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+{"text":"\\section{Introduction}\n\nSince two-dimensional models with a continuous global symmetry\nwere recognized to be asymptotically free \\cite{free}\nthey became a famous laboratory for testing many ideas\nand methods before applying them to more complicated gauge theories.\nIn this paper we follow this common way and present a method,\ndifferent from the conventional perturbation theory (PT), \nwhich allows to investigate\nthese models in the weak coupling region.\nThe conventional PT is one of the main technical\ntools of the modern physics. In spite of the belief of the most\nof community that this method gives the correct asymptotic expansion\nof such theories as $4D$ QCD or $2D$ spin systems with continuous global\nsymmetry, the recent discussion of this problem\n\\cite{superinst}-\\cite{QA} has shown that it is rather far from\nunambiguous solution. \nIndeed, for the PT to be applicable it is necessary\nthat a system under consideration would possess a well ordered\nground state. In two dimensional models, like the $O(N)$-sigma models\nthe MW theorem guarantees the absence of such a state in \nthe thermodynamic limit (TL) however small coupling constant is \\cite{MW}.\nThen, the usual argument in support of the PT is that locally\nthe system is ordered and the PT is not supposed to be used for the calculation\nof long-distance observables. On the other hand it should reproduce the correct\nbehaviour of the fixed-distance correlations as well as\nall thermodynamical functions which can be expressed via\nshort-range observables. The example of $1D$ models shows that\neven this is not always true \\cite{rossi}, \nso why should one believe in correctness of the conventional PT in $2D$?\nIn fact, the only way to justify the PT is to\nprove that it gives the correct asymptotic expansion of \nnonperturbatively defined models in the TL.\nNow, it was shown in \\cite{superinst}\nthat PT results in $2D$ nonabelian models depend on the boundary\nconditions (BC) used to reach the TL.\nThis result could potentially imply that the low-temperature limit and\nthe TL do not commute in nonabelian models. Actually, the main \nargumentation of \\cite{superinst} regarding the failure\nof the PT expansion is based\non the fact that the conventional PT is an expansion around\na broken vacuum, i.e. the state which simply does not exist in\nthe TL of $2D$ models. According to \\cite{superinst},\nthe ground state of these systems can be described\nthrough special configurations -- the so-called gas of\nsuperinstantons (SI) and the correct expansion should take into account\nthese saddle points. \nAt the present stage it is rather unclear how one could\nconstruct an expansion in the SI background. Fortunately, there exists\nother, more eligible way to construct the low-temperature expansion\nwhich respects the MW theorem and is apriori not the expansion around \nthe broken vacuum. We develop this method on an example of\nthe $2D$ $SU(N)\\times SU(N)$ principal chiral model\nwhose partition function (PF) is given by\n\\begin{equation}\nZ = \\int\\prod_xDU_x\n\\exp \\left [\\beta \\sum_{x,n} {\\mbox {Re Tr}} U_xU_{x+n}^+\\right] ,\n\\label{pfsun}\n\\end{equation}\n\\noindent\nwhere $U_x\\in SU(N)$, $DU_x$ is the invariant measure and we\nimpose the periodic BC. The basic idea is the following.\nAs was rigorously proven, the conventional PT \ngives an asymptotic expansion which is uniform\nin the volume for the abelian $XY$ model \\cite{XYPT}. \nOne of the basic theorems which underlies the proof states \nthat the following inequality holds in the $3D$ $XY$ model\n\\begin{equation}\n< \\exp (\\sqrt{\\beta}A(\\phi_x)) > \\ \\leq \\ C \\ ,\n\\label{ineq3D}\n\\end{equation}\n\\noindent\nwhere $C$ is $\\beta$-independent and $A(\\phi +2\\pi)=\\phi$. $\\phi_x$\nis an angle parametrizing the action of the $XY$ model\n$S=\\sum_{x,n}\\cos (\\phi_x - \\phi_{x+n})$. It follows that at large\n$\\beta$ the Gibbs measure is concentrated around $\\phi_x\\approx 0$ \nproviding a possibility to construct an expansion around $\\phi_x=0$.\nThis inequality is not true in $2D$ in the thermodynamic limit\nbecause of the MW theorem, however the authors of \\cite{XYPT} prove \nthe same inequality for the link angle, i.e.\n\\begin{equation}\n< \\exp (\\sqrt{\\beta}A(\\phi_l)) > \\ \\leq \\ C \\ , \\ \n\\phi_l=\\phi_x-\\phi_{x+n} \\ ,\n\\label{ineq2D}\n\\end{equation}\n\\noindent\nwhere the expectation value refers to infinite volume limit.\nThus, in $2D$ the Gibbs measure at large $\\beta$ is concentrated\naround $\\phi_l\\approx 0$ and the asymptotic series can be\nconstructed expanding the action in powers of $\\phi_l$. In the abelian\ncase such an expansion is equivalent to the expansion around $\\phi_x=0$\nbecause i) the action depends only on the difference $\\phi_x - \\phi_{x+n}$\nand ii) the integration measure is flat, $DU_x=d\\phi_x$. \n\nIn $2D$ nonabelian models, again because of the MW theorem, one has to expect \nin the TL something alike to (\\ref{ineq2D}), namely\n\\begin{equation}\n< \\exp (\\sqrt{\\beta}{\\mbox {arg}}A({\\mbox {Tr}}V_l)) >\n\\ \\leq \\ C \\ , \\ V_l=U_xU_{x+n}^+ \\ .\n\\label{nabineq2D}\n\\end{equation}\n\\noindent\nDespite there is not a rigorous proof of (\\ref{nabineq2D}), \nthat such (or similar) inequality holds in $2D$ nonabelian models \nis intuitively clear and should follow from the chessboard \nestimates \\cite{chest} and from the Dobrushin-Shlosman proof of the MW theorem\n\\cite{MW} which shows that spin configurations are distributed uniformly\nin the group space in the TL. Namely, the probability $p(\\xi )$ \nthat ${\\mbox {Tr}}(V_l-I)\\leq -\\xi$ is bounded by \n\\begin{equation}\np(\\xi ) \\leq O(e^{-b\\beta\\xi}) \\ , \\ \\beta\\to\\infty \\ ,\n\\label{chestlink}\n\\end{equation}\n\\noindent\nif the volume is sufficiently large, $b$ is a constant. \nThus, until $\\xi\\leq O((\\sqrt{\\beta})^{-1})$ is not satisfied,\nall configurations are exponentially suppressed.\nThis is equivalent to the statement that the Gibbs \nmeasure at large $\\beta$ is concentrated around $V_l\\approx I$, therefore\n(\\ref{nabineq2D}) or its analog holds. \nIn what follows it is assumed that (\\ref{nabineq2D}) is correct, \nhence $V_l=I$ is the only saddle point for the invariant \nintegrals\\footnote{It follows already from (\\ref{chestlink}).\nWhat is important in (\\ref{nabineq2D}) is a factor $\\sqrt{\\beta}$,\notherwise the very possibility of the expansion in $1\/\\beta$\nbecomes problematic.}. \nThus, the correct asymptotic expansion, if exists,\nshould be given via an expansion around $V_l=I$, similarly to the\nabelian model. If the conventional PT gives the correct asymptotics, it\nmust reproduce the series obtained expanding around $V_l=I$.\nHowever, neither i) nor ii) holds in the nonabelian models, therefore\nit is far from obvious that two expansions indeed coincide.\nLet us parametrize $V_l=\\exp (i\\omega_l)$ and $U_x=\\exp (i\\omega_x)$.\nConsider the following expansion\n\\begin{equation}\nV_l = \\exp \\left [i\\omega_l \\right ] \\sim \nI + \\sum_{n=1}\\frac{1}{(\\beta)^{n\/2}}\\frac{(i\\omega_l)^n}{n!} \\ .\n\\label{ltexp}\n\\end{equation}\n\\noindent\nThe standard PT states that to calculate the asymptotic expansion\none has to re-expand this series as\n\\begin{equation}\n\\omega_l = \\omega_x-\\omega_{x+n}+ \\sum_{k=1}\n\\frac{1}{(\\beta)^{k\/2}}\\omega_l^{(k)} \\ ,\n\\label{ltosres}\n\\end{equation}\n\\noindent\nwhere $\\omega_l^{(k)}$ are to be calculated from the definition\n$U_xU_{x+n}^+=\\exp (i\\omega_l)$.\nThis is presumably true in a finite volume where one can fix\nappropriate BC (like the Dirichlet ones), or to break down \nthe global symmetry by fixing the global gauge \n(on the periodic lattice). Then, making $\\beta$ sufficiently large\none forces all the spin matrices to fluctuate around $U_x\\approx I$,\ntherefore the substitution (\\ref{ltosres}) is justified.\nWe do not see how this procedure could be justified when\nthe volume increases and fluctuations of $U_x$ spread up\nover the whole group space. In other words, it is not clear why\nin the series\n$$\n\\omega_l = \\omega_x-\\omega_{x+n}+ \\sum_{k=1}\\omega_l^{(k)} \\ \n$$\nthe term $\\omega_l^{(k+1)}$ is suppressed as $\\beta^{-1\/2}$\nrelatively to the term  $\\omega_l^{(k)}$.\nIt is only (\\ref{ltexp}) which remains correct\nin the large volume limit and takes into account \nall the fluctuations contributing at a given order of \nthe low-temperature expansion.\n\nIt is a purpose of the present paper to develop\nan expansion around $V_l=I$ aiming to calculate the asymptotic\nseries for nonabelian models.\nFirst of all, one has to give a precise mathematical meaning to\nthe expansion (\\ref{ltexp}). It is done in the next section. \n\n\\section{Link representation for the partition and correlation functions}\n\nTo construct an expansion of the Gibbs measure \nand the correlation functions using (\\ref{ltexp}) \nwe use the so-called link representation for the partition and \ncorrelation functions. First, we make a change of variables\n$V_l=U_xU_{x+n}^+$ in (\\ref{pfsun}). PF becomes\n\\begin{equation}\nZ = \\int \\prod_l dV_l\n\\exp \\left[ \\beta \\sum_l {\\mbox {Re Tr}} V_l + \\ln J(V) \\right] \\ ,\n\\label{lPF}\n\\end{equation}\n\\noindent\nwhere the Jacobian $J(V)$ is given by \\cite{linkrepr}\n\\begin{equation}\nJ(V) = \\int \\prod_xdU_x\\prod_l\n\\left[ \\sum_r d_r \\chi_r \\left( V^+_lU_xU^+_{x+n} \\right) \\right]\n= \\prod_p \\left[ \\sum_r d_r \\chi_r \\left( \\prod_{l\\in p}V_l \\right) \\right].\n\\label{jacob}\n\\end{equation}\n\\noindent\n$\\prod_p$ is a product over all plaquettes of $2D$ lattice,\nthe sum over $r$ is sum over all representations of $SU(N)$, \n$d_r=\\chi_r(I)$ is the dimension of $r$-th representation. \nThe $SU(N)$ character $\\chi_r$ depends on a product of the link \nmatrices $V_l$ along a closed path (plaquette in our case):\n\\begin{equation}\n\\prod_{l\\in p}V_l = V_n(x)V_m(x+n)V_n^+(x+m)V_m^+(x) \\ .\n\\label{prod}\n\\end{equation}\n\\noindent\nThe expression $\\sum_r d_r \\chi _r(\\prod_{l\\in p}V_l)$ is the\n$SU(N)$ delta-function which reflects the fact that the product of\n$U_xU^+_{x+n}$ around plaquette equals $I$ (original model has\n$L^2$ degrees of freedom, $L^2$ is a number of sites; since a number\nof links on the $2D$ periodic lattice is $2L^2$, the Jacobian must\ngenerate $L^2$ constraints)\\footnote{Strictly speaking, on the periodic\nlattice one has to constraint two holonomy operators, i.e. closed\npaths winding around the whole lattice. We do not expect such global \nconstraints to influence the TL in $2D$ (see Discussion).}. \nThe solution of the constraint\n\\begin{equation}\n\\prod_{l\\in p}V_l = I\n\\label{constr}\n\\end{equation}\n\\noindent\nis a pure gauge $V_l=U_xU_{x+n}^+$, so that two forms of the PF\nare exactly equivalent.\n\nThe corresponding representation for the abelian $XY$ model reads\n\\begin{equation}\nZ_{XY} = \\int\\prod_ld\\phi_l\n\\exp \\left [\\beta \\sum_l\\cos\\phi_l \\right ]\\prod_pJ_p \\ ,\n\\label{PFLxy}\n\\end{equation}\n\\noindent\nwhere the Jacobian is given by the periodic delta-function\n\\begin{equation}\nJ_p = \\sum_{r=-\\infty}^{\\infty} e^{ir\\phi_p} \\ , \\\n\\phi_p=\\phi_n(x)+\\phi_m(x+n)-\\phi_n(x+m)-\\phi_m(x+n) \\ .\n\\label{PFLxyJ}\n\\end{equation}\n\\noindent\n\nConsider two-point correlation function\n\\begin{equation}\n\\Gamma (x,y) = < {\\mbox {Tr}} \\ U_xU_y^+ > \\ ,\n\\label{corf}\n\\end{equation}\n\\noindent\nwhere the expectation value refers to the ensemble defined in (\\ref{pfsun}). \nLet $C_{xy}$ be some path connecting points $x$ and $y$. Inserting \nthe unity $U_zU_z^+$ in every site $z\\in C_{xy}$ one gets\n\\begin{equation}\n\\Gamma (x,y) = < {\\mbox {Tr}} \\prod_{l\\in C_{xy}} (U_xU_{x+n}^+) > =\n< {\\mbox {Tr}} \\prod_{l\\in C_{xy}} W_l > \\ ,\n\\label{corf1}\n\\end{equation}\n\\noindent\nwhere $W_l = V_l$ if along the path $C_{xy}$ the link $l$ goes in\nthe positive direction and $W_l = V_l^+$, otherwise.  \nThe expectation value in (\\ref{corf1}) refers now to the ensemble\ndefined in (\\ref{lPF}). Obviously, it does not depend on the path $C_{xy}$\nwhich can be deformed for example to the shortest path between sites $x$ \nand $y$.\n\nIn this representation the series (\\ref{ltexp}) acquires a well\ndefined meaning, therefore the expansion of the action,\nof the invariant measure, etc. can be done.\n\n\\section{$XY$ model: Weak coupling expansion of the free energy}\n\n\nIn this section we prove that for the abelian XY model \nthe large-$\\beta$ expansion in the link\nrepresentation gives the same results as the conventional PT \nin the thermodynamic limit. We consider only the free energy but \nthe generalization for the correlation functions is straightforward.\n\nThe first step is a standard one, i.e. we rescale \n$\\phi\\to\\frac{\\phi}{\\sqrt{\\beta}}$ and make an expansion \n\\begin{equation}\n\\exp \\left[ \\beta\\cos\\frac{\\phi}{\\sqrt{\\beta}} \\right] = \\exp \n\\left[ \\beta -\\frac{1}{2}(\\phi)^2 \\right] \n\\left[ 1 + \\sum_{k=1}^{\\infty} (\\beta)^{-k} \n\\sum_{l_1,..,l_k}\\frac{a_1^{l_1}...a_k^{l_k}}{l_1!...l_k!} \\right] \\ ,\n\\label{cos}\n\\end{equation}\n\\noindent  \nwhere $l_1+2l_2+...+kl_k=k$ and\n\\begin{equation}\na_k = (-1)^{k+1}\\frac{\\phi^{2(k+1)}}{(2k+2)!} \\ .\n\\label{a_k}\n\\end{equation}\n\\noindent\nIn addition to this perturbation one has to extend the integration region\nto infinity. We do not treat this second perturbation, as usually supposing \nthat all the corrections from this perturbation go down exponentially with \n$\\beta$ (in the abelian case it can be proven rigorously \\cite{XYPT}). \nIt is more convenient now to go to a dual lattice identifying \nplaquettes of the original lattice with its center, i.e. $p\\to x$.\nLet $l=(x;n)$ be a link on the dual lattice.\nIntroducing sources $h_l$ for the link field, one then finds \n\\begin{equation}\nZ_{XY}(\\beta >> 1) = e^{\\beta DL^2 - L^2\\ln \\beta}\n\\prod_{l} \\left[ 1+\\sum_{k=1}^{\\infty}\\frac{1}{\\beta^k}\nA_k\\left( \\frac{\\partial^2}{\\partial h_l^2} \\right) \\right] M(h_l) \\ .\n\\label{asxyGll}\n\\end{equation}\n\\noindent\nCoefficients $A_k$ are defined in (\\ref{cos}) and (\\ref{a_k}).\nThe generating functional $M(h_l)$ is given by\n\\begin{eqnarray}\nM(h_l) = \\sum_{r_x=-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\n\\prod_ld\\phi_l \\exp \n\\left[-\\frac{1}{2}\\sum_l \\phi_l^2 \n+ i\\sum_l \\frac{\\phi_l}{\\sqrt{\\beta}} (r_x-r_{x+n}) \n+\\sum_l\\phi_l h_l  \\right] .\n\\label{Gfunxy}\n\\end{eqnarray}\n\\noindent\nSums over representations are treated by the Poisson resummation formula.\nThe Gaussian ensemble appears in terms of the fluctuations of $r$-fields.\nThe integral over zero mode of $r$-field is not Gaussian and leads to \na delta-function in the Poisson formula. Thus, the zero mode \ndecouples from the expansion.\nAll the corrections to the integrals over representations in the\nPoisson formula fall down exponentially with $\\beta$ so that \nthe generating functional becomes\n\\begin{equation}\nM(h_l) = \\exp \n\\left[ \\frac{1}{4} \\sum_{l,l^{\\prime}}h_lG_{ll^{\\prime}}h_{l^{\\prime}} \\right] \\ ,\n\\label{Gfunxy1}\n\\end{equation}\n\\noindent\nwhere we have introduced the following function which we term \n``link'' Green function \n\\begin{equation}\nG_{ll^{\\prime}} = 2\\delta_{l,l^{\\prime}} - G_{x,x^{\\prime}} -\nG_{x+n,x^{\\prime}+n^{\\prime}} + G_{x,x^{\\prime}+n^{\\prime}} + G_{x+n,x^{\\prime}} \\ .\n\\label{Gll1}\n\\end{equation}\n\\noindent\n$G_{x,x^{\\prime}}$ is a ``standard'' Green function on the periodic lattice\n\\begin{equation}\nG_{x,x^{\\prime}} = \\frac{1}{L^2} \\sum_{k_n=0}^{L-1}\n\\frac{e^{\\frac{2\\pi i}{L}k_n(x-x^{\\prime})_n}}\n{D-\\sum_{n=1}^D\\cos \\frac{2\\pi}{L}k_n} \\ , \\ k_n^2\\ne 0 \\ .\n\\label{Gxx}\n\\end{equation}\n\\noindent\nNormalization is such that $G_{ll}=1$.\nAs far as we could check, the expression (\\ref{asxyGll}) reproduces  \nthe well known asymptotics of the free energy of the $XY$ model \nin two dimensions. For example, the first order coefficient of the \nfree energy being expressed in $G_{ll^{\\prime}}$ reads\n\\begin{equation}\nC_1 = \\frac{1}{64 L^2} \\sum_{l} G^2_{ll} = \\frac{1}{32} \\ .\n\\label{C1Gll}\n\\end{equation}\n\\noindent\nLet us add some comments. In the standard expansion to avoid the zero mode problem\none has to fix appropriate BC, like Dirichlet ones or to fix a global gauge\nif one works on the periodic lattice \\cite{unitgauge}. In the present scheme\nthe zero mode decouples automatically due to using $U(1)$ delta-function\nwhich takes into account the periodicity of the integrand in link angles.\nMore important observation is that it is allowed\nto take the TL already in the formula (\\ref{asxyGll}):\nsince the generating functional\ndepends only on the link Green function which is infrared finite, \nthe uniformity of the expansion in the volume follows immediately.  \nThis is a direct consequence of the fact that the Gibbs measure of \nthe $XY$ model is a function of the gradient $\\phi_l$ only. \n\n\n\\section{Weak coupling expansion in the $SU(2)$ model}\n\nWe turn now to the nonabelian models. As the simplest example we analyze\nthe $SU(2)$ principal chiral model. The method developed here \nhas straightforward generalization to arbitrary $SU(N)$ or $SO(N)$ model \nand we shall present it elsewhere. \n\n\\subsection{Representation for the partition function}\n\nWe take the standard form for the $SU(2)$ link matrix which is the most suitable\nfor the weak coupling expansion\n\\begin{equation}\nV_l = \\exp [i\\sigma^k\\omega_k(l)] \\ ,\n\\label{su2mtr}\n\\end{equation}\n\\noindent\nwhere $\\sigma^k, k=1,2,3$ are Pauli matrices.\nLet us define\n\\begin{equation}\nW_l = \\left[ \\sum_k\\omega^2_k(l) \\right]^{1\/2}\n\\label{Wl}\n\\end{equation}\n\\noindent\nand similarly\n\\begin{equation}\nW_p = \\left[ \\sum_k\\omega^2_k(p) \\right]^{1\/2} ,\n\\label{Wp}\n\\end{equation}\n\\noindent\nwhere $\\omega_k(p)$ is a plaquette angle defined as\n\\begin{equation}\nV_p = \\prod_{l\\in p}V_l =  \\exp \\left [ i\\sigma^k\\omega_k(p) \\right ] \\ .\n\\label{plangle}\n\\end{equation}\n\\noindent\nThe exact relation between link angles $\\omega_k(l)$ and the plaquette\nangle $\\omega_k(p)$ is given in the Appendix B. Then, the partition function\n(\\ref{lPF}) can be exactly rewritten to the following form appropriate\nfor the weak coupling expansion\n\\begin{eqnarray}\nZ = \\int \\prod_l \\left[ \\frac{\\sin^2W_l}{W^2_l}\\prod_kd\\omega_k(l) \\right]\n\\exp \\left[ 2\\beta\\sum_l\\cos W_l \\right] \\prod_x \\frac{W_x}{\\sin W_x} \n\\nonumber     \\\\\n\\prod_x \\sum_{m(x)=-\\infty}^{\\infty}\\int\\prod_kd\\alpha_k(x)\n\\exp \\left[ -i\\sum_k\\alpha_k(x)\\omega_k(x) + 2\\pi im(x)\\alpha (x) \\right] \\ ,\n\\label{PFwk}\n\\end{eqnarray}\n\\noindent\nwhere we have introduced auxiliary field $\\alpha_k(x)$ and\n\\begin{equation}\n\\alpha (x) = \\left[ \\sum_k\\alpha^2_k(x) \\right]^{1\/2} .\n\\label{Ax}\n\\end{equation}\n\\noindent\nThe representation for the Jacobian (\\ref{D14}) used here can be \ninterpreted as an analog of the $U(1)$ periodic delta-function \n(\\ref{PFLxyJ}): this is the $SU(2)$ delta-function which is periodic\nwith respect to a length of the vector $\\vec{\\omega}(x)$.\nWe give a derivation of the partition function (\\ref{PFwk})\nin the Appendix A. In rewriting the final formula of that derivation\n(\\ref{D19}), we went over to the dual lattice identifying \nthe links of the original lattice with dual links and the original\nplaquettes with dual sites located in the center of the original\nplaquettes. \n\n\n\\subsection{General expansion}\n\nTo perform the weak coupling expansion \nwe proceed in a standard way, i.e. first we make the substitution\n\\begin{equation}\n\\omega_k(l)\\to (2\\beta)^{-1\/2}\\omega_k(l) \\ , \\\n\\alpha_k(x)\\to (2\\beta)^{1\/2}\\alpha_k(x) \n\\label{subst}\n\\end{equation}\n\\noindent\nand then expand the integrand of (\\ref{PFwk}) \nin powers of fluctuations of the link fields.\nWe would like to give here some technical details of the expansion\nwhich could be useful for a future use. \nIt is straightforward to get the following power series:\n\\begin{enumerate}\n\n\\item Action\n\n\\begin{equation}\n\\exp \\left[ 2\\beta\\cos\\frac{W_l}{\\sqrt{2\\beta}} \\right] = \\exp \n\\left[ 2\\beta -\\frac{1}{2}(W_l)^2 \\right] \n\\left[ 1 + \\sum_{k=1}^{\\infty} (2\\beta )^{-k} \n\\sum_{l_1,..,l_k} \\frac{a_1^{l_1}...a_k^{l_k}}{l_1!...l_k!} \\right] \\ ,\n\\label{actexp}\n\\end{equation}\n\\noindent  \nwhere $l_1+2l_2+...+kl_k=k$ and\n\\begin{equation}\na_k = (-1)^{k+1}\\frac{W_l^{2(k+1)}}{(2k+2)!} \\ .\n\\label{a_kW}\n\\end{equation}\n\\noindent\n\n\\item Invariant measure\n\n\\begin{equation}\n\\frac{\\sin^2W_l}{W^2_l} = 1 +\n\\sum_{k=1}^{\\infty}\\frac{(-1)^k}{(2\\beta )^k}C_kW_l^{2k} \\ , \\ \nC_k = \\sum_{n=0}^k \\frac{1}{(2n+1)!(2k-2n+1)!} \\ .\n\\label{invMexp}\n\\end{equation}\n\\noindent\n\n\\item Contribution from Jacobian I\n\n\\begin{equation}\n\\frac{W_x}{\\sin W_x} = \n1 + \\sum_{k=1}^{\\infty}\\frac{J_k}{(2\\beta )^k}W_x^{2k} \\ , \\\nJ_k = 2\\frac{2^{2k-1}}{(2k)!}\\mid B_{2k} \\mid \\ ,\n\\label{J1exp}\n\\end{equation}\n\\noindent\nwhere $B_{2k}$ are Bernoulli numbers.\n\n\\item Contribution from Jacobian II\n\n\\begin{eqnarray}\n\\alpha_k(x)\\omega_k(x) = \\alpha_k(x)\\left[ \\omega^{(0)}_k(x) +\n\\sum_{n=1}^{\\infty} \\frac{\\omega^{(n)}_k(x)}{(2\\beta )^{n\/2}} \\right] \n\\ , \\\\ \\nonumber\n\\exp \\left[ -i\\sum_k\\alpha_k(x)\\omega_k(x) \\right] = \n\\exp \\left[ -i\\sum_k\\alpha_k(x)\\omega^{(0)}_k(x) \\right] \\\\  \\nonumber\n\\left[ 1 + \\sum_{q=1}^{\\infty}\\frac{(-i)^q}{q!} \\left( \\sum_k\\alpha_k(x) \n\\sum_{n=1}^{\\infty} \\frac{\\omega^{(n)}_k(x)}{(2\\beta )^{n\/2}} \\right)^q \\right] \\ .\n\\label{J2exp}\n\\end{eqnarray}\n\\noindent\n\n\\end{enumerate}\nUsing the relations (\\ref{D9})-(\\ref{D12}) one can calculate \n$\\omega^{(n)}_k(x)$ up to an arbitrary order in $n$. In particular, \n$\\omega^{(0)}_k(x)$ is given by (see Fig.1 for our notations of\ndual links)\n\\begin{equation}\n\\omega^{(0)}_k(x) = \\omega_k(l_3) + \\omega_k(l_4) - \n\\omega_k(l_1) - \\omega_k(l_2) \\ .\n\\label{w0}\n\\end{equation}\n\\noindent\nWe shall use an obvious property\n\\begin{equation}\n\\sum_x\\alpha_k(x)\\omega^{(0)}_k(x) = \n\\sum_l\\omega_k(l) \\left [\\alpha_k(x+n) - \\alpha_k(x) \\right ] \\ , \\ l=(x;n) \\ .