diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjgkw" "b/data_all_eng_slimpj/shuffled/split2/finalzzjgkw" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjgkw" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nOne of the main tasks of the LHC is to search for Supersymmetry\n(SUSY)~\\cite{mssm}. \nThe Minimal Supersymmetric Standard Model (MSSM) predicts two scalar\npartners for all Standard Model (SM) fermions as well as fermionic\npartners to all SM bosons. \nOf particular interest are the scalar partners of the heavy SM\nquarks, the scalar top quarks, $\\tilde{t}_i$ ($i = 1,2$) and scalar bottom\nquarks $\\tilde{b}_j$ ($j = 1,2$) due to their large Yukawa couplings. \nA scalar top quark $\\tilde{t}_i$ has many possible\ndecay modes, depending on the mass patterns of the SUSY particles. \nAmong those decay modes are the decays to a scalar bottom quark, \n$\\tilde{b}_j$, and a charged Higgs boson, $H^+$, or\n $W$~boson, $W^+$, \n\\begin{align}\n\\label{stsbH}\n&\\tilde{t}_i \\to \\tilde{b}_j H^+ \\quad (i,j = 1,2)~, \\\\\n\\label{stsbW}\n&\\tilde{t}_i \\to \\tilde{b}_j W^+ \\quad (i,j = 1,2)~.\n\\end{align}\nIf these channels are kinematically allowed they can even be dominant if\n(most of) the other decay modes are kinematically forbidden.\nConsequently, these processes can constitute a large part of\nthe total stop decay width, and, in case of decays to a Higgs boson, they\ncan serve as a source of charged Higgs bosons in cascade decays at the LHC.\n \nFor a precise prediction of the partial decay widths corresponding to\n\\refeq{stsbH} and \\refeq{stsbW}, at least the one-loop level contributions\nhave to be taken into account.\nThis in turn requires a renormalization of the relevant sectors,\nespecially a simultaneous renormalization of the top and bottom\nquark\/squark sector. \nDue to the $SU(2)_L$ invariance of the left-handed scalar top and\nbottom quarks, these two sectors cannot be treated independently.\nWithin the framework of the MSSM with complex parameters (cMSSM) \nwe analyze various bottom quark\/squark sector renormalization\nschemes, while we \napply a commonly used on-shell renormalization scheme for the\ntop quark\/squark sector throughout all the investigations.\nSpecial attention is payed to ``perturbativity'', i.e.\\ the loop\ncorrections should not be enhanced by large counterterm contributions\nresulting from an inappropriate renormalization scheme.\nThis turns out to be a constraint that is very difficult to fulfill over\nthe whole cMSSM parameter range, where it is especially \ndifficult to achieve this simultaneously for small and large values of\n$\\tan \\beta$.\n\nHigher-order corrections to scalar fermion decays have been evaluated in\nvarious analysis over the last\ndecade. The simultaneous renormalization \nof the top and the bottom quark\/squark sector was taken\ninto account only in a relatively small subset.\nIn \\citeres{squark_q_V_als,stopsbot_phi_als} stop and sbottom decays,\nincluding the ones to charged Higgs and SM\ngauge bosons, have been \nevaluated at \\order{\\alpha_s} within the MSSM with real parameters\n(rMSSM). The numerical investigation \nwas restricted to relatively low $\\tan \\beta$ values. These\ncalculations are implemented in the program {\\tt SDECAY}~\\cite{sdecay}.\nA similar analysis in\n\\citere{sbot_stop_Hpm_altb} included electroweak one-loop corrections,\nwhere again only relatively low $\\tan \\beta$ values were considered. \nThe decays of Higgs bosons to scalar fermions, including the charged\nHiggs decays, at the full one-loop level within the rMSSM was presented in\n\\citeres{A_sferm_sferm_full,H_sferm_sferm_full}, indicating very\nlarge one-loop corrections \nfor large $\\tan \\beta$. An effective Lagrangian approach in the rMSSM \nfor these types of decays was given in \\citere{squark_eff}, with a\nnumerical analysis for $\\tan \\beta = 5$. \n\nThe renormalization of the top and bottom quark\/squark\nsector has been analyzed also in the context of other calculations \nin the past. A comparison of different renormalization schemes within \nthe rMSSM was performed in \\citeres{dissHR,mhiggsFDalbals}, focusing on\nlarge $\\tan \\beta$. One of the renormalization schemes considered therein had\nbeen used before within the calculation of the two-loop bottom\nquark\/squark contributions to the neutral Higgs boson masses~\\cite{sbotrenold}\nwhich are important for large $\\tan \\beta$~values. Within the cMSSM a\nrenormalization was presented in \\citere{dissTF}, however without an\nanalysis of its practicability. In \\citeres{dissHR,mhcMSSM2L} the top and\nbottom quark\/squark sector was renormalized within the cMSSM, but only\nthe QCD part needed for the presented calculation was considered. Thus\nno complete top and bottom quark\/squark sector renormalization has been\nperformed within the cMSSM. Recently a renormalization of nearly all\nsectors of the rMSSM appeared~\\cite{FawziRen}. In this analysis,\nhowever, the main focus has been on gauge parameter independence.\n\nComplex phases, as assumed here in the cMSSM, can be relevant for\ncollider observables and possibly extracted from experimental data. \nScalar top quark branching ratios at a linear collider are discussed in \n\\citere{Bartl:2003pd}. Concerning LHC measurements, triple products\ninvolving the decay of scalar \ntop or bottom quarks are analyzed in\n\\citeres{Bartl:2004jr,Ellis:2008hq,Deppisch:2009nj,Deppisch:2010nc}.\nFinally, rate asymmetries are examined in \\citere{Eberl:2009xe}.\nDepending on assumptions about the LHC performance it might be\npossible to extract information on the phases of $M_1$, $A_t$ and $A_b$\nat the LHC.\n\n\nIn this paper we analyze the renormalization of the full top\nand bottom quark\/squark sector in the cMSSM. We show analytically \n(and numerically) why\ncertain renormalization schemes \nfail for specific parts of the parameter space. Finally, we explore the\none-loop effects \nfor the decays (\\ref{stsbH},\\ref{stsbW}) for important parts of the\ncMSSM parameter space in the favored\nrenormalization scheme. We present numerical results showing the size\nof the one-loop corrections, especially including small and large $\\tan \\beta$.\nThe evaluation of the partial decay widths of the scalar top quarks are\nbeing implemented into the Fortran code \n{\\tt FeynHiggs}~\\cite{feynhiggs,mhiggslong,mhiggsAEC,mhcMSSMlong}. \nA numerical analysis of {\\em all} scalar top quark decay modes,\ninvolving a renormalization of {\\em all} relevant sectors will be \npresented elsewhere~\\cite{Stop2decay}.\n\n\n\n \n\n\n\n\n\n\\section{The generic structure of the quark\/squark sector}\n\\label{sec:generic}\n\nThe decay channels (\\ref{stsbH},\\ref{stsbW}) are calculated at the full\none-loop level (including hard QED and QCD radiation). This requires the\nrenormalization of several sectors of the cMSSM as discussed below. \nThe sectors not discussed in detail are renormalized as follows:\n\\begin{itemize}\n\\item \nThe gauge and Higgs sector renormalization has been performed following\n\\citere{mhcMSSMlong}. The gauge boson masses, $M_W$ and $M_Z$, as well\nas the mass of the \ncharged Higgs boson, $M_H^\\pm$, has been defined on-shell while the sine squared\nof the weak mixing\nangle, $s_{\\mathrm{w}}^2$, is defined via the gauge boson masses,\n $s_{\\mathrm{w}}^2 = 1 - M_W^2\/M_Z^2$. The $Z$ factors for \nthe $W$~boson field are also determined within an on-shell scheme while\nthe $Z$ factors of the charged Higgs boson field are given by a linear\ncombination of the $\\overline{\\text{DR}}$ $Z$ factors of the Higgs\ndoublets (see \\citere{mhcMSSMlong}). An additional finite $Z$ factor is\nintroduced to fulfill on-shell conditions for the external charged\n$H^\\pm$~field. \n $\\tan \\beta$ is defined as $\\overline{\\text{DR}}$\nparameter. \n\\item\nThe Higgs mixing parameter $\\mu$ has been renormalized via an\non-shell (OS) procedure for the neutralino and chargino\nsector~\\cite{dissTF,diplTF}. \n\\item\nFor the renormalization of the electromagnetic charge we require that the\nrenormalized $ee\\gamma$-vertex in the Thomson limit is not changed by\nhigher order corrections with respect to the corresponding tree-level\nvertex~\\cite{denner}. \n\\end{itemize}\nA detailed description of our renormalization of all sectors will be\ngiven in \\citere{Stop2decay}.\n\n\\bigskip\nIn the following we focus on the top and bottom quark\/squark sector.\nThe bilinear part of the Lagrangian with top and bottom squark fields,\n$\\tilde{t}$ and $\\tilde{b}$, \n\\begin{align}\n{\\cal L}_{\\tilde{t}\/\\tilde{b}\\text{ mass}} &= - \\begin{pmatrix}\n{{\\tilde{t}}_{L}}^{\\dagger}, {{\\tilde{t}}_{R}}^{\\dagger} \\end{pmatrix}\n\\matr{M}_{\\tilde{t}}\\begin{pmatrix}{\\tilde{t}}_{L}\\\\{\\tilde{t}}_{R}\n\\end{pmatrix} - \\begin{pmatrix} {{\\tilde{b}}_{L}}^{\\dagger},\n{{\\tilde{b}}_{R}}^{\\dagger} \\end{pmatrix}\n\\matr{M}_{\\tilde{b}}\\begin{pmatrix}{\\tilde{b}}_{L}\\\\{\\tilde{b}}_{R} \n\\end{pmatrix}~,\n\\end{align}\ncontains the stop and sbottom mass matrices\n$\\matr{M}_{\\tilde{t}}$ and $\\matr{M}_{\\tilde{b}}$,\ngiven by \n\\begin{align}\\label{Sfermionmassenmatrix}\n\\matr{M}_{\\tilde{q}} &= \\begin{pmatrix} \n M_{\\tilde Q_L}^2 + m_q^2 + M_Z^2 c_{2 \\beta} (T_q^3 - Q_q s_\\mathrm{w}^2) & \n m_q X_q^* \\\\[.2em]\n m_q X_q &\n M_{\\tilde{q}_R}^2 + m_q^2 +M_Z^2 c_{2 \\beta} Q_q s_\\mathrm{w}^2\n\\end{pmatrix} \n\\end{align}\nwith\n\\begin{align}\\label{kappa}\nX_q &= A_q - \\mu^*\\kappa~, \\qquad \\kappa = \\{\\cot\\beta, \\tan\\beta\\} \n \\quad {\\rm for} \\quad q = \\{t, b\\\n~.\n\\end{align}\n$M_{\\tilde Q_L}^2$ and $M_{\\tilde{q}_R}^2$ are the soft SUSY-breaking mass\nparameters. $m_q$ is the mass of the corresponding quark.\n$Q_{{q}}$ and $T_q^3$ denote the charge and the isospin of $q$, and\n$A_q$ is the trilinear soft SUSY-breaking parameter.\nThe mass matrix can be diagonalized with the help of a unitary\n transformation ${\\matr{U}}_{\\tilde{q}}$, \n\\begin{align}\\label{transformationkompl}\n\\matr{D}_{\\tilde{q}} &= \n\\matr{U}_{\\tilde{q}}\\, \\matr{M}_{\\tilde{q}} \\, \n{\\matr{U}}_{\\tilde{q}}^\\dagger = \n\\begin{pmatrix} m_{\\tilde{q}_1}^2 & 0 \\\\ 0 & m_{\\tilde{q}_2}^2 \\end{pmatrix}~, \\qquad\n{\\matr{U}}_{\\tilde{q}}= \n\\begin{pmatrix} U_{\\tilde{q}_{11}} & U_{\\tilde{q}_{12}} \\\\ \n U_{\\tilde{q}_{21}} & U_{\\tilde{q}_{22}} \\end{pmatrix}~. \n\\end{align}\nThe scalar quark masses, $m_{\\tilde{q}_1}$ and $m_{\\tilde{q}_2}$, will always be mass\nordered, i.e.\\ \n$m_{\\tilde{q}_1} \\le m_{\\tilde{q}_2}$:\n\\begin{align}\nm_{\\tilde{q}_{1,2}}^2 &= \\frac{1}{2} \\left( M_{\\tilde{Q}_L}^2 + M_{\\tilde{q}_R}^2 \\right) \n + m_q^2 + \\frac{1}{2} T_q^3 M_Z^2 c_{2\\beta} \\nonumber \\\\\n&\\quad \\mp \\frac{1}{2} \\sqrt{\\left[ M_{\\tilde{Q}_L}^2 - M_{{\\tilde{q}_R}}^2 \n + M_Z^2 c_{2\\beta} (T_q^3 - 2 Q_q s_\\mathrm{w}^2) \\right]^2 + 4 m_q^2 |X_q|^2}~.\n\\label{MSbot}\n\\end{align}\n\n\\smallskip\nThe parameter renormalization can be performed as follows, \n\\begin{align}\n\\matr{M}_{\\tilde{q}} &\\to \\matr{M}_{\\tilde{q}} + \\delta\\matr{M}_{\\tilde{q}}\n\\end{align}\nwhich means that the parameters in the mass matrix $\\matr{M}_{\\tilde{q}}$ \nare replaced by the renormalized parameters and a counterterm. After the\nexpansion $\\delta\\matr{M}_{\\tilde{q}}$ contains the counterterm part,\n\\begin{align}\\label{proc1a}\n\\delta\\matr{M}_{\\tilde{q}_{11}} &= \\delta M_{\\tilde Q_L}^2 + 2 m_q \\delta m_q \n- M_Z^2 c_{2 \\beta}\\, Q_q \\, \\delta s_\\mathrm{w}^2 + (T_q^3 - Q_q s_\\mathrm{w}^2) \n ( c_{2 \\beta}\\, \\delta M_Z^2 + M_Z^2\\, \\delta c_{2\\beta})~, \\\\\\label{proc1b}\n\\delta\\matr{M}_{\\tilde{q}_{12}} &= (A_q^* - \\mu \\kappa)\\, \\delta m_q \n+ m_q (\\delta A_q^* - \\mu\\, \\delta \\kappa - \\kappa \\, \\delta \\mu)~, \\\\\\label{proc1c}\n\\delta\\matr{M}_{\\tilde{q}_{21}} &=\\delta\\matr{M}_{\\tilde{q}_{12}}^*~, \\\\\\label{proc1d}\n\\delta\\matr{M}_{\\tilde{q}_{22}} &= \\delta M_{\\tilde{q}_R}^2 \n+ 2 m_q \\delta m_q + M_Z^2 c_{2 \\beta}\\, Q_q \\, \\delta s_\\mathrm{w}^2\n+ Q_q s_\\mathrm{w}^2 ( c_{2 \\beta}\\, \\delta M_Z^2+ M_Z^2\\, \\delta c_{2 \\beta})\n\\end{align}\nwith $\\kappa$ given in \\refeq{kappa}.\n\nAnother possibility for the parameter renormalization is to start out\nwith the physical parameters which corresponds to\nthe replacement:\n\\begin{align} \\label{proc2}\n\\matr{U}_{\\tilde{q}}\\, \\matr{M}_{\\tilde{q}} \\, \n{\\matr{U}}_{\\tilde{q}}^\\dagger &\\to\\matr{U}_{\\tilde{q}}\\, \\matr{M}_{\\tilde{q}} \\, \n{\\matr{U}}_{\\tilde{q}}^\\dagger + \\matr{U}_{\\tilde{q}}\\, \\delta \\matr{M}_{\\tilde{q}} \\, \n{\\matr{U}}_{\\tilde{q}}^\\dagger =\n\\begin{pmatrix} m_{\\tilde{q}_1}^2 & Y_q \\\\ Y_q^* & m_{\\tilde{q}_2}^2 \\end{pmatrix} +\n\\begin{pmatrix}\n\\delta m_{\\tilde{q}_1}^2 & \\delta Y_q \\\\ \\delta Y_q^* & \\delta m_{\\tilde{q}_2}^2\n\\end{pmatrix}~,\n\\end{align}\nwhere $\\delta m_{\\tilde{q}_1}^2$ and $\\delta m_{\\tilde{q}_2}^2$ are the counterterms \n of the squark masses squared. $\\delta Y_q$ is the\n counter\\-term\\footnote{The unitary \n matrix $\\matr{U}_{\\tilde{q}}$ can be expressed by a mixing angle\n $\\theta_{\\tilde{q}}$ and\n a corresponding phase $\\varphi_{\\tilde{q}}$. Then the\n counterterm $\\delta Y_q$ can be related to the counterterms of the\n mixing angle and the phase (see \\citere{mhcMSSM2L}).