34&2&0.553&&5.990\\\\ \\hline\n\\hline &&&46&1&19.316&36&1&3.361&36&1&0.571&&5.882\\\\ \\hline\n\\end{array}$\n\\vspace{5mm}\n\\caption{Eigenvalues of the Magic Carpet}\n\\end{center}\n\\end{adjustwidth}\n\\end{table}\n\n\n\\begin{table}[!ht]\n\\begin{adjustwidth}{-.8in}{-.5in}\n\\begin{center}\n$\\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \n\\hline\n\\multicolumn{3}{|c|}{\\text {Level 1}}\n& \\multicolumn{3}{|c|}{\\text{Level 2}}\n& \\multicolumn{3}{|c|}{\\text{Level 3}}\n& \\multicolumn{3}{|c|}{\\text{Level 4}}\\\\\n\\hline\n\\#  & \\text{Mult} & \\text{Eiv} &  \\# & \\text{Mult} & \\text{Eiv}& \\# & \\text{Mult} & \\text{Eiv}& \\# & \\text{Mult} & \\text{Eiv}\\\\\n\\hline 1 & 1 & 0 & 1 & 1 & 0 & 1&1&0&1&1&0\\\\  \\hline\n\\hline 2 &1&57.600&2 & 1& 70.701 &2&1&72.041&2&1&71.974\\\\ \\hline\n\\hline 3&1&65.459&3&2&109.553&3&2&115.511&3&2&114.756\\\\ \\hline\n\\hline 4&2&96.000&5&1&110.497&5&1&120.146&5&1&120.795\\\\ \\hline\n\\hline 6&1&120.141&6&1&204.800&6&1&227.868&6&1&231.693\\\\ \\hline\n\\hline &&&7&2&211.239&7&2&241.978&7&2&245.953\\\\ \\hline\n\\hline &&&9&1&242.373&9&1&258.799&9&1&258.704\\\\ \\hline\n\\hline &&&10&1&269.530&10&1&291.629&10&1&290.581\\\\ \\hline\n\\hline &&&11&1&290.936&11&1&342.0&11&1&347.959\\\\ \\hline\n\\hline &&&12&2&319.840&12&2&375.214&12&2&374.467\\\\ \\hline\n\\hline &&&14&3&368.640&14&2&422.201&14&2&423.121\\\\ \\hline\n\\hline &&&17&2&388.136&16&1&424.783&16&1&431.174\\\\ \\hline\n\\hline &&&19&1&415.633&17&1&448.0801&17&1&456.843\\\\ \\hline\n\\hline &&&20&1&418.938&18&1&452.486&18&1&461.037\\\\ \\hline\n\\hline &&&21&1&461.277&19&1&535.914&19&1&555.996\\\\ \\hline\n\\hline &&&22&1&464.809&20&1&608.567&20&1&630.487\\\\ \\hline\n\\hline &&&23&2&483.192&21&2&616.822&21&2&637.366\\\\ \\hline\n\\hline &&&25&1&491.521&23&1&655.682&23&1&689.711\\\\ \\hline\n\\hline &&&26&1&569.094&24&2&680.226&24&2&710.011\\\\ \\hline\n\\hline &&&27&2&573.394&26&1&685.060&26&1&722.426\\\\ \\hline\n\\hline &&&29&1&601.112&27&2&701.143&27&2&739.204\\\\ \\hline\n\\hline &&&30&11&614.400&29&1&707.183&29&1&767.555\\\\ \\hline\n\\hline &&&41&1&764.436&30&1&726.975&30&1&768.899\\\\ \\hline\n\\hline &&&42&1&764.727&31&2&746.101&32&2&793.059\\\\ \\hline\n\\hline &&&43&1&768.901&33&1&814.798&33&1&870.066\\\\ \\hline\n\\hline &&&44&2&781.840&34&2&868.857&34&2&928.283\\\\ \\hline\n\\hline &&&46&1&791.191&36&1&881.136&36&1&958.650\\\\ \\hline\n\\end{array}$\n\\vspace{5mm}\n\\caption{Renormalized Eigenvalues of the Magic Carpet}\n\\end{center}\n\\end{adjustwidth}\n\\end{table}\n\nIn Table 5.1 we show the eigenvalues for levels $m=1,2,3,4$ and their ratios. The ratio values suggest that the eigenvalue renormalization factor should be around 6.4. In Table 5.2 we show the renormalized eigenvalues (multiplied by $(6.4)^m$). Because 6.4 is smaller than the measure renormalization factor 8, this suggests that the energy renormalization factor would have to be around 1.25. Since this is greater than one, it would imply that points have zero capacity and functions of