\n\\label{resum}\n\\end{equation}\n\\noindent\nIntroducing now the external sources $h_k(l)$ coupled to the link field \n$\\omega_k(l)$ and $s_k(x)$ coupled to the auxiliary field $\\alpha_k(x)$\nand adjusting the definitions\n\\begin{equation}\n\\omega_k(l) \\to \\frac{\\partial}{\\partial h_k(l)} \\ , \\ \n\\alpha_k(x) \\to \\frac{\\partial}{\\partial s_k(x)} \\ ,\n\\label{deriv}\n\\end{equation}\n\\noindent\nwe get finally the following formal weak coupling expansion \nfor the PF (\\ref{PFwk})\n\\begin{eqnarray}\nZ = C(2\\beta ) Z(0,0) \\prod_l\\left[ \\left( 1 + \\sum_{k=1}^{\\infty} (2\\beta )^{-k} \n\\sum_{l_1,..,l_k}^{\\prime}\\frac{a_1^{l_1}...a_k^{l_k}}{l_1!...l_k!} \\right) \n\\left( 1 + \\sum_{k=1}^{\\infty}\\frac{(-1)^k}{(2\\beta )^k}C_kW_l^{2k} \n\\right) \\right]  \\\\  \\nonumber\n\\prod_x \\left[ \\left(1 + \\sum_{k=1}^{\\infty}\\frac{J_k}{(2\\beta )^k}W_x^{2k} \\right) \n\\left( 1 + \\sum_{q=1}^{\\infty}\\frac{(-i)^q}{q!} \\left( \\sum_k\\alpha_k(x) \n\\sum_{n=1}^{\\infty} \\frac{\\omega^{(n)}_k(x)}{(2\\beta )^{n\/2}} \\right)^q \\right) \\right]\n\\ M(h,s) \\ ,\n\\label{PFwkexp}\n\\end{eqnarray}\n\\noindent\nwhere\n\\begin{equation}\nC(\\beta ) = \\exp\\left[ 2\\beta L^2 - \\frac{3}{2}L^2\\ln\\beta \\right] \\ .\n\\label{preexp}\n\\end{equation}\n\\noindent\nAs usually, one has to put $h_k=s_k=0$ after taking all the derivatives. \n$M(h,s)$ is a generating functional which we study in the next subsection.\nIt is obvious that the ground state satisfies\n\\begin{equation}\n<(\\omega^{(0)}_k(x))^p> = 0  \\ , \\  p=1,2,...  \\ ,\n\\label{mainst}\n\\end{equation}\n\\noindent\nprecisely like the abelian model.\n\n\n\\subsection{Generating functional and zero modes}\n\nHere we are going to study the generating functional $M(h,s)$ given by\n\\begin{equation}\nM(h,s) = \\frac{Z(h,s)}{Z(0,0)} \\ ,\n\\label{GF}\n\\end{equation}\n\\noindent\nand\n\\begin{eqnarray}\nZ(h,s) = \\int_{-\\infty}^{\\infty} \\prod_{x,k} d\\alpha_k(x)\n\\int_{-\\infty}^{\\infty} \\prod_{l,k} d\\omega_k(l) \n\\exp \\left[ -\\frac{1}{2}\\omega^2_k(l) \n-i\\omega_k(l)[\\alpha_k(x+n) - \\alpha_k(x) ] \\right]  \\nonumber  \\\\   \n\\sum_{m(x)=-\\infty}^{\\infty}\n\\exp \\left[ 2\\pi i\\sqrt{2\\beta}\\sum_xm(x)\\alpha (x) +\n\\sum_{l,k}\\omega_k(l)h_k(l) + \\sum_{x,k}\\alpha_k(x)s_k(x) \\right] .\n\\label{GF1}\n\\end{eqnarray}\n\\noindent\nAs in the abelian case we expect that integrals over zero modes\nof the auxiliary field are not Gaussian and should lead to some constraint\non the sums over $m_x$. To see this, we put $h_k=s_k=0$ and integrate out\nthe link fields. Partition function becomes\n\\begin{equation}\nZ(0,0) =\\sum_{m(x)=-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \\prod_{x,k} d\\alpha_k(x)\n\\exp \\left[ -\\alpha_k(x)G_{x,x^{\\prime}}^{-1}\\alpha_k(x^{\\prime})\n+ 2\\pi i\\sqrt{2\\beta}m(x)\\alpha (x) \\right] \\ , \n\\label{GF00}\n\\end{equation}\n\\noindent\nwith $G_{x,x^{\\prime}}$ given in (\\ref{Gxx}) and sum over repeating indices is\nunderstood here and in what follows. \nIt is clear that in this case the zero modes should be controlled via integration \nover the radial component of the vector $\\vec{\\alpha}(x)$. To see this, we change\nto the spherical coordinates and treat only a constant mode in the angle variables\n(since the zero mode problem could only arise from this configuration). One has\n\\begin{eqnarray}\nZ(0,0) \\sim \\sum_{m_x=-\\infty}^{\\infty} \\int_0^{\\infty} \n\\prod_x \\alpha^2_x d\\alpha_x\n\\exp \\left[ -\\alpha_xG_{x,x^{\\prime}}^{-1}\\alpha_{x^{\\prime}}\n+ 2\\pi i\\sqrt{2\\beta}m_x\\alpha_x \\right]   \\nonumber  \\\\\n\\sim \\sum_{m_x=-\\infty}^{\\infty} \\delta \\left( \\sum_xm_x \\right) \n\\exp \\left[ -2\\pi^2\\beta \\sum_{x,x^{\\prime}}m_x\nG_{x,x^{\\prime}} m_{x^{\\prime}}\\right] + O(m_x^2) \\ ,\n\\label{GF0mod}\n\\end{eqnarray}\n\\noindent\nand we used the notation $\\alpha_x$ for the radial component of the vector \n$\\vec{\\alpha}(x)$. Since the zero mode of the radial component in the $x$\nspace is\n$$\n\\alpha (p=0) = \\left( \\frac{1}{L^D}\\sum_k(\\sum_x\\alpha_k(x))^2 \\right)^{1\/2}\n$$ \none has to omit the zero mode from the Green function in each term \nof the sum over $k$ in the integrand of (\\ref{GF00}).\nAs in the abelian case the integration over zero modes produces \ndelta-function in (\\ref{GF0mod}). Since however only $m_x=0$ for all $x$\ncontribute to the asymptotics of the free energy and fixed distance correlation\nfunction (other values of $m_x$ being exponentially suppressed) \nwe have to put $m_x=0$ omitting at the same time all zero modes. \nCalculating resulting Gaussian integrals we come to\n\\begin{equation}\nM(h,s) = \\exp \n\\left[ \\frac{1}{4}s_k(x)G_{x,x^{\\prime}}s_k(x^{\\prime}) +\n\\frac{i}{2}s_k(x)D_l(x)h_k(l) +\n\\frac{1}{4}h_k(l)G_{ll^{\\prime}}h_k(l^{\\prime}) \\right] \\ ,\n\\label{GFfin}\n\\end{equation}\n\\noindent\nwhere $G_{ll^{\\prime}}$ was introduced in (\\ref{Gll1}) and\n\\begin{equation}\nD_l(x^{\\prime}) = G_{x,x^{\\prime}} - G_{x+n,x^{\\prime}} \\ , \\ l=(x,n) \\ .\n\\label{Dxl}\n\\end{equation}\n\\noindent\nFrom (\\ref{GFfin}) one can deduce the following simple rules\n\\begin{eqnarray}\n<\\omega_k(l)\\omega_n(l^{\\prime})> =\n\\frac{\\delta_{kn}}{2}G_{ll^{\\prime}} \\ , \\ \n<\\alpha_k(x)\\alpha_n(x^{\\prime})> =\n\\frac{\\delta_{kn}}{2}G_{xx^{\\prime}} \\ ,  \\nonumber  \\\\ \n- i<\\omega_k(l)\\alpha_n(x^{\\prime})> =\n\\frac{\\delta_{kn}}{2}D_l(x^{\\prime}) \\ .\n\\label{Frules}\n\\end{eqnarray}\n\\noindent\nWe describe some simple properties of the functions $G_{ll^{\\prime}}$\nand $D_l(x^{\\prime})$ in the Appendix C.\nThe expansion (\\ref{PFwkexp}), representation (\\ref{GFfin}) for the \ngenerating functional and rules (\\ref{Frules}) \nare main formulas of this section which allow \nto calculate the weak coupling expansion of both the free energy and any\nshort-distance observable. Let us now comment on the infrared\nfinitness of the expansion.\nIt follows from the representation for the generating functional\nthat all expectation values of the link fields are expressed only via\nthe link Green functions $G_{ll^{\\prime}}$ and $D_l(x)$ which are infrared finite\nby construction. All combinations of auxiliary fields which contain\nodd overall powers of the fields are expressed only via $D_l(x)$\nand, therefore are also infrared finite. However, even powers\ninclude $G_{x,x^{\\prime}}$ and the infrared finitness is not provided\nautomatically. In particular, it means that unlike $XY$ \nmodel we are not allowed to take the TL at this stage.\n\nOur last comment concerns the partition function (\\ref{GF00}). We believe it \ncan be regarded as an analog of the corresponding expression in the $XY$ model,\ni.e. this is a nonabelian analog of the so-called ``spin-wave--vortex'' representation\nfor the partition function. One can see that the nonabelian model\nis not factorized into three abelian components and is periodic in\nthe length of the vector $\\vec{\\alpha}(x)$ rather than in \nits components $\\alpha_k(x)$. \n\n\\subsection{First order coefficient of the correlation function}\n\nAs the simplest example we would like to calculate the first\norder coefficient of the correlation function (\\ref{corf1}).\nExpanding (\\ref{corf1}) in $1\/\\beta$ one has\n\\begin{equation}\n\\Gamma (x,y) = 1 - \\frac{1}{4\\beta}\n< \\sum_{k=1}^3\\left (\\sum_l\\omega_k(l) \\right )^2 > + O(\\beta^{-2}) = \n1 - \\frac{3}{8\\beta}\n\\sum_{l,l^{\\prime}\\in C^d_{xy}} G_{ll^{\\prime}} + O(\\beta^{-2}) \\ ,\n\\label{GLXY1}\n\\end{equation}\n\\noindent\nwhere $C^d_{xy}$ is a path dual to the path $C_{xy}$, i.e consisting\nof the dual links which are orthogonal to the original\nlinks $l,l^{\\prime}\\in C_{xy}$.\nThe form of $G_{ll^{\\prime}}$ ensures independence of $\\Gamma (x,y)$\nof a choice of the path $C_{xy}$. After some algebra it is easy to get\nthe result\n\\begin{equation}\n\\Gamma (x,y) = 1 - \\frac{3}{4\\beta} D(x-y) \\ , \\\nD(x) = \\frac{1}{L^2} \\sum_{k_n=0}^{L-1}\n\\frac{1 - e^{\\frac{2\\pi i}{L}k_n x_n}}\n{D-\\sum_{n=1}^D\\cos \\frac{2\\pi}{L}k_n} \\ , \\ k_n^2\\ne 0 ,\n\\label{Dx}\n\\end{equation}\n\\noindent\nwhich coincides with the result of the conventional PT.\n\n\n\\section{First order coefficient of the free energy}\n\nThe main result of our study is the first order\ncoefficient of the $SU(2)$ free energy \n\\begin{equation}\nF=\\frac{1}{2L^2}\\ln Z = 2\\beta - \\frac{3}{4}\\ln\\beta +\n\\frac{1}{2\\beta L^2}C^1 + O(\\beta^{-2}) \\ .\n\\label{fren}\n\\end{equation}\n\\noindent\nThere are four contributions at this order to $C^1$\n\\begin{equation}\nC^1 = C^1_{ac} + C^1_{meas} + C^1_{J1} +  C^1_{J2} \\ .\n\\label{frensum}\n\\end{equation}\n\\noindent\nContribution from the action (\\ref{actexp}) is given by\n\\begin{equation}\n\\frac{1}{2L^2}C^1_{ac}=\\frac{5}{128L^2}\\sum_lG^2_{ll}=\n\\frac{5}{64} \\ .\n\\label{c1ac}\n\\end{equation}\n\\noindent\nContribution from the measure (\\ref{invMexp}) is given by\n\\begin{equation}\n\\frac{1}{2L^2}C^1_{meas}= - \\frac{1}{8L^2}\\sum_lG_{ll}\n= - \\frac{1}{4} \\ .\n\\label{c1meas}\n\\end{equation}\n\\noindent\nContribution from third brackets in (\\ref{PFwkexp})\nis proportional to $[\\omega_k^{(0)}]^2$ and equals 0,\nbecause of (\\ref{mainst}). There are two contributions\nfrom the expansion of the Jacobian (\\ref{J2exp}). The first one\nis given by the expectation value of the operator\n$-i\\sum_x\\sum_{k=1}^3\\alpha_k(x)\\omega_k^{(2)}(x)$.\n$\\omega_k^{(2)}(x)$ is given in the Appendix B (\\ref{wkx2}). \nFrom the form of the generating functional (\\ref{GFfin}) one can see that\nthe expectation value of this operator depends only on\nlink Green functions $G_{ll^{\\prime}}$ and $D_l(x^{\\prime})$.\nOne gets after long but straightforward algebra\\footnote{Our previous\nversion suffered from incorrect sign in this term which led to \nwrong final result.}\n\\begin{eqnarray}\n\\frac{1}{2L^2}C^1_{J1}=\\frac{1}{4L^2}\\sum_l\\sum_xD_l(x) \\nonumber  \\\\\n(\\ \\frac{1}{2}\\sum_{i=1}^4(\\delta_{ll_i}(G_{l_3l_i}+G_{l_4l_i}-\nG_{l_1l_i}-G_{l_2l_i})+G_{l_il_i}(\\delta_{ll_1}+\\delta_{ll_2}-\n\\delta_{ll_3}-\\delta_{ll_4}))+ \\nonumber  \\\\\n\\delta_{ll_1}(G_{l_3l_4}+2G_{l_3l_2}+2G_{l_4l_2})+ \n\\delta_{ll_2}(G_{l_3l_4}-G_{l_3l_1}-G_{l_4l_1})+ \\nonumber  \\\\\n\\delta_{ll_3}(G_{l_4l_1}+G_{l_4l_2}-G_{l_1l_2})-\n\\delta_{ll_4}(2G_{l_3l_1}+2G_{l_3l_2}+G_{l_1l_2}) \\ ) \\ ,\n\\label{c1J1Gr}\n\\end{eqnarray}\n\\noindent\nwhere links $l_i$ are defined in Appendix C (see Fig.1), $l=(x,n)$.\nIn terms of standard $D$-functions defined in Appendix C the result reads\n\\begin{equation}\n\\frac{1}{2L^2}C^1_{J1}=\\frac{1}{4}[6-2D(2,0)-D(1,1)]=\n\\frac{1}{2}+\\frac{3}{2\\pi} \\ .\n\\label{c1J1res}\n\\end{equation}\n\\noindent\nThe second term is given by the operator\n$< \\frac{1}{2}\\left\n(\\sum_x\\sum_{k=1}^3\\alpha_k(x)\\omega_k^{(1)}(x)\\right )^2 >$.\n$\\omega_k^{(1)}(x)$ is given in the Appendix B (\\ref{wkx1}).\nIn terms of Green functions it reads\n\\begin{equation}\n\\frac{1}{2L^2}C^1_{J2} = - \\ \\frac{3}{16} (Q^{(1)} + Q^{(2)}) \\ ,\n\\label{c1J2res}\n\\end{equation}\n\\noindent\n\\begin{equation}\nQ^{(1)}= \\frac{1}{2L^2}\\sum_{x,x^{\\prime}}\n\\sum_{i<j}^4\\sum_{i^{\\prime}<j^{\\prime}}^4\nG_{x,x^{\\prime}} \n(G_{l_i,l^{\\prime}_{i^{\\prime}}}G_{l_j,l^{\\prime}_{j^{\\prime}}} -\nG_{l_i,l^{\\prime}_{j^{\\prime}}}G_{l_j,l^{\\prime}_{i^{\\prime}}}) \\ ,\n\\label{Q1xx}\n\\end{equation}\n\\noindent\n\\begin{eqnarray}\nQ^{(2)}=\\frac{1}{2L^2}\\sum_{x,x^{\\prime}}\n\\sum_{i<j}^4\\sum_{i^{\\prime}<j^{\\prime}}^4\n(G_{l_i,l^{\\prime}_{i^{\\prime}}}D_{l^{\\prime}_{j^{\\prime}}}(x) \nD_{l_{j}}(x^{\\prime}) + \nG_{l_j,l^{\\prime}_{j^{\\prime}}}D_{l^{\\prime}_{i^{\\prime}}}(x) \nD_{l_{i}}(x^{\\prime}) -  \\nonumber  \\\\\nG_{l_i,l^{\\prime}_{j^{\\prime}}}D_{l^{\\prime}_{i^{\\prime}}}(x) \nD_{l_{j}}(x^{\\prime}) - \nG_{l_j,l^{\\prime}_{i^{\\prime}}}D_{l^{\\prime}_{j^{\\prime}}}(x) \nD_{l_{i}}(x^{\\prime})) \\ . \n\\label{Q2xx}\n\\end{eqnarray}\n\\noindent\nLink $l_i$ ($l^{\\prime}_{j^{\\prime}}$) refers to one of four\nlinks attached to a given site $x$ ($x^{\\prime}$).\nWe performed both analytical and numerical studies of last expressions.\nDetails are given in the Appendix C. We find\n\\begin{equation}\nQ^{(1)}= \\frac{1}{2}\n\\label{Q1res}\n\\end{equation}\n\\noindent\nand\n\\begin{equation}\nQ^{(2)}= 1+\\frac{8}{\\pi} \\ .\n\\label{Q2res}\n\\end{equation}\n\\noindent\nCollecting all coefficients we finally obtain \n\\begin{equation}\n\\frac{1}{2L^2}C^1 = \\frac{3}{64} \\ ,\n\\label{C1res}\n\\end{equation}\n\\noindent\nwhich coincides with the result of the conventional PT\n\\cite{rossi}.\n\n\n\\section{Discussion}\n\nIn this paper we propose to use\nan invariant link formulation to investigate some properties \nof $2D$ models in the weak coupling region. \nWe argued that this approach is more suitable for calculation\nof asymptotic expansions of invariant functions in cases\nwhen the Mermin-Wagner theorem forbids spontaneous symmetry\nbreaking in the thermodynamic limit. \nWe have found that both in the abelian $XY$ model and in non-abelian\n$SU(2)$ model our results for the first order coefficients of \nthe free energy and correlation function \nagree with the standard PT expansion in the TL.\nIn our next paper \\cite{corrf} we show that the second\norder coefficient of the correlation function also\nagrees with the conventional PT. \n\nNow we can return to the question raised in the Introduction,\nnamely whether conventional PT gives uniform asymptotic\nexpansion for non-abelian models. It is well known for a long\ntime that it is no so in one-dimensional non-abelian models.\nIn \\cite{corrf} we address this question in the link\nformulation and show that non-uniformity in this case\noriginates from the expansion of holonomy operator\nwhich imposes certain global condition on the configurations\nof link matrices. While this global condition itself \nvanishes in true TL, it survives large volume limit if \nthe low-temperature expansion is performed in a finite volume.\nIn $2D$ there are two such operators which we did not\nconsider in the present paper. The reason being\nthat the contributions from these holonomy operators\nare suppressed as $O(1\/L)$. Since in $2D$ one could \nencounter only logarithmic divergences it is rather\nunlikely that holonomies are relevant for the TL.\nIn this sense there is no similarity between\none and two dimensional models. Nevertheless, \none cannot exclude the possibility of non-uniformity\nof the low-temperature expansion arising from\nthe remainder to the PT series \\cite{QA}.\nThis problem is extremely hard to resolve by means\nof the standard approaches, see for instance \\cite{QUQ}.  \nContrary, in the link formulation we are able to\ncalculate the exponential remainder at a given order\nof the low-temperature expansion. Therefore,\nthe problem of the infrared finitness of the remainder\ncan be addressed explicitly. Such investigations\nare presently in progress.\n\n\n\\vspace{0.5cm} \n\\begin{center}\n{\\bf Acknowledgements.} \n\\end{center}\n\nWe are grateful to J.~Polonyi who found time to go\nthrough many details of the expansion presented here\nand for many encouraging discussions.\nWe would like to thank V.~Miransky and V.~Gusynin for interesting\ndiscussions and healthy critics on different stages of this work.\nOur special thanks to B.~Rusakov for the explanation of his\npaper \\cite{linkrepr} regarding the calculation of the Jacobian $J(V)$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe increasing amounts of data usage by mobile network subscribers imply the need for higher throughputs and higher network capacities. The existing and emerging mobile communication networks, such as 4G long term evolution (LTE)-Advanced and 5G New Radio (NR), are designed to meet these needs and requirements \\cite{DAHLMAN201857}. Carrier aggregation (CA) is one of the key techniques that was introduced in LTE-Advanced to support higher throughput requirements, where multiple component carriers (CCs) at one or multiple LTE bands are aggregated together to form larger transmission bandwidth, and also to facilitate efficient utilization of the available radio spectrum \\cite{3GPP_36850, LTE, Iwamura}. In this work, we particularly focus on the case where the aggregated CCs are at different bands, commonly referred to as inter-band CA.\n\nIn general, modern radio systems employing wideband multicarrier waveforms are vulnerable to practical analog circuit implementation related challenges and imperfections. One of these challenges is the so called passive intermodulation (PIM) that can severely limit the performance of frequency division duplexing (FDD) based systems. Such PIM is typically generated in the passive components of the radio frequency (RF) transceiver front-end, such as duplexer, diplexer, multiplexer or antenna selection switches. Moreover, the nonlinear junctions typically caused by poor RF connection or the presence of dirt over the metal surfaces in the radio components can also generate PIM. As a consequence, unwanted nonlinear distortion products are created due to the intermodulation of the transmit CCs that generally appear at the specific intermodulation (IM) sub-bands. Depending on the used bands and adopted CC center-frequencies, some of these nonlinear PIM products may appear in one or more of the receiver operating bands as illustrated in Fig. 1. Furthermore, since the PIM is generated in the radio transceiver front-end at or after the duplexer filter, it leaks directly into the receiver and may lead to receiver desensitization. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.4]{Fig1.pdf}\n\\caption{{Spectral illustration of the unwanted PIM products with interband CA of Band 1 and Band 3 at mobile device side. In this example, some of the PIM products hit the Band 1 receiver.}}\\label{spectra}\n\\end{figure}\n\nA concrete example case, in terms of exact LTE bands and frequencies, is given in Fig.~\\ref{spectra} illustrating uplink inter-band CA transmission at Band 1 (1920-1980 MHz) and Band 3 (1710-1780 MHz). As shown in the figure, the upper third-order IM sub-band (IM3) falls within the Band 1 downlink, reflecting thus the self-interference problem due to PIM. Other LTE bands that can experience similar problems are, e.g., B3+B8, B2+B4, B5+B7, as discussed and acknowledged also in many inter-band CA related 3GPP technical documents, such as \\cite{3GPP_36101}, \\cite{121131}. In general, the problem of PIM-induced self-interference is not only limited to UE devices but can actually be even more pronounced in the base station (BS) transceiver systems \\cite{1418994}, \\cite{8077999} where, in addition to internal PIM sources, external sources such as metal objects in the antenna near field and reflections from nearby buildings can cause self-interference. Therefore, PIM can be a big concern also for network vendors and operators. \n\nObvious solutions to avoid or reduce the PIM-induced self-interference are to either reduce the transmit signal power or to allow for a degradation in the receiver reference sensitivity level. At the UE side, these approaches are referred to as the maximum power reduction (MPR) and maximum sensitivity degradation (MSD), respectively. However, these approaches impact negatively the UL link budget and throughputs \\cite{Adnan_MTT} and are thus not the most appealing solutions. Alternatively, one could argue to utilize higher quality RF components with good linearity characteristics, however, this may considerably raise the overall radio implementation costs and size.\n\nSome recent works have addressed digital cancellation of PIM. Specifically, Dabag et al. \\cite{Dabag_2} considered third-order PIM cancellation caused by the antenna switch by devising a multiple input single output (MISO) canceller to suppress the frequency-selective PIM with time delay differences between different transmit signals. While showing promising results, the associated parameter estimation complexity is very high. Then, in \\cite{Dabag_1}, digital cancellation of second-order PIM due to a diplexer is pursued. In general, these reference techniques do not take into account the nonlinear behavior of the individual PAs in the transmitter chains and the memory effects of the PAs. Since the transmit CCs are distorted in a nonlinear fashion by each of the individual PAs before entering a PIM nonlinearity, this implies that the self-interference is in fact a combination of two nonlinearities. This has been identified recently in \\cite{zeeshan_SIPS} and \\cite{Zeeshan_MTT_submit}, where different digital cancellers for joint mitigation of PA and PIM nonlinerities are proposed and experimented. \n\nIn this paper, we develop advanced digital cancellation solutions for suppressing the PIM with memory effects while exclude the PA nonlinearities for simplicity. In addition to PA memory, the proposed method can also account for different mutual time delays of the transmit signals before entering the PIM source, while can also account for memory along and after the PIM generation stage. For presentation simplicity, we focus primarily on modeling and digital cancellation of third-order PIM. \nThe performance of the developed method is evaluated through practical RF measurements, adopting commercial LTE\/LTE-Advanced UE transceiver modules and RF components. Moreover, we also address in the performance evaluations and RF measurements an additional practical case\nwhere the UE is equipped with a diversity RX chain. Specifically, we show that \\emph{PIM coupled over-the-air} from main transceiver to the diversity RX can also be a real problem. Although such PIM coupling over the air can basically be avoided by improving the isolation between the antennas, the isolation is in practice limited by the compact size of the mobile devices. The proposed digital cancellation solution is shown to be able to efficiently suppress the PIM appearing in the main RX branch as well as in the diversity RX branch. Thus, in general, the proposed solution can relax the RF components' linearity requirements and improve the receiver sensitivity by effectively suppressing the PIM, and is applicable in both main RX and diversity RX branches.\n\nThe rest of the paper is organized as follows. In Section II, we address baseband equivalent modeling of third-order PIM at RX band under various sources of memory. The corresponding digital cancellation solution and the necessary parameter estimation procedures are presented in Section III. Then, the RF measurement results are reported and analyzed in Section IV. Finally, Section V concludes the paper.