} to the squark\n mixing parameter $Y_q$ (which vanishes\n at tree level, $Y_q = 0$, and corresponds to the \n off-diagonal entries in $\\matr{D}_{\\tilde{q}} =\\matr{U}_{\\tilde{q}}\\,\n \\matr{M}_{\\tilde{q}} \\, \n{\\matr{U}}_{\\tilde{q}}^\\dagger$, see~\\refeq{transformationkompl}). Using\n\\refeq{proc2} \n one can express $\\delta\\matr{M}_{\\tilde{q}}$ by the counterterms $\\delta m_{\\tilde{q}_1}^2$,\n $\\delta m_{\\tilde{q}_2}^2$ and $\\delta Y_q$. Especially for $\\delta\\matr{M}_{\\tilde{q}_{12}}$\n one yields\n\\begin{align}\\label{dMsq12physpar}\n\\delta\\matr{M}_{{\\tilde{q}}_{12}} &=\nU^*_{\\tilde{q}_{11}} U_{\\tilde{q}_{12}}\n(\\delta m_{\\tilde{q}_1}^2 - \\delta m_{\\tilde{q}_2}^2) +\nU^*_{\\tilde{q}_{11}} U_{\\tilde{q}_{22}} \\delta Y_q + U_{\\tilde{q}_{12}}\nU^*_{\\tilde{q}_{21}} \\delta Y_q^*~.\n\\end{align}\nIn the following the relation given by \\refeq{proc1b} and\n\\refeq{dMsq12physpar} will be used to express either $\\delta Y_q$, $\\delta\nA_q$ or $\\delta m_q$ by the other counterterms.\n\nFor the field renormalization the following procedure is applied,\n\\begin{align}\n\\begin{pmatrix} \\tilde{q}_1 \\\\ \\tilde{q}_2 \\end{pmatrix} &\\to \n \\left( \\id + \\frac{1}{2} \\delta\\matr{Z}_{\\tilde{q}} \\right) \n \\begin{pmatrix} \\tilde{q}_1 \\\\ \\tilde{q}_2 \\end{pmatrix} \n \\quad {\\rm with} \\quad\n\\delta\\matr{Z}_{\\tilde{q}} = \\begin{pmatrix} \n \\delta Z_{\\tilde{q}_{11}} & \\delta Z_{\\tilde{q}_{12}} \\\\\n \\delta Z_{\\tilde{q}_{21}} & \\delta Z_{\\tilde{q}_{22}} \n \\end{pmatrix}~.\n\\end{align}\n\nThis yields for the renormalized self-energies\n\\begin{align}\n\\hat{\\Sigma}_{\\tilde{q}_{11}}(k^2) &= \\Sigma_{\\tilde{q}_{11}}(k^2) \n + \\frac{1}{2} (k^2 - m_{\\tilde{q}_1}^2) (\\dZ{\\tilde{q}_{11}} + \\dZ{\\tilde{q}_{11}}^*)\n - \\dem_{\\tilde{q}_1}^2~, \\\\\n\\hat{\\Sigma}_{\\tilde{q}_{12}}(k^2) &= \\Sigma_{\\tilde{q}_{12}}(k^2)\n + \\frac{1}{2} (k^2 - m_{\\tilde{q}_1}^2) \\dZ{\\tilde{q}_{12}}\n + \\frac{1}{2} (k^2 - m_{\\tilde{q}_2}^2) \\dZ{\\tilde{q}_{21}}^* \n - \\delta Y_q~, \\\\\n\\hat{\\Sigma}_{\\tilde{q}_{21}}(k^2) &= \\Sigma_{\\tilde{q}_{21}}(k^2)\n + \\frac{1}{2} (k^2 - m_{\\tilde{q}_1}^2) \\dZ{\\tilde{q}_{12}}^*\n + \\frac{1}{2} (k^2 - m_{\\tilde{q}_2}^2) \\dZ{\\tilde{q}_{21}} \n - \\delta Y_q^*~, \\\\\n\\hat{\\Sigma}_{\\tilde{q}_{22}}(k^2) &= \\Sigma_{\\tilde{q}_{22}}(k^2) \n + \\frac{1}{2} (k^2 - m_{\\tilde{q}_2}^2) (\\dZ{\\tilde{q}_{22}} + \\dZ{\\tilde{q}_{22}}^*)\n - \\dem_{\\tilde{q}_2}^2~.\n\\end{align}\nIn order to complete the quark\/squark sector renormalization also for the\ncorresponding quark (i.e. its mass, $m_q$, and the quark\nfield, $q$) renormalization constants have to be introduced:\n\\begin{align}\nm_q &\\to m_q + \\delta m_q~,\\\\\n\\omega_{\\mp} \\,q &\\to (1 + \\frac{1}{2}\\dZ{q}^{L\/R})\\, \\omega_{\\mp} \\,q~\n\\end{align}\nwith $\\delta m_q$ being the quark mass counterterm and $\\dZ{q}^L$ and\n$\\dZ{q}^R$ being the $Z$~factors of the left-handed and the right-handed\ncomponent of the quark field $q$, respectively. $\\omega_{\\mp} =\n\\frac{1}{2}(\\id \\mp \\gamma_5)$\nare the left- and right-handed projectors, respectively.\nThen the renormalized self energy, $\\Hat{\\Sigma}_{q}$, can be decomposed\ninto left\/right-handed and scalar left\/right-handed parts, \n${\\hat{\\Sigma}}_q^{L\/R}$ and ${\\hat{\\Sigma}}_q^{SL\/SR}$, respectively,\n\\begin{align}\\label{decomposition}\n \\Hat{\\Sigma}_{q} (k) &= \\not\\! k\\, \\omega_{-} \\Hat{\\Sigma}_q^L (k^2)\n + \\not\\! k\\, \\omega_{+} \\Hat{\\Sigma}_q^R (k^2)\n + \\omega_{-} \\Hat{\\Sigma}_q^{SL} (k^2) \n + \\omega_{+} \\Hat{\\Sigma}_q^{SR} (k^2)~,\n\\end{align}\nwhere the components are given by\n\\begin{align}\n\\Hat{\\Sigma}_q^{L\/R} (k^2) &= {\\Sigma}_q^{L\/R} (k^2) \n + \\frac{1}{2} (\\dZ{q}^{L\/R} + {\\dZ{q}^{L\/R}}^*)~, \\\\\n\\Hat{\\Sigma}_q^{SL} (k^2) &= {\\Sigma}_q^{SL} (k^2) \n - \\frac{m_q}{2} (\\dZ{q}^L + {\\dZ{q}^R}^*) - \\delta m_q~, \\\\\n\\Hat{\\Sigma}_q^{SR} (k^2) &= {\\Sigma}_q^{SR} (k^2) \n - \\frac{m_q}{2} (\\dZ{q}^R + {\\dZ{q}^L}^*) - \\delta m_q~.\n\\end{align}\nNote that $\\Hat{\\Sigma}_q^{SR} (k^2) = {\\Hat{\\Sigma}_q^{SL} (k^2)}^*$ \nholds due to ${\\cal CPT}$ invariance.\n\n\n\n\\section{Field renormalization of the quark\/squark sector}\n\nWe first discuss the field renormalization of the top and bottom\nquark\/squark sector and turn to the parameter renormalization in the next\nsection \\ref{sec:stop}.\n\nThe field renormalization, meaning the determination of the $Z$~factors,\nis done within an on-shell scheme for squarks and quarks. We impose\n equivalent renormalization conditions for the top as well as for\n the bottom quark\/squark sector: \n\\begin{itemize}\n\n\\item[(a)]\nThe diagonal $Z$~factors of the squark fields are determined such that\nthe real part of the residua of propagators is set to unity, \n\\begin{align}\n\\label{residuumStopOS}\n\\widetilde\\re \\frac{\\partial\\, \\hat{\\Sigma}_{\\tilde{q}_{ii}}(k^2)}{\\partial\\, k^2}\n \\Big|_{k^2 = m_{\\tilde{q}_i}^2} &= 0 \\qquad (i = 1,2)~.\n\\end{align}\nThis condition fixes the real parts of the diagonal $Z$~factors to\n\\begin{align}\n\\mathop{\\mathrm{Re}}\\dZ{\\tilde{q}_{ii}} = - \\widetilde\\re \\frac{\\partial\\, \\Sigma_{\\tilde{q}_{ii}}(k^2)}{\\partial\\, k^2}\n \\Big|_{k^2 = m_{\\tilde{q}_i}^2} \\qquad (i = 1,2)~.\n\\end{align}\n$\\widetilde\\re$ above denotes the real part with respect to\ncontributions from the loop integral, but leaves the complex\ncouplings unaffected.\n\nThe imaginary parts of the diagonal $Z$~factors are so far undetermined\nand are set to zero, \n\\begin{align}\n\\mathop{\\mathrm{Im}} \\dZ{\\tilde{q}_{ii}} &= 0 \\qquad (i = 1,2)~.\n\\end{align}\nThis is possible since they do not contain divergences.\n\n\\item[(b)]\nFor the non-diagonal $Z$~factors of the squark fields we impose the\ncondition that for \non-shell squarks no transition from one squark to the other occurs, \n\\begin{align}\n\\widetilde\\re\\hat{\\Sigma}_{\\tilde{q}_{12}}(m_{\\tilde{q}_1}^2) &= 0~, \\\\\n\\widetilde\\re\\hat{\\Sigma}_{\\tilde{q}_{12}}(m_{\\tilde{q}_2}^2) &= 0~.\n\\end{align}\nThis yields\n\\begin{align}\n\\dZ{\\tilde{q}_{12}} &= + 2 \\frac{\\widetilde\\re\\Sigma_{\\tilde{q}_{12}}(m_{\\tilde{q}_2}^2) - \\delta Y_q}\n {(m_{\\tilde{q}_1}^2 - m_{\\tilde{q}_2}^2)}~, \\nonumber \\\\\n\\dZ{\\tilde{q}_{21}} &= - 2 \\frac{\\widetilde\\re\\Sigma_{\\tilde{q}_{21}}(m_{\\tilde{q}_1}^2) - \\delta Y_q^*}\n {(m_{\\tilde{q}_1}^2 - m_{\\tilde{q}_2}^2)}~.\n\\label{dZstopoffdiagOS}\n\\end{align}\nThe counterterm $\\delta Y_q$ is determined in the corresponding parameter\nrenormalization scheme. This means the non-diagonal $Z$~factors of the\nsquark fields do also depend on the choice of the parameter\nrenormalization scheme.\n\n\\item[(c)] \nThe quark fields are also defined via an on-shell condition. We impose\n\\begin{align}\\label{ZquarkOS}\n\\lim_{k^2\\rightarrow m_q^2}\\frac{ \\not\\! k + m_q}{k^2 - m_q^2} \\widetilde\\re\n\\Hat{\\Sigma}_{q} (k) u(k) &= 0~,\\quad\\ \\lim_{k^2\\rightarrow m_q^2} \\bar{u}(k)\\widetilde\\re\n\\Hat{\\Sigma}_{q} (k)\\frac{ \\not\\! k + m_q}{k^2 - m_q^2} = 0~,\n\\end{align}\nwhere $u(k)$, $\\bar{u}(k)$ are the spinors of the external fields. \nThis yields\n\\begin{align}\n\\mathop{\\mathrm{Re}} \\dZ{q}^{L\/R} &= - \\widetilde\\re \\Big\\{ {\\Sigma}_q^{L\/R} (m_q^2) \\\\ \n&\\quad + m_q^2 \\left[ {{\\Sigma}_q^{L}}'(m_q^2) + {{\\Sigma}_q^{R}}'(m_q^2) \\right]\n + m_q \\left[ {{\\Sigma}_q^{SL}}'(m_q^2) \n + {{\\Sigma}_q^{SR}}'(m_q^2) \\right] \\Big\\}~, \\nonumber \\\\\nm_{q} \\left( \\mathop{\\mathrm{Im}} \\dZ{q}^L - \\mathop{\\mathrm{Im}}\\dZ{q}^R \\right) &= \n i\\, \\widetilde\\re\\left\\{ {\\Sigma}_q^{SR}(m_q^2) - {\\Sigma}_q^{SL}(m_q^2) \\right\\}\n = 2 \\mathop{\\mathrm{Im}} \\left\\{ \\widetilde\\re {\\Sigma}_q^{SL}(m_q^2) \\right\\}~,\n\\end{align}\nwith $\\Sigma'(k^2) \\equiv \\frac{\\partial \\Sigma(k^2)}{\\partial k^2}$. \nChoosing also $\\mathop{\\mathrm{Im}} \\dZ{q}^L = - \\mathop{\\mathrm{Im}}\\dZ{q}^R$, the imaginary parts of the\n$Z$~factors can be expressed as\n\\begin{align}\n\\mathop{\\mathrm{Im}} \\dZ{q}^{L\/R} &= \\pm \\frac{i}{2\\, m_q} \n \\widetilde\\re \\left\\{ {\\Sigma}_q^{SR}(m_q^2) - {\\Sigma}_q^{SL}(m_q^2) \\right\\}\n = \\pm \\frac{1}{m_q} \\mathop{\\mathrm{Im}} \\left\\{ \\widetilde\\re {\\Sigma}_q^{SL}(m_q^2) \\right\\}~.\n\\end{align}\nNote that the renormalization condition \\refeq{ZquarkOS} can only be \nfully satisfied if the corresponding quark mass is defined as on-shell, \ntoo.\n\nThe $Z$~factors of the quark fields are not needed for the calculation\nof the considered decay modes of the scalar top quarks (see, however,\n\\citere{Stop2decay}). \n\n\\end{itemize}\n\n\n\\section{Parameter renormalization of the top and bottom \nquark\/squark sector}\n\\label{sec:stop}\n\n\nWithin the top and bottom quark\/squark sector nine real parameters are\ndefined: The real\nsoft SUSY-breaking parameters $M_{\\tilde Q_L}^2$, $M_{{\\tilde{t}}_R}^2$\nand $M_{{\\tilde{b}}_R}^2$, the complex trilinear couplings $A_t$ and\n$A_b$ and the top and bottom Yukawa couplings $y_t$ and $y_b$ which both\ncan be chosen to be real. ($\\mu$ and $\\tan \\beta$ as well as the\ngauge boson masses and the weak mixing angle are determined within other\nsectors, see the beginning of \\refse{sec:generic}). Note that the\nsoft SUSY-breaking parameter $M_{\\tilde Q_L}^2$ is the same in the top\nas well as in the bottom squark sector due to the \n$SU(2)_L$ invariance of the \nleft-handed fields.\nAs in \\citeres{dissHR,mhcMSSM2L}, instead of choosing the five\nquantities $M_{\\tilde Q_L}^2$, $M_{{\\tilde{t}}_R}^2$,\n$M_{{\\tilde{b}}_R}^2$ and $y_t$, $y_b$ the\nsquark masses $m_{\\tilde{t}_1}^2$, $m_{\\tilde{t}_2}^2$, $m_{\\tilde{b}_2}^2$ as well as the top and\nbottom quark masses $m_t$, $m_{b}$ were taken as independent parameters. \n\nIf a regularization scheme is applied which does not break the symmetries\nof the model, it is sufficient to use counterterms which respects the\nunderlying symmetries. Such counterterms are\n generated by multiplicative\nrenormalization of\nparameters and fields of the MSSM. The parameter counterterms can be fixed by\nas many renormalization conditions as independent parameters exist\n\\cite{MSSMrenormierung}.\nConcerning the top and bottom quark\/squark sector we have to set\nnine renormalization conditions to define all indepedent parameters.\n\n For the renormalization of the top quark\/squark sector we follow\n\\citeres{dissHR,mhcMSSM2L} but we also include electroweak contributions.%\n\nWe impose five renormalization conditions, (A)--(E), to fix the\nparameters of the top quark\/squark sector:\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(A)] The top-quark mass is determined via an on-shell condition,\n yielding the one-loop counterterm $\\delta m_t$:\n\\begin{align}\\label{dmt}\n\\delta m_t &= \\frac{1}{2} \\widetilde\\re \\left\\{ \n m_t \\left[\\Sigma_t^L (m_t^2) + \\Sigma_t^R (m_t^2) \\right] \n + \\left[ \\Sigma_t^{SL} (m_t^2) + \\Sigma_t^{SR} (m_t^2) \\right] \\right\\}~.\n\\end{align}\n\\item[(B), (C)]\nThe two top squark masses are also defined on-shell, yielding the real\ncounterterms \n\\begin{align}\n\\label{dmst}\n\\dem_{\\tilde{t}_i}^2 &= \\widetilde\\re\\Sigma_{\\tilde{t}_{ii}}(m_{\\tilde{t}_i}^2) \\qquad (i = 1,2)~.\n\\end{align}\n\n\\item[(D), (E)]\nFinally, the non-diagonal entry in the matrix of \\refeq{proc2} is fixed\nas \n\\begin{align}\n\\delta Y_t &= \\frac{1}{2} \\widetilde\\re \n \\left\\{ \\Sigma_{\\tilde{t}_{12}}(m_{\\tilde{t}_1}^2) + \\Sigma_{\\tilde{t}_{12}}(m_{\\tilde{t}_2}^2) \\right\\}~,\n\\end{align}\nwhich corresponds to two seperate conditions as $\\delta Y_t$ is complex.\n\\end{itemize}\n\\end{itemize}\n\n\nThe counterterm of the trilinear coupling $\\delta A_t$ is then given via the\nrelation of \\refeqs{proc1b} and \\eqref{dMsq12physpar} as:\n\\begin{align}\n\\delta A_t &= \\frac{1}{m_t}\\bigl[U_{\\tilde{t}_{11}} U_{\\tilde{t}_{12}}^*\n (\\delta m_{\\tilde{t}_1}^2 - \\delta m_{\\tilde{t}_2}^2) \n + U_{\\tilde{t}_{11}} U_{\\tilde{t}_{22}}^* \\delta Y_t^*\n + U_{\\tilde{t}_{12}}^* U_{\\tilde{t}_{21}} \\delta Y_t \n - (A_t - \\mu^* \\cot\\beta)\\, \\dem_t \\bigr] \\nonumber \\\\\n&\\quad + (\\delta\\mu^* \\cot\\beta - \\mu^* \\cot^2\\beta\\, \\delta\\!\\tan\\!\\beta\\,)~.\n\\end{align}\nThe definition of $\\delta\\!\\tan\\!\\beta\\,$ and $\\delta\\mu$ is indicated in\n\\refse{sec:generic}. \n\n\\bigskip\nFor the bottom quark\/squark sector we are left with four independent\nparameters which are not defined yet. We choose the following\nfour renormalization conditions, (i)--(iv):\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(i)] The $\\tilde{b}_2$~mass is defined on-shell:\n\\begin{align}\\label{sbotzOS}\n\\dem_{\\tilde{b}_2}^2 &= \\widetilde\\re\\Sigma_{\\tilde{b}_{22}}(m_{\\tilde{b}_2}^2)~.