finite energy do not have to be continuous, in contrast to all PCF fractals, SC and PG. We also observe that the spectral data agrees exactly with the data in \\cite{kform} for the approximations to the zero-forms Laplacian on MC. In fact the approximate graph Laplacians are identical. \n\nAs in the case of OG, it appears that the only multiplicities in the spectrum of $ -\\Delta $ will be one and two, as the higher multiplicities in the spectrum of $-\\Delta_m$ occur high up in the spectrum and will not survive in the limit. The only noticeable spectral gap in $-\\Delta_m$ occur near $\\lambda=15$ and again will not survive in the limit. There may be smaller spectral gaps that survive in the limit, especially since the average separation of eigenvalues goes to zero. This remains to be investigated.\nIn Figures 5.7 and 5.8 we show the graph of some eigenfunction on the parameter circle. Note that the Peano curve does not respect the symmetries of the MC, so these graphs do not show the kind of symmetry found in the cases of PG or OG. In Figure 5.9 we show graphs of the eigenvalue counting function and the Weyl ratio at levels 2,3, and 4. \n\n\\newpage\n\\newpage\n\n\\section{The Torus and the Triangle}\nThe Peano curves to the torus, $T_0$, and the triangle, $T_r$, yield graph approximations that are identical to the standard lattice graph approximations. For $T_o$, the graph $\\Gamma_m$ has vertices that we may identify with the points $\\left(\\frac{j}{3^m},\\frac{k}{3^m}\\right) \\mod 1$ with $0\\leq j,\\ k\\leq3^m$. The neighbors are the four points $\\left(\\frac{j\\pm 1}{3^m},\\frac{k\\pm 1}{3^m}\\right)$. Thus the Laplacian $-\\Delta_m$ is \n\\begin{equation}\n-\\Delta_m u\\left(\\frac{j}{3^m},\\frac{k}{3^m}\\right)=\\sum\\left(u\\left(\\frac{j}{3^m},\\frac{k}{3^m}\\right)-u\\left(\\frac{j\\pm 1}{3^m},\\frac{k\\pm 1}{3^m}\\right)\\right).\n\\end{equation}\nOf course the formula looks different in terms of the parameterization $\\gamma(t)$ for $t=\\frac{n}{2\\cdot9^m}$ with identifications. The eigenvalues will be exactly the same, while the eigenfunctions, $e^{2\\pi i(px+qy)}$ for $(p,q)\\in\\mathbb{Z}^2$ (with eigenvalue $4\\pi^2(p^2+q^2)$), will have a different appearance as a function of $t$. Aside from the zero-eigenspace, all eigenvalues have multiplicities equal to a multiple of 4 (if $p\\neq q$, then multiplicity is at least eight, including $(\\pm p,\\pm q)$ and $(\\pm q, \\pm p)$). Because the Peano curve does not respect the dihedral-4 symmetries of the torus, it is difficult to separate out specific eigenfunctions within each eigenspace, so we have been unable to ``interpret\" the graphs of eigenfunctions for this example. It is obvious from $(6.1)$ that $9^m\\Delta_m\\to -\\Delta$. In Table 6.1 we show the eigenvalues of $\\Delta_m$ and their ratios. In Table 6.2 we show the eigenvalues normalized by multiplication by $\\frac{9^m}{4\\pi^2}$. The deviation from the expected integer values $p^2+q^2$ is small enough at the low end of the spectrum to confirm the convergence, but it rapidly grows out of hand as the eigenvalues increase. This just confirms that this finite difference method has rather poor accuracy.