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.42]{Fig2.pdf}\n\\caption{{The considered inter-band CA FDD transceiver architecture at UE side.}}\\label{trans_architecture}\n\\end{figure}\n\n\n\\section{Baseband Equivalent Models for Passive Intermodulation at RX Band}\nIn this section, we present the relevant signal models for describing the PIM observed in the RX chain, particularly at digital baseband. A block diagram describing the considered FDD radio transceiver system supporting inter-band CA is shown in Fig.~\\ref{trans_architecture}. It is assumed that the adopted bands have dedicated TX-RX chains, each, while on the transmit direction the CCs are combined together in a duplexer or a multiplexer \\cite{121984}. The PIM then occurs in the passive RF components due to cross-modulation between the aggregated transmit signals, and may appear in one or more of the configured receiver bands causing nonlinear self-interference. \n\nGenerally speaking, as shown in \\cite{zeeshan_SIPS} and \\cite{Zeeshan_MTT_submit}, the TX PAs can also cause nonlinear distortion to the individual transmit carriers, leading to spectral regrowth around the main transmit carriers. Such nonlinear distortion in the transmit CCs can then affect the overall characteristics of the PIM-induced self-interference and its cancellation. However, in this paper, we restrict our attention to linear transmit chains and PAs, thus basically assuming that the PAs are properly linearized, e.g., through digital pre-distortion (DPD). It can, however, be argued that the transmit carriers experience some filtering or linear distortion effects before they are combined and thus experience the PIM nonlinearity. Additionally, there can also potentially be multiple PIM sources in the transceiver. As a result, the overall TX-RX PIM coupling path can be viewed as a system of nonlinearities with memory effects. Keeping this in view, we consider two cases for PIM modeling. In the first case, we consider linear and memoryless PAs and pursue frequency-selective behavioral modeling of the nonlinear passive components. In the second case, we also adopt the memory or linear distortion modeling of the individual transmit CCs,  prior to the actual PIM stage. As we later demonstrate with practical RF measurements, the latter model can facilitate more accurate PIM modeling and enhanced cancellation.\n\nFor presentation purposes and notational simplicity, we focus on third-order PIM effects, while more elaborate higher-order cases as well as coexisting cascaded nonlinearities are addressed in \\cite{Zeeshan_MTT_submit}.\n\n\\subsection{Baseband Equivalent PIM Model: Memoryless TX Chains}\\label{sec:modelA}\nLet us denote the complex baseband waveforms of the two transmit CCs\nby $s_1[n]$ and $s_2[n]$, respectively. The signals are up-converted to their respective RF frequencies and amplified by the PAs. The corresponding aggregated RF signal at the multiplexer\/duplexer output, after combining the carriers, can then be expressed as\n\\begin{equation}\\label{TXsignal}\n\ts_{\\mathrm{RF}}[n] = \\Re{\\alpha_1s_1[n] e^{j \\omega_1 n}} + \\Re{\\alpha_2s_2[n] e^{j\\omega_2 n}},\n\\end{equation}\nwhere $\\alpha_1$ and $\\alpha_2$ denote the complex voltage gains of the two PAs, and $\\omega_1$ and $\\omega_2 < \\omega_1$ are the angular center frequencies of the individual CCs after up-conversion. This signal then travels towards the antenna, however, due to nonlinear passive components, unwanted PIM products of the transmit signal are created. Assuming that the upper third-order IM sub-band, i.e. $2\\omega_1-\\omega_2$, lies in the downlink frequency band, similar to Fig. 1, the baseband equivalent complex PIM waveform appearing in the RX band reads then\n\\begin{equation}\\label{PIM}\n\ts_{\\mathrm{PIM}}[n] = \\sum_{l=-L_1}^{L_2} \\gamma_l s_1[n-l]^2s_2^*[n-l] \n\\end{equation}\nwhere $\\gamma_l$ denote the impulse response coefficients of the third-order nonlinear term modeling the memory of the PIM generation mechanism, while $L_1$ and $L_2$ are the numbers of pre-cursor and post-cursor memory taps in the PIM model, respectively. The total memory length of the PIM generation stage is then $L_1 + L_2 + 1$.\n\n\\subsection{Baseband Equivalent PIM Model: TX Chains with Memory}\\label{sec:modelB}\nWe next generalize the PIM modeling to the case where the individual TX component carrier signals are subject to memory or linear distortion prior to the PIM stage.\nWhen complemented with a memory polynomial model for the actual PIM generation stage, this allows for versatile memory modeling, for example in cases where there are mutually different delays along the TX paths, or more generally the frequency responses of the two TX chain are different and both contain memory.\n\nTo this end, the two transmit carriers travel through their independent TX chains before arriving at the PIM source, and are subject to linear filtering effects described by the impulse responses $\\alpha_{1,m}$ and $\\alpha_{2,m}$, respectively.\nThe corresponding RF signal model for the combined signal, prior to the PIM stage, then reads\n\\begin{equation}\\label{TXsignal2}\n\\begin{split}\n\ts_{\\mathrm{RF}}[n] = \\Re{e^{j \\omega_1 n}\\sum_{m=-M_1}^{M_2} \\alpha_{1,m}s_1[n-m]} \\\\\n\t+ \\Re{e^{j\\omega_2 n}\\sum_{m=-M_1}^{M_2} \\alpha_{2,m}s_1[n-m]},\n\\end{split}\n\\end{equation}\nwhere $M_1, M_2$ are the numbers of the input pre- and post-cursor memory taps for the TX carriers, with the total input memory length being $M_1+M_2+1$.\nWhen the above combined signal with memory is subject to a third-order PIM nonlinearity with additional memory, the baseband equivalent PIM waveform at own RX band can be written as shown in \\eqref{equation_mem_PIM}, next page, where $\\gamma_{k_{11}, \\cdots, k_{1M}, k_{21}, \\cdots, k_{2M}}$ are the effective total memory coefficients for the term defined by the parameters $k_{1M}, k_{21}, \\cdots, k_{2M}$, and $M = M_1+M_2$. Note that in this model, the memory lengths of the transmit CCs and the PIM nonlinearity can be independently chosen, resulting in an overall flexible model from the modeling and cancellation perspective.\n\nTo give a concrete example, Table \\ref{table:Basis} shows the basis function samples stemming from the two signal models presented in this section, with short memory orders of $L_1=L_2=1$ and $M_1=M_2=1$ for presentation simplicity. As can be observed, the signal model under TX chains with memory has altogether $42$ basis functions, opposed to $3$ basis functions obtained in the memoryless TX chain case. As a consequence of the different sample delays between the CCs in the basis functions, the more complicated signal model has the potential of better estimating and cancelling also the impacts of any potential timing mismatch errors as well as overall frequency responses in the TX chains. This is achieved, however, at the cost of an increased computational complexity.\n\\begin{figure*}[ht]\n\\begin{equation}\\begin{split}\\label{equation_mem_PIM}\n    s_{\\mathrm{PIM}}[n] = &\\sum_{k_{11}=0}^{2}\\sum_{k_{12}=0}^{2-k_{11}}\\cdots\\sum_{k_{1M}=0}^{2-\\sum_{i=1}^{M-1}k_{1i}}\n\\sum_{k_{21}=0}^{1}\\sum_{k_{22}=0}^{1-k_{21}}\\cdots\\sum_{k_{2M}=0}^{1-\\sum_{i=1}^{M-1}k_{2i}} \\sum_{l=-L_1}^{L_2}\\gamma_{l,k_{11}, \\cdots, k_{1M}, k_{21}, \\cdots, k_{2M}} \\times \\\\\n&s_1[n-l+M_1]^{k_{11}}s_1[n-l+M_1-1]^{k_{12}}\\cdots s_1[n-l-M_2]^{2-\\sum_{i=1}^{M}k_{1i}} \\times \\\\ \n&s_2^*[n-l+M_1]^{k_{21}}s_2^*[n-l+M_1-1]^{k_{22}}\\cdots s_2^*[n-l-M_2]^{1-\\sum_{i=1}^{M}k_{2i}}.\n\\end{split}\n\\end{equation}\n\\vspace{2mm}\n\\hrulefill\n\\end{figure*}\n\\setlength{\\arrayrulewidth}{0.2mm}\n\\setlength{\\tabcolsep}{3pt}\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{table*}[h]\n\\caption{Example basis functions of the two considered PIM models}\n\\label{table:Basis}\n\\centering\n\\begin{tabular}{ | p{3cm} | p{14cm} | }\n\\hline\n \\textbf{Signal model}  & \\textbf{Basis functions when $L_1=L_2=1$} \\\\\n \\hline\n Memoryless TX Chains & {\\centering{}$s_1[n+1]^2 s_2^*[n+1]$, $s_1[n]^2 s_2^*[n]$, $s_1[n-1]^2 s_2^*[n-1]$\\\\ \\vspace{-10pt}}\\\\\n \\hline\n TX Chains with Memory, $M_1=M_2=1$ & {\\centering{} $s_1[n-2]^2 s_2^*[n-2]$, $s_1[n-1]^2 s_2^*[n-1]$, $s_1[n]^2 s_2^*[n]$, $s_1[n-2]^2 s_2^*[n-1]$, $s_1[n-1]^2 s_2^*[n]$, $s_1[n]^2 s_2^*[n+1]$,\\\\$s_1[n-2]^2 s_2^*[n]$, $s_1[n-1]^2 s_2^*[n+1]$,$s_1[n]^2 s_2^*[n+2]$, $s_1[n-1] s_1[n-2] s_2^*[n-2]$, $s_1[n] s_1[n-1] s_2^*[n-1]$,\\\\$s_1[n+1] s_1[n] s_2^*[n]$, $s_1[n-1] s_1[n-2] s_2^*[n-1]$, $s_1[n] s_1[n-1] s_2^*[n]$, $s_1[n+1] s_1[n] s_2^*[n+1]$,\\\\$s_1[n-1] s_1[n-2] s_2^*[n]$, $s_1[n] s_1[n-1] s_2^*[n+1]$, $s_1[n+1] s_1[n] s_2^*[n+2]$, $s_1[n] s_1[n-2] s_2^*[n-2]$,\\\\$s_1[n+1] s_1[n-1] s_2^*[n-1]$, $s_1[n+2] s_1[n] s_2^*[n]$, $s_1[n] s_1[n-2] s_2^*[n-1]$, $s_1[n+1] s_1[n-1] s_2^*[n]$,\\\\$s_1[n+2] s_1[n] s_2^*[n+1]$, $s_1[n] s_1[n-2] s_2^*[n]$, $s_1[n+1] s_1[n-1] s_2^*[n+1]$, $s_1[n+2] s_1[n] s_2^*[n+2]$,\\\\$s_1[n] s_1[n-1] s_2^*[n-2]$, $s_1[n+1] s_1[n] s_2^*[n-1]$, $s_1[n+2] s_1[n+1] s_2^*[n]$, $s_1[n+2] s_1[n+1] s_2^*[n+2]$,\\\\$s_1[n+2]^2 s_2^*[n+1]$, $s_1[n+2]^2 s_2^*[n+2]$, $s_1[n]^2 s_2^*[n-2]$, $s_1[n+1]^2 s_2^*[n-1]$, $s_1[n+2]^2 s_2^*[n]$,\\\\$s_1[n-1]^2 s_2^*[n-2]$, $s_1[n]^2 s_2^*[n-1]$, $s_1[n+1]^2 s_2^*[n]$, $s_1[n+1]^2 s_2^*[n+1]$, $s_1[n+1]^2 s_2^*[n+2]$,\\\\$s_1[n+2] s_1[n+1] s_2^*[n+1]$\\\\ \\vspace{-10pt}}\\\\\n \\hline\n \\end{tabular}\n \\end{table*}\n \n\n\n\n\n\\section{Proposed Digital PIM Canceller and Parameter Estimation}\nThe proposed digital PIM canceller builds directly on the derived baseband signal models described in the previous section. Specifically, the canceller re-generates the PIM samples, using either (\\ref{PIM}) or (\\ref{equation_mem_PIM}), and then subtracts the estimated PIM samples from the actual received baseband signal. In general, the baseband complex samples of the component carriers, $s_1[n]$ and $s_2[n]$, are known in the transceiver. However, the equivalent model parameters, i.e., the $\\gamma$ variables\nwhich act as the complex weights of the basis function samples are unknown and thus must be estimated.\n\nRegarding the parameter estimation, the baseband PIM signal models in (\\ref{PIM}) and (\\ref{equation_mem_PIM}), and thus the canceller processing structures, are in fact linear in the parameters, i.e., the $\\gamma$ variables.\nThus, the parameters can be straight-forwardly estimated with any standard estimator for linear signal models, such as linear least-squares (LS), recursive least squares (RLS), or least mean squares (LMS) \\cite{Haykin}.\n\nFor presentation purposes, we next switch to vector-matrix notations and consider a block of $N$ samples of the received signal. We express the samples of the PIM for the corresponding time duration, stacked into a vector, as\n\\begin{align}\n\t\\mathbf{s}_{\\mathrm{PIM}} = \\bm{A} \\bm{\\theta},\n\\end{align}\nwhere $\\bm{A}$ denotes the data matrix that collects the samples of the different nonlinear basis functions while $\\bm{\\theta}$ is a vector containing the corresponding unknown coefficients. The structure of the data matrix $\\bm{A}$ depends obviously on which of the two canceller structures is deployed. Using these notations, the LS parameter estimator reads \n\\begin{align}\n\\widehat{\\bm{\\theta}} = \\left(\\bm{A}^H \\bm{A}\\right)^{-1} \\bm{A}^H\\mathbf{y}_{\\mathrm{RX}},\n\\label{eq:alpha_estimates}\n\\end{align}\nwhere $(.)^H$ denotes the Hermitian transpose, and $\\mathbf{y}_{\\mathrm{RX}}$ is the vector of the received complex baseband samples. \n\nAfter obtaining the estimates of the coefficients, they are used to create a baseband replica of the PIM-induced self-interference in the digital front-end of the radio, during the actual online operation, which is subtracted from the received signal sample by sample.\nThe cancelled signal during the transceiver online operation can thus formally be expressed as\n\\begin{align}\n    y_\\mathrm{RX,canc}[n] = y_{\\mathrm{RX}}[n] - \\bm{a}^T[n] \\hat{\\bm{\\theta}},\n\\end{align}\nwhere $\\bm{a}^T[n]$ refers to the row vector containing the basis function samples corresponding to the particular time instant $n$.\n\nIn general, the basic approach as described above is that the parameter estimation is carried out offline. In such operating approach, the parameter estimation must then be repeated periodically as the exact PIM characteristics can  change over time, e.g., due to changes in the temperature or the transmit power. However, since the PIM is primarily caused by radio transceiver internal front-end components, it is likely that the estimation needs to be repeated only fairly seldom.\nAlternatively, recursive least squares type of adaptive parameter estimation and tracking approach can also be deployed meaning that the parameters are continuously adapted, sample by sample, during the normal online operation. In this case, it is to be noted that the actual received signal acts as noise from the parameter estimation point of view. In both cases, periodic or continuous parameter estimation, the actual cancellation is anyway running continuously in real-time, using the online transmit data.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.42]{Fig3.pdf}\n\\caption{{Illustration of the RF measurement setup, showing the relevant transmitter and receiver units as well as the RF front-end components.}}\n\\label{meas_setup}\n\\end{figure}\n\n\\section{RF Measurement Results}\n\\subsection{Measurement Setup and Settings}\nThe performance of the proposed PIM cancellation method is now evaluated through practical RF measurements. A block diagram of the corresponding measurement setup is shown in Fig.~\\ref{meas_setup}, while the relevant measurement parameters are listed in Table~\\ref{table:simu}.\n\\begin{table}[t]\n\\setlength{\\tabcolsep}{5pt}\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Basic RF measurement parameters}\n\\label{table:simu}\n\\centering\n\\begin{tabular}{|c||c|}\n\\hline\n\\textbf{Parameter} & \\textbf{Value}\\\\\n\\hline\nBandwidths of the TX CCs & 5 MHz\\\\\n\\hline\nTotal transmit power & 24 dBm\\\\\n\\hline\nPost PA loss & 4 dB\\\\\n\\hline\nDuplexer insertion loss & 3 dB\\\\\n\\hline\nSwitch insertion loss & 1 dB\\\\\n\\hline\nRX center frequency & 2140 MHz\\\\\n\\hline\nLS parameter learning sample size & 90000\\\\\n\\hline\nNumber of PIM pre-cursor taps $(L_1)$ & 3\\\\\n\\hline\nNumber of PIM post-cursor taps $(L_2)$ & 4\\\\\n\\hline\nNumber of TX\/PA pre-cursor taps $(M_1)$ & 0 or 1\\\\\n\\hline\nNumber of TX\/PA post-cursor taps $(M_2)$ & 0 or 1\\\\\n\\hline\\end{tabular}\n\\end{table}\n The measurement setup includes the Analog Devices AD9368 $2\\times 1$ transceiver board to generate the Band 1 and Band 3 CC transmit signals. These signals are next amplified using two separate Skyworks SKY77643-21 PAs and combined in a multiplexer TDK B8690, which has separate TX and RX ports and a common antenna port. The aggregated TX signal is then fed to an Infineon BGS12PL6 switch and a TDK DPX162690DT-8022B2 diplexer. The diplexer output port is connected to an antenna. In addition, to implement a diversity RX chain, an additional B1 duplexer connected to an additional antenna is placed in close proximity of the main transceiver. The signals at the B1 RX ports are fed to the RF input of the National Instrument (NI) PXIe-5645R vector signal transceiver (VST), which is used here for down-conversion and digitization of the received signals for further processing. A host processor with MATLAB is used for post-processing the captured data and for the proposed algorithm evaluation. Block least-squares is used for parameter learning, without actual RX signal. Different sets of transmit signal samples are always used, for parameter learning and for evaluating the cancellation performance. All the measurements are performed in an electromagnetic compatibility (EMC) chamber to avoid any external interferences that might affect the results.\n\\subsection{Measurement Results for Main RX Branch}\nIn this section, we first present the cancellation results for the main RX branch. The bandwidths of the uplink CCs are assumed to be 5 MHz. When adopting the full transmit power of $+24$ dBm, the essential power spectral density (PSD) curves of the observed PIM before and after the digital cancellation are shown in Fig.~\\ref{Main_RX_PSD}, containing both canceller cases of with and without memory in the TX chains. Notice that for illustration purposes, the PSD curves are all referenced to the actual receiver RF frequencies despite the digital canceller operates at baseband. For one, these results clearly indicate that the PIM induced self-interference is significantly above the receiver noise floor when employing state-of-the-art radio components, and can thus cause RX desensitization.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.5]{Fig4.pdf}\n\\caption{{Measured PSD curves of the PIM distortion at the main RX branch of Band 1 without and with digital cancellation. The TX CC bandwidths at the LTE Bands 1 and 3 are $5\\;$MHz each, and the aggregated TX power is +$24\\;$dBm.}}\n\\label{Main_RX_PSD}\n\\end{figure}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.5]{Fig5.pdf}\n\\caption{{The measured powers of the PIM distortion at the main RX branch of Band 1 without and with digital cancellation as functions of the aggregated transmit power. The TX CC bandwidths at the LTE Bands 1 and 3 are $5\\;$MHz each.}}\n\\label{CancCurves_MainRX}\n\\end{figure}\nHowever, the proposed digital cancellation solutions are able to suppress the PIM induced self-interference by up to 21 dB or so. Moreover, it can also be noticed that the memory modeling of the PAs, or TX chains overall, prior to the PIM generation stage can further improve the cancellation performance by ca. 1 dB.\n \nNext, Fig.~\\ref{CancCurves_MainRX} shows the behavior of the PIM-induced self-interference power at Band 1 main RX as a function of the TX power, without and with digital cancellation. The results suggest that the self-interference is already significant even at lower TX powers of some $+14\\;$dBm, and can heavily degrade the system performance as the TX power is increased. It can also be observed that the proposed cancellation approach can guarantee interference-free reception for the TX powers up to $+20\\;$dBm, thus substantially extending the usable transmit power range without the risk of receiver desensitization.\n\\subsection{Measurement Results for Diversity RX Branch}\nIn this subsection, we continue the RF measurements and demonstrate that the PIM-induced distortion can also couple over-the-air and thus cause self-interference, e.g., to a diversity RX branch. To demonstrate this, a diversity RX antenna is placed at a distance of $5\\;$cm from the main RX antenna, which provides an antenna isolation of around $10\\;$dB. Fig.~\\ref{diversity_Rx} shows then the observed PIM interference coupled over the air from the main transceiver into the diversity RX with the aggregated transmit power of $+24$ dBm. Again, the self-interference is substantial and clearly above the noise floor, and can thus degrade the receiver sensitivity. It can also be observed that the proposed digital cancellation approach is able to suppress the OTA-coupled self-interference efficiently very close to the noise floor. Finally, Fig.~\\ref{CancCurvesdiversity_Rx} shows the PIM-induced distortion power at the diversity RX as a function of the aggregated TX power, evidencing that the proposed digital cancellation is able to efficiently suppress the self-interference close to the noise floor.\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.5]{Fig6.pdf}\n\\caption{{\nMeasured PSD curves of the PIM distortion at the diversity RX branch of Band 1 without and with digital cancellation. The TX CC bandwidths at the LTE Bands 1 and 3 are $5\\;$MHz each, and the aggregated TX power is +$24\\;$dBm.\n}}\n\\label{diversity_Rx}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.5]{Fig7.pdf}\n\\caption{{\nThe measured powers of the PIM distortion at the diversity RX branch of Band 1 without and with digital cancellation as functions of the aggregated transmit power. The TX CC bandwidths at the LTE Bands 1 and 3 are $5\\;$MHz each.\n}}\n\\label{CancCurvesdiversity_Rx}\n\\end{figure}\n\n\\subsection{Further Discussion}\nWhile the presented results clearly demonstrate the modeling and cancellation capability of the proposed technique, it can be observed that there is still some residual interference present after the digital cancellation. This is particularly so when larger transmit powers are adopted. The main reason for this is that the developed cancellers consider only third-order PIM while neglect the higher-order distortion terms. The role of such higher-order components is more and more essential when the transmit power increases. Another reason is, despite elementary PA linearization, that also PA induced nonlinear distortion has an impact on the exact PIM waveform.\nTherefore, developing advanced techniques for joint compensation of PA-induced and PIM nonlinearities, with memory effects, and such that also higher-order nonlinear terms are taken into account \\cite{Zeeshan_MTT_submit} forms an important research topic for our future work. \n\n\n\\section{Conclusion}\\label{sec:conc}\nIn this paper, a novel digital cancellation solution was presented to estimate and cancel the self-interference resulting from the nonlinear passive components in CA-based radio transceivers. Along the self-interference modeling and the corresponding cancellation processing, the memory effects of the TX chains and PAs were also taken into account. The performance of the proposed digital cancellers was tested and analyzed with  actual RF measurements using real-life transceivers and RF components for UE devices, demonstrating excellent self-interference suppression. The presented measurement results also show that PIM can couple over the air into the diversity RX branch and thus cause significant interference in the diversity RX. Based on the obtained results, the proposed digital cancellation approach is an effective solution to suppress the PIM, both in the main RX as well as in the diversity RX chains. Such novel digital cancellation solutions can relax the linearity requirements of the passive RF components, while also enabling efficient utilization of the RF spectrum. Extending the developed cancellation solutions to accommodate higher-order nonlinear products as well as co-existing cascaded nonlinearities were identified as important future research topics.