\n\\end{align}\n\\item[(ii)--(iv)] These three renormalization conditions are chosen\n according to the different renormalization conditions listed in\n \\refta{tab:RS} and to the corresponding subsections \n \\ref{sec:OS}--\\ref{sec:AbOS_ReYbOS}. \n They yield the counterterms $\\delta m_{b}$, $\\delta A_b$\n and $\\delta Y_b$ where only three of these five real counterterms are\n independent (counting each of the complex counterterms, $\\delta A_b$\n and $\\delta Y_b$, as two real counterterms). The two dependent\n counterterms can be expressed as a combination of the other ones.\n\\end{itemize}\n\\end{itemize}\nApplying these renormalization conditions fixes the counterterms\ngenerated by multiplicative renormalization which fulfill the\nsymmetry relations \\cite{MSSMrenormierung}.\n\n\\begin{table}[ht!]\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c||c|c|}\n\\hline\nscheme & $m_{\\tilde b_{1,2}}$ & $m_{b}$ & $A_b$ & $Y_b$ & Sect. & name \n \\\\ \\hline\\hline\n{\\small analogous to the $t\/\\tilde{t}$ sector:} \n ``OS'' & OS & OS & & OS \n& \\ref{sec:OS} & RS1 \\\\ \\hline\n``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'' & OS & \\ensuremath{\\overline{\\mathrm{DR}}}\\ & \\ensuremath{\\overline{\\mathrm{DR}}}\\ & \n& \\ref{sec:mbDRbar_AbDRbar} & RS2 \\\\ \\hline\n``$m_{b},\\, Y_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'' & OS & \\ensuremath{\\overline{\\mathrm{DR}}}\\ & & \\ensuremath{\\overline{\\mathrm{DR}}}\\ \n& \\ref{sec:mbDRbar_YbDRbar} & RS3 \\\\ \\hline\n``$m_{b}$~\\ensuremath{\\overline{\\mathrm{DR}}}, $Y_b$~OS'' & OS & \\ensuremath{\\overline{\\mathrm{DR}}}\\ & & OS\n& \\ref{sec:mbDRbar_YbOS} & RS4 \\\\ \\hline\n``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS'' & OS & & \\ensuremath{\\overline{\\mathrm{DR}}} & $\\mathop{\\mathrm{Re}} Y_b$:\\, OS\n& \\ref{sec:AbDRbar_ReYbOS} & RS5 \\\\ \\hline\n``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS'' & OS & & vertex & $\\mathop{\\mathrm{Re}} Y_b$:\\, OS\n& \\ref{sec:AbOS_ReYbOS} & RS6 \\\\ \\hline\n\\end{tabular}\n\\caption{Summary of the six renormalization schemes for the\n $b\/\\tilde{b}$~sector investigated below. Blank entries indicate dependent\n quantities. $\\mathop{\\mathrm{Re}} Y_b$ denotes that only the real part of \n $Y_b$ is renormalized on-shell, while the imaginary part is a\n dependent parameter. The rightmost columns indicates the section that\n contains the detailed description of the respective renormalization\n and the abbreviated notation used in our analysis.}\n\\label{tab:RS}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{center}\n\\end{table}\n\n\n\nWhile the $\\tilde{b}_2$~mass is defined on-shell, the\n$\\tilde{b}_1$~mass receives a shift due to the radiative corrections: \n\\begin{align}\nm_{\\tilde{b}_{1,{\\rm OS}}}^2 &= m_{\\tilde{b}_1}^2 + \\left(\n \\dem_{\\tilde{b}_1}^2 - \\widetilde\\re\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1}^2)\\right)~.\n\\end{align}\nThe term in parentheses is the shift from $m_{\\tilde{b}_1}^2$ to the on-shell mass\nsquared. The value of \n$m_{\\tilde{b}_1}^2$ is derived from the diagonalization of the \nsbottom mass matrix, see \\refeq{transformationkompl}, and $\\dem_{\\tilde{b}_1}^2$\nis defined as a dependent\nquantity~\\cite{hr,mhiggsFDalbals}. $m_{\\tilde{b}_{1,{\\rm OS}}}^2$ is the on-shell\n$\\tilde{b}_1$~mass squared. In ~\\citere{hr} the size of the shift was\nanalyzed while in~\\citere{mhiggsFDalbals} bottom squarks appeared only as\n``internal'' particles, i.e.\\ as particles inside the loop\ndiagrams. Concerning the scalar top quark decay, \\refeqs{stsbH} and\n\\eqref{stsbW}, we are now dealing with \nscalar bottom quarks as ``external'' particles, which are defined as\nincoming or outgoing particles. These ``external'' particles should\nfulfill on-shell properties. At this point there are two options to \nproceed:\n\\begin{itemize}\n\\item[(${\\cal O} 1$)] \n The first option is to use different mass values, $m_{\\tilde{b}_1}$ and\n $m_{\\tilde{b}_{1,{\\rm OS}}}$, for the ``internal'' and the ``external'' particles,\n respectively, which can cause problems for charged particles as,\n for instance, scalar bottom quarks (see below).\n\\item[(${\\cal O} 2$)] \n The second option is to impose a further renormalization\n condition which ensures that the $\\tilde{b}_1$~mass is on-shell:\n\\begin{align}\\label{sboteOS}\n \\dem_{\\tilde{b}_1}^2 &=\\widetilde\\re\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1}^2)~.\n\\end{align}\nIn this case the input has to be chosen such that the symmetry\nrelations are fulfilled at the one-loop level.\n\\end{itemize}\n\nAs mentioned above, the option~(${\\cal O} 1$) leads to a problem. \nThe IR-divergences originating from the loop diagrams involve the\n``inner'' (i.e.\\ tree-level) \nmass $m_{\\tilde{b}_1}$. These have to cancel with the real\nBremsstrahlung IR-divergences, which are evaluated with the help of the\n``external'' (i.e.\\ one-loop on-shell) mass $m_{\\tilde{b}_{1,{\\rm OS}}}$, which is inserted into \nthe tree-level diagram (the result can, as usual, be expressed with the\nhelp of the Soft Bremsstrahlung (SB) factor $\\delta_{\\rm SB}$:\n${\\cal M}_{\\rm tree} \\times \\delta_{\\rm SB}$, see \\citere{denner}). \nDue to the two different sets of masses the IR-divergences do not\ncancel. \nOne way out would be the use of tree-level masses in all diagrams\ncontributing to the part $2 \\mathop{\\mathrm{Re}} \\{{\\cal M}_{\\rm tree} {\\cal M}_{\\rm loop}\\}$,\ni.e.\\ in all loop diagrams and in the hard and soft Bremsstrahlung diagrams. \nHowever, this would lead to inconsistencies in the evaluation of the complete\nloop corrected amplitude squared\n$\\propto (|{\\cal M}_{\\rm tree}|^2 + 2 \\mathop{\\mathrm{Re}} \\{{\\cal M}_{\\rm tree} {\\cal M}_{\\rm loop}\\} )$ \ndue to the different masses entering the phase space evaluation.\nA consistent phase space integration requires the use of the same\n``external'' masses for all outgoing particles in all parts of \nthe calculation.\n\n\nTo circumvent the problem of the non-cancellation of IR-divergences we\nchoose the option~(${\\cal O} 2$) and impose the further renormalization \ncondition \\refeq{sboteOS}. This requires to choose an input that restores \nthe symmetries. Relating $(\\matr{M}_{\\tilde q})_{11}$ of \n\\refeq{Sfermionmassenmatrix} and \n$(\\matr{U_{\\tilde{q}}}^{\\dagger} \\matr{D}_{\\tilde{q}} \\matr{U_{\\tilde{q}}})_{11}$\nwith $\\matr{D}_{\\tilde{q}}$ of \\refeq{transformationkompl} yields an\nexpression for the soft SUSY-breaking parameter $M_{\\tilde{Q}_L}^2$\n(depending on the squark flavor),\n\\begin{align}\nM_{\\tilde{Q}_L}^2(\\tilde{q}) = |U_{\\tilde{q}_{11}}|^2 m_{\\tilde{q}_1}^2\n + |U_{\\tilde{q}_{12}}|^2 m_{\\tilde{q}_2}^2 \n - M_Z^2 c_{2\\beta} (T_q^3 - Q_q s_\\mathrm{w}^2) - m_{q}^2\n\\end{align}\nwith $\\tilde{q} = \\{\\tilde{t}, \\tilde{b}\\}$. Requiring the $SU(2)_L$ relation\nto be valid at the one-loop level induces the following shift in \n$M^2_{\\tilde{Q}_L}$ (see also\n\\citeres{squark_q_V_als,stopsbot_phi_als,dr2lA}): \n\\begin{align}\nM_{\\tilde{Q}_L}^2(\\tilde{b}) = M_{\\tilde{Q}_L}^2(\\tilde{t}) \n + \\delta M_{\\tilde{Q}_L}^2(\\tilde{t}) - \\delta M_{\\tilde{Q}_L}^2(\\tilde{b})\n\\label{MSbotshift}\n\\end{align}\nwith\n\\begin{align}\n\\delta M_{\\tilde{Q}_L}^2(\\tilde{q}) &= |U_{\\tilde{q}_{11}}|^2 \\dem_{\\tilde{q}_1}^2\n + |U_{\\tilde{q}_{12}}|^2 \\dem_{\\tilde{q}_2}^2\n - U_{\\tilde{q}_{22}} U_{\\tilde{q}_{12}}^* \\delta Y_q\n - U_{\\tilde{q}_{12}} U_{\\tilde{q}_{22}}^* \\delta Y_q^* - 2 m_{q} \\dem_{q} \\nonumber \\\\\n&\\quad + M_Z^2\\, c_{2\\beta}\\, Q_q\\, \\delta s_\\mathrm{w}^2 \n - (T_q^3 - Q_q s_\\mathrm{w}^2) (c_{2\\beta}\\, \\delta M_Z^2 + M_Z^2\\, \\delta c_{2\\beta})~.\n\\label{MSbotshift-detail}\n\\end{align}\nIn other words, everywhere in the calculation the masses and mixing\nmatrix elements coming \nfrom the diagonalization of the bottom squark mass matrix, see\n\\refeq{transformationkompl}, are used with $M_{\\tilde{Q}_L}^2(\\tilde{b})$\nincluding the above shift as in \\refeq{MSbotshift}. \nIn this way the problems concerning UV- and IR-finiteness are\navoided. (An exception is the field renormalization of the $W$-boson\nfield: In the corresponding selfenergies the $SU(2)_L$ relation is needed\nat tree-level to ensure UV-finiteness. In this case, tree-level bottom\nsquark masses are used.)\n\nThe various renormalization schemes, following the general\nchoice~(${\\cal O} 2$), are summarized in \\refta{tab:RS} and\noutlined in detail in the following subsections.\n\n\nComparing with the literature, several of the renormalization\nschemes (or variants of them) have been used to calculate higher-order\ncorrections to squark or Higgs decays.\nThe older calculations of the loop corrections have all been\nperformed in the rMSSM. \n\n\\begin{itemize}\n\n\\item\nA renormalization scheme employing an ``OS'' renormalization for\n$m_{b}$ and $Y_b$ was used in \n\\citeres{stopsbot_phi_als,sbot_stop_Hpm_altb} for the calculation of\nstop and sbottom decays. (The calculation of\n\\citere{stopsbot_phi_als} is also implemented in\n\\citere{sdecay}.) In order to check our\nimplementation given in \\refse{sec:OS} we calculated the decay \n$\\tilde{b}_{1,2} \\to \\tilde{t}_1 H^{-}$ (see \\refse{sec:calc} for our set-up) and\nfound good agreement with \\citere{stopsbot_phi_als}.\n\n\\item\nA renormalization scheme similar to the real version of RS2,\ni.e.\\ ``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}''\nhas been employed in \\citere{H_sferm_sferm_full} for the\ncalculation of Higgs decays to scalar fermions. In the scalar top and\nthe Higgs sector they apply an on-shell scheme \n(partially \nbased on \\citeres{sbot_top_cha_alt,sfermprod_alf}), \nwhich differs in some points from our renormalization scheme.\n\n\\item\nAn on-shell scheme was also used in \\citere{sferm_f_V_full} (based on\n\\citeres{sbot_top_cha_alt,squark_q_chi_full}) to evaluate the decay\n$\\tilde{f} \\to \\tilde{f}' V$ ($V = W^\\pm, Z$). \n\n\\item\nIn \\citere{stop_stop_H_alt}, as a starting point, an on-shell renormalization\nscheme was used for the calculation of the electroweak corrections to\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{t}_1 \\phi)$, ($\\phi = h, H, A$).\nTo improve the calculation, the parameters $m_{b}$, $m_t$, $A_t$ and $A_b$ have\nalso been used as running parameters.\n\n\n\n\\item\nOther ``early'' papers considered QCD corrections to various scalar\nquark decays~\\cite{squark_q_chi_als,squark_q_gl_als,stop_top_gl_als}. \nThey mostly employed an on-shell \nscheme for the quark\/squark masses and the squark mixing angle \n$\\theta\\kern-.15em_{\\tilde{q}}$, where the counterterm to the mixing angle is\n$\\delta \\theta\\kern-.15em_{\\tilde{q}} \\propto \\delta Y_q$.\n\n\\item\nThe renormalization scheme ``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS'' is the\ncomplex version of the renormalization used in\n\\citeres{sbotrenold,mhiggsFDalbals} for the \n\\order{\\alpha_b\\alpha_s} corrections to the neutral Higgs boson self-energies\nand thus to the lightest MSSM Higgs boson mass, $M_h$.\n\n\\end{itemize}\n\n\n\\bigskip\nIn the following subsections we define in detail the various\nrenormalization schemes. As explained before and indicated in\n\\refta{tab:RS} the two bottom squark\nmasses are renormalized on-shell in all the schemes, as in \n\\refeqs{sbotzOS} and \\eqref{sboteOS}, and taking into account\nthe shift of $M_{\\tilde Q_L}^2(\\tilde{b})$ in \\refeq{MSbotshift}. \nWithin the subsections only the remaining conditions and\nrenormalization constants are defined explicitly\n(where $\\delta \\mu$ and $\\delta\\!\\tan\\!\\beta\\,$ are\ndefined within the chargino\/neutralino sector and the Higgs sector,\nrespectively, in all the different renormalization schemes and are not\ndiscussed any further).\n\n\n\n\n\\subsection{On-shell (RS1)}\n\\label{sec:OS}\n\nThis renormalization scheme is analogous to the OS scheme employed for\nthe top quark\/squark sector.\n\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(ii)]\nThe bottom-quark mass is defined OS, yielding the one-loop \ncounterterm $\\delta m_{b}$:\n\\begin{align} \n\\label{dmb_OS}\n\\dem_{b} = \\frac{1}{2} \\widetilde\\re \\left\\{\n m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]\n + \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SL} (m_{b}^2) \\right] \\right\\}~.\n\\end{align}\n\n\\item[(iii), (iv)]\nWe choose an OS renormalization condition for the non-diagonal\nentry in the matrix of \\refeq{proc2}, analogous to the one\napplied in the top quark\/squark sector, setting\n\\begin{align}\n\\delta Y_b = \\frac{1}{2} \\widetilde\\re \\left\\{\n \\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2) + \\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2) \\right\\}~.