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=.50,trim = 0mm 0mm 0mm 0mm, clip]{Figure61}\n\\caption*{FIGURE 6.1: $x$ is a boundary point, with two boundary neighbors $z_1$,$z_2$ and two interior neighbors $y_1$,$y_2$ in the triangle. We add on two virtual triangles (indicated by dotted lines) with new vertices $y_1'$ and $y_2'$, and even reflection makes $u(y_1')=u(y_1)$ and $u(y_2')=u(y_2)$}\n\\end{center}\n\\end{figure}\n\n\\begin{table}\n\\begin{center}\n$\\begin{array}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{3}{|c|}{\\text{Level 1}}\n& \\multicolumn{3}{|c|}{\\text{Level 2}}\\\\\n\\hline\n\\# & \\text{Mult} & \\text{Eiv} & \\# & \\text{Mult} & \\text{Eiv}\\\\ \\hline\n1 & 1 & 0 & 1 & 1 & 0\\\\ \\hline\n2 & 4 & 3 & 2 & 4 & 0.4679\\\\ \\hline\n6 & 4 & 6 & 6 & 4 & 0.9358\\\\ \\hline\n&&& 10 & 4 & 1.6527\\\\ \\hline\n&&&14 & 8 & 2.1206\\\\ \\hline\n&&&22 & 7 & 3\\\\ \\hline\n&&&29 &  4 & 3.3051\\\\ \\hline\n&&&33 &  8 & 3.4679\\\\ \\hline\n&&&41 & 4 & 3.8793\\\\ \\hline\n&&&45 & 8 & 4.3473\\\\ \\hline\n&&&53 & 8 &4.6527\\\\ \\hline\n\\end{array}$\n\\vspace{1mm}\n\n\\caption*{TABLE 6.1: Eigenvalues of the Torus}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n$\\begin{array}{|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{3}{|c|}{\\text{Level 1}}\n& \\multicolumn{3}{|c|}{\\text{Level 2}}\\\\\n\\hline\n\\# & \\text{Mult} & \\text{Eiv} & \\# & \\text{Mult} & \\text{Eiv}\\\\ \\hline\n1 & 1 & 0 & 1 & 1 & 0\\\\ \\hline\n2 & 4 & 1 & 2 & 4 & 1\\\\ \\hline\n6 & 4 & 2 & 6 & 4 & 2\\\\ \\hline\n&&& 10 & 4 & 3.5320\\\\ \\hline\n&&&14 & 8 & 4.5320\\\\ \\hline\n&&&22 & 7 & 6.4114\\\\ \\hline\n&&&29 &  4 & 7.0641\\\\ \\hline\n&&&33 &  8 & 7.4114\\\\ \\hline\n&&&41 & 4 & 8.2908\\\\ \\hline\n&&&45 & 8 & 9.2908\\\\ \\hline\n&&&53 & 8 &9.9435\\\\ \\hline\n\\end{array}$\n\\vspace{1mm}\n\n\\caption*{TABLE 6.2: Renormalized eigenvalues of the Torus}\n\\end{center}\n\\end{table}\n\nThe situation for the triangle is much better. The Peano curve again produces graph approximations which are identical to the triangular lattice graphs. At level $m$ we subdivide each side of the triangle into $2^m$ equal segments, and the intersection points of the lines joining the vertices on the sides are the vertices of the graph. Interior points have four neighbors (the two interior neighbors carry twice the conductance of the boundary neighbors), and the three corners of the triangle have two neighbors. The approximate Laplacian may be written\n\n\\begin{equation}\n-\\Delta_m u(x)=6 u(x)-\n\\begin{cases} \n\\sum_{y\\underset{m} \\sim x} u(y) &  $\\text{$x$ an interior point}$ \\\\\n2\\sum_{y\\underset{m}\\sim x} u(y)+\\sum_{z\\underset{m} \\sim x} u(z) &\\text{$x$ a boundary point}\\\\ & \\text{$z$ its boundary neighbors}\\\\\n\\\\\n3\\sum_{z\\underset{m}\\sim x} u(z) & \\text{$x$ a corner point.