\n\\section*{Acknowledgment}\nThis work was supported by the Academy of Finland (under the grants 304147 and 301820), the Finnish Funding Agency for Innovation (Tekes, under the project ``5G Transceivers for Base Stations and Mobile Devices, 5G TRX''), Nokia Bell Labs, RF360, Pulse Finland, Sasken Finland, and Huawei Technologies Finland. The work was also supported by the Tampere University of Technology Graduate School.\n\n\n\\IEEEtriggeratref{5}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe continuous double auction (CDA) is a dominant market mechanism\nused to store and match orders and to facilitate trading in most\nmodern equity markets\n\\cite{Smith-Farmer-Gillemot-Krishnamurthy-2003-QF}. In most of the\norder driven markets, there are two kinds of basic orders, called\nmarket orders and limit orders. A market order is submitted to buy\nor sell a number of shares at the market quote which results in an\nimmediate transaction, while a limit order is placed to buy (or\nsell) a number of shares below (or above) a given price. All the\nlimit orders that fail to result in an immediate transaction are\nstored in a queue called {\\em{limit-order book}}. Buy limit orders\nare called bids while sell limit orders are called asks or offers.\n{\\em{Best bid}} price $b(t)$ and {\\em{best ask}} (or {\\em{best\noffer}}) price $a(t)$ are the highest buying price and the lowest\nselling price at any time $t$ in the limit-order book. The best bid\n(or ask) is called the {\\em{same best}} for buy (or sell) orders,\nwhile the best ask (or bid) is called the {\\em{opposite best}} for\nbuy (or sell) orders. A limit order causes an immediate transaction\nif the associated limit price penetrates the opposite best price.\nSuch kind of limit orders are called {\\em{marketable limit orders}}\nor {\\em{effective market orders}} and other limit orders are termed\n{\\em{effective limit orders}}. In the Chinese stock market, only\nlimit orders were permitted in the placement of orders before July\n1, 2006.\n\nIt is a dynamic process concerning the limit-order book. Effective\nlimit orders accumulate in the book while effective market orders\ncause transactions and remove the limit orders according to their\nprice and the time they arrive. Effective limit orders can also be\nremoved by cancelation for a variety of reasons. Unveiling the\ndynamics of order placement and cancelation will deepen our\nunderstanding of the microscopic mechanism of price formation and\nallow us to reproduce remarkably many key features of common stocks\nsuch as the probability distribution of returns\n\\cite{Zovko-Farmer-2002-QF,Gabaix-Gopikrishnan-Plerou-Stanley-2003-Nature,Bouchaud-Gefen-Potters-Wyart-2004-QF,Plerou-Gopikrishnan-Gabaix-Stanley-2004-QF,Farmer-Patelli-Zovko-2005-PNAS,Malevergne-Pisarenko-Sornette-2005-QF,Bouchaud-Kockelkoren-Potters-2006-QF,Mike-Farmer-2007-JEDC}.\n\nThe difference between best-ask price and best-bid price,\n$s(t)=a(t)-b(t)$, is the bid-ask spread. Numerous work has been\ncarried out to explore the different components of the bid-ask\nspread \\cite{Stoll-1989-JF,Huang-Stoll-1997-RFS}. On the other hand,\nthere are several groups studying the statistical properties of the\nbid-ask spread time series for different stock markets. Farmer\n{\\em{et al.}} reported that the bid-ask spread defined by\n$\\ln[a(t)]-\\ln[b(t)]$ on the London Stock Exchange follows power-law\ndistribution in the tail\n\\begin{equation}\nP(>s) \\sim s^{-\\zeta}~,\n\\end{equation}\nwhere the exponent $\\zeta=3.03\\pm0.41$ ranging from 2.4 to 3.9\n\\cite{Farmer-Gillemot-Lillo-Mike-Sen-2004-QF,Mike-Farmer-2007-JEDC},\nwhich is well consistent with the inverse cubic law\n\\cite{Gopikrishnan-Meyer-Amaral-Stanley-1998-EPJB,Gabaix-Gopikrishnan-Plerou-Stanley-2003-PA,Gabaix-Gopikrishnan-Plerou-Stanley-2003-Nature}.\nIn addition, Mike and Farmer found that the spread possesses long\nmemory with the Hurst index being $0.75<H<0.85$\n\\cite{Mike-Farmer-2007-JEDC}. Plerou {\\em{et al.}} adopted the 116\nmost frequently traded stocks on the New York Stock Exchange over\nthe two-year period 1994-1995 to investigate the coarse-grained\nbid-ask spread over a time interval $\\Delta{t}$ and found that the\ntail distribution decays as a power law with a mean tail exponent of\n$\\zeta=3.0\\pm0.1$ and the spread after removing the intraday pattern\nexhibits long memory with $H=0.73\\pm0.01$\n\\cite{Plerou-Gopikrishnan-Stanley-2005-PRE}. Qualitatively similar\nresults were found by Cajueiro and Tabak in the Brazilian equity\nmarket where the mean tail exponent is $\\zeta=2.18$ ranging from\n1.18 to 2.97 and the Hurst index is $H=0.68\\pm0.08$ varying from\n0.52 to 0.89 \\cite{Cajueiro-Tabak-2007-PA}.\n\nDue to the fast development of the economy of China and the\nincreasing huge capitalization of its stock market, more concerns\nare attracted to study the emerging Chinese stock market. In order\nto reduce the market risks and speculation actions, the Chinese\nstock market adopts $t+1$ trading system, which does not allow\ntraders to sell and buy stocks on the same day, and no market orders\nwere permitted until July 1, 2006, which may however consume the\nliquidity of the market and cause the spread to show different\nproperties when compared to other stock markets. In this work, we\ninvestigated the probability distribution, long memory, and presence\nof multifractal nature of the bid-ask spread using limit-order book\ndata on the Shenzhen Stock Exchange (SSE) in China.\n\n\nThe rest of this paper is organized as follows. In\nSec.~\\ref{s1:database}, we describe in brief the trading rules of\nthe Shenzhen Stock Exchange and the database we adopt. Section\n\\ref{s1:definition} introduces three definitions of the bid-ask\nspread and investigates the intraday pattern in the spread. The\ncumulative distributions of the spreads for different definitions\nare discussed in Sec.~\\ref{s1:PDF}. We show in Sec.~\\ref{s1:memory}\nthe long memory of the spread based on the detrended fluctuation\nanalysis (DFA) quantified by the estimate of the Hurst index. In\nSec.~\\ref{s1:MFA}, we perform multifractal analysis on the bid-ask\nspread time series. The last section concludes.\n\n\n\\section{SSE trading rules and the data set}\n\\label{s1:database}\n\nOur analysis is based on the limit-order book data of a liquid stock\nlisted on the Shenzhen Stock Exchange. SSE was established on\nDecember 1, 1990 and has been in operation since July 3, 1991. The\nsecurities such as stocks, closed funds, warrants and Lofs can be\ntraded on the Exchange. The Exchange is open for trading from Monday\nto Friday except the public holidays and other dates as announced by\nthe China Securities Regulatory Commission. With respect to\nsecurities auction, opening call auction is held between 9:15 and\n9:25 on each trading day, followed by continuous trading from 9:30\nto 11:30 and 13:00 to 15:00. The Exchange trading system is closed\nto orders cancelation during 9:20 to 9:25 and 14:57 to 15:00 of each\ntrading day. Outside these opening hours, unexecuted orders will be\nremoved by the system. During 9:25 to 9:30 of each trading day, the\nExchange is open to orders routing from members, but does not\nprocess orders or process cancelation of orders.\n\nAuction trading of securities is conducted either as a call auction\nor a continuous auction. The term ``call auction'' (from 9:15 to\n9:25) refers to the process of one-time centralized matching of buy\nand sell orders accepted during a specified period in which the\nsingle execution price is determined according to the following\nthree principles: (i) the price that generates the greatest trading\nvolume; (ii) the price that allows all the buy orders with higher\nbid price and all the sell orders with lower offer price to be\nexecuted; and (iii) the price that allows either buy orders or sell\norders to have all the orders identical to such price to be\nexecuted.\n\nThe term ``continuous auction'' (from 9:25 to 11:30 and from 13:00\nto 15:00) refers to the process of continuous matching of buy and\nsell orders on a one-by-one basis and the execution price in a\ncontinuous trading is determined according to the following\nprinciples: (i) when the best ask price equals to the best bid\nprice, the deal is concluded at such a price; (ii) when the buying\nprice is higher than the best ask price currently available in the\ncentral order book, the deal is concluded at the best ask price; and\n(iii) when the selling price is lower than the best bid price\ncurrently available in the central order book, the deal is executed\nat the best bid price. The orders which are not executed during the\nopening call auction automatically enter the continuous auction.\n\nThe tick size of the quotation price of an order for A\nshares\\footnote{{\\textbf{A shares}} are common stocks issued by\nmainland Chinese companies, subscribed and traded in Chinese RMB,\nlisted in mainland Chinese stock exchanges, bought and sold by\nChinese nationals. A-share market was launched in 1990.} is RMB 0.01\nand that for B shares\\footnote{{\\textbf{B shares}} are issued by\nmainland Chinese companies, traded in foreign currencies and listed\nin mainland Chinese stock exchanges. B shares carry a face value\ndenominated in Renminbi. The B Share Market was launched in 1992 and\nwas restricted to foreign investors before February 19, 2001. B\nshare market has been opened to Chinese investors since February 19,\n2001.} is HKD 0.01. Orders are matched and executed based on the\nprinciple of {\\em{price-time priority}} which means priority is\ngiven to a higher buy order over a lower buy order and a lower sell\norder is prioritized over a higher sell order; The order sequence\nwhich is arranged according to the time when the Exchange trading\nsystem receives the orders determines the priority of trading for\nthe orders with the same prices.\n\nWe studied the data from the limit-order book of the stock SZ000001\n(Shenzhen Development Bank Co., LTD) in the whole year of 2003. The\nlimit-order book recorded high-frequency data whose time stamps are\naccurate to 0.01 second. The size of the data set is $3,925,832$,\nincluding $12,965$ invalid orders, $122,034$ order submissions and\ncancelations in the opening call auction, $47,576$ order submissions\nand cancelations during the cooling period (9:25-9:30), and\n$3,743,257$ valid events during the continuous auction. In\ncontinuous auction, there are $317,015$ cancelations of buy orders\nand $274,929$ cancelations of sell orders, $889,700$ effective\nmarket orders, and $2,261,613$ effective limit orders. Table\n\\ref{Tb:1} shows a segment taken from the limit-order book recorded\non 2003\/07\/09. The seven columns stand for order size, limit price,\ntime, best bid, best ask, transaction volume, and buy-sell\nidentifier (which identifies whether a record is a buy order, sell\norder, or a cancelation). For a cancelation record, the limit price\nis set to be zero.\n\n\n\\begin{table}[htp]\n\\caption{A segment of the limit-order book}\\label{Tb:1}\n\\begin{center}\n\\begin{tabular}{rrrrrrrrrrrrr}\n  \\hline\n 1400 &0    &9390015&11.33&11.34&0&31\\\\\n 1000 &11.48&9390016&11.33&11.34&0&29\\\\\n 400  &11.65&9390311&11.33&11.34&0&29\\\\\n 400  &0    &9390317&11.33&11.34&0&30\\\\\n 1000 &11.33&9390365&11.33&11.34&0&26\\\\\n 6000 &11.33&9390408&11.33&11.34&6000&23\\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\section{Defining bid-ask spread}\n\\label{s1:definition}\n\nThe literature concerning the bid-ask spread gives different\ndefinitions\n\\cite{Farmer-Patelli-Zovko-2005-PNAS,Mike-Farmer-2007-JEDC,Stoll-1989-JF,Huang-Stoll-1997-RFS,Farmer-Gillemot-Lillo-Mike-Sen-2004-QF,Plerou-Gopikrishnan-Stanley-2005-PRE,Cajueiro-Tabak-2007-PA,Roll-1984-JF,Daniels-Farmer-Gillemot-Iori-Smith-2003-PRL,Wyart-Bouchaud-Kockelkoren-Potters-Vettorazzo-2006}.\nIn this section, we discuss three definitions according to sampling\ntime when best bid prices and best ask prices are selected to define\nthe spread. Some definitions are based on the transaction time,\nwhile the others are based on the physical time. The latter scheme\nis actually a coarse-graining of the data within a given time\ninterval.\n\n\\subsection{Definition I}\n\nThe first definition of the bid-ask spread used in this work is the\nabsolute or relative difference between the best ask price and the\nbest bid price right before the transaction, that is,\n\\begin{subequations}\\label{Eq:s12}\n\\begin{equation}\n  s(t) = a(t) - b(t)\n\\end{equation}\nfor absolute difference or\n\\begin{equation}\n  s(t)=\\log_{10} [a(t)] -\\log_{10} [b(t)]\n\\end{equation}\n\\end{subequations}\nfor relative difference. This was used to analyze the stocks on the\nLondon Stock Exchange\n\\cite{Farmer-Gillemot-Lillo-Mike-Sen-2004-QF,Mike-Farmer-2007-JEDC}.\nThe size of the spread time series is $895,606$.\n\n\\subsection{Definition II}\n\nThe best ask price or the best bid price may change due to the\nremoval of all shares at the best price induced by an effective\nmarket order, or the placement of an limit order inside the spread,\nor the cancelation of all limit orders at the best bid\/ask price.\nHence the bid-ask spread does not always change when a transaction\noccurs, and it nevertheless changes without transaction. This\nsuggests to introduce an alternative definition of the spread which\nconsiders the absolute or relative difference between the best bid\nprice and the best ask price whensoever it changes. The expressions\nof definition II are the same as those in Eq.~(\\ref{Eq:s12}) except\nthat they have different definitions for the time $t$. The size of\nthe spread time series is $142,913$.\n\n\n\n\\subsection{Definition III}\n\nObviously, the time in the first two definitions are on the basis of\n``event''. An alternative definition considers the average bid-ask\nspread over a time interval  $\\Delta{t}$\n\\cite{Mcinish-Wood-1992-JF}. In this definition, the bid-ask spread\nis the average difference between the best ask the best bid when\ntransactions occur over a fixed time interval\n\\cite{Plerou-Gopikrishnan-Stanley-2005-PRE}:\n\\begin{equation}\ns(t) = \\frac{1}{N}\\sum_{i=1}^N{s_i(t)}~,~~~s_i(t) = a_i(t) -\nb_i(t)~,\n \\label{Eq:s3}\n\\end{equation}\nwhere $a_i(t)$ and $b_i(t)$ are the best ask and bid prices in the\ntime interval $(t-{\\Delta} t,t]$, and $N$ is the total number of\ntransaction in the interval and is a function of $t$ and\n$\\Delta{t}$. We use ${\\Delta} t= 1$, $2$, $3$, $4$, and $5$\nminute(s) to calculate the average spreads.\n\n\\subsection{Intraday pattern}\n\n\nIn most modern financial markets, the intraday pattern exists\nextensively in many financial variables\n\\cite{Wood-McInish-Ord-1985-JF,Harris-1986-JFE,Admati-Pfleiderer-1988-RFS},\nincluding the bid-ask spread \\cite{Mcinish-Wood-1992-JF}. The\nperiodic pattern has significance impact on the detection of long\nmemory in time series\n\\cite{Hu-Ivanov-Chen-Carpena-Stanley-2001-PRE}. To the best of our\nknowledge, the investigation of the presence of intraday pattern in\nthe spreads of Chinese stocks is lack.\n\n\nFigure~\\ref{Fig:autocorrelation} shows the autocorrelation function\n$\\langle{s(t)s(t+{\\ell})}\\rangle$ as a function of the time lag\n$\\ell$ for the average bid-ask spread calculated from definition III\nwith linear best bids and asks. We note that the results are very\nsimilar when logarithmic prices are adopted in the definition. We\nsee that there are spikes evenly spaced along multiples of 245 min,\nwhich is exactly the time span of one trading day. What is\ninteresting is that Fig.~\\ref{Fig:autocorrelation} indicates that\nthe average spread also possesses half-day periodicity.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig-Autocorrelation.eps}\n\\caption{Autocorrelation function $\\langle{s(t)s(t+{\\ell})}\\rangle$\nof the average bid-ask spread calculated from definition III with\nthe time interval $\\Delta{t}=1$ min. Note that one trading day\ncontains 245 trading minutes in the Chinese stock market.}\n\\label{Fig:autocorrelation}\n\\end{center}\n\\end{figure}\n\n\n\nIn order to quantify the intraday pattern, we introduce a variable\n$A(t)$, which is defined as the average bid-ask spread at time $t$\nfor all the trading days, that is,\n\\begin{equation}\nA(t) = \\frac{1}{M}\\sum^{M}_{j=1}{s^j(t)}~,\n \\label{Eq:intraday}\n\\end{equation}\nwhere $M$ is the number of trading days in the data set and $s^j(t)$\nis the bid-ask spread at time $t$ of day $j$. The spread $S(t)$\nafter removing the intraday pattern reads\n\\cite{Plerou-Gopikrishnan-Stanley-2005-PRE}\n\\begin{equation}\nS(t) = s(t) \/ A(t)~.\n \\label{Eq:S}\n\\end{equation}\nFigure~{\\ref{Fig:intraday}} illustrates the intraday pattern of the\nbid-ask spread with ${\\Delta} t = 1$ minute. The overall plot shows\nan evident L-shaped pattern, which is consistent with the one-day\nperiodicity shown in the autocorrelation function in\nFig.~\\ref{Fig:autocorrelation}. After the opening call auction, the\nspread $A(t)$ widens rapidly and reaches its maximum 0.0183 at the\nend of the cooling auction (9:30)\\footnote{In 2003, there were three\nbest prices at each side disposed in 9:25 and remained unchanged\nduring the cooling period. Hence, the spreads shown in\nFigure~{\\ref{Fig:intraday}} during this period are virtually\ngenrated according to the trading mechanism.}. Then it decreases\nsharply in fifteen minutes and becomes flat at a level of\n$0.0112\\pm0.0008$ afterwards till 11:30. At the begin of continuous\nauction in the afternoon, $A(t)$ abruptly rises to 0.0133 and drops\ndown to a stable level within about ten minutes which maintains\nuntil the closing time 15:00. Therefore, there are two L-shaped\npatterns each day, which suggests that the wide spread is closely\nrelated to the opening of the market. The intraday pattern makes no\ndifference when we use ${\\Delta} t = 2$, $3$, $4$, and $5$ minutes.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig-Spread_intraday.eps}\n\\caption{Intraday pattern in the bid-ask spread with $\\Delta{t}=1$\nmin. The spread reaches its maximum at the end of the cooling period\nat 9:30.} \\label{Fig:intraday}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\section{Probability distribution}\n\\label{s1:PDF}\n\n\nThe cumulative distributions of the bid-ask spread of stocks in\ndifferent stock markets decay as power laws with the tail exponent\nclose to 3 for the major western markets\n\\cite{Farmer-Gillemot-Lillo-Mike-Sen-2004-QF,Plerou-Gopikrishnan-Stanley-2005-PRE,Mike-Farmer-2007-JEDC}\nand much smaller and more heterogeneous in an emerging market\n\\cite{Cajueiro-Tabak-2007-PA}. Similar behavior is found in the\nChinese stock market. Figure~\\ref{Fig:s12:cdf} presents the\ncomplementary cumulative distribution $P(\\geqslant{s})$ of the\nspreads using definition I and II, where linear prices are used.\nSince the minimum spread equals to the tick size 0.01, the abscissa\nis no less than -2 in double logarithmic coordinates and\n$P(\\geqslant0.01)=1$ for both definitions. The proportion of\n$s=0.01$ in the first definition is much greater than in the second\ndefinition such that the $P(\\geqslant{s})$ for the second defintion\ndrops abruptly for small spreads $s$. The two distributions decay as\npower laws with exponents $\\zeta_{\\rm{I}} = 2.57 \\pm 0.06$ for\ndefinition I and $\\zeta_{\\rm{II}} = 2.30 \\pm 0.05$ for definition\nII. When logarithmic prices are utilized, the spreads also follow\npower-law tail distributions with $\\zeta_{\\rm{I}} = 2.67 \\pm 0.03$\nfor definition I and $\\zeta_{\\rm{II}} = 2.42 \\pm 0.04$ for\ndefinition II. Not much difference in the corresponding tail\nexponents $\\zeta_{\\rm{I}}$ and $\\zeta_{\\rm{II}}$ was found for\nlogarithmic and linear prices.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig-CD_12.eps}\n\\caption{Empirical complementary cumulative distribution of the\nspreads calculated from definitions I and II using linear prices.}\n\\label{Fig:s12:cdf}\n\\end{center}\n\\end{figure}\n\n\nFigure~\\ref{Fig:s3:cdf} illustrates the complementary cumulative\ndistributions of the average spreads over time interval $\\Delta{t}=\n1$, $2$, $3$, $4$, and $5$ minute(s) calculated from definition III\nwith linear prices. The average spreads have power-law tails with\nthe exponents equal to $\\zeta_{\\rm{III},1} = 2.99 \\pm 0.04$,\n$\\zeta_{\\rm{III},2} = 3.00 \\pm 0.04$, $\\zeta_{\\rm{III},3} = 3.00 \\pm\n0.05$, $\\zeta_{\\rm{III},4} = 2.95 \\pm 0.05$, and $\\zeta_{\\rm{III},5}\n= 2.97 \\pm 0.06$. Similarly, for logarithmic prices, we find similar\npower-law tail distributions with $\\zeta_{\\rm{III},1} = 3.07 \\pm\n0.06$, $\\zeta_{\\rm{III},2} = 2.95 \\pm 0.05$, $\\zeta_{\\rm{III},3} =\n3.00 \\pm 0.04$, $\\zeta_{\\rm{III},4}= 2.97 \\pm 0.07$, and\n$\\zeta_{\\rm{III},5} = 2.98 \\pm 0.07$. We find that all the tail\nexponents $\\zeta_{{\\rm{III}},\\Delta{t}}$ for both linear and\nlogarithmic prices are very close to three and are independent to\nthe time interval $\\delta{t}$, showing a nice inverse cubic law.\nThis is well in agreement with the results in the NYSE case for\n$\\Delta{t}=15$, $30$, and $60$ min\n\\cite{Plerou-Gopikrishnan-Stanley-2005-PRE}.\n\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig-CD_3.eps}\n\\caption{Empirical complementary cumulative distributions of the\naverage spreads calculated from definition III with time intervals\n${\\Delta} t= 1$, $2$, $3$, $4$, and $5$ min using linear prices. The\nmarkers represent the real data and the solid lines are the best\nfits in the scaling ranges. The curves with $\\Delta{t}>1$ has been\ntranslated vertically for clarity.} \\label{Fig:s3:cdf}\n\\end{center}\n\\end{figure}\n\nThere are also significant discrepancies. Comparing the cumulative\ndistributions in Fig.