\n\\label{dYb_OS}\n\\end{align}\n\\end{itemize}\n\\end{itemize}\nThe conditions (i)--(iv) fix all independent parameters and their\nrespective counterterms. \nAnalogous to the calculation of the counterterm of the trilinear\ncoupling $A_t$, relating \\refeq{proc1b} and \\refeq{dMsq12physpar} yields\nthe following condition for $\\delta A_b$, \n\\begin{align}\\label{Ab_OS}\n\\deA_b &= \\frac{1}{m_{b}} \\bigl[U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n (\\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2) \n + U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\delta Y_b^*\n + U_{\\tilde{b}_{12}}^* U_{\\tilde{b}_{21}} \\delta Y_b \n - (A_b - \\mu^*\\tan \\beta)\\, \\dem_{b} \\bigr] \\nonumber \\\\\n&\\quad + (\\delta\\mu^* \\tan \\beta + \\mu^*\\, \\delta\\!\\tan\\!\\beta\\,)\n\\end{align} \nwith $\\dem_{\\tilde{b}_1}^2$ and $\\dem_{\\tilde{b}_2}^2$ given in \\refeqs{sboteOS} and \n\\eqref{sbotzOS}, respectively. \n\n\n\n\n\\subsection{\\boldmath{$m_{b}$ \\ensuremath{\\overline{\\mathrm{DR}}}\\ } and \\boldmath{$A_b$ \\ensuremath{\\overline{\\mathrm{DR}}}\\ } (RS2)}\n\\label{sec:mbDRbar_AbDRbar}\n\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(ii)]\nThe bottom-quark mass is defined \\ensuremath{\\overline{\\mathrm{DR}}}, yielding the one-loop\n counterterm $\\delta m_{b}$:\n\\begin{align} \n\\dem_{b} = \\frac{1}{2} \\widetilde\\re \\left\\{\n m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]_{\\rm div}\n+ \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SR} (m_{b}^2) \\right]_{\\rm div} \\right\\}~.\n\\end{align}\nThe $|_{\\rm div}$ terms are the ones proportional to\n$\\Delta = 2\/\\varepsilon - \\gamma_{\\rm E} + \\log(4 \\pi)$, when using dimensional\nregularization\/reduction in $D = 4 - \\varepsilon$ dimensions; $\\gamma_{\\rm E}$ is\nthe Euler constant.\n\n\\item[(iii), (iv)]\nThe complex parameter $A_b$ is renormalized \\ensuremath{\\overline{\\mathrm{DR}}},\n\\begin{align}\n\\deA_b &= \\frac{1}{m_{b}} \\Bigl[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n \\left( \\widetilde\\re\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1}^2)|_{\\rm div} \n -\\widetilde\\re\\Sigma_{\\tilde{b}_{22}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right) \\nonumber \\\\\n&\\quad + \\frac{1}{2}\\, U_{\\tilde{b}_{12}}^* U_{\\tilde{b}_{21}} \n \\left( \\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2)|_{\\rm div}\n +\\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right) \\nonumber \\\\\n&\\quad + \\frac{1}{2}\\, U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \n \\left( \\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2)|_{\\rm div}\n +\\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right)^* \\nonumber \\\\\n&\\quad - \\frac{1}{2}(A_b - \\mu^* \\tan \\beta)\\, \n\\widetilde\\re \\bigl\\{\n m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]_{\\rm div} \\nonumber \\\\\n&\\qquad + \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SR} (m_{b}^2) \\right]_{\\rm\n div} \\bigr\\} \n \\Bigr] \n + \\delta\\mu^*|_{\\rm div} \\tan \\beta + \\mu^*\\, \\delta\\!\\tan\\!\\beta\\,~. \n\\end{align} \n\\end{itemize}\n\\end{itemize}\n\nAll independent parameters are defined by the conditions (i)--(iv) and\nthe corresponding counterterms are determined.\nSolving \\refeqs{proc1b} and\n(\\ref{dMsq12physpar}) for $\\delta Y_b$ yields\n\\begin{align}\n\\delta Y_b &= \\frac{1}{|U_{\\tilde{b}_{11}}|^2 - |U_{\\tilde{b}_{12}}|^2} \\Big[ \n U_{\\tilde{b}_{11}} U_{\\tilde{b}_{21}}^* \n (\\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2) \\nonumber \\\\\n&\\quad + m_{b} \\Big( U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \n \\left( \\deA_b^* - \\mu\\, \\delta\\!\\tan\\!\\beta\\, - \\tan \\beta\\, \\delta\\mu \\right) \n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \n \\left( \\deA_b - \\mu^* \\delta\\!\\tan\\!\\beta\\, - \\tan \\beta\\, \\delta\\mu^* \\right) \\Big) \\nonumber \\\\\n&\\quad + \\left( U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* (A_b^* - \\mu \\tan \\beta)\n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* (A_b - \\mu^* \\tan \\beta) \\right)\\, \\dem_{b}\n\\Big]~,\n\\label{dYb_mbDRbar_AbDRbar}\n\\end{align}\nwhere $\\dem_{\\tilde{b}_1}^2$ and $\\dem_{\\tilde{b}_2}^2$ are given in \\refeqs{sboteOS} and \n\\eqref{sbotzOS}, respectively. \n\n\n\n\n\\subsection{\\boldmath{$m_{b}$ \\ensuremath{\\overline{\\mathrm{DR}}}\\ } and \\boldmath{$Y_b$ \\ensuremath{\\overline{\\mathrm{DR}}}\\ } (RS3)}\n\\label{sec:mbDRbar_YbDRbar}\n\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(ii)]\nThe bottom-quark mass is defined \\ensuremath{\\overline{\\mathrm{DR}}}, yielding the one-loop\n counterterm $\\delta m_{b}$:\n\\begin{align} \n\\label{dmb_mbDRbar_YbDRbar}\n\\dem_{b} = \\frac{1}{2} \\widetilde\\re \\left\\{\n m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]_{\\rm div}\n+ \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SR} (m_{b}^2) \\right]_{\\rm div} \\right\\}~.\n\\end{align}\n\n\\item[(iii), (iv)]\nThe complex counterterm $\\delta Y_b$ is determined via a \\ensuremath{\\overline{\\mathrm{DR}}}\\\nrenormalization condition, setting\n\\begin{align}\n\\delta Y_b = \\frac{1}{2} \\widetilde\\re \\left\\{\n \\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2)|_{\\rm div} + \\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right\\}~.\n\\label{dYb_mbDRbar_YbDRbar}\n\\end{align}\n\\end{itemize}\n\\end{itemize}\n\nAs in Sect.~\\ref{sec:OS}, the renormalization conditions \n(ii), (iii) and (iv) fix the\ncounterterms $\\delta m_{b}$ and $\\delta Y_b$, respectively. Together with the\nrenormalization conditions for $\\dem_{\\tilde{b}_1}^2$ and\n$\\dem_{\\tilde{b}_2}^2$ (see \\refeq{sboteOS} and \\refeq{sbotzOS}, respectively),\n$\\delta A_b$ is given by the linear \ncombination of these counterterms as\n\\begin{align}\n\\deA_b &= \\frac{1}{m_{b}} \\bigl[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n (\\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2) \n + U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\delta Y_b^*\n + U_{\\tilde{b}_{12}}^* U_{\\tilde{b}_{21}} \\delta Y_b \n - (A_b - \\mu^*\\tan \\beta)\\, \\dem_{b} \\bigr] \\nonumber \\\\\n&\\quad + (\\delta\\mu^* \\tan \\beta + \\mu^*\\, \\delta\\!\\tan\\!\\beta\\,)~,\n\\label{dAb_mbDRbar_YbDRbar}\n\\end{align}\nwhich, of course, shows the same analytical dependence of the independent\ncounterterms as $\\deA_b$ in \\refeq{Ab_OS} in \\refse{sec:OS}.\n\n\n\n\\subsection{\\boldmath{$m_{b}$ \\ensuremath{\\overline{\\mathrm{DR}}}\\ } and \\boldmath{$Y_b$} on-shell (RS4)}\n\\label{sec:mbDRbar_YbOS}\n\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(ii)]\nThe bottom-quark mass is defined \\ensuremath{\\overline{\\mathrm{DR}}}, yielding the one-loop\n counterterm $\\delta m_{b}$:\n\\begin{align} \n\\label{dmb_mbDRbar_YbOS}\n\\dem_{b} = \\frac{1}{2} \\widetilde\\re \\left\\{\n m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]_{\\rm div}\n+ \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SR} (m_{b}^2) \\right]_{\\rm div} \\right\\}~.\n\\end{align}\n\n\\item[(iii), (iv)]\nThe complex counterterm $\\delta Y_b$ is fixed by an on-shell \nrenormalization condition, as in Sect.~\\ref{sec:OS},\n\\begin{align}\\label{dYb_mbDRbar_YbOS}\n\\delta Y_b = \\frac{1}{2} \\widetilde\\re \\left\\{\n \\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2) + \\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2) \\right\\}~.\n\\end{align}\n\\end{itemize}\n\\end{itemize}\nAs in Sect.~\\ref{sec:OS} and in \\refse{sec:mbDRbar_YbDRbar}, the\n renormalization conditions (i)--(iv) fix the \ncounterterms $\\dem_{\\tilde{b}_2}^2$, $\\delta m_{b}$ and $\\delta Y_b$. The further\nrenormalization condition \\refeq{sboteOS} determines the counterterm\n$\\dem_{\\tilde{b}_1}^2$. Analogous to Sect.~\\ref{sec:OS} and to\n\\refse{sec:mbDRbar_YbDRbar}, \n$\\delta A_b$ can be expressed in terms of these counterterms, \n\\begin{align}\n\\deA_b &= \\frac{1}{m_{b}} \\bigl[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n (\\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2) \n + U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\delta Y_b^*\n + U_{\\tilde{b}_{12}}^* U_{\\tilde{b}_{21}} \\delta Y_b \n - (A_b - \\mu^*\\tan \\beta)\\, \\dem_{b} \\bigr] \\nonumber \\\\\n&\\quad + (\\delta\\mu^* \\tan \\beta + \\mu^*\\, \\delta\\!\\tan\\!\\beta\\,)~,\n\\end{align}\nwhich, of course, has the same form as in \\refeqs{Ab_OS} and\n\\eqref{dAb_mbDRbar_YbDRbar}. \n\n\n\n\n\\subsection{\\boldmath{$A_b$ \\ensuremath{\\overline{\\mathrm{DR}}}\\ } and \\boldmath{$\\mathop{\\mathrm{Re}} Y_b$} on-shell (RS5)}\n\\label{sec:AbDRbar_ReYbOS}\n\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(ii)]\nIn the subsections \\ref{sec:OS}--\\ref{sec:mbDRbar_YbOS} the second\nrenormalization condition defines the bottom quark mass. In this scheme,\nwe choose an on-shell renormalization condition for the real part of the\ncounterterm $\\delta Y_b$ which determines $\\mathop{\\mathrm{Re}} \\delta Y_b$ as following\n\\begin{align}\n\\mathop{\\mathrm{Re}} \\delta Y_b = \\frac{1}{2} \\mathop{\\mathrm{Re}} \\left\\{\n \\widetilde\\re{\\Sigma}_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2) +\n \\widetilde\\re{\\Sigma}_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2) \\right\\}~.\n\\label{dReYb_AbDRbar_ReYbOS}\n\\end{align}\n\n\n\\item[(iii), (iv)]\nThe complex $A_b$ parameter is defined \\ensuremath{\\overline{\\mathrm{DR}}}\n\\begin{align}\n\\deA_b &= \\frac{1}{m_{b}} \\Bigl[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n \\left( \\widetilde\\re\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1}^2)|_{\\rm div} \n -\\widetilde\\re\\Sigma_{\\tilde{b}_{22}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right) \\nonumber \\\\\n&\\quad + \\frac{1}{2} U_{\\tilde{b}_{12}}^* U_{\\tilde{b}_{21}} \n \\left( \\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2)|_{\\rm div}\n +\\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right) \\nonumber \\\\\n&\\quad + \\frac{1}{2} U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \n \\left( \\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2)|_{\\rm div}\n +\\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right)^* \\nonumber \\\\\n&\\quad - \\frac{1}{2}(A_b - \\mu^* \\tan \\beta)\\, \n\\widetilde\\re \\bigl\\{\n m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]_{\\rm div} \\nonumber\\\\\n&\\qquad + \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SR} (m_{b}^2) \\right]_{\\rm\n div} \\bigr\\} \n + \\delta\\mu^*|_{\\rm div} \\tan \\beta + \\mu^*\\, \\delta\\!\\tan\\!\\beta\\,~. \n\\label{dAb}\n\\end{align} \n\\end{itemize}\n\\end{itemize}\n\nWith the conditions (i)--(iv) the independent counterterms $\\delta\nm_{\\tilde{b}_2}^2$, $\\mathop{\\mathrm{Re}} \\delta Y_b$ and $\\delta A_b$ are determined, \nand $\\dem_{\\tilde{b}_1}^2$ is given by \\refeq{sboteOS}. The missing\ncounterterms $\\delta m_{b}$ and $\\mathop{\\mathrm{Im}} \\delta Y_b$ can be expressed by the\nindependent counterterms. Relating \\refeq{proc1b}, here explicitly\nwritten as\n\\begin{align}\\label{proc1bexpl}\n(\\delta \\matr{M}_{\\tilde{b}})_{12} &= (A_b^* - \\mu\\tan \\beta)\\, \\dem_{b} \n + m_{b} \\left( \\deA_b^*\n - \\mu\\, \\delta\\!\\tan\\!\\beta\\, - \\delta\\mu \\tan \\beta \\right)~,\n\\end{align}\nand \\refeq{dMsq12physpar}, here with $\\delta Y_b$ explicitly split into a\nreal and an imaginary part\n\\begin{align}\\label{dMsq12physparsplit}\n(\\delta \\matr{M}_{\\tilde{b}})_{12} &= \n U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{12}} (\\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2) \\nonumber \\\\\n&\\quad\n + U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{22}} (\\mathop{\\mathrm{Re}} \\delta Y_b + i \\mathop{\\mathrm{Im}} \\delta Y_b)\n + U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* (\\mathop{\\mathrm{Re}} \\delta Y_b - i \\mathop{\\mathrm{Im}} \\delta Y_b)~,\n\\end{align}\nresults in the two equations\n\\begin{align}\n\\label{eq:dM12R}\n\\mathop{\\mathrm{Re}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\}\\, \\dem_{b}\n&= - m_{b} \\mathop{\\mathrm{Re}} \\deA_b - \\mathop{\\mathrm{Re}} \\delta S \n - \\mathop{\\mathrm{Im}} \\left\\{ U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{22}} \n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\right\\} \\mathop{\\mathrm{Im}} \\delta Y_b~, \\\\\n\\label{eq:dM12I}\n\\mathop{\\mathrm{Im}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\}\\, \\dem_{b} \n&= + m_{b} \\mathop{\\mathrm{Im}} \\deA_b - \\mathop{\\mathrm{Im}} \\delta S \n + \\mathop{\\mathrm{Re}} \\left\\{ U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{22}} \n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\right\\} \\mathop{\\mathrm{Im}} \\delta Y_b\n\\end{align}\nwith\n\\begin{align}\n\\delta S &= - m_{b}\\, (\\mu\\,\\delta\\!\\tan\\!