}\n\\end{cases}\n\\end{equation}\n\n\\begin{table}\n\\begin{adjustwidth}{-1.3in}{-1in}\n\\begin{center}\n$\\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c||c|c|c|} \n\\hline\n\\multicolumn{3}{|c|}{\\text {Level 1}}\n& \\multicolumn{3}{|c|}{\\text{Level 2}}\n& \\multicolumn{3}{|c|}{\\text{Level 3}}\n& \\multicolumn{4}{|c|}{\\text{Level 4}}\n& \\multicolumn{3}{|c|}{\\text{Ratios}}\\\\\n\\hline\n\\#  & \\text{Mult} & \\text{Eiv} &  \\# & \\text{Mult} & \\text{Eiv}& \\# & \\text{Mult} & \\text{Eiv}& \\# & \\text{Mult} & \\text{Eiv}& \\text{Norm. Eiv} & \\frac{\\lambda_1}{\\lambda_2}&\\frac{\\lambda_2}{\\lambda_3} &\\frac{\\lambda_3}{\\lambda_4} \\\\\n\\hline 1 & 1 & 0 & 1 & 1 & 0 & 1&1&0&1&1&0&&&\\\\  \\hline\n\\hline 2 &2&0.5120 \t&2 & 2& 0.1347 &2&2&0.0341 &2&2&0.0086&1&3.8&3.95&3.97 \\\\ \\hline\n\\hline 4&1&1.3333&4&1&0.3905&4&1&0.1015 &4&1&0.0256&2.976&3.41&3.85&3.96 \\\\ \\hline\n\\hline 5&2&1.6667 &5&2&0.5120 &5&2&0.1347&5&2&0.0341&3.965&3.26&3.8&3.95\\\\ \\hline\n\\hline 7&2&2.3333 &7&2&0.8502 &7&2&0.2328 &7&2&0.0595&6.918&2.74&3.65&3.91 \\\\ \\hline\n\\hline 9&3&2.6667 &9&2&1.0572 &9&2&0.2968 &9&2&0.0764&8.883&2.52&3.56&3.88 \\\\ \\hline\n\\hline 12&2&2.8214 &11&1&1.3333 &11&1&0.3905 &11&1&0.1015&11.802&2.12&3.41&3.85\\\\ \\hline\n\\hline 14&2&3.0000 &12&2&1.4227 &12&2&0.4213&12&2&0.1098&12.767&2.11&3.38&3.84 \\\\ \\hline\n\\hline &&&14&2&1.6667 &14&2&0.5120 &14&2&0.1348&15.674&&3.26&3.8 \\\\ \\hline\n\\hline &&&16&2&1.8619 &16&2&0.5999 &16&2&0.1595&18.456&&3.1&3.76 \\\\ \\hline\n\\hline &&&18&2&2.0000 &18&2&0.6576&18&2&0.1759&20.453&&3.04&3.74\\\\ \\hline\n\\hline &&&20&2&2.2323 &20&2&0.7697 &20&2&0.2085 &24.244&&2.9&3.69\\\\ \\hline\n\\hline &&&22&1& 2.2761 &22&1&0.8231 &22&1&0.2247 &26.127&&2.77&3.66\t\\\\ \\hline\n\\hline &&&23&2&2.3333 &23&2&0.8502 &23&2&0.2328&27.069&&2.74&3.65 \\\\ \\hline\n\\hline &&&25&2&2.4832&25&2&0.9298 &25&2&0.2569&29.872&&2.67&3.62 \\\\ \\hline\n\n\\end{array}$\n\\vspace{5mm}\n\\caption*{TABLE 6.3: Eigenvalues of the Triangle}\n\\end{center}\n\\end{adjustwidth}\n\\end{table}\n\n\n\nNote that this is an approximation to the Neumann Laplacian on $T_r$, since even reflection across a boundary line to a virtual neighboring triangle transforms \n\\[2\\sum_{y\\underset{m}\\sim x} u(y)+\\sum_{z\\underset{m} \\sim x} u(z)\\]\ninto the sum of $u$ at the six neighboring vertices in the larger configuration(Figure 6.1).\nThen we have \n\\begin{equation}-4^m\\Delta_m\\to \\frac{3}{2}\\Delta \\text{(Neumann boundary conditions) }\\end{equation}\n(the factor $\\frac{3}{2}$ comes from $\\left(\\frac{\\partial}{\\partial  x}\\right)^2+\\left(\\frac{1}{2}\\frac{\\partial}{\\partial x}+\\frac{\\sqrt{3}}{2}\\frac{\\partial}{\\partial y}\\right)^2+\\left(\\frac{1}{2}\\frac{\\partial}{\\partial x}-\\frac{\\sqrt{3}}{2}\\frac{\\partial}{\\partial y}\\right)^2=\\frac{3}{2}\\Delta$). In Table 6.3 we show the eigenvalues at different levels and their ratios, and on level 4 the eigenvalues normalized by multiplication by $4^m\\frac{2}{3}\\left(\\frac{3}{4\\pi}\\right)^2$, to be compared with the integer values $p^2+q^2+pq$.