~\\ref{Fig:s3:cdf} and that on the NYSE\n\\cite{Plerou-Gopikrishnan-Stanley-2005-PRE}, significant differences\nare observed. The distribution of the spreads on the SSE decays much\nfaster than that on the NYSE for small spreads. In other words, the\nproportion of small spreads is much larger on China's SSE. Possible\ncauses include the absence of market orders, no short positions, the\nmaximum percentage of fluctuation (10\\%) in each day, and the $t+1$\ntrading mechanism in the Chinese stock markets on the one hand and\nthe hybrid trading system containing both specialists and\nlimit-order traders in the NYSE on the other hand. The exact cause\nis not clear for the time being, which can however be tested when\nnew data are available after the introduction of market orders in\nJuly 1, 2006. Moreover, the PDF's in SSE drop abruptly after the\npower-law parts for the largest spreads, which is not observed in\nthe NYSE case \\cite{Plerou-Gopikrishnan-Stanley-2005-PRE}.\n\n\n\n\\section{Long memory}\n\\label{s1:memory}\n\nAnother important issue about financial time series is the presence\nof long memory, which can be characterized by its Hurst index $H$.\nIf $H$ is significantly larger than $0.5$ the time series is viewed\nto possess long memory. Long memory can be defined equivalently\nthrough autocorrelation function $C(\\ell) \\sim \\ell^{-{\\gamma}}$ and\nthe power spectrum $p(\\omega)\\sim\\omega^{-\\eta}$, where the\nautocorrelation exponent ${\\gamma}$ is related to the Hurst index\n$H$ by $\\gamma=2-2H$\n\\cite{Kantelhardt-Bunde-Rego-Havlin-Bunde-2001-PA,Maraun-Rust-Timmer-2004-NPG},\nand the power spectrum exponent $\\eta$ is given by $\\eta=2H-1$\n\\cite{Talkner-Weber-2000-PRE,Heneghan-McDarby-2000-PRE}.\n\nThere are many methods proposed for estimating the Hurst index such\nas the rescaled range analysis (RSA)\n\\cite{Hurst-1951-TASCE,Mandelbrot-Ness-1968-SIAMR,Mandelbrot-Wallis-1969a-WRR,Mandelbrot-Wallis-1969b-WRR,Mandelbrot-Wallis-1969c-WRR,Mandelbrot-Wallis-1969d-WRR},\nfluctuation analysis (FA)\n\\cite{Peng-Buldyrev-Goldberger-Havlin-Sciortino-Simons-Stanley-1992-Nature},\ndetrended fluctuation analysis (DFA)\n\\cite{Peng-Buldyrev-Havlin-Simons-Stanley-Goldberger-1994-PRE,Hu-Ivanov-Chen-Carpena-Stanley-2001-PRE,Kantelhardt-Bunde-Rego-Havlin-Bunde-2001-PA},\nwavelet transform module maxima (WTMM) method\n\\cite{Holschneider-1988-JSP,Muzy-Bacry-Arneodo-1991-PRL,Muzy-Bacry-Arneodo-1993-JSP,Muzy-Bacry-Arneodo-1993-PRE,Muzy-Bacry-Arneodo-1994-IJBC},\ndetrended moving average (DMA)\n\\cite{Alessio-Carbone-Castelli-Frappietro-2002-EPJB,Carbone-Castelli-Stanley-2004-PA,Carbone-Castelli-Stanley-2004-PRE,Alvarez-Ramirez-Rodriguez-Echeverria-2005-PA,Xu-Ivanov-Hu-Chen-Carbone-Stanley-2005-PRE},\nto list a few. We adopt the detrended fluctuation analysis.\n\nThe method of detrended fluctuation analysis is widely used for its\neasy implementation and robust estimation even for a short time\nseries\n\\cite{Taqqu-Teverovsky-Willinger-1995-Fractals,Montanari-Taqqu-Teverovsky-1999-MCM,Heneghan-McDarby-2000-PRE,Audit-Bacry-Muzy-Arneodo-2002-IEEEtit}.\nThe idea of DFA was invented originally to investigate the\nlong-range dependence in coding and noncoding DNA nucleotides\nsequence\\cite{Peng-Buldyrev-Havlin-Simons-Stanley-Goldberger-1994-PRE}\nand then applied to various fields including finance. The method of\nDFA consists of the following steps.\n\nStep 1: Consider a time series $x(t)$, $t=1,2,\\cdots,N$. We first\nconstruct the cumulative sum\n\\begin{equation}\nu(t) = \\sum_{i = 1}^{t}{x(i)}, ~~t=1,2,\\cdots,N~.\n \\label{Eq:DFA_u}\n\\end{equation}\n\nStep 2: Divide the series $u(t)$ into $N_\\ell$ disjoint segments\nwith the same length $\\ell$, where $N_\\ell = [N\/\\ell]$. Each segment\ncan be denoted as $u_v$ such that $u_v(i) = u(l + i)$ for\n$1\\leqslant{i}\\leqslant{\\ell}$, and $l = (v - 1)\\ell$. The trend of\n$u_v$ in each segment can be determined by fitting it with a linear\npolynomial function $\\widetilde{u}_v$. Quadratic, cubic or higher\norder polynomials can also be used in the fitting procedure while\nthe simplest function could be linear. In this work, we adopted the\nlinear polynomial function to represent the trend in each segment\nwith the form:\n\\begin{equation}\n\\widetilde{u}_v(i) = ai+b~,\n \\label{Eq:DFA_wu}\n\\end{equation}\nwhere $a$ and $b$ are free parameters to be determined by the least\nsquares fitting method and $1\\leqslant{i}\\leqslant\\ell$.\n\nStep 3: We can then obtain the residual matrix $\\epsilon_{v}$ in\neach segment through:\n\\begin{equation}\n\\epsilon_{v}(i)=u_{v}(i)-\\widetilde{u}_{v}(i)~,\n\\end{equation}\nwhere $1\\leqslant{i}\\leqslant{\\ell}$. The detrended fluctuation\nfunction $F(v,\\ell)$ of the each segment is defined via the sample\nvariance of the residual matrix $\\epsilon_{v}$ as follows:\n\\begin{equation}\nF^2(v,\\ell) = \\frac{1}{\\ell}\\sum_{i = 1}^{\\ell}[\\epsilon_{v}(i)]^2~.\n \\label{Eq:DFA_F1}\n\\end{equation}\nNote that the mean of the residual is zero due to the detrending\nprocedure.\n\nStep 4: Calculate the overall detrended fluctuation function\n$F(\\ell)$, that is,\n\\begin{equation}\nF^2(\\ell) = \\frac{1}{N_\\ell}\\sum_{v = 1}^{N_\\ell}F^2(v,\\ell)~.\n \\label{Eq:DFA_F2}\n\\end{equation}\n\nStep 5: Varying the value of $\\ell$, we can determine the scaling\nrelation between the detrended fluctuation function $F(\\ell)$ and\nthe size scale $\\ell$, which reads\n\\begin{equation}\nF(\\ell) \\sim \\ell^{H}~,\n \\label{Eq:DFA_H}\n\\end{equation}\nwhere $H$ is the Hurst index of the time series\n\\cite{Taqqu-Teverovsky-Willinger-1995-Fractals,Kantelhardt-Bunde-Rego-Havlin-Bunde-2001-PA}.\n\nFigure~\\ref{Fig:dfa} plots the detrended fluctuation function\n$F(\\ell)$ of the bid-ask spreads from different definitions using\nlinear prices. The bottom $F(\\ell)$ curve is for the average spread\nafter removing the intraday pattern. All the curves show evident\npower-law scaling with the Hurst indexes $H_{\\rm{I}} = 0.91 \\pm\n0.01$ for definition I, $H_{\\rm{II}} = 0.92 \\pm 0.01$ for definition\nII, $H_{\\rm{III}} = 0.75 \\pm 0.01$ for definition III, and\n$H_{\\rm{III}} = 0.77 \\pm 0.01$ for definition without intraday\npattern, respectively. Quite similar results are obtain for\nlogarithmic prices where $H_{\\rm{I}} = 0.89 \\pm 0.01$ for definition\nI, $H_{\\rm{II}} = 0.91 \\pm 0.01$ for definition II, $H_{\\rm{III}} =\n0.77 \\pm 0.01$ for definition III, and $H_{\\rm{III}} = 0.76 \\pm\n0.01$ for definition III without intraday pattern. The two Hurst\nindexes for definitions I and II are higher than their counterparts\non the London Stock Exchange where ``even time'' is adopted\n\\cite{Mike-Farmer-2007-JEDC}. It is interesting to note that the\npresence of intraday pattern does not introduce distinguishable\ndifference in the Hurst index and the two indexes for definition III\nare also very close to those of average spreads in the Brazilian\nstock market and on the New York Stock Exchange where real time is\nused\n\\cite{Plerou-Gopikrishnan-Stanley-2005-PRE,Cajueiro-Tabak-2007-PA}.\nDue to the large number of data used in the analysis, we argue that\nthe bid-ask spreads investigated exhibit significant long memory.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig-DFA.eps}\n\\caption{Detrended fluctuation function $F(\\ell)$ for the spreads\nobtained from three definition with linear prices. The curves have\nbeen shifted vertically for clarity.} \\label{Fig:dfa}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\section{Multifractal analysis}\n\\label{s1:MFA}\n\nIn this section, we investigate whether the time series of bid-ask\nspread obtained from definition III possesses multifractal nature.\nThe classical box-counting algorithm for multifractal analysis is\nutilized and described below\n\\cite{Halsey-Jensen-Kadanoff-Procaccia-Shraiman-1986-PRA}.\n\nConsider the spread time series $S(t)$, $t=1,2,\\cdots,N$. First, we\ndivide the series $S(t)$ into $N_\\ell$ disjoint segments with the\nsame length $\\ell$, where $N_\\ell = [N\/\\ell]$. Each segment can be\ndenoted as $S_v$ such that $S_v(i) = S(l + i)$ for\n$1\\leqslant{i}\\leqslant{\\ell}$, and $l = (v - 1)\\ell$. The sum of\n$S_v$ in each segment is calculated as follows,\n\\begin{equation}\n\\Gamma(v,\\ell) = \\sum_{i = 1}^{\\ell}{S_v(i)},\n~~v=1,2,\\cdots,N_\\ell~.\n \\label{Eq:MF_F}\n\\end{equation}\nWe can then calculate the $q$th order partition function $\\Gamma(q;\n\\ell)$ as follows,\n\\begin{equation}\n\\Gamma(q; \\ell) = \\sum_{v = 1}^{N_\\ell}[\\Gamma(v,\\ell)]^q~.\n \\label{Eq:MF_Fq}\n\\end{equation}\nVarying the value of $\\ell$, we can determine the scaling relation\nbetween the partition function $\\Gamma(q; \\ell)$ and the time scale\n$\\ell$, which reads\n\\begin{equation}\n\\Gamma(q; \\ell) \\sim \\ell^{\\tau(q)}~.\n \\label{Eq:DFA_M_h}\n\\end{equation}\n\nFigure~\\ref{Fig:mffq} illustrates the power-law scaling dependence\nof the partition function $\\Gamma(q; \\ell)$ of the bid-ask spreads\nafter removing the intraday pattern in definition III for different\nvalues of $q$, where both linear prices and logarithmic prices are\ninvestigated. The continuous lines are the best linear fits to the\ndata sets. The collapse of the data points on the linear lines\nindicates evident power-law scaling between $\\Gamma(q; \\ell)$ and\n$\\ell$. The slopes $\\tau(q)$ of the fitted lines are $\\tau(-4) =\n-5.02 \\pm 0.01$, $\\tau(-2) = -3.01 \\pm 0.01$, $\\tau(0) = -1.01 \\pm\n0.01$, $\\tau(2) = 0.99 \\pm 0.01$, and $\\tau(4) = 2.98 \\pm 0.01$ for\nlogarithmic prices and $\\tau(-4) = -5.02 \\pm 0.01$, $\\tau(-2) =\n-3.01 \\pm 0.01$, $\\tau(0) = -1.01 \\pm 0.01$, $\\tau(2) = 0.99 \\pm\n0.01$, and $\\tau(4) = 2.98 \\pm 0.01$ for linear prices. We notice a\nnice relation $\\tau(q)=q-1$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=7cm]{Fig-MFDFA_Fs.eps}\n\\caption{Log-log plots of the partition function $\\Gamma(q; \\ell)$\nof the bid-ask spreads calculated from definition III with the\nintraday pattern removed for five different values of $q$. Both\nlinear and logarithmic prices are investigated (shown in the\nlegend). The markers stand for the results calculated from the real\ndata and the continuous lines are the best fits. The plots for $q =\n-2, 0, 2$, and $4$ are shifted upwards for clarity.}\n \\label{Fig:mffq}\n\\end{center}\n\\end{figure}\n\nQuantitatively similar results are obtained when the intraday\npattern is not removed. The scaling exponents are $\\tau(-4) = -5.03\n\\pm 0.01$, $\\tau(-2) = -3.01 \\pm 0.01$, $\\tau(0) = -1.01 \\pm 0.01$,\n$\\tau(2) = 0.99 \\pm 0.01$, and $\\tau(4) = 2.96 \\pm 0.01$ for\nlogarithmic prices and $\\tau(-4) = -5.03 \\pm 0.01$, $\\tau(-2) =\n-3.02 \\pm 0.01$, $\\tau(0) = -1.01 \\pm 0.01$, $\\tau(2) = 0.99 \\pm\n0.01$, and $\\tau(4) = 2.97 \\pm 0.01$ for linear prices. Again, we\nobserve that $\\tau(q)=q-1$.\n\nIn the standard multifractal formalism based on partition function,\nthe multifractal nature is characterized by the scaling exponents\n$\\tau(q)$. It is easy to obtain the generalized dimensions $D_q=\n{\\tau}(q)\/(q - 1)$\n\\cite{Grassberger-1983-PLA,Hentschel-Procaccia-1983-PD,Grassberger-Procaccia-1983-PD}\nand the singularity strength function $\\alpha(q)$, the multifractal\nspectrum $f(\\alpha)$ via the Legendre transform\n\\cite{Halsey-Jensen-Kadanoff-Procaccia-Shraiman-1986-PRA}:\n$\\alpha(q) = {\\rm d}{\\tau}(q)\/{\\rm d}q$ and $f(q) = q{\\alpha} -\n{\\tau}(q)$.\n\nFigure~\\ref{Fig:falpha:tau} shows the multifractal spectrum\n$f(\\alpha)$ and the scaling function $\\tau(q)$ in the inset for\nlinear and logarithmic prices. One finds that the two $\\tau(q)$\ncurves are linear and $\\tau(q)=q-1$, which is the hallmark of the\npresence of monofractality, not multifractality. The strength of the\nmultifractality can be characterized by the span of singularity\n$\\Delta\\alpha=\\alpha_{\\max}-\\alpha_{\\min}$. If $\\Delta\\alpha$ is\nclose to zero, the measure is almost monofractal. The maximum and\nminimum of $\\alpha$ can be reached when $q\\to\\pm\\infty$, which can\nnot be achieved in real applications. However, $\\Delta\\alpha$ can be\napproximated with great precision with mediate values of $q$. The\nsmall value of $\\Delta\\alpha$ shown in Fig.~\\ref{Fig:falpha:tau}\nindicates a very narrow spectrum of singularity. Indeed, one sees\nthat $f(\\alpha)\\approx1$ and $\\alpha\\approx1$ for all values of $q$.\nWe thus conclude that there is no multifractal nature in the bid-ask\nspread investigated.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig-MFDFA_fAlpha_tau.eps}\n\\caption{Multifractal function $f({\\alpha})$ of the spreads in\ndefinition III with the intraday pattern removed. Inset: Scaling\nexponents ${\\tau}(q)$ of partition functions as a function of $q$.\nFor clarity, the $\\tau(q)$ curve for logarithmic price is shifted\nupwards by 1.}\n \\label{Fig:falpha:tau}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{s7:conclusion}\n\nThe bid-ask spread defined by the difference of the best ask price\nand the best bid price is considered as the benchmark of the\ntransaction cost and a measure of the market liquidity. In this\npaper, we have carried out empirical investigations on the\nstatistical properties of the bid-ask spread using the limit-order\nbook data of a stock SZ000001 (Shenzhen Development Bank Co., LTD)\ntraded on the Shenzhen Stock Exchange within the whole year of 2003.\nThree different definitions of spread are considered based on event\ntime at transaction level and on fixed interval of real time.\n\nThe distributions of spreads at transaction level decay as power\nlaws with tail exponents well below 3. In contrast the average\nspread in real time fulfils the inverse cubic law for different time\nintervals $\\Delta{t}= 1$, $2$, $3$, $4$, and $5$ min. We have\nperformed the detrended fluctuation analysis on the spread and found\nthat the spread time series exhibits evident long-memory, which is\nin agreement with other stock markets. However, an analysis using\nthe classic textbook box-counting algorithm does not provide\nevidence for the presence of multifractality in the spread time\nseries. To the best of our knowledge, this is the first time to\ncheck the presence of multifractality in the spread.\n\nOur analysis raises an intriguing open question that is not fully\naddressed. We have found that the spread possesses a\nwell-established intraday pattern composed by a large L-shape and a\nsmall L-shape separated by the noon closing of the Chinese stock\nmarket. This feature will help to understand the cause of the wide\nspread at the opening of the market, which deserves further\ninvestigation.\n\n\\begin{acknowledgement}\nWe are grateful to Dr. Tao Wu for fruitful suggestions. This work\nwas partially supported by the National Natural Science Foundation\nof China (Grant No. 70501011) and the Fok Ying Tong Education\nFoundation (Grant No. 101086).\n\\end{acknowledgement}\n\n\\bibliographystyle{h-elsevier3}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStudies in linguistics are often characterized by small and unbalanced data sets.\nThis is particularly true for quantitative sociolinguistic studies that investigate the use of non-standard speech forms since such forms often occur at much lower token frequencies than standard speech forms (e.g.\\ Buschfeld 2020).\nThe present paper outlines ways to statistically meet the problems of low token frequencies and unbalanced data sets.\nDrawing on child data from Singapore and England, collected by one of the authors, we discuss a new resampling method which builds on the traditional method of repeated random undersampling but enhances it in two important respects.\nWe evaluate the outcome of undersampling by means of two criteria, i.e.\\ interpretability of the results and predictive power.\nWe therefore assess potential models from both their linguistic and statistical perspectives, which, to our best knowledge, is so far an unprecedented approach.\n\nIn this paper, we first introduce the linguistic study, i.e.\\ its data and research questions, in Section~\\ref{sec:2}. Section~\\ref{sec:3} outlines the general statistical approach we take, i.e.\\ classification rules and, in particular, conditional inference trees, and discusses their power and value against the background of notions such as prediction and interpretation, unbalanced classes, and re- and undersampling. In Section~\\ref{sec:int}, we conflate the statistical and linguistic perspectives. We show how and why interpretability and predictive power are crucial criteria for linguistic studies and how these criteria can be jointly utilized. To this end, we introduce our newly developed statistical approach {\\bf PrInDT}. The results of our study are presented and discussed in Section~\\ref{sec:5} on the basis of the three best trees generated by {\\bf PrInDT}. Last but not least, we look into ensembles of our {\\bf PrInDT}-generated trees and show how they can add important information to the findings of individual trees. We finish with some general statements that point the way ahead for both statistical and linguistic research.\n\n\\section{Linguistic Hypotheses and Data}\\label{sec:2}\n\nThe data were collected for a large-scale research project investigating the acquisition of English as a first language (L1) in Singapore (Buschfeld 2020).\nSingapore English (SingE) is one of the most extensively researched second language (L2) varieties of English.\nHowever, for an ever-increasing number of children, it is (one of) their home language(s). According to the Straits Times, one of the leading newspapers in Singapore, the number of primary one students who speak mostly English at home has risen to around 70\\% in all three major ethnic groups in Singapore, i.e.\\ the Chinese, Indian, and Malay (Chan, 2020).\nTo place L1 SingE on the map of L1 varieties of English and investigate the linguistic characteristics of SingE as well as the acquisitional route Singaporean children take in their linguistic development, we compare the data to data collected from monolingual and bi-\/multilingual children in England.\nThe relevant hypotheses for the study on subject pronoun realization read as follows (Buschfeld 2020, 85):\n\\begin{itemize}\n\\itemsep0pt\n\\item[]{\\bf Hypothesis 1}: All children (from Singapore and from England) drop subject pronouns in their early acquisitional stages. The amount of zero subjects is higher in the Singapore group due to the differences in input (SingE behaves similar to null-subject languages, at least at a surface level).\n\\item[]{\\bf Hypothesis 2}: In later acquisitional stages, the Singaporean children drop subject pronouns at a much higher rate than children acquiring English in a traditional native English setting. The children from England ultimately realize subjects as obligatory sentence constituents.\n\\end{itemize}\nTo sum up our hypotheses, we expect to find both age-related and ethnicity-related differences for the realization of subject pronouns.\n\nThe data were elicited systematically in video-recorded task-directed dialogue between the researcher and the children, consisting of several parts: a grammar elicitation task, a story retelling task, elicited narratives, and free interaction. The recorded material was orthographically transcribed and manually coded for the realization of subject pronouns (realized vs. zero):\n\nThe following {\\bf examples} illustrate the respective SingE variants (Buschfeld 2020, 144):\n\\begin{itemize}\n\\itemsep0pt\n\\item[1.]\tResearcher: [\\ldots ] what do you do with your friends? Do you play with them?\\\\\nChild: [ {\\bf \\o} I] Play with them. Sometimes drawing. [\\ldots]\\\\\nChild: Sometimes [ {\\bf \\o} WE] play some fun things.\n\\item[2.]\tChild: I think in MH370, I think they can find because [ {\\bf \\o} IT] is easy to go there [\\ldots].\n\\end{itemize}\n\n\\noindent The forms in Examples 1 and 2 are variably used by children acquiring English as an L1 in Singapore. They are also typical for adult SingE and thus part of the input the children receive.\n\nThe aim of our analysis is to find prediction rules for the use of subject pronouns (realized vs. zero) by means of extra- and intralinguistic variables.\nThe {\\bf extralinguistic variables} considered as independent variables in the statistical analysis are \\textsc{ethnicity} (ETH), \\textsc{age} (AGE), \\textsc{sex} (SEX), \\textsc{linguistic background} (LiBa), and \\textsc{mean length of utterance} (MLU). \\textsc{pronoun} (PRN) is taken into account as the {\\bf intralinguistic variable}.\nWe aim to determine whether any of these variables has a statistically significant influence on the results.\nAll in all, we extracted 6146 tokens of the values of the subject pronoun variable, each with the full set of extra- and intra-linguistic variables. 528 of these are realized as zero pronouns.\n\n\\section{Statistical Modeling}\\label{sec:3}\n\nStatistical modeling has long found its way into quantitative linguistics, but still, we would like to first address one of the central questions raised by such endeavors:\nWhat can we expect from statistical modeling?\n\nFrom a statistical perspective, the answer is quite straightforward. Statistical analyses can yield two kinds of insights. Description elicits information from the data sample. Inference generalizes from a sample to a population.\nBoth description and inference aim at the {\\bf interpretation} of properties of either the observed data (description) or of a more general population (inference).\nInference often comprises the {\\bf prediction} of such properties for the more general population.\n\nIn order to approximate the relationships between a class variable (e.g.\\ pronoun type with possible values `zero' and `realized') and influential variables like extra- and intra-linguistic variables, {\\bf classification rules} can be employed. Such rules can be used either for description or inference.\nNote that models never depict reality, but are only a hopefully adequate approximation thereof.\nTo construct such models, we need \\, n \\, observations (tokens) comprising a value of the class variable and of all influential variables ({\\bf learning sample}).\n\n\\subsection{Conditional Inference Trees}\\label{page:lowfit}\n\nA variety of different methods exists for constructing classification rules.\nIn the present paper, we concentrate on so-called {\\bf decision trees}, which\ncombine different decisions on individual variables into a set of rules.\nThe type of tree most often used in linguistics is called {\\bf conditional inference tree} ({\\bf c-tree}) (cf., e.g., Tagliamonte\/Baayen, 2012; Gries, 2020). Such trees include only those decisions that significantly improve the correct prediction about the realization of the dependent variable and employ statistical tests for inference. Therefore, such trees are geared towards generalization \/ prediction.