\\beta\\, + \\delta\\mu \\tan \\beta)\n - U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{12}} (\\delta m_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2) \\nonumber \\\\\n&\\quad - \\left( U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{22}} \n + U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\right) \\mathop{\\mathrm{Re}} \\delta Y_b~,\n\\end{align}\nwhere $\\delta m_{\\tilde{b}_1}^2$ and $\\dem_{\\tilde{b}_2}^2$ are given by \\refeq{sboteOS} and\n\\refeq{sbotzOS}. \n\n\\medskip\nThe above two equations, \n(\\ref{eq:dM12R}) and (\\ref{eq:dM12I}), \ncan be solved for $\\mathop{\\mathrm{Im}}\\delta Y_b$ and $\\dem_{b}$, yielding\\\\[1em]\n\\begin{align}\\label{dmb_AbDRbar_ReYbOS}\n\\delta m_{b} &= \\frac{b_r c_i - b_i c_r}{a_r b_i - a_i b_r}~,\\\\[2mm]\n\\label{dImYb_AbDRbar_ReYbOS}\n\\mathop{\\mathrm{Im}}\\delta Y_b &= \\frac{a_i c_r - a_r c_i}{a_r b_i - a_i b_r}\n\\end{align}\nwith\n\\begin{align}\na_r &= \\mathop{\\mathrm{Re}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\}~, \\\\\na_i &= \\mathop{\\mathrm{Im}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\}~, \\\\\nb_r &= + \\mathop{\\mathrm{Im}} \\left\\{ U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{22}} \n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\right\\}~, \\\\\nb_i &= - \\mathop{\\mathrm{Re}} \\left\\{ U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{22}} \n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\right\\}~, \\\\\nc_r &= + m_{b} \\mathop{\\mathrm{Re}} \\deA_b + \\mathop{\\mathrm{Re}}\\delta S~, \\\\\nc_i &= - m_{b} \\mathop{\\mathrm{Im}} \\deA_b + \\mathop{\\mathrm{Im}}\\delta S~.\n\\end{align}\n\n\n\n\n\\subsection{\\boldmath{$A_b$} via vertex and \\boldmath{$\\mathop{\\mathrm{Re}} Y_b$ on-shell} (RS6)}\n\\label{sec:AbOS_ReYbOS}\n\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(ii)]\nAn on-shell renormalization condition is imposed for the real part of\nthe counterterm $\\delta Y_b$ which determines $\\mathop{\\mathrm{Re}} \\delta Y_b$ as\n\\begin{align}\n\\mathop{\\mathrm{Re}} \\delta Y_b = \\frac{1}{2} \\mathop{\\mathrm{Re}} \\left\\{\n \\widetilde\\re{\\Sigma}_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2) +\n \\widetilde\\re{\\Sigma}_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2) \\right\\}~.\n\\label{dReYb_AbOS_ReYbOS}\n\\end{align}\n\n\\item[(iii), (iv)]\nThe renormalization conditions introduced here are analogous to \nthe prescriptions used in \\citeres{dissHR,sbotrenold,mhiggsFDalbals}, \nbut extended to the complex MSSM.\nThe complex parameter $A_b$ is renormalized via the vertex\n$A\\, \\tilde{b}_1^\\dagger \\tilde{b}_2$, denoting the renormalized vertex as \n$\\hat\\Lambda(p_{A}^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2)$, see \\reffi{fig:vertex}.\n\\vspace{-2ex}\n\\begin{figure}[htb!]\n\\begin{center}\n\\setlength{\\unitlength}{1pt}\n\\begin{picture}(350, 180)\n\\DashArrowLine(160,105)(195,125){5}\n\\DashArrowLine(195,055)(160,075){5}\n\\DashLine(80,90)(140,90){5}\n\\put(65,85){$A$}\n\\put(200,50){$\\tilde{b}_2$}\n\\put(200,125){$\\tilde{b}_1$}\n\\put(240,85){$\\Hat{=} \\quad\n i\\, \\hat{\\Lambda}(p_{A}^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2)$}\\,\n\\GCirc(140,90){20}{.6}\n\\end{picture}\n\\vspace{-2ex}\n\\caption{The renormalized vertex \n$\\hat\\Lambda(p_{A}^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2)$.}\n\\label{fig:vertex}\n\\end{center}\n\\end{figure}\n\nThe tree-level vertex $A\\, \\tilde{b}_1^\\dagger \\tilde{b}_2$, denoted as $V_{A\\,\n \\tilde{b}_1^\\dagger \\tilde{b}_2}$, is given as \n\\begin{align}\\nonumber\nV_{A\\, \\tilde{b}_1^\\dagger \\tilde{b}_2} = \\frac{i e\\,m_{b}}{2M_Ws_\\mathrm{w} \\cos \\beta} \n\\Bigl[&\n U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* (\\mu \\cos \\be_{\\rm n} + A_b^* \\sin \\be_{\\rm n}) \n\\nonumber \\\\\n- & U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* (\\mu^* \\cos \\be_{\\rm n} + A_b \\sin \\be_{\\rm n})\n\\Bigr]~,\n\\end{align}\nwhere $\\be_{\\rm n}$ is the mixing angle of the ${\\cal CP}$-odd Higgs boson fields\nwith $\\be_{\\rm n} = \\beta$ at tree-level. Note that in our renormalization\nprescription we do not renormalize the mixing angles but only $\\tan\n\\beta$ appearing in the Lagrangian before the transformation of the \n${\\cal CP}$-odd Higgs boson fields into mass eigenstate fields is performed.\nThe renormalized vertex reads,\n\\begin{align}\n& \\hat\\Lambda(p_A^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2) \\; = \\;\n \\Lambda(p_A^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2) \\; \n + \\frac{i e\\,m_{b}}{2M_Ws_\\mathrm{w}} \\Bigg\\{ \\nonumber \\\\\n&\\qquad\\; \\tan \\beta \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\deA_b^*\n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\delta A_b \\right] \n + \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\delta\\mu\n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\delta \\mu^* \\right] \\nonumber \\\\\n&\\quad + \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\left( \\mu + \\tbA_b^* \\right)\n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\left( \\mu^* + \\tbA_b \\right) \n \\right] \\nonumber \\\\\n&\\qquad \\times \\left[ \\frac{\\dem_{b}}{m_{b}} + \\frac{1}{2} (\\delta \\bar Z_{\\tilde{b}_{11}}^* \n + \\delta\\bar Z_{\\tilde{b}_{22}} + \\dZ{AA}) \n + \\sin \\beta\\,\\cos \\beta\\, \\delta\\!\\tan\\!\\beta\\, \\right] \\nonumber \\\\\n&\\quad + \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* (\\mu + \\tbA_b^*)\n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* (\\mu^* + \\tbA_b)\n \\right] \\left( \\dZ{e} - \\frac{\\deM_W^2}{2\\,M_W^2}\n - \\frac{\\des_\\mathrm{w}}{s_\\mathrm{w}} \\right) \\nonumber \\\\ \n&\\quad + i \\mathop{\\mathrm{Im}} \\left\\{ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^* \n \\left( \\mu + \\tbA_b^* \\right)\n \\right\\} \\dZ{\\tilde{b}_{12}}\n + i \\mathop{\\mathrm{Im}} \\left\\{ U_{\\tilde{b}_{21}} U_{\\tilde{b}_{22}}^* \n \\left( \\mu + \\tbA_b^* \\right)\n \\right\\} \\dZ{\\tilde{b}_{21}}^* \\nonumber \\\\\n&\\quad - \\frac{1}{2} \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\left( A_b^* - \\mu \\tan \\beta \\right)\n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\left( A_b - \\mu^* \\tan \\beta \\right)\n \\right] \\dZ{AG}\n\\Bigg\\}~.\n\\end{align}\nThe off-diagonal $Z$~factors are determined according to\n\\refeq{dZstopoffdiagOS}, \n\\begin{align}\n\\dZ{\\tilde{b}_{12}} &= + 2\\, \n\\frac{\\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2) - \\mathop{\\mathrm{Re}}\\delta Y_b - i \\mathop{\\mathrm{Im}} \\delta Y_b}\n {(m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2)} \\; \n=: \\dZ{\\tilde{b}_{12}}^{\\rm c} - \\frac{2 i \\mathop{\\mathrm{Im}} \\delta Y_b}{m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2}~, \\nonumber\\\\\n\\dZ{\\tilde{b}_{21}} &= - 2\\,\n\\frac{\\widetilde\\re\\Sigma_{\\tilde{b}_{21}}(m_{\\tilde{b}_1}^2) - \\mathop{\\mathrm{Re}} \\delta Y_b + i \\mathop{\\mathrm{Im}} \\delta Y_b}\n {(m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2)} \\;\n=: \\dZ{\\tilde{b}_{21}}^{\\rm c} - \\frac{2 i \\mathop{\\mathrm{Im}} \\delta Y_b}{m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2}~.\n\\end{align}\nIntroducing appropriate abbreviations we get\n\\begin{align}\n& \\hat\\Lambda(p_A^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2) \\; = \\;\n \\Lambda(p_A^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2) \\; + \\\\\n&\\quad\\;\\, \\frac{i e}{2M_Ws_\\mathrm{w}} \\Big\\{ \nm_{b} \\tan \\beta\\; (U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\deA_b^* \n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\deA_b) + \\delta M \n + i\\, U_Y\\, \\mathop{\\mathrm{Im}} \\delta Y_b \\nonumber \\\\\n&\\quad + \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\left( \\mu + \\tbA_b^* \\right)\n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\left( \\mu^* + \\tbA_b \\right) \n \\right] \n (\\dem_{b} + \\delta Z_{\\rm d}) \\Big\\} + \\delta Z_{\\rm o} \\nonumber\n\\end{align}\n\nwith\n\n\\begin{align}\n\\delta M &= m_{b} \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\delta\\mu \n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\delta\\mu^* \n \\right]~, \\\\[2mm]\nU_Y &= \\frac{4\\, i\\, m_{b}}{m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2}\n \\mathop{\\mathrm{Im}} \\Big\\{ U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{12}} \\left( \\mu^* + \\tbA_b \\right)\n \\Big\\}~, \\\\[2mm]\n\\dZ{\\rm d} &= m_{b} \\left[ \\frac{1}{2} (\\delta \\bar Z_{\\tilde{b}_{11}}^* \n + \\delta \\bar Z_{\\tilde{b}_{22}} + \\dZ{AA} )\n + \\sin \\beta\\,\\cos \\beta\\, \\delta\\!\\tan\\!\\beta\\, \\right]~,\\\\[2em]\n\\dZ{\\rm o} &= \\frac{i e\\, m_{b}}{2M_Ws_\\mathrm{w}} \\Bigg\\{ \\nonumber \\\\\n&\\quad \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* (\\mu + \\tbA_b^*)\n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* (\\mu^* + \\tbA_b)\n \\right] \\left( \\dZ{e} - \\frac{\\deM_W^2}{2\\,M_W^2}\n - \\frac{\\des_\\mathrm{w}}{s_\\mathrm{w}} \\right) \\nonumber \\\\\n&\\quad + i \\mathop{\\mathrm{Im}} \\left\\{ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^* \n \\left( \\mu + \\tbA_b^* \\right)\n \\right\\} \\dZ{\\tilde{b}_{12}}^{\\rm c}\n + i \\mathop{\\mathrm{Im}} \\left\\{ U_{\\tilde{b}_{21}} U_{\\tilde{b}_{22}}^* \n \\left( \\mu + \\tbA_b^* \\right)\n \\right\\} \\dZ{\\tilde{b}_{21}}^{{\\rm c}\\,*} \\nonumber \\\\\n&\\quad - \\frac{1}{2} \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\left( A_b^* - \\mu \\tan \\beta \\right)\n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\left( A_b - \\mu^* \\tan \\beta \\right)\n \\right] \\dZ{AG}\n\\Bigg\\}~. \n\\end{align}\nThe renormalization condition reads~\\cite{dissHR,sbotrenold}\n\\begin{align}\n\\widetilde\\re\\hat\\Lambda(0, m_{\\tilde{b}_1}^2, m_{\\tilde{b}_1}^2) + \\widetilde\\re\\hat\\Lambda(0, m_{\\tilde{b}_2}^2, m_{\\tilde{b}_2}^2) = 0~,\n\\end{align}\nwhich corresponds to the two conditions\n\\begin{align}\n\\label{eq:ReLam}\n\\mathop{\\mathrm{Re}} \\left\\{ \\widetilde\\re\\hat\\Lambda(0, m_{\\tilde{b}_1}^2, m_{\\tilde{b}_1}^2) \n + \\widetilde\\re\\hat\\Lambda(0, m_{\\tilde{b}_2}^2, m_{\\tilde{b}_2}^2) \\right\\} = 0~, \\\\\n\\label{eq:ImLam}\n\\mathop{\\mathrm{Im}} \\left\\{ \\widetilde\\re\\hat\\Lambda(0, m_{\\tilde{b}_1}^2, m_{\\tilde{b}_1}^2) \n + \\widetilde\\re\\hat\\Lambda(0, m_{\\tilde{b}_2}^2, m_{\\tilde{b}_2}^2) \\right\\} = 0~.\n\\end{align}\n\n\\end{itemize}\n\\end{itemize}\n\nThe conditions (i)--(iv), are sufficient to fix all independent parameters\nand their respective counterterms. As in Sect.~\\ref{sec:AbDRbar_ReYbOS},\nrelating \\refeqs{proc1bexpl} and \\eqref{dMsq12physparsplit}, one\nderives \\refeqs {eq:dM12R} and \\eqref{eq:dM12I} which can also be\nwritten in the form\n\\begin{align}\n\\label{eq:dM12R_AbOS_ReYbOS}\n\\mathop{\\mathrm{Re}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\}\\, \\dem_{b} \n&= - m_{b} \\mathop{\\mathrm{Re}} \\deA_b - \\mathop{\\mathrm{Re}} \\delta S + \\mathop{\\mathrm{Im}} U_+ \\mathop{\\mathrm{Im}} \\delta Y_b~, \\\\\n\\label{eq:dM12I_AbOS_ReYbOS}\n\\mathop{\\mathrm{Im}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\}\\, \\dem_{b} \n&= + m_{b} \\mathop{\\mathrm{Im}} \\deA_b - \\mathop{\\mathrm{Im}} \\delta S + \\mathop{\\mathrm{Re}} U_- \\mathop{\\mathrm{Im}} \\delta Y_b\n\\end{align}\nwith\n\\begin{align}\n\\delta S &= -m_{b} (\\mu\\,\\delta\\!\\tan\\!\\beta\\, + \\delta\\mu \\tan \\beta)\n - U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{12}} (\\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2)\n - \\left( \\mathop{\\mathrm{Re}} U_+ - i \\mathop{\\mathrm{Im}} U_- \\right) \\mathop{\\mathrm{Re}} \\delta Y_b~, \\\\\nU_{\\pm} &= U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\pm U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^*~.