\nHowever, now we are able to make sense of the graphs of the eigenfunctions as a function of the circle parameter $t$. In fact we will argue that this alternate way of visualizing eigenfunctions offers some appealing advantages to the rather awkward view of functions defined on $T_r$. The dihedral-3 symmetry group acting on $T_r$ is not completely respected by the Peano curve, but the subgroup of rotations $\\left(\\text{through angles }0,\\ \\frac{2\\pi}{3}, \\ \\frac{4\\pi}{3}\\right)$ is. A rotation through the angle $\\frac{2\\pi}{3}$ amounts to the translation $t\\to t+\\frac{1}{3}$, so any function invariant under the rotation subgroup is represented by a function periodic of period $\\frac{1}{3}$, a property instantly visible from the graph.\n\n\n\nThere is also a different symmetry, not part of the dihedral-3 symmetry group, that plays an important role in miniaturization. Take any Neumann eigenfunction $u$ on $T_r$ with eigenvalue $\\lambda_j$, shrink it to the subtriangle $\\frac{1}{2}\\ T_r$ by dilation, and reflect it in each of the interior sides to the remaining subtriangles. Because of the Neumann boundary conditions this miniaturization produces an eigenfunction $\\tilde u$ with eigenvalue $4\\lambda$. In the circle parameterization $\\tilde u (t)=u(4t)$, so we obtain a function that is periodic of period $\\frac{1}{4}$. Of course if the initial $u$ was rotation invariant, then the period of $\\tilde u$ is $\\frac{1}{12}$. By iterating miniaturization we may obtain functions that have period $\\frac{1}{4^n}$ or $\\frac{1}{3\\cdot 4^n}$. All these periods are immediately apparent from the graphs, shown in Figure 6.2.\n\nBut we can say much more precisely where these periods occur. The Neumann spectrum of $T_r$ is well-known.  Suppose the triangle has side length 1 and corners at $(0,0),\\ \\left(\\frac{\\sqrt{3}}{2},\\frac{1}{2}\\right)$ and $\\left(\\frac{\\sqrt{3}}{2},\\frac{-1}{2}\\right)$. For every pair $(p,q)$ of non-negative integers define\n\\begin{equation}\nu(p,q)=\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\n\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\n\\quad\\quad\\quad\n\\end{equation}\n\\quad$e(p,q)+e(-p,-q)+e(-p,p+q)+e(q,-p-q)+e(-p-q,p)+e(p+q,-q),$\\\\\n\\\\\nwhere\n\\begin{equation}\nu(p,q)(x)=e^{2\\pi i(pv+qw)\\cdot x}\\text{ for }v=\\left(\\frac{1}{\\sqrt{3}},\\frac{1}{3}\\right),\\ w=\\left(0,\\frac{2}{3}\\right).\n\\end{equation}\nThen $u(p,q)$ is a Neumann eigenfunction with eigenvalue $\\left(\\frac{4\\pi}{3}\\right)^2(p^2+q^2+pq)$, and these are the only ones. Note that when $p=q$ we obtain an eigenspace of multiplicity one, and for $p\\neq q$ the functions $u(p,q)$ and $u(q,p)$ span an eigenspace of multiplicity two. Coincidences where $p^2+q^2+pq=(p')^2+(q')^2+p'q'$ may lead to higher multiplicities, but this does not change the narrative substantially.