\n\nAn example of a c-tree for subject realization can be found in Figure~\\ref{fig:simple}.\nThe tree has to be interpreted as follows: decision variables and p-values of the individual decisions are presented in the `node' corresponding to the decision. e.g.\\ PRN and p < 0.001 in node 1. The terminal nodes indicate the relative frequency of each class in the node, e.g.\\ 20\\% of zeros and 80\\% of realized subject pronouns in node 4. The class with the highest frequency is predicted for each token assigned to the node. This is of particular interest for tokens not used for rule construction, i.e.\\ {\\bf prediction}. Note that the tree in Figure~\\ref{fig:simple} predicts only one of the two classes, namely `realized', i.e.\\ the zero class is never predicted. The statistical repercussions and approaches to meet this problem are discussed in the following section.\n\n\\begin{figure}[h]\n\\centering\n\\vspace{-0.6cm}\n\\includegraphics[scale=0.4]{zeroTreeSimple.pdf}\\\\\n\\raggedright prediction: realized \\hspace{0.9cm} realized \\hspace{1.7cm} realized \\hspace{1.65cm} realized\\\\\n\\caption{Simple c-tree for pronoun realization.}\n\\label{fig:simple}\n\\end{figure}\n\n\\subsection{Evaluating trees with unbalanced classes}\\label{subsec:eval}\n\nEvaluation of classification rules is indispensable to assess their suitability.\nThe standard evaluation measure for classification rules is\\\\\n\n\\hspace{0.6cm}{\\bf overall accuracy} = (no. of correct predictions) \/ (no. of tokens) .\\\\\n\n\\noindent For the tree in Figure~\\ref{fig:simple}, the overall accuracy is 0.914, i.e.\\ very high. This is caused by the strong imbalance between the two classes. The zero class is extremely small (528 tokens) whereas the larger class contains more than ten times as many tokens (5618). Although the smaller class is never predicted, the accuracy is very high but clearly misleading. Two things can be done to improve the predictive power of the model: the model quality has to be assessed in a different way and the high imbalance between the classes has to be resolved (cf. Section~\\ref{subsec:under}).\nFor unbalanced classes model quality can be assessed more adequately:\n\\begin{equation*}\n\\begin{split}\n\\text{\\bf balanced accuracy} & =  \\text{mean of accuracies in the two classes} \\\\\n& =  {(\\text{accuracy(zero)} + \\text{accuracy(realized)})\/2} .\n\\end{split}\n\\end{equation*}\n\\noindent For our example, the value of the {balanced accuracy} = ({\\bf 0} + 1.0)\/2 = {\\bf 0.5} confirms that the high overall accuracy is misleading, since the average of the accuracies of the two classes amounts to only 50\\%. The low balanced accuracy and the fact that the smaller class is totally ignored in the predictions render the tree unsuitable for linguistic application.\n\n\\subsection{Resampling}\n\nThe suitability of a tree-like classification rule strongly depends on whether it is to be used for description or prediction. Do we only want to describe the data set at hand or do we want to predict the behavior of an unknown subject?\n\nIf classification rules are used for prediction, their {\\bf predictive power} has to be assessed. In order to simulate prediction, we divide our learning sample (i.e.\\ the full data set) into training and test sets. This is called {\\bf resampling}. Different resampling methods exist. In this paper, we utilize two resampling methods: hold-out and subsampling.\n\nThe simplest way of resampling is called {\\bf hold-out}, where one part of the observations (say 1\/3) is randomly withheld from rule construction (training). Subsequently, a rule is constructed on the training set. This rule is then tested on the hold-out and the test accuracy is used for evaluation. The disadvantage of this method is that the hold-out is generated only once and may not be fully representative of the data set.\n\n{\\bf Subsampling} solves this problem by repeating this procedure B times (e.g., B = 200). Then, for each observation (token) the class most often predicted by the B different rules is compared to the observed value of the class variable in the actual data set. The resulting accuracy is used as the evaluation measure for the predictive power of all the B trees generated in the subsampling process.\nIn the following, we introduce a subsampling method which meets the problem of unbalanced classes.\n\n\\subsection{Undersampling}\\label{subsec:under}\n\nIn order to construct a classification rule with acceptable balanced accuracy, we stochastically reduce the larger class, i.e.\\ we apply the method of undersampling.\n{\\bf Undersampling} takes, in the simplest case, the full sample of the smaller class together with a small sample of the larger class for constructing a rule (training). This procedure is repeated B times (cf. Weiss 2004).\n\nFor the {\\bf evaluation of undersampling}, we combine goodness of fit and predictive power. The idea is to compute the overall balanced accuracy on all observations of both classes.\nFor the smaller class, the accuracy is computed on the training sample (fit),\ni.e.\\ the full data sample of the small class.\nFor the larger class, it is computed not\nonly on the training sample (fit), but also on the hold-out from rule construction (prediction).\nThis way, rules from different training samples can be adequately compared and the best\nrule can be identified by looking for the highest balanced accuracy.\n\nIn our example, the smaller class (zero) comprises 528 tokens. For undersampling, we decided to subsample the larger class for training. To create a data set with roughly equally-sized classes, we randomly selected 9\\% of the larger class, i.e.\\ we kept 506 tokens for training. We repeated the process of random subsampling B = 1001 times.\n\n\\section{Statistics meets Linguistics}\\label{sec:int}\n\n\\subsection{Limits on interpretability}\\label{page:uninter}\n\nUnfortunately, undersampling might produce trees with high accuracies that are linguistically uninterpretable.\nFigure~\\ref{fig:uninter} shows such a tree with a balanced accuracy of 0.6898. We later see that this balanced accuracy is only marginally (< 0.02) lower than the best interpretable tree we found (Figure~\\ref{fig:best}). The problem in the uninterpretable tree in Figure~\\ref{fig:uninter} is the split in node 4. Here, the Singapore Chinese (\\textit{S\/C}) cluster with the ancestral English (\\textit{E\/a}) children. This contradicts the linguistic, typologically motivated expectation\nto find differences in pronoun realization between these two\ngroups. First of all, the Chinese languages the Singapore Chinese\nchildren speak as additional languages all allow for zero subjects and it is widely\naccepted that languages acquired bilingually influence each other\nstructurally. Therefore, the Chinese children have a stronger inclination towards zero\nsubjects than monolingual children growing up in England.\nFor the Indian children, the linguistic situation is not as clear since the Indian languages spoken by the children do not as unambiguously prefer the zero subject option as the Chinese languages. If any significant differences are to be found between the\ngroups, these clearly have to be found between the Chinese children and the\nmonolingual ancestral English ones. Therefore, a combination of the two ethnicities in the same group of values has to be excluded.\n\n\\begin{figure}[thp]\n\\centering\n\\hspace*{-0.5cm}\\includegraphics[angle=90,scale=0.5]{treeT7.pdf}\\\\\n\\caption{Uninterpretable tree from undersampling.}\n\\label{fig:uninter}\n\\end{figure}\n\n\nTo make sure that the resulting tree is linguistically interpretable, we have identified the following combinations of values of our variables as not permissible in splits of interpretable trees:\\\\\n\n\\noindent {\\bf Excluded groups of values in our linguistic example}: \\\\\nETH == \\{E\/a, S\/C\\},\\\\ETH == \\{E\/a, E\/m, S\/C\\},\\\\ ETH == \\{E\/a, E\/migr, S\/C\\},\\\\\nETH == \\{E\/a, E\/m, E\/migr, S\/C\\},\\\\\nETH == \\{E\/a, E\/m, E\/migr, S\/C, S\/I\\},\\\\ ETH == \\{E\/a, E\/m, E\/migr, S\/C, S\/m\\},\\\\\nETH == \\{E\/a, E\/m, S\/C, S\/I\\},\\\\ ETH == \\{E\/a, E\/migr, S\/C, S\/I\\},\\\\\nETH == \\{E\/a, S\/C, S\/I\\},\\\\ MLU == \\{1, 3\\}\\\\\n\n\\noindent The above restrictions mainly concern \\textsc{ethnicity} excluding all combinations entailing \\{E\/a, S\/C\\} for the reasons stated above. Moreover, splits with MLU are restricted by excluding the combination of MLU groups 1 and 3. The decision to exclude this combination is again linguistically motivated. Group 1 contains children of 45 months and younger. Language acquisition research has shown that in this age group, children often produce subjectless sentences, not only in Singapore but also in England (e.g. Roeper \\& Rohrbacher 2000). The children in group 3 are all 7 years and older. The children from England in this group have thus clearly moved beyond the zero-subject stage. The children from Singapore still use zero subjects, due to the reasons outlined earlier, but to a lesser extent. This is also due to the decreasing acquisition effect but also since they enter formal schooling and thus more standard language exposure from age 7 onwards. Therefore, the children in MLS groups 1 and 3 show completely different inclinations towards zero subjects.\n\n\\subsection{Interpretability meets predictive power}\n\nOn the basis of the above deliberations, we are able to generate trees with both predictive power and interpretability if the following criteria are met:\n\\begin{itemize}\n\\itemsep0pt\n\\item[] {\\bf Interpretability}: no excluded grouping is included in the tree.\n\\item[] {\\bf Predictive power}: the balanced accuracy is `high enough', e.g.\\ greater than a threshold $c$.\n\\end{itemize}\n\n\\noindent We propose two kinds of tree-like approaches that meet these criteria. As a first step, we use that tree from undersampling which is interpretable and has the highest balanced accuracy. As a second step, we employ {\\bf ensembles of rules} that meet the criteria. Such ensembles are defined in the following way:\n\nThe B rules from undersampling are assessed by the two criteria defined above.\nTo assess predictive power, in our example, the threshold $c$ is set to the value of the median of balanced accuracies in undersampling.\nAll rules from undersampling meeting the two criteria are combined to a so-called {\\bf ensemble} of rules.\nFor each observation, the ensemble of rules predicts that class which is most often `predicted' by the individual trees in the ensemble. These predictions are the basis for calculating balanced accuracies.\n\n\\section{Results}\\label{sec:5}\n\nAll results were created by means of the software R (R Core Team 2019). The decision trees were generated by the R-package `party'. Note that, since the significance limit of 0.01 is used, the trees are smaller than for the standard limit of 0.05. This facilitates straightforward interpretability in order to illustrate our line of argumentation. The following results are generated by means of the R-function PrInDT, that implements our newly developed procedure.\\footnote{The source code of this function can be requested from the first author by e-mail.}\n\nIn Sections 5.1 and 5.2, we show and discuss the three best trees meeting the two criteria discussed in the previous sections. In Section 5.3, we present the results from the ensembles.\n\n\\subsection{The three best individual trees}\\label{subsec:5.1}\n\nThe three best trees presented in Figures~\\ref{fig:best}--\\ref{fig:best3} have balanced accuracies of 0.702, 0.701, and 0.700, respectively. We argue that, for a linguistic study, this is at least an acceptable if not high accuracy (cf. Winter 2020, 77, for a similar line of argumentation). Furthermore, all three trees fulfill the interpretability criterion and reveal a similar structure.\n\nIn all three trees, the most prominent split separates the pronoun \\textit{it} from the rest (\\textit{he, I, she, they, we, you} singular\\footnote{Plural \\textit{you} does not exist in the data.}). If further significant splits are modeled for \\textit{it} (Figures~\\ref{fig:best} and \\ref{fig:best2}), the variables ETH and AGE are involved. The ETH split mainly separates the Singaporean children from the children growing up in England. The second-best tree (Figure~\\ref{fig:best2}) further shows an AGE-related split (node 10) for the ancestral English and mixed-English children. Those children approximately four and a half years and younger use considerably more zero pronouns than the older ones in these groups (Figure~\\ref{fig:best2}; nodes 11 and 12).\n\nFor the realization of the remaining pronouns, MLU, LiBa, and AGE play a role in all three trees. MLU is the most prominent variable for the rest of the pronouns and splits the children into the outliers (OL)\\footnote{Four of the children in the Singapore cohort were identified as outliers on the basis of the MLU-categorization since the syntactic complexity of their utterances does not match the expected age-related performance.} and group 1, i.e. those children younger than 45 months, and the two groups of older children (MLU groups 2 and 3). In all three trees, no further splits occur for groups 2 and 3. The group 1 and OL children are split into further subgroups by LiBa. ETH determines significant splits of the data in the best and second-best tree (Figures~\\ref{fig:best} and \\ref{fig:best2}) and SEX has a significant impact in the best and third-best tree (Figure~\\ref{fig:best} and \\ref{fig:best3}).\n\nLooking into the extreme frequencies in the manifestations of our dependent variable, the trees reveal the following: In the best tree (Figure~\\ref{fig:best}), the highest share of zero pronouns can be found for multilingual male children (node 10). In the second-best tree (Figure~\\ref{fig:best2}), all multilingual children show a particularly high share of zero pronouns (node 8). The third-best tree (Figure~\\ref{fig:best3}) again shows that male gender and multilingual linguistic background are predictors for high rates of zero pronouns, but only for the pronouns \\textit{he, I, she}, and \\textit{they} (node 8).\n\nExtreme frequencies of realized subject pronouns can be found for the following sub-rules. In the best tree (Figure~\\ref{fig:best}), monolingual children older than 28 months exclusively use the realized variant (node 8). Furthermore, the realized variant clearly dominates for pronoun \\textit{you} singular (node 4).\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[angle=90,scale=0.5]{tree1.pdf}\\\\\n\\caption{Best interpretable tree from undersampling.}\n\\label{fig:best}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[angle=90,scale=0.48]{tree2.pdf}\\\\\n\\caption{Second-best interpretable tree from undersampling.}\n\\label{fig:best2}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[angle=90,scale=0.48]{tree3.pdf}\\\\\n\\caption{Third-best interpretable tree from undersampling.}\n\\label{fig:best3}\n\\end{figure}\n\\noindent In the second-best tree (Figure~\\ref{fig:best2}), again, monolingual children older than 28 months exclusively use the realized variant (node 7). MLU groups 2 and 3 also clearly favor the realized variant (node 3). In the third-best tree (Figure~\\ref{fig:best3}), those male children that are multilingual use exceptionally high proportions of realized \\textit{we} and \\textit{you} singular (node 7) and again, MLU groups 2 and 3 make exceptionally high use of the realized variant (node 3).\n\n\\subsection{Discussion of trees}\\label{subsec:5.2}\n\nTo summarize, the three best trees from our {\\bf PrInDT} analysis show that all considered variables (ETH, AGE, SEX, LiBa, MLU, PRN) have an effect on the data to different degrees and in different combinations of sub-rules. The intralinguistic variable PRN has the strongest effect on the results in all three trees. Zero pronouns most prominently occur for the pronoun \\textit{it}, this means either for the semantically empty dummy \\textit{it} (\"It is raining\") or the referential \\textit{it} (\"My cat, it is black\").\n\nMLU, LiBa, and AGE play a role in all three trees in very similar ways: MLU always determines a split into group 1 and the outliers on the one hand and the groups 2 and 3 on the other, with the former being more prone to using zero forms than the latter.\nAGE plays a different role for the pronoun \\textit{it} and the group of all other pronouns. For \\textit{it}, AGE splits the English ancestral and mixed children at 55 months. For the rest of the pronouns and the OL and young children (group 1), AGE splits the children at 28 months. Therefore, hypothesis 1 is confirmed: AGE and MLU are strong predictors for the realization of subject pronouns.\n\nHypothesis 2 is also confirmed: ETH clearly guides the realization of subject pronouns in that Singaporean children make significantly stronger use of zero subjects throughout the age groups (at least in two of the three best trees).\n\nFurther findings that go beyond our initial hypotheses have revealed that SEX has a significant impact on the realization of subject pronouns in the best and the third-best trees, with the male children showing a stronger inclination towards using the zero variant than the female children. This is not surprising either, as it is a common finding in sociolinguistic research that women often use more standard speech forms than men.\n\nAs we have seen, the trees not only have satisfactory if not high balanced accuracies, they are also fully interpretable and consistent in their linguistic findings. This nicely illustrates the methodological power of our {\\bf PrInDT} approach. Note that a clear-cut threshold for linguistic studies does not exist and depends on the individual data set (cf. Section~\\ref{subsec:5.3}).\n\n\\subsection{Ensembles of decision trees}\\label{subsec:5.3}\n\nLet us now consider four possible ensembles of interpretable trees with acceptable predictive power.\\footnote{Note that random forests, as often used in linguistics, are another type of ensembles.} Figure~\\ref{fig:ba} illustrates the distribution of the balanced accuracies of all 1001 trees from undersampling. Note that the range of balanced accuracies is quite small (from 0.6755 to 0.7021). In particular, this means that none of the trees has a totally unacceptable predictive power.\n\n\\begin{figure}[H]\n\\centering\n\\vspace{-0.9cm}\n\\includegraphics[width=0.75\\textwidth]{histba.pdf}\\\\\n\\vspace{-0.3cm}\n\\caption{Histogram of balanced accuracies from undersampling; the median is represented by the red vertical line.}\n\\label{fig:ba}\n\\end{figure}\n\\vspace{-0.2cm}\n\\noindent In this paper, we consider\n\\begin{itemize}\n\\itemsep0pt\n\\item[a)] the ensemble comprising only the three interpretable trees with the highest balanced accuracies (discussed in Section~\\ref{subsec:5.1}),\n\\item[b)] the ensemble consisting of {\\bf all} interpretable trees from undersampling,\n\\item[c)] the ensemble including all interpretable trees with a predictive power greater than the median of the balanced accuracies of all trees from undersampling (c = 0.6904; cf. Figure~\\ref{fig:ba}).\n\\end{itemize}\n\n\\begin{table}[H]\n\\centering\n\\caption{No. of trees and balanced accuracies for the different ensembles}\n\\label{tab:acc}\n\\vspace{0.3cm}\n    \\begin{tabular}{l|rr}\n    ensemble & trees & accuracy \\\\\n    \\hline\n    a) & 3   & 0.699\\\\\n    b) & 941 & 0.690\\\\\n    c) & 137 & 0.696\n  \\end{tabular}\n\\end{table}\n\n\\noindent Table~\\ref{tab:acc} summarizes the ensembles and their respective accuracies. Note that only 60 of the 1001 trees turned out to be uninterpretable (cf. b)). Overall, ensemble a), which comprises the three interpretable trees with the best balanced accuracies, comes with the {\\bf best balanced accuracy} (0.699) of the ensembles.\nThe second-best ensemble with an accuracy of 0.696 is ensemble c), which consists of 137 trees, which have greater balanced accuracies than the median of all balanced accuracies.\nEnsemble b), which consists of all interpretable trees from undersampling, includes \n941 trees and has the lowest accuracy of the ensembles (0.690). Still, its accuracy is close to the median. \n\n\\subsection{Discussion of ensembles}\n\nFinally, we would like to argue that ensembles can add important information to the findings of individual trees. Small samples in resampling represent only small parts of the data set. Therefore, trees based on these samples also only depict parts of the reality. Ensembles ensure that as many of these parts are considered as determined by the number of repetitions (in our example 3, 941, and 137). It is further interesting to investigate how strongly the accuracies vary when the underlying samples are changing as this gives some indication of the robustness of the findings.\n\nFrom a linguistic perspective, working with ensembles makes sense for three reasons. Similar to the first statistical reason outlined above, working with trees trained on resampled data sets runs the risk of neglecting important parts of the data. Ensembles are more representative of the data set and thus of the overall population. Furthermore, randomly selected ensembles mirror the randomness of the data collected for a linguistic study. In both cases, only parts of the overall population can be captured with basically everybody or every token having an equal chance of being selected. Moreover, ensembles account for the nature of sociolinguistic variation, since it is neither fully random nor deterministic. As the last 60 years of sociolinguistic research have clearly shown, linguistic variation is guided by intra- and extralinguistic factors but at the same time shows inter- and even intraspeaker variation (e.g. Meyerhoff 2019, 12). Our study once more confirms these findings.\nEnsembles, therefore, aptly represent the strong systematicity of linguistic data on the one hand while at the same time leaving room for randomness and variation.\nThe one shortcoming of ensembles for linguistic studies, however, is that the interpretability of the model is blurred since it is difficult to synthesize the interpretation of a high number of trees.