\n\\end{align}\n$\\dem_{\\tilde{b}_1}^2$ and $\\dem_{\\tilde{b}_2}^2$\nare fixed by \\refeqs{sboteOS} and \\eqref{sbotzOS}. \n\n\\medskip\nThe above four equations (\\ref{eq:ReLam}), (\\ref{eq:ImLam}),\n(\\ref{eq:dM12R_AbOS_ReYbOS}) and \n(\\ref{eq:dM12I_AbOS_ReYbOS}), \ncan be solved for $\\mathop{\\mathrm{Re}}\\delta A_b$, $\\mathop{\\mathrm{Im}}\\delta A_b$, $\\mathop{\\mathrm{Im}}\\delta Y_b$ and\n$\\dem_{b}$. Though, we still consider $\\mathop{\\mathrm{Re}}\\delta A_b$ and $\\mathop{\\mathrm{Im}}\\delta A_b$ as\nindependent counterterms we first calculate $\\mathop{\\mathrm{Im}}\\delta Y_b$ and\n$\\dem_{b}$ in dependence of $\\mathop{\\mathrm{Re}}\\delta A_b$ and $\\mathop{\\mathrm{Im}}\\delta A_b$ for\neconomically solving the systems of equations. The solution for $\\mathop{\\mathrm{Im}}\\delta\nY_b$ and $\\dem_{b}$ is\\\\[1em]\n\\begin{align}\n\\label{dmb_AbOS_ReYbOS}\n\\delta m_{b} &= \\frac{d_i f_r - d_r f_i}{e_r f_i - e_i f_r}~,\\\\[2mm]\n\\mathop{\\mathrm{Im}}\\delta Y_b &= \\frac{d_r e_i - d_i e_r}{e_r f_i - e_i f_r}\n\\end{align}\nwith\n\\begin{align}\nd_r &= 2 \\tan \\beta \\left( \\mathop{\\mathrm{Im}} U_+ \\mathop{\\mathrm{Im}} \\delta S - \\mathop{\\mathrm{Re}} U_- \\mathop{\\mathrm{Re}} \\delta S \\right) \\\\\n&\\quad + 2 \\mathop{\\mathrm{Re}} \\left[ \\frac{M_W s_\\mathrm{w}}{i\\,e} \n \\left( 2 \\dZ{\\rm o} + \\widetilde\\re\\Lambda(0, m_{\\tilde{b}_1}^2, m_{\\tilde{b}_1}^2) \n + \\widetilde\\re\\Lambda(0, m_{\\tilde{b}_2}^2, m_{\\tilde{b}_2}^2) \\right)\n + \\delta M + \\dZ{\\rm d} U_m \\right]~, \\nonumber \\\\\nd_i &= -2 \\tan \\beta \\left( \\mathop{\\mathrm{Re}} U_+ \\mathop{\\mathrm{Im}} \\delta S + \\mathop{\\mathrm{Im}} U_- \\mathop{\\mathrm{Re}} \\delta S \\right) \\\\\n&\\quad + 2 \\mathop{\\mathrm{Im}} \\left[ \\frac{M_W s_\\mathrm{w}}{i\\,e} \n \\left( 2 \\dZ{\\rm o} + \\widetilde\\re\\Lambda(0, m_{\\tilde{b}_1}^2, m_{\\tilde{b}_1}^2) \n + \\widetilde\\re\\Lambda(0, m_{\\tilde{b}_2}^2, m_{\\tilde{b}_2}^2) \\right)\n + \\delta M + \\dZ{\\rm d} U_m \\right]~, \\nonumber \\\\\ne_r &= +2 \\tan \\beta \\left[ \\mathop{\\mathrm{Im}} U_+ \\mathop{\\mathrm{Im}} \\left\\{ A_b^* - \\mu\\tan \\beta \\right\\} \n -\\mathop{\\mathrm{Re}} U_- \\mathop{\\mathrm{Re}} \\left\\{ A_b^* - \\mu\\tan \\beta \\right\\} \n \\right] + 2 \\mathop{\\mathrm{Re}} U_m~,\\\\\ne_i &= -2 \\tan \\beta \\left[ \\mathop{\\mathrm{Re}} U_+ \\mathop{\\mathrm{Im}} \\left\\{ A_b^* - \\mu\\tan \\beta \\right\\} \n +\\mathop{\\mathrm{Im}} U_- \\mathop{\\mathrm{Re}} \\left\\{ A_b^* - \\mu\\tan \\beta \\right\\} \n \\right] + 2 \\mathop{\\mathrm{Im}} U_m~, \\\\\nf_r &= - 2 \\mathop{\\mathrm{Im}} U_Y~, \\\\\nf_i &= 2 \\tan \\beta \\left( |U_{\\tilde{b}_{11}}|^2 - |U_{\\tilde{b}_{12}}|^2 \\right)\n\\end{align}\nand\n\\begin{align}\n\\label{def:Um}\nU_m &= U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* (A_b^* \\tan \\beta + \\mu)\n - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* (A_b \\tan \\beta + \\mu^*)~.\n\\end{align}\nFrom the \\refeqs{eq:dM12R_AbOS_ReYbOS} and \n\\eqref{eq:dM12I_AbOS_ReYbOS} we immediately obtain $\\delta A_b$ as\n\\begin{align}\n\\mathop{\\mathrm{Re}}\\deA_b &= \\ed{m_{b}} \\left[ + \\mathop{\\mathrm{Im}}\\delta Y_b \\mathop{\\mathrm{Im}} U_+ - \\mathop{\\mathrm{Re}} \\delta S\n - \\dem_{b} \\mathop{\\mathrm{Re}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\} \\right]~, \\\\\n\\mathop{\\mathrm{Im}}\\deA_b &= \\ed{m_{b}} \\left[ - \\mathop{\\mathrm{Im}}\\delta Y_b \\mathop{\\mathrm{Re}} U_- + \\mathop{\\mathrm{Im}} \\delta S\n + \\dem_{b} \\mathop{\\mathrm{Im}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\} \\right]~.\n\\end{align}\n\nFinally the $\\bar{Z}$~factors in $\\hat{\\Lambda}$ have to be\ndetermined. The following \ncondition is used \n\\begin{align}\n\\widetilde\\re \\hat{\\Sigma}_{\\tilde{b}_{ii}}(m_{\\tilde{b}_1}^2) - \n\\widetilde\\re \\hat{\\Sigma}_{\\tilde{b}_{ii}}(m_{\\tilde{b}_2}^2) = 0 \\qquad (i = 1,2)~.\n\\end{align} \nThis condition results in the following $\\bar{Z}$~factors\n\\begin{align}\n\\delta\\bar{Z}_{\\tilde{b}_{ii}} = \n-\\frac{\\widetilde\\re \\Sigma_{\\tilde{b}_{ii}}(m_{\\tilde{b}_1}^2) - \n \\widetilde\\re \\Sigma_{\\tilde{b}_{ii}}(m_{\\tilde{b}_2}^2)}{m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2} \n \\qquad (i = 1,2)~,\n\\end{align}\nwhich guarantees the IR finiteness of the renormalized vertex\n$\\hat{\\Lambda}$~\\cite{sbotrenold}.\\\\ \n\n\n\\smallskip\nAnother subtlety has to be explained here:\ndue to the fact that we have infrared divergent $C$-functions\nat $p_1 = 0$ in $\\Lambda(p_1^2=0,p^2,p^2)$, we must deal with vanishing \nGram-determinants. \nTherefore we follow \\citere{cfunc} (and references therein) and \nreplace the corresponding $C$-functions by well behaving linear \ncombinations of $B$-functions. \nDetails can be found in the appendix.\n\n\n\n\\subsection{Parameter definition}\n\n\nThe input parameters in the $b\/\\tilde{b}$ sector have to correspond to the\nchosen renormalization scheme. We start by defining the bottom quark\nmass, where the \nexperimental input is the SM \\ensuremath{\\overline{\\mathrm{MS}}}\\ mass \\cite{pdg},\n\\begin{align}\n\\label{def:mbMB}\nm_{b}^{\\ensuremath{\\overline{\\mathrm{MS}}}}(m_{b}) & = 4.2 \\,\\, \\mathrm{GeV}~.\n\\end{align}\nThe value of $m_{b}^{\\ensuremath{\\overline{\\mathrm{MS}}}}(\\mu_R)$ (at the renormalization scale \n$\\mu_R$) is calculated from $m_{b}^{\\ensuremath{\\overline{\\mathrm{MS}}}}(m_{b})$ at the three loop\nlevel following the prescription given in~\\citere{RunDec}.\n\nAn ``on-shell'' mass is derived from the \\ensuremath{\\overline{\\mathrm{MS}}}\\ mass via\n\\begin{align}\nm_{b}^{\\mathrm{OS}} &= m_{b}^{\\ensuremath{\\overline{\\mathrm{MS}}}}(\\mu_R) \\; \n \\left[ 1 + \\frac{\\alpha_s^{\\ensuremath{\\overline{\\mathrm{MS}}}}(\\mu_R)}{\\pi} \n \\left( \\frac{4}{3} + 2\\, \\ln \\frac{\\mu_R}{m_{b}^{\\ensuremath{\\overline{\\mathrm{MS}}}}(\\mu_R)} \\right) \n \\right]~.\n\\end{align}\nThe $\\ensuremath{\\overline{\\mathrm{DR}}}$ bottom quark mass is calculated iteratively from%\n\\footnote{\nIn case of complex $\\Delta_b$ the replacement $(1 + \\Delta_b) \\to |1 + \\Delta_b|$\nshould be performed~\\cite{komplexDb}.}\n\\begin{align}\n\\label{eq:mbDR}\nm_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}} &= \\frac{m_{b}^{\\mathrm{OS}} (1 + \\Delta_b) + \\dem_{b}^{\\mathrm{OS}} - \\dem_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}}}\n {1 + \\Delta_b}\n\\end{align}\nwith an accuracy of $|1 - (m_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}})^{(n)}\/(m_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}})^{(n-1)}| < 10^{-5}$\nreached in the $n$th step of the iteration.\nThe bottom quark mass of a special renormalization scheme is then obtained\nfrom \n\\begin{align}\\label{eq:mbcorr}\nm_{b} &= m_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}} + \\dem_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}} - \\dem_{b}~.\n\\end{align}\nHere we have used\n\\begin{align}\n\\dem_{b}^{\\mathrm{OS}} &= \\frac{1}{2} \\widetilde\\re \\left\\{\n m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]\n + \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SL} (m_{b}^2) \\right] \\right\\}~, \\nonumber \\\\[2mm]\n\\dem_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}} &= \\frac{1}{2} \\widetilde\\re \\left\\{\n m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]_{\\rm div}\n+ \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SR} (m_{b}^2) \\right]_{\\rm div} \\right\\}~, \n\\end{align}\nand $\\dem_{b}$ as given in \\refses{sec:OS}--\\ref{sec:AbOS_ReYbOS}.\nThe quantity $\\Delta_b$~\\cite{deltab1,deltab2} resums the \\order{(\\alpha_s\\tan \\beta)^n}\nand \\order{(\\alpha_t\\tan \\beta)^n} terms and is given by \n\\begin{align}\n\\Delta_b &= \\frac{2\\alpha_s(m_t)}{3\\pi} \\, \\tan \\beta \\, M_3^* \\, \\mu^* \\,\n I(m_{\\tilde{b}_1}^2, m_{\\tilde{b}_2}^2, m_{\\tilde{g}}^2) \\;\n + \\frac{\\alpha_t(m_t)}{4\\pi} \\, \\tan \\beta \\, A_t^* \\, \\mu^* \\, \n I(m_{\\tilde{t}_1}^2, m_{\\tilde{t}_2}^2, |\\mu|^2)\n\\end{align}\nwith\n\\begin{align}\nI(a, b, c) &= - \\frac{a b\\, \\ln(b\/a) + a c\\, \\ln(a\/c) + b c\\, \\ln(c\/b)}\n {(a - c) (c - b) (b - a)}~.\n\\end{align}\nHere $\\alpha_t$ is defined in terms of the top Yukawa coupling \n$y_t(m_t) = \\sqrt{2} m_t(m_t)\/v$ as\n$\\alpha_t(m_t) = y_t^2(m_t)\/(4\\pi)$ with \n$v = 1\/\\sqrt{\\sqrt{2}\\, G_F} = 246.218 \\,\\, \\mathrm{GeV}$\nand \n$m_t(m_t)\\approx m_t\/(1-\\frac{1}{2\\,\\pi} \\alpha_t(m_t) +\\frac{4}{3\\,\\pi}\\alpha_s(m_t))$.\n$M_3$ is the soft SUSY-breaking parameter\nfor the gluinos, with the gluino mass given as $m_{\\tilde{g}} := |M_3|$.\n\n\\newpage\n\n\n\n\n\n\n\n\\section{Renormalization scheme analysis}\n\\label{sec:RSana}\n\n\\subsection{Calculation of loop diagrams}\n\\label{sec:calc}\n\nIn this section we give the relevant details about the calculation of the\nhigher-order corrections to the decay channels (\\ref{stsbH},\\ref{stsbW}). \nSample diagrams are shown in \\reffis{fig:fdsbotHpm}, \\ref{fig:fdsbotW}. \nNot shown are the diagrams for real (hard or soft) photon and gluon\nradiation (which, however, can become numerically very important). \nThey are obtained from the corresponding tree-level diagrams\nby attaching a photon (gluon) to the electrically (color) charged\nparticles. The internal, in a generical way depicted particles in\n\\reffis{fig:fdsbotHpm}, \\ref{fig:fdsbotW} are labeled as follows:\n$F$ can be a SM fermion, a chargino or neutralino or a gluino, $S$\ncan be a sfermion \nor a Higgs boson, $V$ can be a photon $\\gamma$, a $Z$ or $W^\\pm$ boson or a\ngluon $g$. \nNot shown are the diagrams with a gauge boson (Goldstone $G^\\pm$)--Higgs\nselfenergy \ncontribution on the external Higgs boson leg that can appear in \nthe decay $\\tilde{t}_2 \\to \\tilde{b}_i H^+$.\nOn the other hand, in our calculation, the wave function corrections for\n$\\tilde{t}_2 \\to \\tilde{b}_i W^+$ vanish as all the external particle fields are\nrenormalized on-shell.\n\nThe diagrams and corresponding amplitudes have been obtained with the\nprogram {\\em FeynArts}~\\cite{feynarts}. \nThe further evaluation has been performed with \n{\\em FormCalc}~\\cite{formcalc}. As regularization scheme for the UV-divergences we\nhave used constrained differential renormalization~\\cite{cdr}, \nwhich has been shown to be equivalent to \ndimensional reduction~\\cite{dred} at the one-loop\\ level~\\cite{formcalc}. \nThus the employed regularization preserves SUSY~\\cite{dredDS,dredDS2}. \nIt was checked that all UV-divergences cancel in the final result.\n\nThe IR-divergences from diagrams with an internal photon or gluon have\nto cancel with the ones from the corresponding real soft radiation.\nIn the case of QED we have included the soft photon contribution\nfollowing the description given in \\citere{denner}. \nIn the case of QCD we have modified this prescription by replacing the\nproduct of electric charges by the appropriate combination of color\ncharges (linear combination of $C_A$ and $C_F$ times $\\alpha_s$).\nMore details will be given in \\citere{Stop2decay}.\nUsing the sbottom masses at the one-loop level, see \\refse{sec:stop},\nwe found cancellation beyond one-loop order\nof the related IR and UV divergences for\nthe decay $\\tilde{t}_2 \\to \\tilde{b}_i H^+$, and a cancellation, as required, at the\none-loop level for the decay $\\tilde{t}_2 \\to \\tilde{b}_i W^+$.%\n\\footnote{Using tree-level masses yields a cancellation of IR divergences\nbeyond one-loop order also for $\\tilde{t}_2 \\to \\tilde{b}_i W^+$.}\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.90\\textwidth]{fdsbotHpm}\n\\caption{\nGeneric Feynman diagrams for the decay \n$\\tilde{t}_2 \\to \\tilde{b}_i H^+$ ($i = 1,2$).\n$F$ can be a SM fermion, a chargino or neutralino or a gluino, $S$\ncan be a \nsfermion or a Higgs boson, $V$ can be a $\\gamma$, $Z$, $W^\\pm$ or $g$. \nNot shown are the diagrams with a $W^+$--$H^+$ or $G^+$--$H^+$ transition\ncontribution on the external Higgs boson leg. \n}\n\\label{fig:fdsbotHpm}\n\\vspace{2cm}\n\\includegraphics[width=0.90\\textwidth]{fdsbotW}\n\\caption{\nGeneric Feynman diagrams for the decay \n$\\tilde{t}_2 \\to \\tilde{b}_i W^+$ ($i = 1,2$).\n$F$ can be a SM fermion, a chargino or neutralino or a gluino, $S$ can be a\nsfermion or a Higgs boson, $V$ can be a $\\gamma$, $Z$, $W^\\pm$ or $g$. \n}\n\\label{fig:fdsbotW}\n\\end{center}\n\\end{figure}\n\n\nFor completness we show here also the formulas that have been\nused to calculate the tree-level decay widths:\n\\begin{align}\n\\Gamma^{\\rm tree}(\\tilde{t}_2 \\to \\tilde{b}_i H^+) &= \\frac{|C(\\tilde{t}_2, \\tilde{b}_i, H^+)|^2\\,\n \\lambda^{1\/2}(m_{\\tilde{t}_2}^2,m_{\\tilde{b}_i}^2,M_{H^\\pm}^2)}\n {16\\, \\pi\\, m_{\\tilde{t}_2}^3}\\qquad (i = 1,2)~, \\\\\n\\Gamma^{\\rm tree}(\\tilde{t}_2 \\to \\tilde{b}_i W^+) &= \\frac{|C(\\tilde{t}_2, \\tilde{b}_i, W)|^2\\,\n \\lambda^{3\/2}(m_{\\tilde{t}_2}^2,m_{\\tilde{b}_i}^2,M_W^2)}\n {16\\, \\pi\\, M_W^2\\, m_{\\tilde{t}_2}^3}\\qquad (i = 1,2)~,\n\\end{align}\nwhere $\\lambda(x,y,z) = (x - y - z)^2 - 4yz$ and the couplings \n$C(a, b, c)$ can be found in the {\\em FeynArts}~model files~\\cite{feynarts-mf}.\nThe bottom-Yukawa couplings generically are enhanced with $\\tan \\beta$.\n\n\n\\newpage\n\\pagebreak\n\\clearpage\n\n\n\n\\subsection{Numerical examples for the six renormalization schemes}\n\\label{sec:numpar}\n\nWe start our analysis by showing some representative numerical\nexamples. We evaluate the tree-level results and the one-loop correction\nfor $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ including wave function corrections. \nThe parameters are chosen according to the two scenarios, S1\\ and S2, \nshown in \\refta{tab:para}.\\footnote{It should be noted that we do not\n include any further \nshifts in the parameters than the one given in \\refeq{MSbotshift}. \nCorrespondingly, the values for the parameters $A_b$ and $M_{\\tilde{b}_R}$ in\n\\refta{tab:para} do not reflect the actual values for the input\nparameters with respect to the \nchosen renormalization scheme. For example, the\n$\\tilde{b}_2$~mass --- though considered as an input in the\nrenormalization scheme and defined as on-shell mass --- receives a shift\ngoing from tree- to one-loop level when starting out with the values in\n\\refta{tab:para} and including only the shift \\refeq{MSbotshift}. \nTo circumvent this shift of the $\\tilde{b}_2$~mass, additional shifts to the\ntree-level values of \n$A_b$ and $M_{\\tilde b_R}$ would be required (depending on the\nrenormalization scheme).