\n\n\\begin{table}\n\\begin{adjustwidth}{-1.3in}{-1in}\n\\begin{center}\n\n\\vspace{5mm}\n\n\\begin{tabular}{  c c c c }\n$\\text{Eigenfunction \\# 2 \\& 3} $ &   $\\text{Eigenfunction \\# 4}$ \\\\\n\\includegraphics[height=1.25in,width=1.75in]{TriEig2Plus} \\includegraphics[height=1.25in,width=1.75in]{TriEig2Min} &\n \\includegraphics[height=1.25in,width=1.75in]{TriEig4} \\\\\n\\\\\n$ \\text{Eigenfunction \\# 5 \\& 6}  $ & $ \\text{Eigenfunction \\# 7 \\& 8} $ \\\\\n\\includegraphics[height=1.25in,width=1.75in]{TriEig5Plus} \\includegraphics[height=1.25in,width=1.75in]{TriEig5Min}&\n\\includegraphics[height=1.25in,width=1.75in]{TriEig7_PlusLev4} \\includegraphics[height=1.25in, width=1.75in]{TriEig7_MinLev4}\\\\\n\\\\ \n\n$\\text{Eigenfunction \\# 9 \\& 10}$ & $\\text{Eigenfunction \\#11}$\\\\\n\\includegraphics[height=1.25in, width=1.75in]{TriEig9_PlusLev4} \\includegraphics[height=1.25in, width=1.75in]{TriEig9_MinLev4}&  \\includegraphics[height=1.25in,width=1.75in]{TriangleEig11} \n\\\\\n$\\text{Eigenfunction \\# 14 \\& 15}$ \\\\\n \\includegraphics[height=1.25in,width=1.75in]{TriEig14Plus}  \\includegraphics[height=1.25in, width=1.75in]{TriEig14Min}\\\\\n\\\\\n\n\\end{tabular}\n\\end{center}\n\\caption*{FIGURE 6.2: Eigenfunctions of the Triangle at Level 4}\n\\end{adjustwidth}\n\n\\end{table}\n\nThe $p=q$ multiplicity one space transforms according to the $1+$ representation (symmetric with respect to reflections) and the formula simplifies to $u(p,p)=2\\cos 2\\pi p\\frac{\\sqrt{3}}{3}x\\cos 2\\pi p y+\\cos 2\\pi p\\frac{2\\sqrt{3}}{3}x$, and so $u(2^n(2^l+1),2^n(2^l+1))$ also transforms according to the $1+$ representation and has period $\\frac{1}{3\\cdot 4^n}$. This is seen in $\\# 4$ corresponding to$(1,1)$ and $\\# 11$ $(2,2)$. When $p\\neq q$ and both are odd then the multiplicity two space transforms according to the 2 representation and has no periodicity. When $p$ is even and $q$ is odd, then there are two cases: unless $q=p\\pm 3$, it is again the 2 representation, seen in $\\#7$ $\\&$ $8$ $(2,1)$,but if $q=p\\pm 3$ then the space breaks up into a direct sum of a $1+$ and a $1-$ representation, and both have period $\\frac{1}{3}$, seen in $\\# 9 \\& 10$ $(0,3)$. In fact, taking the sum and difference of (6.4) for $(p+3,p)$ and $(p,p+3)$ yields the formulas \n\\begin{equation}\n\\cos 2\\pi \\frac{p+3}{\\sqrt{3}} x\\cos 2\\pi(p+1)y+\\cos 2\\pi \\frac{p}{\\sqrt{3}} x\\cos 2\\pi(p+2)y+\\cos 2\\pi \\frac{2p+3}{\\sqrt{3}} x\\cos 2\\pi y\n\\end{equation}\nfor the $1+$ function and \n\\begin{equation}\n\\cos 2\\pi \\frac{p+3}{\\sqrt{3}} x\\sin 2\\pi(p+1)y+\\cos 2\\pi \\frac{p}{\\sqrt{3}} x\\sin 2\\pi(p+2)y+\\cos 2\\pi \\frac{2p+3}{\\sqrt{3}} x\\sin 2\\pi y\n\\end{equation}\nfor the $1-$ function. Finally, if $p$ and $q$ are both even with $(p,q)=2^n(p',q')$ with at least one of $p',q'$ odd, then the space is the $n$-fold iterated miniaturization of the $(p',q')$ space, with the same representations and the period multiplied by $\\frac{1}{4^n}$. These behaviors are seen in $\\#$ $5$ $\\&$ $6$ $(2,0)$ and $\\#$ $14$ $\\&$ $15$ $(4,0)$. \n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}