\n\n\\section{Conclusion}\n\nIn this paper, we have seen that statistical models and their evaluation should be integral parts of linguistic analyses, but only if the following crucial points are taken into consideration:\n\\begin{itemize}\n\\itemsep0pt\n\\item[1.]\t Perfectly interpretable trees with allegedly clear and strong linguistic findings can be misleading due to extremely low predictive power (see Figure~\\ref{fig:simple}).\n\\item[2.]\tTrees with high accuracies can be linguistically uninterpretable (see Figure~\\ref{fig:uninter}).\n\\end{itemize}\nTo study and illustrate this finding, we have developed and presented the statistical method {\\bf PrInDT} ({\\bf Pr}ediction and {\\bf In}terpretation with {\\bf D}ecision {\\bf T}rees),\nwhich prescribes interpretability and high predictive power as properties for trees generated by resampling. Either the individual interpretable trees with highest predictive power or a whole ensemble of interpretable trees with `high enough' predictive power are then used for the prediction of relationships between linguistic variables. We successfully applied this method to find prediction rules for the use of subject pronouns (realized vs. zero) depending on extra- and intralinguistic variables.\n\nWe would like to conclude with some general remarks on statistical modeling, not only but in particular in linguistics:\n\\begin{description}\n\\itemsep0pt\n\\item[\\bf Predictive power:] {\\bf Bad models can never be interpreted.} Therefore, evaluation is a crucial step of model building. Significant splits alone do not qualify a model for usage. Interpretation of models with a bad fit or predictive power intrinsically carries the risk of drawing the wrong conclusions.\n\\item[\\bf Interpretability of trees and ensembles:] {\\bf Use interpretable models only.} Restricting models by means of prior knowledge, i.e.\\ findings from earlier studies, or on the basis of theoretical assumptions enhances the interpretability of trees and the homogeneity of ensembles. This way, all trees are in accordance with prior knowledge or theory. If, despite of this, an ensemble turns out to have low predictive power, then our data do not match the underlying theoretical assumptions and the model cannot be interpreted (see Predictive power above).\n\\end{description}\n\n\\noindent Last but not least, we would like to stress that our method {\\bf PrInDT} is not restricted to undersampling but can be applied to any ensemble, e.g. to random forests and other more general resampling methods. Therefore, as a next step we would also like to withhold (small) parts of the smaller class from the training of the trees. This way, the smaller class is also predicted for some tokens. Moreover, in order to facilitate the interpretation of ensembles, we will implement a ranking of the importance of predictors in ensembles, similar to the ranking of variables in random forests.\n\n\\section{References}\n\\begin{hangparas}{.25in}{1}\nBuschfeld, S. (2020), `Children's English in Singapore: Acquisition, Properties, and Use', Routledge.\n\nChan, M. (2020), `English, mother tongue and the Singapore identity', The Straits Times, url: https:\/\/www.straitstimes.com\/opinion\/english-mother-tongue-and-the-spore-identity.\n\nGries, S.Th. (2020), `On classification trees and random forests in corpus linguistics: Some words of caution and suggestions for improvement', Corpus Linguistics and Ling. Theory 16(3): 617--647.\n\nMeyerhoff, M. (2019), `Introducing Sociolinguistics', 3rd edn; Routledge.\n\nR Core Team (2019), `R: A language and environment for statistical\n  computing'. R Foundation for Statistical Computing, https:\/\/www.R-project.org\/.\n\nRoeper, T., Rohrbacher, B. (2000). `Null subjects in early child language and the economy of projection'. In S. Powers, C. Hamann (Eds.), Acquisition of Scrambling and Cliticization, 345--397. Berlin: Springer.\n\nTagliamonte, S.A., Baayen, R.H. (2012), `Models, forests, and trees of York English: Was\/were variation as a case study for statistical practice', Language Variation and Change 24, 135--178.\n\nWeiss, G.M. (2004), `Mining with rarity: A unifying framework', ACM SIGKDD Explorations 6: 7--19.\n\nWinter, B. (2020), `Statistics for Linguists. An Introduction using R', Routledge.\n\\end{hangparas}\n\n\\end{document} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \n\nThe understanding to non-perturbative dynamics of supersymmetric gauge theory\nhas made rapid progress in recent years following the \nseminal contribution by Seiberg and Witten$\\cite{sw}$, combining the ideas \nof holomorphicity$\\cite{sei}$ and duality$\\cite{mo}$.\nThe web of arguments leading to the explicit results consists of a skillful\ncombination of perturbative and nonperturbative arguments, formal\nconsiderations and physical reasoning. It should be checked by explicit\ncomputations, whenever possible, that no unexpected failure of these \narguments occurs. \nIn a recent work we have made an investigation in this\ndirection$\\cite{ref4}$ and this paper is intended as a review.\n\nThe starting point in Seiberg and Witten's work is the low-energy\neffective action of an $N=2$ supersymmetric Yang-Mills theory \nwith the gauge group $SU(2)$ of the following form,\n\\begin{eqnarray}\n\\Gamma =\\frac{1}{16\\pi}\\mbox{Im}{\\int}d^4xd^2{\\theta}d^2\\widetilde{\\theta}\n\\left[ \\frac{1}{2}{\\tau}{\\Psi}^2\n+\\frac{i}{2\\pi}{\\Psi}^2\\,\\ln\\frac{{\\Psi}^2}{{\\Lambda}^2}\n+\\sum_{n=1}^{\\infty}A_n\n\\left(\\frac{{\\Lambda}^2}{{\\Psi}^2}\\right)^{2n}{\\Psi}^2\\right] \\, ,\n\\label{eq1}\n\\end{eqnarray}\nwhere $\\displaystyle \\tau=\\frac{\\theta}{2\\pi}+\\frac{4\\pi i}{g^2}$\nis the modular parameter and ${\\Psi}$ the $N=2$ chiral superfield describing\nthe light degrees of freedom. The logarithmic term represents \nthe one-loop perturbative result and was first obtained by Di Vecchia \net al.${\\cite{dmnp}}$ in a calculation where they coupled the gauge \nsuperfield to an $N=2$ matter supermultiplet and integrated out the \nlatter. Subsequently, Seiberg$\\cite{sei}$ used the anomalous transformation \nbehaviour under $U(1)_R$ and holomorphicity to argue that the full \nlow-energy effective action should take the form ({\\ref{eq1}), where \nthe infinite series arises from nonperturbative instanton contributions.\nThe Seiberg-Witten solution $\\cite{sw}$ gives the explicit form of this \npart of $\\Gamma$.\n\nThe form ({\\ref{eq1}) has been confirmed by calculations in $N=1$ superspace\nand in harmonic superspace, extending the result to nonleading terms in the\nnumber of derivatives $\\cite{dgr,pw,ov,ma,ke}$. Independent confirmation has\nbeen obtained from $M$-theory $\\cite{oo}$. \n\nOur intention is\nto check the perturbative part of the effective \naction in Wess-Zumino gauge by a very down-to-earth calculation. In the Higgs \nphase of the theory, the $SU(2)$ gauge symmetry breaks down to $U(1)$, \nand the super-Higgs mechanism splits the supermultiplet into a massive \none and a massless one. The effective action of the massless fields \nshould be obtained by integrating out the heavy fields. In comparison\nwith other approaches, our method is quite conventional and is along\nthe lines of the standard definition of the low-energy effective theory. \n\nIt should be emphasized that this conventional calculation is very\ncomplicated. Even this modest programme we cannot carry out fully. \nWhat we have actually accomplished is the \ncomputation of the heavy fermion determinant. Reassuringly, we find that the form\n(\\ref{eq1}) is reproduced. Although no unexpected surprises were unearthed\nby our calculation, we still hope that it has some pedagogical value in\nshowing explicitly how the effective action arises. \n\nThe outline of this review is as follows. \nIn section 2 we describe the model and exhibit the Higgs mechanism. Section 3\ncontains the computation of the heavy fermion determinant using the constant\nfield approximation. The detailed calculations of the fermion eigenvalues and\ntheir degeneracies, which contain certain subtle points, are given \nin Appendix B. In section 4 we present a discussion of the results. \nIn the pedagogical vein of this paper, we give in Appendix A the \ncomponent form of the low-energy effective action (\\ref{eq1}).    \n\n\n\\section{Splitting of $N=2$ Supermultiplet }\n\n\nThe classical action of \n$N=2$ supersymmetric $SU(2)$ Yang-Mills theory is$\\cite{dhd}$, \n\\begin{eqnarray}\nS&=& {\\int}d^4 x\\left[-\\frac{1}{4}G_{\\mu\\nu}^aG^{\\mu\\nu a}\n+D_{\\mu}{\\varphi}^{\\dagger a}D^{\\mu}{\\varphi}^a\n+i\\overline{\\psi}^a{\\gamma}^{\\mu}D_{\\mu}^{ab}{\\psi}^b\\right.\\nonumber\\\\[2mm]\n&&+\\left. \\frac{ig}{\\sqrt{2}}{\\epsilon}^{abc}\\overline{\\psi}^c\n[(1-{\\gamma}_5){\\varphi}^a\n+(1+{\\gamma}_5){\\varphi}^{\\dagger a}]{\\psi}^b\n+\\frac{g^2}{2}{\\epsilon}^{abc}{\\epsilon}^{ade}\n{\\varphi}^b{\\varphi}^{\\dagger c}{\\varphi}^{d}{\\varphi}^{\\dagger e}\\right]\\,,\n\\label{eq2} \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nG_{\\mu\\nu}^a&=&{\\partial}_{\\mu}K_{\\nu}^a-{\\partial}_{\\nu}K_{\\mu}^a\n-g{\\epsilon}^{abc}K_{\\mu}^bK_{\\nu}^c,~~\n D_{\\mu}{\\varphi}^a={\\partial}_{\\mu}{\\varphi}^a\n-g {\\epsilon}^{abc}K_{\\mu}^b{\\varphi}^c,\n\\nonumber\\\\[2mm]\n{\\varphi}^a&=&\\frac{1}{\\sqrt{2}}(S^a+iP^a),~~ \n{\\varphi}^{\\dagger a}=\\frac{1}{\\sqrt{2}}(S^a-iP^a), ~~a=1,2,3\\,.\\nonumber\n\\end{eqnarray}\n The bosonic part of the action (\\ref{eq2}) \nis just the Georgi-Glashow model\nin the Bogomol'nyi-Prasad-Sommerfield (BPS) limit. \nIn addition to the fermionic term and Yukawa interaction term, this action\n has the scalar potential\n\\begin{eqnarray}\nV(\\varphi)=-\\frac{g^2}{2}{\\epsilon}^{abc}{\\epsilon}^{ade}\n{\\varphi}^b{\\varphi}^{\\dagger c}\n{\\varphi}^d{\\varphi}^{\\dagger e}\n{\\equiv}g^2\\mbox{Tr}\\left([\\varphi,{\\varphi}^{\\dagger}]\\right)^2\\,.\n\\end{eqnarray}\nThe unbroken supersymmetry requires that in the ground state the\nscalar potential must vanish, which leads to\n\\begin{eqnarray}\n[\\varphi,{\\varphi}^{\\dagger}]&=&0\\,.\n\\label{eq:h2}\n\\end{eqnarray}\n(\\ref{eq:h2}) means that ${\\varphi}^{\\dagger}$ and ${\\varphi}$ \ncommute. Since the theory is gauge invariant, we can always choose$\\cite{dhd}$  \n\\begin{eqnarray}\n\\langle S^a \\rangle =v{\\delta}^{a3},~~\\langle P^a \\rangle =0 \\, ,\n\\label{eq:c1}\n\\end{eqnarray}\nwhere $v$ is a real constant.   \nFor $v{\\neq}0$ the theory is in the Higgs phase\nand exhibits a spontaneous breaking of the gauge symmetry.\nIn a unitary gauge \n\\begin{eqnarray}\nS^{T}=\\left(0,0, S+v \\right)  \\, .\n\\label{ug} \n\\end{eqnarray}\nThe corresponding classical Lagrangian can be written as follows,\n\\begin{eqnarray}\n{\\cal L}={\\cal L}_V+{\\cal L}_S+{\\cal L}_P+{\\cal L}_F+{\\cal L}_Y\\, ,\n\\end{eqnarray}\nwhere ${\\cal L}_V$, ${\\cal L}_S$, ${\\cal L}_P$, ${\\cal L}_F$ and ${\\cal L}_Y$\ndenote respectively the vector field, the scalar field,\nthe scalar potential, \nthe fermionic and the Yukawa interaction parts, \n\\begin{eqnarray}\n{\\cal L}_V&=&  -\\frac{1}{4}({\\partial}^{\\mu}A^{\\nu}\n-{\\partial}^{\\nu}A^{\\mu}) ({\\partial}_{\\mu}A_{\\nu}-{\\partial}_{\\nu}A_{\\mu}) \n-\\frac{1}{2}({\\partial}_{\\mu}W_{\\nu}^+-{\\partial}_{\\nu}W_{\\mu}^+) \n({\\partial}^{\\mu}W^{-\\nu}-{\\partial}^{\\nu}W^{-\\mu})\\nonumber  \\\\[2mm]\n&&-i g[({\\partial}^{\\mu}W_{\\nu}^+W_{\\mu}^-\n-{\\partial}^{\\mu}W_{\\nu}^-W_{\\mu}^+)A^{\\nu}+\n({\\partial}_{\\nu}W_{\\mu}^-W^{+\\mu}\n-{\\partial}_{\\nu}W^+_{\\mu}W^{\\mu -})A^{\\nu}\\nonumber\\\\[2mm]\n&&+({\\partial}^{\\mu}A^{\\nu}\n-{\\partial}^{\\nu}A^{\\mu})W_{\\mu}^+W_{\\nu}^-]\n+g^2(-W_{\\mu}^+W^{-\\mu}A_{\\nu}A^{\\nu}+W_{\\mu}^+W_{\\nu}^-A^{\\mu}A^{\\nu})\n\\nonumber\\\\\n&&+ \\frac{g^2}{2}W^{+\\mu}W^{-\\nu}(W^+_{\\mu}W^-_{\\nu}-W^-_{\\mu} W^+_{\\nu});\n\\label{eq8}\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\cal L}_S&=&\\frac{1}{2}\\partial_{\\mu}P\\partial^{\\mu}P\n+\\frac{1}{2}\\partial_{\\mu}S\\partial^{\\mu}S+\\partial_{\\mu}P^+\\partial^{\\mu}P^-\n+igA^{\\mu}(\\partial_{\\mu}P^-P^+-\\partial_{\\mu}P^+P^-)\\nonumber\\\\\n&&+igP(\\partial^{\\mu}P^+W_{\\mu}^--\\partial^{\\mu}P^-W_{\\mu}^+)+\nig\\partial^{\\mu}P(W_{\\mu}^+P^--W_{\\mu}^-P^+)+g^2P^2W^{+\\mu}W^-_{\\mu}\\nonumber\\\\\n&&+g^2(S+v)^2W^{+\\mu}W^-_{\\mu}+g^2A^{\\nu}A_{\\nu}P^{+}P^--g^2(W_{\\mu}^+P^--\nW_{\\mu}^-P^+)A^{\\mu}P\\nonumber\\\\\n&&-\\frac{g^2}{2}(W^{\\mu +}P^--W^{\\mu -}P^+)^2.\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\cal L}_P=g^2(S+v)^2P^+P^-,\n\\end{eqnarray}\nwhere the various quantities are defined as follows:\n\\begin{eqnarray}\n&& W_{\\mu}^+{\\equiv}\\frac{1}{\\sqrt{2}}(K_{\\mu}^{1}\n-iK_{\\mu}^{2})~,~\nW_{\\mu}^-{\\equiv}\\frac{1}{\\sqrt{2}}(K_{\\mu}^{1}\n+iK_{\\mu}^{2})~,~K_{\\mu}^3 {\\equiv}A_{\\mu} \\,;\\nonumber\\\\\n&&P^+ {\\equiv}\\frac{1}{\\sqrt{2}}(P^1-iP^2)~,~P^-{\\equiv}\n \\frac{1}{\\sqrt{2}}(P^1+iP^2)~,~P^3{\\equiv}P.\n\\end{eqnarray}\nThe above Lagrangians  clearly show that $W_{\\mu}^{\\pm}$ and $P^{\\pm}$\nbecome massive with mass $m{\\equiv}|gv|$ while $A_{\\mu}$, $S$ and $P$\nremain massless.\n\nUp to some total derivative terms, the bosonic part of the Lagrangian \ncan be rewritten in the following form:\n\\begin{eqnarray} \n{\\cal L}_B&=&{\\cal L}_V+{\\cal L}_S+{\\cal L}_P\\nonumber\\\\\n&=&-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}+\\frac{1}{2}\\partial_{\\mu}P\\partial^{\\mu}P\n+\\frac{1}{2}\\partial_{\\mu}S\\partial^{\\mu}S+\\frac{1}{2}\nW^{+\\mu}\\left[\n{\\eta}_{\\mu\\nu}D^{\\dagger\\alpha}D_{\\alpha}-D_{\\nu}^{\\dagger}D_{\\mu}\n-igF_{\\mu\\nu}\\right]W^{-\\nu}\\nonumber\\\\[2mm]\n&&+\\frac{1}{2}W^{-\\mu}\\left[\n{\\eta}_{\\mu\\nu}D^{\\alpha}D^{\\dagger}_{\\alpha}\n-D_{\\nu}D_{\\mu}^{\\dagger}+igF_{\\mu\\nu}\\right]W^{+\\nu}\n+g^2[P^2+(S+v)^2]W_{\\mu}^+W^{\\mu -}\\nonumber\\\\\n&&+\\frac{1}{2}P^+(-\\partial^{\\mu}\\partial_{\\mu}\n+2igA_{\\mu}\\partial^{\\mu}+g^2A_{\\mu}A^{\\mu})P^-\n+\\frac{1}{2}P^-(-\\partial^{\\mu}\\partial_{\\mu}\n-2igA_{\\mu}\\partial^{\\mu}+g^2A_{\\mu}A^{\\mu})P^+\n\\nonumber\\\\\n&&+\\frac{1}{2}W_{\\mu}^+(-igP\\partial^{\\mu}+ig\\partial^{\\mu}P-g^2A_{\\mu}P)P^-\n+\\frac{1}{2}P^-(igP\\partial^{\\mu}+2ig\\partial^{\\mu}P-g^2A_{\\mu}P)W_{\\mu}^+\n\\nonumber\\\\\n&&+\\frac{1}{2}P^+(-2ig\\partial^{\\mu}P-igP\\partial^{\\mu}-g^2A^{\\mu}P)W_{\\mu}^-\n+\\frac{1}{2}W_{\\mu}^-(-ig\\partial^{\\mu}P+igP\\partial^{\\mu}-g^2A^{\\mu}P)P^+\n\\nonumber\\\\\n&&+ \\frac{g^2}{2}W^{+\\mu}W^{-\\nu}(W^+_{\\mu}W^-_{\\nu}-W^-_{\\mu} W^+_{\\nu})\n-\\frac{g^2}{2}(W^+_{\\mu}P^--W^-_{\\mu}P^+)^2\\nonumber\\\\ \n&=&-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}\n+\\partial_{\\mu}{\\phi}^{*}\\partial^{\\mu}{\\phi}\n+\\frac{1}{2}W^{+\\mu}{\\Delta}_{\\mu\\nu}W^{-\\nu}\n+\\frac{1}{2}W^{-\\mu}{\\Delta}^{\\dagger}_{\\mu\\nu}W^{+\\nu}\n+\\frac{1}{2}P^+\\Delta P^-\\nonumber\\\\\n&&+\\frac{1}{2}P^-\\Delta^{\\dagger}P^+\n+\\frac{1}{2}W^{+\\mu}\\Delta_{\\mu}P^-\n+\\frac{1}{2}P^-\\widetilde{\\Delta}_{\\mu}W^{+\\mu}\n+\\frac{1}{2}P^+\\widetilde{\\Delta}_{\\mu}^{\\dagger}W^{-\\mu}\n+\\frac{1}{2}W^{-\\mu}\\Delta_{\\mu}^{\\dagger}P^+\\nonumber\\\\\n&&+ \\frac{g^2}{2}W^{+\\mu}W^{-\\nu}(W^+_{\\mu}W^-_{\\nu}-W^-_{\\mu} W^+_{\\nu})\n-\\frac{g^2}{2}(W^+_{\\mu}P^--W^-_{\\mu}P^+)^2 \\, ,\n\\end{eqnarray} \nwhere \n\\begin{eqnarray} \n{\\Delta}_{\\mu\\nu}\n&{\\equiv}&{\\eta}_{\\mu\\nu}D^{\\dagger\\alpha}D_{\\alpha}-D_{\\nu}^{\\dagger}D_{\\mu}\n-igF_{\\mu\\nu}+g^2|\\sqrt{2}\\phi+v|^2{\\eta}_{\\mu\\nu},\\nonumber\\\\[2mm]\n{\\Delta}_{\\mu\\nu}^{\\dagger}\n&=&{\\eta}_{\\mu\\nu}D^{\\alpha}D_{\\alpha}^{\\dagger}\n-D_{\\nu}D_{\\mu}^{\\dagger}\n+igF_{\\mu\\nu}+g^2|\\sqrt{2}\\phi+v|^2{\\eta}_{\\mu\\nu};\\nonumber\\\\[2mm]\n\\Delta_{\\mu}&{\\equiv}&-igP\\partial^{\\mu}+ig\\partial^{\\mu}P-g^2A_{\\mu}P,~\n\\widetilde{\\Delta}_{\\mu}{\\equiv}igP\\partial^{\\mu}\n+2ig\\partial^{\\mu}P-g^2A_{\\mu}P,\n\\nonumber\\\\\n\\Delta_{\\mu}^{\\dagger}&=&-ig\\partial^{\\mu}P+igP\\partial^{\\mu}-g^2A^{\\mu}P,\n~\\widetilde{\\Delta}_{\\mu}^{\\dagger}\n=-2ig\\partial^{\\mu}P-igP\\partial^{\\mu}-g^2A^{\\mu}P;\\nonumber\\\\\n\\Delta &=& -\\partial^{\\mu}\\partial_{\\mu}\n+2igA_{\\mu}\\partial^{\\mu}+g^2A_{\\mu}A^{\\mu},~\n\\Delta^{\\dagger}=-\\partial^{\\mu}\\partial_{\\mu}\n-2igA_{\\mu}\\partial^{\\mu}+g^2A_{\\mu}A^{\\mu}; \\nonumber\\\\\nD_{\\mu}&=&\\partial_{\\mu}-igA_{\\mu}, ~D_{\\mu}^{\\dagger}\n=\\partial_{\\mu}+igA_{\\mu}; ~\\phi{\\equiv}\\frac{1}{\\sqrt{2}}(S+iP).\t\n\\end{eqnarray}\n\nTo explicitly show that the spinor fields split into massive and massless ones, \nwe need some operations on ${\\cal L}_F$ and ${\\cal L}_Y$. The fermionic \npart is\n\\begin{eqnarray}\n{\\cal L}_F&=&i\\overline{\\psi}^1{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\psi}^1+\ni\\overline{\\psi}^2{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\psi}^2\n+i\\overline{\\psi}^3{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\psi}^3\\nonumber\\\\[2mm]\n&&+ \\frac{g}{\\sqrt{2}}\\overline{\\psi}^1(W_{\\mu}^+-W_{\\mu}^-)\n{\\gamma}^{\\mu}{\\psi}^3+ig \\overline{\\psi}^1A_{\\mu}{\\gamma}^{\\mu}{\\psi}^2\n\\nonumber\\\\[2mm]\n&&+ig \\overline{\\psi}^1A_{\\mu}{\\gamma}^{\\mu}{\\psi}^2+\n \\frac{ig}{\\sqrt{2}}\\overline{\\psi}^2\n{\\gamma}^{\\mu}(W_{\\mu}^++W_{\\mu}^-){\\psi}^3\n\\nonumber\\\\[2mm]\n&&-\\frac{ig}{\\sqrt{2}}\\overline{\\psi}^3{\\gamma}^{\\mu}\n(W_{\\mu}^++W_{\\mu}^-){\\psi}^2-\n\\frac{g}{\\sqrt{2}}\\overline{\\psi}^3{\\gamma}^{\\mu}\n(W_{\\mu}^+-W_{\\mu}^-){\\psi}^1 \\, .\n\\label{fp}\n\\end{eqnarray}\nAs for the Yukawa part, we first write it in terms of chiral spinors,\n\\begin{eqnarray}\n{\\cal L}_Y=i\\sqrt{2}gf^{abc}\\overline{\\psi}_L^c{\\varphi}^a{\\psi}_R^b+i\n\\sqrt{2}gf^{abc}\\overline{\\psi}_R^c{\\varphi}^{\\dagger  a}{\\psi}_L^b \\, ,\n\\label{yu}\n\\end{eqnarray}\nwhere ${\\psi}_L=\\displaystyle \\frac{1}{2}(1-{\\gamma}_5){\\psi}$ and \n${\\psi}_R=\\displaystyle \\frac{1}{2}(1+{\\gamma}_5){\\psi}$.\nIn the unitary gauge, Eq.(\\ref{yu}) becomes\n\\begin{eqnarray}\n{\\cal L}_Y&=&ig(\\sqrt{2}{\\phi}+v)\n(\\overline{\\psi}_L^2{\\psi}_R^1-\\overline{\\psi}_L^1{\\psi}_R^2)\n+ig (\\sqrt{2}{\\phi}^*+v) (\\overline{\\psi}_R^2{\\psi}_L^1\n-\\overline{\\psi}_R^1{\\psi}_L^2)\\nonumber\\\\\n&&+\\frac{g}{\\sqrt{2}}(P^++P^-)\\left[(\\overline{\\psi}_R^3{\\psi}_L^2\n-\\overline{\\psi}_R^2{\\psi}_L^3)-(\\overline{\\psi}_L^3{\\psi}_R^2\n-\\overline{\\psi}_L^2{\\psi}_R^3)\\right]\\nonumber\\\\\n&&+\\frac{ig}{\\sqrt{2}}(P^+-P^-)\\left[(\\overline{\\psi}_R^1{\\psi}_L^3\n-\\overline{\\psi}_R^3{\\psi}_L^1)-(\\overline{\\psi}_L^1{\\psi}_R^3\n-\\overline{\\psi}_L^3{\\psi}_R^1)\\right] \\, .\n\\end{eqnarray}\nWith the combination\n\\begin{eqnarray}\n{\\Psi}_1{\\equiv} \\frac{1}{\\sqrt{2}}({\\psi}^1+i{\\psi}^2)\\,,\\,\n   {\\Psi}_2{\\equiv} \\frac{1}{\\sqrt{2}}({\\psi}^1-i{\\psi}^2)\\, ,\\,\n{\\Psi}{\\equiv}{\\psi}^3 \\, , \n\\end{eqnarray}\n${\\cal L}_F$ and ${\\cal L}_Y$ can be formulated in these new fields, \n\\begin{eqnarray}\n{\\cal L}_Y&=&\n-g\\overline{\\Psi}_1\\left[\\frac{1}{\\sqrt{2}}(1-\\gamma_5)\\phi\n+\\frac{1}{\\sqrt{2}}(1+\\gamma_5)\\phi^*+v\\right]{\\Psi}_1\\nonumber\\\\\n&&+g\\overline{\\Psi}_2 \\left[\\frac{1}{\\sqrt{2}}(1-\\gamma_5)\\phi\n+\\frac{1}{\\sqrt{2}}(1+\\gamma_5)\\phi^*+v\\right]{\\Psi}_2\n\\nonumber\\\\\n&&- igP^+\\overline{\\Psi}\\gamma_5\\Psi_1+igP^-\\overline{\\Psi}\\gamma_5\\Psi_2\n-ig\\overline{\\Psi}_1 \\gamma_5\\Psi P^-+ig\\overline{\\Psi}_2\\gamma_5\\Psi P^+ \\, ,\n\\\\[2mm]\n{\\cal L}_F&=&i\\overline{\\Psi}_1{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\Psi}_1+\ni\\overline{\\Psi}_2{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\Psi}_2\n+i\\overline{\\Psi}{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\Psi}\\nonumber\\\\[2mm]\n&&+g\\overline{\\Psi}_1{\\gamma}^{\\mu}A_{\\mu}{\\Psi}_1-\ng\\overline{\\Psi}_2{\\gamma}^{\\mu}A_{\\mu}{\\Psi}_2 \\nonumber\\\\[2mm]\n&&+g\\overline{\\Psi}_2{\\gamma}^{\\mu}W_{\\mu}^+{\\Psi}-\ng\\overline{\\Psi}_1{\\gamma}^{\\mu}W_{\\mu}^-{\\Psi}\\nonumber\\\\[2mm]\n&&-g\\overline{\\Psi}{\\gamma}^{\\mu}W_{\\mu}^+{\\Psi}_1+\ng\\overline{\\Psi}{\\gamma}^{\\mu}W_{\\mu}^-{\\Psi}_2\\, .\n\\end{eqnarray}\nSo now the whole classical action is given by the Lagrangian\n\\begin{eqnarray}\n{\\cal L}&=&-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}+\\partial_{\\mu}\\phi^*\n\\partial^{\\mu}\\phi +i\\overline{\\Psi}{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\Psi}+\n\\frac{1}{2}W^{+\\mu}{\\Delta}_{\\mu\\nu}W^{-\\nu}\n+\\frac{1}{2}W^{-\\mu}{\\Delta}^{\\dagger}_{\\mu\\nu}W^{+\\nu}\\nonumber\\\\\n&&+\\frac{1}{2}P^+\\Delta P^-+\\frac{1}{2}P^-{\\Delta}^{\\dagger}P^+\n+\\frac{1}{2}W^{+\\mu}\\Delta_{\\mu}P^-+\\frac{1}{2}P^-\n\\widetilde{\\Delta}_{\\mu}W^{+\\mu}\n\\nonumber\\\\\n&&+\\frac{1}{2}P^+\\widetilde{\\Delta}_{\\mu}^{\\dagger}W^{-\\mu}\n+\\frac{1}{2}W^{-\\mu}\\Delta_{\\mu}^{\\dagger}P^+ \n+\\overline{\\Psi}_1\\Delta_F{\\Psi}_1\n+\\overline{\\Psi}_2\\widetilde{\\Delta}_F{\\Psi}_2\n\\nonumber \\\\[2mm]\n&&- igP^+\\overline{\\Psi}\\gamma_5\\Psi_1+igP^-\\overline{\\Psi}\\gamma_5\\Psi_2\n-ig\\overline{\\Psi}_1 \\gamma_5\\Psi P^-+ig\\overline{\\Psi}_2\\gamma_5\\Psi P^+ \n\\nonumber\\\\[2mm]\n&&+ g\\overline{\\Psi}_2{\\gamma}^{\\mu}W_{\\mu}^+{\\Psi}-\ng\\overline{\\Psi}_1{\\gamma}^{\\mu}W_{\\mu}^-{\\Psi}\n-g\\overline{\\Psi}{\\gamma}^{\\mu}W_{\\mu}^+{\\Psi}_1+\ng\\overline{\\Psi}{\\gamma}^{\\mu}W_{\\mu}^-{\\Psi}_2\\nonumber\\\\\n&&+ \\frac{g^2}{2}W^{+\\mu}W^{-\\nu}(W^+_{\\mu}W^-_{\\nu}-W^-_{\\mu} W^+_{\\nu})\n-\\frac{g^2}{2}(W^+_{\\mu}P^--W^-_{\\mu}P^+)^2 \\, \n\\label{eq20}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\Delta_F&{\\equiv}&i{\\gamma}^{\\mu}D_{\\mu}\n-\\frac{g}{\\sqrt{2}}(1-\\gamma_5)\\phi-\\frac{g}{\\sqrt{2}}(1+\\gamma_5)\\phi^*-gv,\n\\nonumber\\\\\n\\widetilde{\\Delta}_F&{\\equiv}&i{\\gamma}^{\\mu}D_{\\mu}^{\\dagger}\n+\\frac{g}{\\sqrt{2}}(1-\\gamma_5)\\phi +\\frac{g}{\\sqrt{2}}(1+\\gamma_5)\\phi^*+gv.\n\\end{eqnarray}\n\n\\section{Low-energy Effective Action: Calculation of the Fermionic \nDeterminant in Constant Field Approximation}\n\nThe standard definition of the low-energy effective action is given by \n\\begin{eqnarray}\n\\mbox{exp}\\left\\{i\\,{\\Gamma}_{\\rm eff}\n[A_{\\mu},{\\phi},{\\Psi},\\overline{\\Psi}]\\right\\}\n{\\equiv}{\\int} {\\cal D} W_{\\mu}^+ {\\cal D} W_{\\mu}^-\n{\\cal D}\\overline{\\Psi}_1{\\cal D}\\overline{\\Psi}_2 {\\cal D}{\\Psi}_1\n{\\cal D}{\\Psi}_2 {\\cal D}P^+{\\cal D}P^- \\mbox{exp}\n\\left[i\\, \\int \\, d^4 x\\, {\\cal L}\\right]\\, .\n\\end{eqnarray}\nAt tree level\n\\begin{eqnarray}\n{\\Gamma}_{\\rm eff}^{(0)}=S_{\\rm tree}={\\int}\\,d^4x\\left[ -\\frac{1}{4}\nF_{\\mu\\nu}F^{\\mu\\nu}\n+{\\partial}^{\\mu}{\\phi}^*{\\partial}_{\\mu}{\\phi}+\ni\\overline{\\Psi}{\\gamma}^{\\mu}{\\partial}_{\\mu}{\\Psi}\\right]\\,.