\n\\vspace{1mm}}\n\n\n\\begin{table}[t!]\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{|c||r|r|r|r|r|r|r|r|}\n\\hline\nScen.\\ & $M_{H^\\pm}$ & $m_{\\tilde{t}_2}$ & $\\mu$ & $A_t$ & $A_b$ & $M_1$ & $M_2$ & $M_3$ \n\\\\ \\hline\\hline\nS1 & 150 & 600 & 200 & 900 & 400 & 200 & 300 & 800 \n\\\\ \\hline\nS2 & 180 & 900 & 300 & 1800 & 1600 & 150 & 200 & 400 \n\\\\ \\hline\n\\end{tabular}\n\\caption{MSSM parameters for the initial numerical investigation; all\nparameters are in GeV. \nWe always set $m_{b}^{\\ensuremath{\\overline{\\mathrm{MS}}}}(m_{b}) = 4.2 \\,\\, \\mathrm{GeV}$.\nIn our analysis we use \n$M_{\\tilde Q_L}(\\tilde{t}) = M_{\\tilde{t}_R} = M_{\\tilde{b}_R} =: M_{\\rm SUSY}$, where $M_{\\rm SUSY}$\n is chosen such that the above value of $m_{\\tilde{t}_2}$ is realized.\nFor the $\\tilde{b}$~sector the shift in $M_{\\tilde Q_L}(\\tilde{b})$ as defined in\n\\refeq{MSbotshift} is taken into account.\nThe parameters entering the scalar lepton sector and\/or the first two\ngenerations do not play a relevant role in our analysis.\nThe values for $A_t$ and $A_b$ are chosen such that charge- or\ncolor-breaking minima are avoided~\\cite{ccb}.\n}\n\\label{tab:para}\n\\end{center}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\n\n\nSo far we concentrate on the rMSSM: if a scheme shows deficiencies in\nthe rMSSM, the same problems occur in the cMSSM. The final numerical\nexamples in \\refse{sec:numex} will also show complex parameters as well\nas results for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_{1,2} W^+)$.\nIt should be noted that $\\tan \\beta \\lsim 9.6\\, (4.6)$\nis excluded for S1\\ (S2) \ndue to the MSSM Higgs boson searches at LEP~\\cite{LEPHiggsSM,LEPHiggsMSSM}. \nHowever, we are interested in the general behavior of the renormalization \nschemes. If certain features appear in the two numerical scenarios \n(S1\\ and S2) only for experimentally excluded $\\tan \\beta$ values, \nother parameter choices may exhibit these features also in unexcluded \nparts of the MSSM parameter space. \nConsequently, in order to investigate the various renormalization\nschemes on \ngeneral grounds, in the following we show the results for $\\tan \\beta > 1$. \nA similar reasoning applies to the limits on the MSSM parameter space \ndue to SUSY searches. Nevertheless, to avoid completely unrealistic spectra, \nthe following exclusion limits \\cite{pdg} hold in our two\nscenarios:\n\\begin{align}\nm_{\\tilde{t}_1} &> 95 \\,\\, \\mathrm{GeV}, \\;\nm_{\\tilde{b}_1} > 89 \\,\\, \\mathrm{GeV}, \\;\nm_{\\tilde{q}} > 379 \\,\\, \\mathrm{GeV}, \\;\nm_{\\tilde{e}_1} > 73 \\,\\, \\mathrm{GeV}, \\nonumber \\\\\n\\mneu{1} &> 46 \\,\\, \\mathrm{GeV}, \\;\n\\mcha{1} > 94 \\,\\, \\mathrm{GeV}, \\;\nm_{\\tilde{g}} > 308 \\,\\, \\mathrm{GeV} .\n\\end{align}\n\n\nA few examples of the scalar top and bottom quark masses \nat the one-loop level%\n\\footnote{For the scalar top quark masses the\n tree-level and the one-loop values are the same (according to\n our renormalization conditions).}%\n~(using \n$M_{\\tilde{Q}_L}^2(\\tilde{b})$ in \\refeq{MSbotshift} for the\none-loop result) \nin the scenarios S1\\ and S2\\ are \nshown in \\refta{tab:squark}. The values of $m_{\\tilde{t}_2}$ allow copious\nproduction of the heavier scalar top quark at the LHC. For other\nchoices of the \ngluino mass, $m_{\\tilde{g}} > m_{\\tilde{t}_2}$, which would leave no visible effect for\nmost of the decay modes of the $\\tilde{t}_2$, the heavier\nscalar top quark could also be\nproduced from gluino decays at the LHC. \nFurthermore, in S1\\ (even for the nominal value of $m_{\\tilde{t}_2}$ as given in\n\\refta{tab:para}) the production of $\\tilde{t}_2$ at the ILC(1000), i.e.\\ with \n$\\sqrt{s} = 1000 \\,\\, \\mathrm{GeV}$, via $e^+e^- \\to \\tilde{t}_2\\tilde{t}_1$ will be possible,\nwith the subsequent decay modes (\\ref{stsbH}) and (\\ref{stsbW})\nbeing open. The clean environment of the ILC would permit a detailed\nstudy of the scalar top quark decays.\nDepending on the combination of allowed decay\nchannels a determination of the branching ratios at the few per-cent\nlevel might be achievable in the high-luminosity running of the ILC(1000).\nMore details will be discussed elsewhere~\\cite{Stop2decay}.\n\n\n\n\\begin{table}[t!]\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{|c|c||r|r|r|r|}\n\\hline\nScen. & $\\tan \\beta$ & $m_{\\tilde{t}_1}$~~ & $m_{\\tilde{t}_2}$~~ & $m_{\\tilde{b}_1}$~~ & $m_{\\tilde{b}_2}$~~ \n\\\\ \\hline\\hline\n & 2 & 293.391 & 600.000 & 441.987 & 447.168\n\\\\ \\cline{2-6}\nS1 & 20 & 235.073 & 600.000 & 418.824 & 439.226\n\\\\ \\cline{2-6}\n & 50 & 230.662 & 600.000 & 400.815 & 449.638\n\\\\ \\hline\\hline\n & 2 & 495.014 & 900.000 & 702.522 & 707.598\n\\\\ \\cline{2-6}\nS2 & 20 & 445.885 & 900.000 & 678.531 & 695.180\n\\\\ \\cline{2-6}\n & 50 & 442.416 & 900.000 & 628.615 & 697.202\n\\\\ \\hline\n\\end{tabular}\n\\caption{The top and bottom squark masses \n at the one-loop level (see text) in the\n scenarios S1 and S2 and \n at different $\\tan \\beta$ for the numerical investigation; \n all masses are in GeV and rounded to one MeV.\n}\n\\label{tab:squark}\n\\end{center}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\n\n\nLater we will also analyze numerical results for complex input parameters.\nHere it should be noted that the results for physical observables are\naffected only \nby certain combinations of the complex phases of the \nparameters $\\mu$, the trilinear couplings $A_f$, \n$f = \\{u,c,t,d,s,b,e,\\mu,\\tau\\}$, the \ngaugino mass parameters $M_1$, $M_2$,\n$M_3$ and the Higgs soft SUSY breaking parameter\n$m_{12}^2$~\\cite{MSSMcomplphasen,SUSYphases}. \nIt is possible, for instance, to eliminate the phase $\\varphi_{M_2}$ and\nthe phase $\\varphi_{m_{12}^2}$.\nExperimental constraints on the (combinations of) complex phases \narise in particular from their contributions to electric dipole moments of\nheavy quarks~\\cite{EDMDoink}, of the electron and \nthe neutron (see \\citeres{EDMrev2,EDMPilaftsis} and references therein), \nand of the deuteron~\\cite{EDMRitz}. While SM contributions enter \nonly at the three-loop level, due to its\ncomplex phases the MSSM can contribute already at one-loop\norder.\nLarge phases in the first two generations of sfermions can only be \naccommodated if these generations are assumed to be very\nheavy~\\cite{EDMheavy} or large cancellations occur~\\cite{EDMmiracle},\nsee however the discussion in \\citere{EDMrev1,plehnix}.\nA recent review can be found in \\citere{EDMrev3}.\nAccordingly, using the convention that $\\varphi_{M_2} =0$ and\n$\\varphi_{m_{12}^2} =0$, as done in this paper, in particular \nthe phase $\\varphi_\\mu$ is tightly constrained~\\cite{plehnix}, \nwhile the bounds on the phases of the third generation\ntrilinear couplings are much weaker.\nThe phase of $\\mu$ enters in the combinations \n$(\\varphi_{A_{t,b}} + \\varphi_{\\mu} - \\varphi_{m_{12}^2})$. Setting\n$\\varphi_\\mu = 0$ (and $\\varphi_{M_2} =\\varphi_{m_{12}^2} =0$, see above) \nleaves us with $A_t$ and $A_b$ as complex valued\nparameters. Since we are interested in the renormalization of the\n$b\/\\tilde{b}$~sector, in our numerical analysis we will focus on a\ncomplex~$A_b$ and keep $A_t$ real (see, however, \\citere{Stop2decay}).\n\n\n\\begin{table}[t!] \n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{|cc||c||r|r|r||r|r|r|}\n\\hline\n\\multicolumn{3}{|c||}{$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ for S1} & \n\\multicolumn{3}{|c||}{$\\tan \\beta = 2$} &\n\\multicolumn{3}{|c|}{$\\tan \\beta = 50$} \\\\\n\\hline\n& renorm.\\ scheme & $\\mu_R$ & tree & loop & \n$m_{b}$ & tree & loop & $m_{b}$ \n\\\\\n\\hline \\hline\nRS1: &``OS'' & $m_{\\tilde{t}_2}$ & \n0.0017 & -0.0011 & 3.29 & 2.5930 & -53.3469 & 3.84 \\\\ \\hline\nRS2: & ``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'' & $m_{\\tilde{t}_2}$ & \n0.0009 & 0.0002 & 2.38 & 0.9653 & -0.0311 & 2.16 \\\\ \\hline\nRS3: & ``$m_{b},\\, Y_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'' & $m_{\\tilde{t}_2}$ & \n0.0009 & 0.0004 & 2.38 & 0.9484 & -1.5404 & 2.16 \\\\ \\hline\nRS4: & ``$m_{b}$~\\ensuremath{\\overline{\\mathrm{DR}}}, $Y_b$~OS'' & $m_{\\tilde{t}_2}$ & \n0.0009 & 0.0000 & 2.38 & 0.9593 & -0.3411 & 2.16 \\\\ \\hline\nRS5: & ``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS'' & $m_{\\tilde{t}_2}$ & \n------ & ------ & ------ & 0.9399 & -0.0481 & 2.13 \\\\ \\hline\nRS6: & ``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS'' & $m_{\\tilde{t}_2}$ & \n0.0007 & 0.0001 & 2.19 & 0.9390 & -0.0347 & 2.13 \\\\ \\hline\n\\end{tabular}\\\\[3ex]\n\\begin{tabular}{|cc||c||r|r|r||r|r|r|}\n\\hline\n\\multicolumn{3}{|c||}{$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ for S2} & \n\\multicolumn{3}{|c||}{$\\tan \\beta = 2$} &\n\\multicolumn{3}{|c|}{$\\tan \\beta = 50$} \\\\\n\\hline\n& renorm.\\ scheme & $\\mu_R$ & tree & loop & \n$m_{b}$ & tree & loop & $m_{b}$ \n\\\\\n\\hline \\hline\nRS1: & ``OS'' & $m_{\\tilde{t}_2}$ & \n2.0928 & -0.0776 & 3.23 & 8.5163 & -106.9700 & 3.70 \\\\ \\hline\nRS2: & ``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'' & $m_{\\tilde{t}_2}$ & \n2.2171 & -0.1449 & 2.33 & 1.8173 & -0.5125 & 2.11 \\\\ \\hline\nRS3: & ``$m_{b},\\, Y_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'' & $m_{\\tilde{t}_2}$ & \n0.0077 & 0.0582 & 2.33 & 3.1409 & -11.6833 & 2.11 \\\\ \\hline\nRS4: & ``$m_{b}$~\\ensuremath{\\overline{\\mathrm{DR}}}, $Y_b$~OS'' & $m_{\\tilde{t}_2}$ & \n2.2564 & -0.1031 & 2.33 & 2.9230 & -4.5506 & 2.11 \\\\ \\hline\nRS5: & ``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS'' & $m_{\\tilde{t}_2}$ & \n2.2332 & -0.1004 & 2.45 & 2.3018 & 0.2924 & 1.84 \\\\ \\hline\nRS6: & ``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS'' & $m_{\\tilde{t}_2}$ & \n2.2925 & -0.1067 & 2.14 & 2.3558 & -0.0710 & 1.86 \\\\ \\hline\n\\end{tabular}\n\\caption{Examples for tree-level and full one-loop contributions\n (see text) to $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ for S1\\ (upper table) and S2\\ (lower table); \n all values are in GeV (no comparison of the renormalization\n schemes, see text). \n In S1 using RS5 a divergence is reached for $\\tan \\beta = |A_b|\/|\\mu| = 2$ and no\n value can be computed (see text below). The different\n renormalization schemes are listed in \\refta{tab:RS}.\n}\n\\label{tab:numex}\n\\end{center}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\n\n\\bigskip\nWe start our numerical examples with the evaluation of $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ in\nS1\\ and S2\\ for $\\tan \\beta = 2$ and $\\tan \\beta = 50$ as shown in\n\\refta{tab:numex}. The corresponding results as a continuous function of\n$\\tan \\beta$ can be seen in \\reffi{fig:st2sb1H.RS}.\nIt must be emphasized here that the table and the plots do not\nconstitute a {\\em comparison} of the various schemes, but ``only''\nindividual numerical examples that are used to exhibit certain problems\nof the various schemes. A numerical comparison of the schemes requires\nthat the input parameters are converted from one scheme into another,\nsee, for instance, \\citere{mhiggsFDalbals}, which is not performed\nwithin this analysis. In our numerical examples \nthe renormalization scale, $\\mu_R$, has been set to the mass of the\ndecaying particle, i.e.\\ $\\mu_R = m_{\\tilde{t}_2}$. \nIn \\refta{tab:numex} the two main columns, labeled ``$\\tan \\beta = 2$'' and \n``$\\tan \\beta = 50$'', are divided into three columns where ``tree'' contains the \ntree-level results and ``loop'' the one-loop \ncontribution. $m_{b}$ denotes the corrected bottom quark value\ncorresponding to the respective renormalization, see \\refeq{eq:mbcorr}.\n\nThe two values of $\\tan \\beta$ were chosen as an example of a very low \nand a very high value. It should be kept in mind that the low value is\npossibly already in conflict with MSSM Higgs boson \nsearches~\\cite{LEPHiggsSM,LEPHiggsMSSM}, but kept to show an\n``extreme'' example as explained above.\nIt can be seen that RS1, RS3, RS4 and RS5 yield \nrelatively large absolute values of loop contributions with respect to\nthe tree-level \nresult, either for $\\tan \\beta = 2$ {\\em or} for $\\tan \\beta = 50$, at least in one of\nthe two numerical scenarios.\nThis simple example shows that (by choosing a specific scenario) \nalready all except two renormalization schemes fail in part\nof the parameter space. \n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RS.S1.st2sb1H.eps}\n\\hspace{-4mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RS.S2.st2sb1H.eps} \n\\end{tabular}\n\\caption{$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$. \n Full one-loop corrected partial decay widths for the different\n renormalization schemes (no comparison, see text). \n The parameters are chosen according to S1\\ \n in the\n left plot and S2\\ in the right plot. For S1\\ the grey region \n and for S2\\ the dark grey region is excluded by LEP Higgs searches\n (see text).