\n\\end{eqnarray}\nAt one-loop level, the integration over the heavy modes \nwill lead to the determinants of the dynamical operators. \nIn practical calculation we cannot evaluate\nthe determinant exactly. Here we shall\nemploy a technique called constant field \napproximation to compute the determinant, which\nwas invented by Schwinger${\\cite{sch}}$ \nand later was used in\nin \\cite{dmnp} and \\cite{ddds} to extract \nthe anomaly term in $N=2$ supersymmetric Yang-Mills\ntheory and the one-loop effective action of the supersymmetric $CP^{N-1}$ model.\nTo apply this method\n we first rewrite the the quadratic part of the\n classical action (\\ref{eq20}) as \n\\begin{eqnarray}\nS_{\\rm quad}=S_{\\rm tree}+{\\int}d^4x\\left(\\Phi^{\\dagger} M_{bb} \\Phi+\n\\overline{\\widetilde{\\Psi}}M_{fb}\\Phi +\\Phi^{\\dagger}M_{bf}\\widetilde{\\Psi}\n+\\overline{\\widetilde{\\Psi}}M_{ff}\\widetilde{\\Psi}\\right),\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\Phi&{\\equiv}&\\left(\\begin{array}{c} W^{-\\mu}\\\\W^{+\\mu}\\\\P^-\\\\P^+\\end{array}\\right),\n~~\\widetilde{\\Psi}{\\equiv}\\left(\\begin{array}{c}\\Psi_1\\\\ \\Psi_2\\\\\n\\Psi_1\\\\ \\Psi_2\\\\  \\end{array}\\right);\\nonumber\\\\\nM_{bb}&{\\equiv}&\\frac{1}{2}\n\\left(\\begin{array}{cccc}\n{\\Delta}_{\\mu\\nu} & 0 &\\Delta_{\\mu} & 0\\\\\n0 & {\\Delta}^{\\dagger}_{\\mu\\nu} &0 &\\Delta_{\\mu}^{\\dagger} \\\\\n\\widetilde{\\Delta}_{\\nu}^{\\dagger} & 0 &\\Delta &0 \\\\\n0 & \\widetilde{\\Delta}_{\\nu} & 0 & \\Delta^{\\dagger}\n\\end{array}\\right);\\nonumber\\\\\nM_{fb}&{\\equiv}&\\frac{1}{2} \\left(\\begin{array}{cccc}\n-g\\gamma_{\\mu}\\Psi & 0 & -ig\\gamma_5 \\Psi & 0 \\\\\n0 & g\\gamma_{\\mu}\\Psi & 0 & ig\\gamma_5 \\Psi  \\\\\n-g\\gamma_{\\mu}\\Psi & 0 &  -ig\\gamma_5 \\Psi & 0\\\\\n0 & g\\gamma_{\\mu}\\Psi & 0 &  ig\\gamma_5 \\Psi\\\\\n \\end{array}\\right); \\nonumber\\\\\nM_{bf}&{\\equiv}&\\frac{1}{2} \\left(\\begin{array}{cccc}\n-g\\overline{\\Psi}\\gamma_{\\mu} & 0 & -g\\overline{\\Psi}\\gamma_{\\mu} & 0  \\\\\n0 &g\\overline{\\Psi}\\gamma_{\\mu} & 0 & g\\overline{\\Psi}\\gamma_{\\mu} \\\\\n-ig\\overline{\\Psi} \\gamma_5 & 0 &  -ig\\overline{\\Psi} \\gamma_5 & 0\\\\\n0 & ig\\overline{\\Psi} \\gamma_5 & 0 & ig\\overline{\\Psi} \\gamma_5\n\\end{array}\\right);\\nonumber\\\\\nM_{ff}&{\\equiv}&\\frac{1}{2} \\left(\\begin{array}{cccc}\n\\Delta_F & 0 & 0 & 0\\\\\n0 & \\widetilde{\\Delta}_F & 0 & 0 \\\\\n0 & 0 & \\Delta_F & 0 \\\\\n0 & 0 & 0 &\\widetilde{\\Delta}_F\n \\end{array}\\right).\n\\label{eq27x}\n\\end{eqnarray}\nUsing the standard formulas\n\\begin{eqnarray} \nI&=&\\int {\\cal D}b^{\\dagger}{\\cal D}b{\\cal D}\\overline{f}{\\cal D}f\\,\n\\mbox{exp}\\left[\\int (dx)\n\\left(b^{\\dagger}M_{bb}b+\\overline{f}M_{fb}b+b^{\\dagger}M_{bf}f\n+\\overline{f}M_{ff}f\\right)\\right] \\nonumber\\\\[2mm]\n&=&\\int{\\cal D}b^{\\dagger}{\\cal D}b{\\cal D}\\overline{f}{\\cal D}f\\,\n \\mbox{exp}\\left\\{\\int (dx) \\left[b^{\\dagger}(M_{bb}-M_{bf}M_{ff}^{-1}M_{fb})b\n\\right.\\right.\\nonumber\\\\\n&&+\\left.\\left.(\\overline{f}\n+b^{\\dagger}M_{bf}M_{ff}^{-1})M_{ff}(M_{ff}^{-1}M_{fb}b+f)\n\\right]\\right\\}\n\\nonumber\\\\[2mm]\n&=&\\det M_{ff}\\det{}^{-1}(M_{bb}-M_{bf}M_{ff}^{-1}M_{fb}); \\nonumber\\\\\n\\det M&=&\\mbox{exp}\\,\\mbox{Tr}\\,\\ln M,\n\\end{eqnarray}\n$b$ and $f$ representing the general \nbosonic and fermionic fields, respectively,\nwe get\n\\begin{eqnarray}\nZ[A, \\phi,\\Psi,\\overline{\\Psi}]\n&=&\\mbox{exp}\\left\\{i\\,{\\Gamma}_{\\rm eff}\n[A_{\\mu},{\\phi},{\\Psi},\\overline{\\Psi}]\\right\\}\n{\\equiv}{\\int} {\\cal D} W_{\\mu}^+ {\\cal D} W_{\\mu}^-\n{\\cal D}\\overline{\\Psi}_1{\\cal D}\\overline{\\Psi}_2 {\\cal D}{\\Psi}_1\n{\\cal D}{\\Psi}_2 \\mbox{exp}\\left[ iS \\right] \\nonumber\\\\\n&=& \\mbox{exp}\\left[iS_{\\rm tree}\\right]\\det M_{ff}\\det{}^{-1}(M_{bb}\n-M_{bf}M_{ff}^{-1}M_{fb})\\nonumber\\\\\n&=&\\mbox{exp}\\left[iS_{\\rm tree}+\\mbox{Tr}\\ln M_{ff}-\n\\mbox{Tr}\\ln(M_{bb}-M_{bf}M_{ff}^{-1}M_{fb})\\right];\\nonumber\\\\\n{\\Gamma}_{\\rm eff}&=&S_{\\rm tree}-i\\left[\\mbox{Tr}\\ln M_{ff}-\n\\mbox{Tr}\\ln(M_{bb}-M_{bf}M_{ff}^{-1}M_{fb})\\right].\n\\label{eq27} \n\\end{eqnarray}\n\nThe following task is to evaluate the above determinants. \nLet us first consider the fermionic \npart. Since $M_{ff}$  has the form of a reducible matrix,\n\\begin{eqnarray} \n\\det M_{ff}=\\frac{1}{16}(\\det\\Delta_F)^2(\\det\\widetilde{\\Delta}_F)^2\n=\\frac{1}{16}\\mbox{exp}[2(\\mbox{Tr}\\ln\\Delta_F\n+\\mbox{Tr}\\ln\\det\\widetilde{\\Delta}_F)].\n\\end{eqnarray}\nNow we switch on the constant field approximation to work out the eigenvalues\nand eigenvectors of the above operators and further evaluate the determinant.\nWe choose only the third components of the electric\nand magnetic fields to be the constants different from zero,\n\\begin{eqnarray} \n-E_3=F^{03}{\\neq}0, ~~B_3=F^{12}{\\neq}0,\n\\end{eqnarray}\nand $\\phi$ the non-vanishing constant field.\nConsequently, the potential becomes\n\\begin{eqnarray}                \nA^1=-F^{12}x_2, ~~A^3=-F^{30}x_0, ~~A^0=A^2=0.\n\\end{eqnarray}\nTo get the eigenvalues of the operators, it is necessary\nto rotate into Euclidean space,\n\\begin{eqnarray}\n&& x^4=x_4=-ix^0, ~~\\partial_0=\\frac{\\partial}{\\partial x^0}\n=i\\frac{\\partial}{\\partial x^4},\\nonumber\\\\\n&& f^{34}=f_{34}=iF^{30}, ~~f^{12}=f_{12}=F^{12}.\n\\end{eqnarray}\nLet us first consider $\\det \\Delta_F$.\nThe eigenvalue equation for $\\Delta_F$ is\n\\begin{eqnarray}\n\\Delta_F\\psi(x)=\\left[i\\gamma^{\\mu}D_{\\mu}-\\frac{g}{\\sqrt{2}}(1-\\gamma_5)\\phi\n-\\frac{g}{\\sqrt{2}}(1+\\gamma_5)\\phi^* -gv\\right]\\psi(x) \n=\\omega \\psi_1,\n\\label{eq32}\n\\end{eqnarray}\nwhere $\\psi$ is a four-component spinor wave function.\nIn order to get normalizable eigenstates, we consider the system in a\nbox of finite size $L$ in the $x_1$ and $x_3$ directions with periodic\nboundary conditions, so the eigenvector should be of the following form,\n\\begin{eqnarray}\n\\psi(x)=\\frac{1}{L}e^{ip_1{\\cdot}x_1}e^{ip_3{\\cdot}x_3}{\\chi}(x_2,x_4), \n\\nonumber\\\\\np_1=\\frac{2\\pi l}{L}, ~~p_3=\\frac{2\\pi k}{L},~~k,l=\\mbox{integers}.\n\\end{eqnarray}\nTo find the eigenvalues and eigenvectors, \nwe write the operators and the wave function in two-component forms,\n\\begin{eqnarray}\n\\Delta_F=\\left(\\begin{array}{cc} -g(\\sqrt{2}\\phi^*+v){\\bf 1} & \\Delta^-\\\\\n\\Delta^+ & -g(\\sqrt{2}\\phi +v){\\bf 1} \\end{array}\\right),~~~\n\\chi =\\left(\\begin{array}{c}\\chi_{1}\\\\ \\chi_{2}\\end{array}\\right),\n\\end{eqnarray}\nwhere ${\\bf 1}$ is the $2{\\times}2$ identity matrix and\n\\begin{eqnarray}\n\\Delta^{\\pm}=\\partial_4{\\pm}i\\left[\\sigma_1(\\partial_1+igf_{12}x_2)\n+\\sigma_2\\partial_2+\\sigma_3(\\partial_3+igf_{34}x_4)\\right].\n\\end{eqnarray}\nThe eigenvalue equation (\\ref{eq32}) \nis thus reduced to the following set of equations,\n\\begin{eqnarray}\n-g(\\sqrt{2}\\phi^*+v){\\chi}_{1}+ \\Delta^-{\\chi}_{2}\n&=&\\omega {\\chi}_{1},\\nonumber\\\\\n\\Delta^+{\\chi}_{1}-g(\\sqrt{2}\\phi +v){\\chi}_{2}&=&\\omega {\\chi}_{2},\n\\label{eq54}\n\\end{eqnarray}\nand now\n\\begin{eqnarray}\n\\Delta^{\\pm}=\\partial_4{\\mp}[\\sigma_1(p_1+gf_{12}x_2)\n-i\\sigma_2\\partial_2+\\sigma_3(p_3+gf_{34}x_4)].\n\\end{eqnarray}\nA detailed calculation and discussion of the eigenvalues \nare collected in Appendix B.\nWe obtain two series of eigenvalues,\n\\begin{eqnarray}\n\\omega_{\\pm}(m,n)\n=-g\\left[\\frac{(\\phi +\\phi^*)}{\\sqrt{2}} +v\\right]\n{\\pm}\\sqrt{\\frac{1}{2}g^2(\\phi -\\phi^*)^2-2mgf_{12}-2ngf_{34}},\n\\end{eqnarray} \nwhere for $m{\\geq}1$, $n{\\geq}1$ both eigenvalues are doubly degenerate,\nwhile $\\omega_{\\pm}(m.0)$ and $\\omega_{\\pm}(0,n)$ are nondegenerate, \nand for $m=n=0$, there exists only a nondegenerate eigenvalue \n$\\omega_-(0,0)$.\n\nIn a similar way we can solve the eigenvalue equation \n\\begin{eqnarray}\n\\widetilde{\\Delta}_F\\widetilde{\\psi}=\\left[i\\gamma^{\\mu}D_{\\mu}^{\\dagger}\n+\\frac{g}{\\sqrt{2}}(1-\\gamma_5)\\phi \n+\\frac{g}{\\sqrt{2}}(1+\\gamma_5)\\phi^*+gv \\right]\\widetilde{\\psi}\n=\\widetilde{\\omega}\\widetilde{\\psi},\n\\end{eqnarray}\n and obtain the eigenvalues,\n\\begin{eqnarray}\n\\widetilde{\\omega}_{\\pm}(m,n)\n=g\\left[\\frac{(\\phi +\\phi^*)}{\\sqrt{2}} +v\\right]\n{\\pm}\\sqrt{\\frac{g^2}{2}(\\phi -\\phi^* )^2-2m gf_{12}-2n gf_{34}}.\n\\end{eqnarray}\nThe degeneracies of $\\widetilde{\\omega}_{\\pm}(m,n)$, \n$\\widetilde{\\omega}_{\\pm}(m,0)$ and $\\widetilde{\\omega}_{\\pm}(0,n)$ with\n$m{\\geq}1$, $n{\\geq}1$ are the same as those of the $\\omega$s. There\n still only exists a nondegenerate eigenvalue $\\omega_-(0,0)$.    \n\nWith the above eigenvalues\n$\\mbox{Tr}\\ln\\Delta_F$ and $\\mbox{Tr}\\ln\\widetilde{\\Delta}_F$ can be computed\nstraightforwardly, \n\\begin{eqnarray}\n\\mbox{Tr}\\ln\\Delta_F=\\ln\\det\\Delta_F\n=\\ln\\left[\\Pi\\omega_{\\pm (lk)}(m,n)\\right]^r\n=\\sum_{l,k=-\\infty}^{+\\infty}\\sum_{m,n=0}^{\\infty} r\n\\ln\\omega_{\\pm (lk)}(m,n),\n\\end{eqnarray}\nwhere $r$ is the degeneracy of $\\omega_{\\pm}(m,n)$.\nDue to the relation $x_2=2\\pi l\/(gf_{12} L)$ and $x_4=2\\pi k\/(gf_{34} L)$,\nthe summation over the momenta $k$ and $l$ is actually equivalent to\nan integration over $x_2$ and $x_4$. Since the fields are constants,\nthis integration will yield only a Euclidean \nspace volume factor, which tends to infinity\nin the continuous limit ($L{\\rightarrow}\\infty$), \n\\begin{eqnarray}\n\\sum_{l,k}=\\frac{L^2}{4\\pi^2}g^2f_{12}f_{34}{\\int}dx_2dx_4\n=\\frac{V}{4\\pi^2}g^2f_{12}f_{34},\n\\end{eqnarray}\nwhile the Lagrangian will be well defined.\nConsider the degeneracy of each eigenvalue, we have\n\\begin{eqnarray}\n\\mbox{Tr}\\ln\\Delta_F&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\n\\left\\{\\left[\\ln\\omega_-(0,0)+\\sum_{m=1}^{\\infty}\\ln\\omega_+(m,0)\n+\\sum_{n=1}^{\\infty}\\ln\\omega_+(0,n)\\right.\\right.\\nonumber\\\\\n&+&\\left.\\left.2\\sum_{m,n=1}^{\\infty}\\ln\\omega_+(m,n)\n\\right] +\\left[\\sum_{m=1}^{\\infty}\\ln\\omega_-(m,0)+\n\\sum_{n=1}^{\\infty}\\ln\\omega_-(0,n)\n+2\\sum_{m,n=1}^{\\infty}\\ln\\omega_-(m,n)\n\\right]\\right\\}\\nonumber\\\\\n&=&\\frac{V}{4\\pi^2}ge^2f_{12}f_{34}\\left\\{\\ln\\omega_-(0,0)\n+\\sum_{m=1}^{\\infty}\\ln[\\omega_+(m,0)\\omega_-(m,0)]\n\\right.\\nonumber\\\\\n&&+\\left.\\sum_{n=1}^{\\infty}\\ln[\\omega_+(0,n)\\omega_-(0,n)]\n+2\\sum_{m,n=1}^{\\infty}\\ln\\left[\\omega_+(m,n)\\omega_-(m,n)\\right]\n\\right\\} \\nonumber\\\\ \n&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\n\\left\\{\\ln[-g(\\sqrt{2}\\phi +v)]+\n\\sum_{m=1}^{\\infty}\n\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)+2mgf_{12}]\\right.\\nonumber\\\\\n&&+\\sum_{n=1}^{\\infty}\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)\n+2ngf_{34}]\\nonumber\\\\\n&&+\\left.2\\sum_{m,n=1}^{\\infty}\n\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)+2mgf_{12}+2ngf_{34}]\\right\\}.\n\\end{eqnarray}\nSimilarly, we get\n\\begin{eqnarray}\n\\mbox{Tr}\\ln\\widetilde{\\Delta}_F\n&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\\left\\{\\ln\\left[g(\\sqrt{2}\\phi^*+v) \\right]\n+\\sum_{m=1}^{\\infty}\n\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)+2mgf_{12}]\\right.\\nonumber\\\\\n&&+\\sum_{n=1}^{\\infty}\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)+2ngf_{34}]\n\\nonumber\\\\\n&&+\\left.2\\sum_{m,n=1}^{\\infty}\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)\n+2mgf_{12}+2ngf_{34}]\\right\\}.\n\\end{eqnarray}\nThus we finally obtain\n\\begin{eqnarray}\n\\mbox{Tr}\\ln\\Delta_F +\\mbox{Tr}\\ln\\widetilde{\\Delta}_F\n&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\n\\left\\{\\ln\\left[\\frac{\\sqrt{2}\\phi^*+v }{\\sqrt{2}\\phi +v }\n\\right]\\right.\\nonumber\\\\\n&&+2\\sum_{m=1}^{\\infty} \\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)\n+2mgf_{12}]\\nonumber\\\\\n&&+2\\sum_{n=1}^{\\infty}\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)\n+2ngf_{34}]\\nonumber\\\\\n&&+ \\left.4\\sum_{m,n=1}^{\\infty}\\ln [g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)\n+2mgf_{12}+2ngf_{34}]  \\right\\}.\n\\end{eqnarray}\nMaking using of the proper-time regularization,\n\\begin{eqnarray}\n\\ln \\alpha=-\\int^{\\infty}_{{1}\/{\\Lambda^2}}\\frac{ds}{s}e^{-\\alpha s}\n\\end{eqnarray}\nwith $\\Lambda^2$ being the cut-off to regularize the infinite sum,\nwe have\n\\begin{eqnarray}\n&&\\mbox{Tr}\\ln\\Delta_F +\\mbox{Tr}\\ln\\widetilde{\\Delta}_F\n=\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\\left\\{\n\\ln\\left[\\frac{\\sqrt{2}\\phi^*+v}{\\sqrt{2}\\phi+v) }\\right]\n-2\\int^{\\infty}_{{1}\/{\\Lambda^2}}\n\\frac{ds}{s}e^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v) s}\\right.\\nonumber\\\\\n&&{\\times}\\left.\\left[\\sum_{m=1}^{\\infty}e^{-2mgf_{12}s}\n+\\sum_{n=1}^{\\infty}e^{-2ngf_{34}s}\n+2\\sum_{m,n=1}^{\\infty}e^{-(2mgf_{12}+2ngf_{34})s}\\right]\\right\\}\n\\nonumber\\\\\n&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\n\\left\\{\\ln\\left[\\frac{\\sqrt{2}\\phi^*+v }{\\sqrt{2}\\phi+v }\\right]\n-2\\int^{\\infty}_{{1}\/{\\Lambda^2}}\n\\frac{ds}{s}e^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v) s}\n\\left[\\frac{e^{-gf_{34}s}}{\\sinh (gf_{34}s)}\\right.\\right.\\nonumber\\\\\n&&+\\frac{e^{-gf_{12}s}}{\\sinh (gf_{12}s)}\n+\\left.\\left.\\frac{e^{-(gf_{12}+gf_{34})s}}\n{\\sinh (gf_{12}s)\\,\\sinh (gf_{34}s)}\\right]\\right\\}\n\\nonumber\\\\\n&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\n\\left[\\ln\\left(\\frac{\\sqrt{2}\\phi^*+v}{\\sqrt{2}\\phi+v }\\right)\n-\\int^{\\infty}_{{1}\/{\\Lambda^2}}\\frac{ds}{s}\ne^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)s}\\coth(gf_{12}s)\n\\coth(gf_{34}s)\\right]\\nonumber\\\\\n&=&\\frac{V}{4\\pi^2}g^2f_{12}f_{34}\n\\left[\\ln\\left(\\frac{\\sqrt{2}\\phi^*+v}{\\sqrt{2}\\phi+v}\\right)\\right.\\nonumber\\\\\n&&-\\left.\\int^{\\infty}_{{1}\/{\\Lambda^2}}\\frac{ds}{s}\ne^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v) s}\n\\frac{\\cosh[g(f_{12}+f_{34})s]+\\cosh[g(f_{12}-f_{34})s]}\n{\\cosh[g(f_{12}+f_{34})s]-\\cosh[g(f_{12}-f_{34})s]}\\right]\\, ,\n\\label{eq50n}\n\\end{eqnarray}\nwhere we have used\n\\begin{eqnarray}\n\\sum_{m=1}^{\\infty}e^{-2mt}=\\frac{e^{-t}}{2\\sinh t},~~~\n\\cosh (x{\\pm}y)=\\cosh x\\cosh y{\\pm}\\sinh x\\sinh y.\n\\end{eqnarray}\nRotating back to Minkowski space and denoting ${\\bf X}{\\equiv}{\\bf H}+i{\\bf E}$,\nwe write (\\ref{eq50n}) as\n\\begin{eqnarray}\n&&\\mbox{Tr}\\ln\\Delta_F +\\mbox{Tr}\\ln\\widetilde{\\Delta}_F\n=\\frac{V}{4\\pi^2}g^2iE_z H_z\n\\left[\\ln\\left(\\frac{\\sqrt{2}\\phi^*+v }{\\sqrt{2}\\phi+v}\\right)\\right.\n\\nonumber\\\\\n&-&\\left.\\int^{\\infty}_{{1}\/{\\Lambda^2}}\n\\frac{ds}{s}e^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v) s}\n\\frac{ \\cosh[g(H_z+iE_z)s]+\\cosh[g(H_z-iE_z)s]}\n{\\cosh[g(H_z+iE_z)s]-\\cosh[g(H_z-iE_z)s]}\\right]\\nonumber\\\\\n&{\\equiv}&\\frac{V}{4\\pi^2}g^2i{\\bf E}{\\cdot}{\\bf H}\n\\left[\\ln\\left(\\frac{\\sqrt{2}\\phi+v }{\\sqrt{2}\\phi^*+v }\\right)\n\\nonumber\\right.\\\\\n&&-\\left.\\int^{\\infty}_{{1}\/{\\Lambda^2}}\\frac{ds}{s}\ne^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v) s}\n\\frac{ \\cosh[g{\\bf X}s]+\\cosh (g{\\bf X}^*s)}\n{\\cosh (g{\\bf X} s)-\\cosh (g{\\bf X}^*s)}\\right],\n\\label{eq84}\n\\end{eqnarray}\nTo extract the divergence, we must analyze the small-$s$ behaviour of\nthe integrand of (\\ref{eq84}) by\nusing the identities\n\\begin{eqnarray} \ni{\\bf E}{\\cdot}{\\bf H}=\\frac{1}{4}({\\bf X}^2-{\\bf X}^{*2})\n=\\frac{1}{4}F_{\\mu\\nu}\\widetilde{F}^{\\mu\\nu}, ~~\n{\\bf H}^2-{\\bf E}^2=\\frac{1}{2}({\\bf X}^2+{\\bf X}^{*2})\n=\\frac{1}{2}F_{\\mu\\nu}F^{\\mu\\nu},\n\\end{eqnarray}\nand the series expansion near $s\\sim 0$\n\\begin{eqnarray} \n\\frac{ \\cosh[g{\\bf X}s]+\\cosh[g {\\bf X}^*s]}\n{\\cosh[g{\\bf X} s]-\\cosh[g{\\bf X}^*s]}\n&=&\\frac{1}{({\\bf X}^2-{\\bf X}^{*2})}\\left[\\frac{4}{g^2s^2}+\n\\frac{2}{3}({\\bf X}^2+{\\bf X}^{*2})+{\\cal O}(s^2)\\right].\n\\label{eq84x}\n\\end{eqnarray}\nIt can be easily seen from (\\ref{eq84x}) that the integral in \n(\\ref{eq84}) has a quadratic divergence and a logarithmic one. Thus\nthe divergence term can be extracted by writing (\\ref{eq84}) \nas the form,\n\\begin{eqnarray}\n&&\\mbox{Tr}\\ln\\Delta_F +\\mbox{Tr}\\ln\\widetilde{\\Delta}_F\n=\\frac{V}{4\\pi^2}\\left\\{\\frac{1}{4}g^2F_{\\mu\\nu}\\widetilde{F}^{\\mu\\nu}\n \\ln\\left[\\frac{\\sqrt{2}\\phi^*+v}{\\sqrt{2}\\phi +v}\\right]\\right.\\nonumber\\\\\n&-&\\int^{\\infty}_{{1}\/{\\Lambda^2}}ds\n\\left(\\frac{1}{s^3}+\\frac{1}{6}\\frac{1}{s}g^2F_{\\mu\\nu}F^{\\mu\\nu}\\right)\ne^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)s}\\nonumber\\\\\n&-&\\int^{\\infty}_0\\frac{ds}{s^3}\ne^{-g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v) s}\\left[\\frac{1}{4}g^2s^2\nF_{\\mu\\nu}\\widetilde{F}^{\\mu\\nu}\\frac{ \\cosh (g{\\bf X}s)+\\cosh (g{\\bf X}^*s)}\n{\\cosh (g{\\bf X} s)-\\cosh (g{\\bf X}^*s)}\\right.\\nonumber\\\\\n&-&\\left.\\left.\\frac{1}{s^3}\n-\\frac{1}{6}\\frac{1}{s}g^2F_{\\mu\\nu}F^{\\mu\\nu}\\right]\\right\\}.\n\\label{eq86}\n\\end{eqnarray}\nThe second term (\\ref{eq86}) is the UV divergent term, \nso the cut-off $1\/\\Lambda^2$ is preserved to regularize the \nintegral, while the last term is a finite term\nand hence the cut-off has been removed.\n\nNow we turn to the bosonic determinant. From (\\ref{eq27}) we have  \n\\begin{footnotesize}\n\\begin{eqnarray}\n&&M_{ff}^{-1}=2\\left(\\begin{array}{cccc} 1\/{\\Delta}_F & 0 & 0 & 0\\\\\n0 & 1\/\\widetilde{\\Delta}_F & 0 & 0 \\\\\n0 & 0 & 1\/{\\Delta}_F & 0 \\\\\n0 & 0 & 0 & 1\/\\widetilde{\\Delta}_F \\end{array}\\right), ~~\nM_{bb}-M_{bf}M_{ff}^{-1}M_{fb}=\\nonumber\\\\ &&  \\frac{1}{2}\n\\left(\\begin{array}{cccc} \\Delta_{\\mu\\nu}\n-2g^2\\overline{\\psi}{\\gamma}_{\\mu}\\frac{1}{\\widetilde{\\Delta}_F}{\\gamma}_{\\nu}\\psi & 0 &\n\\Delta_{\\mu}-2ig^2\\overline{\\psi}{\\gamma}_{\\mu}\\frac{1}{\\Delta_F}{\\gamma}_5\\psi & 0\\\\\n0 & {\\Delta}^{\\dagger}_{\\mu\\nu}\n-2g^2\\overline{\\psi}{\\gamma}_{\\mu}\\frac{1}{\\widetilde{\\Delta}_F}{\\gamma}_{\\nu}\\psi & 0\n& {\\Delta}^{\\dagger}_{\\mu}\n-2ig^2\\overline{\\psi}{\\gamma}_{\\mu}\\frac{1}{\\widetilde{\\Delta}_F}{\\gamma}_5\\psi\\\\\n\\widetilde{\\Delta}_{\\nu}^{\\dagger}-2ig^2\\overline{\\psi}{\\gamma}_5\n\\frac{1}{\\Delta_F}{\\gamma}_{\\nu}\\psi & 0 & \\Delta+2g^2\\overline{\\psi}{\\gamma}_5\n\\frac{1}{\\Delta_F}{\\gamma}_5\\psi & 0\\\\\n0 &  \\widetilde{\\Delta}_{\\nu}-2ig^2\\overline{\\psi}{\\gamma}_5\n\\frac{1}{\\Delta_F}{\\gamma}_{\\nu}\\psi & 0 & \\Delta^{\\dagger}\n+2g^2\\overline{\\psi}{\\gamma}_5\n\\frac{1}{\\Delta_F}{\\gamma}_5\\psi\n\\end{array}\\right).\\nonumber\\\\\n\\end{eqnarray}\n\\end{footnotesize}\nIn constant field approximation, $\\overline{\\psi}$ and $\\psi$ can be regarded\nGrassman numbers, so we can expand the bosonic determinant only \nto the quartic terms in  $\\overline{\\psi}$ and $\\psi$.\nNow the key problem is how to find the eigenvalues and eigenstates of\nthe operator matrix $M_{bb}-M_{bf}M_{ff}^{-1}M_{fb}$. If they could be worked\nout, then with the eigenvalues and eigenvectors of fermionic \noperator, we can use the technique \ndeveloped in \\cite{dmnp} to evaluate this determinant. Unfortunately,\nit seems to us that in the constant field approximation\nit is very to find the eigenvalues and eigenstates of such \na horrible operator matrix. This difficulty is waiting to be overcome.\n\nDespite the fact that the bosonic part cannot be evaluated, we can see from \n(\\ref{eq27}) and (\\ref{eq86}) that the effective\nLagrangian associated with the fermionic part has already shown\nthe features of the perturbative part of the low-energy effective action.\nFirst, we believe that the quadratic divergence of Eq.(\\ref{eq86})\nwill be canceled owing to the nonrenormalization theorem. \nSecond, for the logarithmic divergence of Eq.(\\ref{eq86}), with\n\\begin{eqnarray}\n\\int^{\\infty}_{{1}\/{\\Lambda^2}}\\frac{ds}{s}\ne^{-g^2(\\sqrt{2}\\phi +v)(\\sqrt{2}\\phi^*+v) s}{\\sim}\n-\\ln\\left[\\frac{g^2(\\sqrt{2}\\phi +v)(\\sqrt{2}\\phi^*+v) }{\\Lambda^2}\\right],\n\\end{eqnarray}\nEq.(\\ref{eq86}) shows that the Wilson effective action has one term\nproportional to\n\\begin{eqnarray}\nF_{\\mu\\nu}F^{\\mu\\nu}\\ln\\left[\n\\frac{g^2(\\sqrt{2}\\phi^*+v)(\\sqrt{2}\\phi+v)}{\\Lambda^2}\\right].\n\\end{eqnarray}\nComparing with the component field form given by (\\ref{eq:last}), \nwe can conclude that the complete calculation should give the \nform (\\ref{eq1}) of the low-energy effective action. One can even guess \nthis from the requirement of\nsupersymmetry since the constant field approximation and the \nproper-time regularization preserve the supersymmetry explicitly.  \nFurther, there is a finite term proportional \nto $F\\widetilde{F}\\ln\\left[(\\sqrt{2}\\phi +v)\/(\\sqrt{2}\\phi^*+v)\\right]$ in\n(\\ref{eq86}). As pointed out in \\cite{dmnp}, this is the \nreflection of the axial $U(1)_R$ anomaly in the effective action. \nThis anomaly term had played a crucial role\nin the nonperturbative analysis\\cite{sei}.\n \n\\section{Summary}\n\nIn summary, we have tried to calculate the perturbative part of\nthe low-energy effective action of \n$N=2$ supersymmetric Yang-Mills\ntheory based on a standard effective field theory technique.\nIt is well known that the Seiberg-Witten effective action is the\ncornerstone for all those new developments in $N=2$ supersymmetric\ngauge theory, and that this effective action has been obtained\nin an indirect way. Therefore, it is worthwhile to\ntry to compute this effective action using a straightforward\nintegration of the heavy degrees of freedom.\n Unfortunately, we have encountered an insurmountable difficulty \nin evaluating the bosonic operator adopting the constant field\napproximation. This prevents us from getting the  complete result and\ngiving a thorough comparison with the form of (\\ref{eq1}). However, \nthe calculation of the fermionic determinant has indeed shown the \nbasic features of the low-energy effective action. This gives \na partial verification of the abstract symmetry analysis in extracting \nthe low-energy effective action. The complete calculation presents \nan interesting problem for further investigation.\n\n\\vspace{8mm}\n\n\\noindent{\\bf Acknowledgments:}\n We acknowledge the financial support by the Academy of Finland \nunder the project No. 37599 and 44129. W.F.C is partially supported  \nby the Natural Sciences and Engineering Research Council of Canada. \n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}