}\n\\label{fig:st2sb1H.RS}\n\\end{center}\n\\end{figure}\n\n\n\n\nMore problems of the renormalization schemes RS1, RS3, RS4 and RS5\nbecome visible in \\reffi{fig:st2sb1H.RS}.\nIn the left (right) plot of \\reffi{fig:st2sb1H.RS} we show the results of\nS1 (S2) as a function of $\\tan \\beta$. For S1\\ the grey region \nand for S2\\ the dark grey region at low values of $\\tan \\beta$ are excluded\nby LEP Higgs searches~\\cite{LEPHiggsMSSM}. \nIt can be seen in \\reffi{fig:st2sb1H.RS} that RS1 and RS3 deviate\nstrongly from the (see the end of \\refse{sec:calc})\nexpected behavior of increasing\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ with growing $\\tan \\beta$ that the other schemes exhibit. The\nsame is observed for RS4 in S2\\ for $\\tan \\beta \\gsim 35$.\nProblems in RS2 are discussed in \\refse{sec:mbAb}, problems in RS6\nhave been found for complex parameters, see \\refse{sec:problem_non_mb}.\nThe various spikes and dips can be understood as follows:\n\\begin{itemize}\n\n\\item\nFor RS3 in S2\\ a ``peak'' appears at $\\tan \\beta \\approx 4.6$ and at \n$\\tan \\beta \\approx 6.2$. This is discussed in \\refse{sec:problem_non_Ab} below.\n\n\\item\nFor RS5 in S1\\ a ``peak'' appears (not visible) at \n$\\tan \\beta = |A_b|\/|\\mu| = 2$.\nThis is caused by large corrections to the bottom quark mass as discussed \nfurther in \\refse{sec:problem_non_mb}. This is also the reason why \nthere is no entry in \\refta{tab:numex} for RS5, S1 at $\\tan \\beta = 2$.\n\n\\item\nFor RS5 in S2\\ a ``peak'' appears at $\\tan \\beta = |A_b|\/|\\mu| = 5.33$.\nThis is caused by large corrections to the bottom quark mass as \ndiscussed further in \\refse{sec:problem_non_mb}.\n\n\\end{itemize}\n\n\n\n\n\\subsection{Generic considerations for the \\boldmath{$b\/\\tilde{b}$} sector\n renormalization (I)}\n\\label{sec:genericI}\n\nAs discussed in \\refse{sec:stop}, a bottom quark\/squark sector\nrenormalization scheme always contains dependent counterterms which can\nbe expressed by the independent ones.\nAccording to our six definitions, this can be\n$\\dem_{b}$, $\\deA_b$ or $\\delta Y_b$. \nA problem can occur when the MSSM parameters are chosen such that the\nindependent counterterms (nearly) drop out of the relation determining\nthe dependent counterterms.\nAs will be shown below, even\nrestricting to the two numerical examples, S1\\ and S2, it is \npossible to find a set of MSSM parameters which show this behaviour for\neach of the chosen\nrenormalization schemes.\nConsequently, it appears to be difficult {\\em by construction} to\ndefine a \nrenormalization scheme for the bottom quark\/squark sector (once\nthe top quark\/squark\nsector has been defined) that behaves well for the full MSSM parameter\nspace. One possible exception could be a pure \\ensuremath{\\overline{\\mathrm{DR}}}\\ scheme, which,\nhowever, is not well suited for processes with external top\nsquarks and\/or bottom squarks. \n\nAssuming that SUSY, and more specifically the MSSM, will be discovered\nat the LHC and its parameters will be measured, the problem will have\ndisappeared. For a specific set of MSSM parameters, renormalization\nschemes can (easily) be found that behave well. \nHowever, due to our ignorance about the actual values of the SUSY\nparameters, scans over large parts of the MSSM parameter space are\nperformed, see also \\refse{sec:numex}. For this kind of analysis a\ncareful choice of the renormalization scheme has to be made.\n\nIn the following subsections we will analyze in more detail, analytically\nand numerically, the deficiencies of the various schemes.\n\n\n\n\n\n\n\\subsection{Problems of the ``OS'' renormalization}\n\\label{sec:problem_OS}\n\nThe ``OS'' renormalization as described in \\refse{sec:OS} does not yield\nreasonable results in perturbative \ncalculations as shown \nalready, e.g., in \\citere{mhiggsFDalbals,sbotrenold}. \nFor the sake of completeness we briefly repeat the results. \nThe ``OS'' scheme of \\refse{sec:OS} is the renormalization scheme\nanalogous to the one used \nin the $t\/\\tilde{t}$ sector and thus would be the ``naive'' choice. It\nincludes an on-shell renormalization \ncondition on the sbottom mixing parameter $Y_b$ that contains the\ncombination $(A_b - \\mu^* \\tan \\beta)$. \nIn parameter regions where $(\\mu\\tan \\beta)$ is much larger than $A_b$, the\ncounterterm $\\deA_b$ receives a very large finite shift when calculated\nfrom the counterterm $\\delta Y_b$. More specifically, $\\deA_b$ as given in\n\\refeq{Ab_OS} contains the contribution\n\\begin{align}\n\\label{dAb_OSproblem}\n\\deA_b = \\frac{1}{m_{b}} \\left[ -(A_b - \\mu^*\\tan \\beta)\\, \\dem_{b} + \\ldots \\right]\n\\end{align}\nthat can give rise to very large corrections to $A_b$.\nThis is also visible in \\reffi{fig:dAb} below, where we show the \nnumerical values of $\\deA_b$ as a function of $\\tan \\beta$ for various \nrenormalization schemes. \nIn \\citere{mhiggsFDalbals} it was shown that, because of\n\\refeq{dAb_OSproblem}, the ``OS'' renormalization yields\nhuge corrections to the lightest MSSM Higgs mass. \nAlso the numerical results shown in \\refta{tab:numex} and \n\\reffi{fig:st2sb1H.RS} show extremely large one-loop corrections \nfor $\\tan \\beta = 50$. \n\nThis problem is (more or less) avoided in the other renormalization\nschemes introduced in \\refta{tab:RS}, where the renormalization\ncondition is applied directly to $A_b$, rather than deriving $\\deA_b$\nfrom a renormalization condition fixing $\\delta Y_b$.\nAlso the renormalization schemes \nRS3 (``$m_{b},\\, Y_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'') and RS4 (``$m_{b}$~\\ensuremath{\\overline{\\mathrm{DR}}}, $Y_b$~OS'')\navoid this severe problem by renormalizing the bottom quark mass \\ensuremath{\\overline{\\mathrm{DR}}}. \n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RC.TB.dA.S1.eps}\n\\hspace{-4mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RC.TB.dA.S2.eps} \n\\end{tabular}\n\\caption{\n Finite parts of $\\deA_b$ in various renormalization schemes. The\n parameters are chosen according to S1\\\n left plot and S2\\ right plot. \n For S1\\ the grey region is excluded \n and for S2\\ the dark grey region is excluded.}\n\\label{fig:dAb}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{Problems of non-\\boldmath{$A_b$} renormalization schemes}\n\\label{sec:problem_non_Ab}\n\nTwo of our schemes, besides the ``OS'' scheme (RS1), do not employ\na renormalization of $A_b$: \nRS3 (``$m_{b},\\, Y_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'') and RS4 (``$m_{b}$~\\ensuremath{\\overline{\\mathrm{DR}}}, $Y_b$~OS''). \nAs argued in \\refse{sec:problem_OS} a huge contribution to $\\deA_b$ as\nevaluated in that section is avoided by the \\ensuremath{\\overline{\\mathrm{DR}}}\\ renormalization of\n$m_{b}$. However, following \\refeq{Ab_OS} with $\\dem_{\\tilde{b}_1}^2$, $\\dem_{\\tilde{b}_2}^2$,\n$\\delta Y_b$ and $\\delta m_{b}$ chosen according to the renormalization\nschemes RS3 and RS4, respectively, one finds for the finite parts of $\\deA_b$:\n\\begin{align}\n\\mbox{RS3} &:~ \\deA_b|_{\\text{fin}} = \\ed{m_{b}} \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n \\left( \\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2 \\right) \\right]_{\\text{fin}} \n + \\ldots~, \\\\\n\\label{dAb_problem}\n\\mbox{RS4} &:~ \\deA_b|_{\\text{fin}} = \\ed{m_{b}} \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n \\left( \\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2 \\right)\n + U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\delta Y_b^*\n + U_{\\tilde{b}_{12}}^* U_{\\tilde{b}_{21}} \\delta Y_b \\right]_{\\text{fin}} + \\ldots~,\n\\end{align}\nwhere the ellipses denote contributions from $\\delta\\mu$ which, however, are\nnot relevant for our argument.\nIt can be seen that still $\\deA_b$ depends on parameters (diagonal and\noff-diagonal sbottom self-energies) that are independent of $A_b$. As\nan example, Higgs boson loops in the sbottom self-energy contain\ncontributions $\\sim \\mu \\tan \\beta$, which can become very large, independently\nof the value of $A_b$. This can be seen in the right plot of\n\\reffi{fig:dAb}, where \nwe show $\\deA_b$ as a function of $\\tan \\beta$ in S2. In both renormalization\nschemes, RS3 and RS4, $\\deA_b$ becomes very large and negative for\nlarge $\\tan \\beta$. This yields the very large and negative loop corrections to\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ shown in the right plot of \\reffi{fig:st2sb1H.RS}. \nIn S1\\ this problem is less pronounced, as can be seen in the left \nplot of \\reffi{fig:dAb} ($\\deA_b$) and \\reffi{fig:st2sb1H.RS} \n($\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$). \n\nBut also for lower $\\tan \\beta$ values, $\\tan \\beta \\lsim 10$, problems can occur. \nThe (finite) ``multiple spike structure'' in RS3 for S2\\ around \n$\\tan \\beta \\approx 5.33$ (for details see the small insert within the \nright plot of \\reffi{fig:dAb}) \nis due to an interplay of top\/chargino contributions to the two \ndiagonal sbottom self-energies, invalidating this scenario also for \nthis part of the parameter space. \n\n\n\n\n\n\\subsection{Problems of an \\boldmath{$m_{b}$}--\\boldmath{$A_b$} \n renormalization}\n\\label{sec:mbAb}\n\nIf $m_{b}$ and $A_b$ are renormalized, the sbottom mixing parameter $Y_b$\nis necessarily a dependent \nparameter, see \\refta{tab:RS}. This situation is realized in the scheme\nRS2 (``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}''), see \\refse{sec:mbDRbar_AbDRbar}.\n$\\delta Y_b$ enters prominently into $\\dZ{\\tilde{b}_{21}}$. For real parameters\nwe have,\n\\begin{align}\n\\dZ{\\tilde{b}_{21}} &=- 2\\, \\frac{\\mathop{\\mathrm{Re}}\\Sigma_{\\tilde{b}_{21}}(m_{\\tilde{b}_2}^2) - \\delta Y_b}\n {m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2}~.\n\\label{dZSbot21}\n\\end{align}\nIn this way $\\delta Y_b$ (or the interplay between $\\delta Y_b$ and\n$\\mathop{\\mathrm{Re}}\\Sigma_{\\tilde{b}_{21}}(m_{\\tilde{b}_2}^2)$) can induce large loop corrections to the\nscalar top quark decay width.\n$\\delta Y_b$ can be decomposed according to \\refeq{dYb_mbDRbar_AbDRbar}\n(concentrating again on the case of real parameters),\n\\begin{align}\n\\delta Y_b &= \\frac{U_{\\tilde{b}_{11}} U_{\\tilde{b}_{21}}}\n {|U_{\\tilde{b}_{11}}|^2 - |U_{\\tilde{b}_{12}}|^2}\n \\left( \\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2 \\right)\n + \\ldots~,\n\\label{dYb_problem1}\n\\end{align}\nwhere the ellipses denote terms with only divergent\ncontributions (due to the chosen renormalization scheme RS2) as well as\nfinite contributions from $\\delta\\mu$, \nwhich, however, do not play a role for our argument.\nFor ``maximal sbottom mixing'', \n$|U_{\\tilde{b}_{11}}| \\approx |U_{\\tilde{b}_{12}}|$,\n$\\delta Y_b$ diverges, and the loop calculation does not yield a reliable\nresult. In our two parameter scenarios, S1\\ and S2, this is not the\ncase. Such a \nlarge sbottom mixing is often associated with large values of $|A_b|$\nthat may be in conflict with charge- or color-breaking minima~\\cite{ccb}.\n\nHowever, in order to show an example with a divergence in $\\delta\nY_b$ we use a modified version of S1\\ with $A_b = 1000 \\,\\, \\mathrm{GeV}$ \n(a value still allowed following \\citere{ccb}).\nIn this scenario at $\\tan \\beta \\approx 37$ we indeed find the case of\n``maximal mixing'' in the scalar bottom sector.\nAs expected this leads to a divergence in $\\delta Y_b$,\nas can be seen in the left plot of\n\\reffi{fig:mbAb}. This divergence propagates into $\\dZ{\\tilde{b}_{21}}$ as\nshown in the right plot of \\reffi{fig:mbAb}.%\n\\footnote{\nThe scalar bottom masses could receive large corrections via \n$M_{\\tilde{Q}_L}^2(\\tilde{b})$ in \\refeq{MSbotshift}, with $\\delta Y_b$ \nentering via \\refeq{MSbotshift-detail}.\n}%\n(Also $\\Sigma_{\\tilde{b}_{21}}$ exhibits a discontinuity due to a sign change\nin $U_{\\tilde{b}}$ for this extreme set of MSSM parameters.)\nThe $\\tan \\beta$ value for which this ``divergence'' occurs depends on the\nchoice of the other MSSM parameters. For (numerical) comparison we\nalso show $\\dZ{\\tilde{t}_{21}}$ for the two scenarios.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RC.TB.dY.eps}\n\\hspace{-4mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RC.TB.dZ.eps} \n\\end{tabular}\n\\caption{\n $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$.\n Left plot: size of $\\delta Y_b$ and $\\mathop{\\mathrm{Re}}\\Sigma_{\\tilde{b}_{21}}(m_{\\tilde{b}_1}^2)$, the\n two contributions to $\\dZ{\\tilde{b}_{21}}$, in RS2.\n Right plot: comparison of the size of $\\dZ{\\tilde{b}_{21}}$ in the scheme RS2\n (``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}''). For both plots the parameters are chosen\n according to S1 (but here with $A_b = 1000$ GeV), S2\\ in \\refta{tab:para}.\n For S1\\ the grey region is excluded and for S2\\ the\n dark grey region is excluded via LEP Higgs searches (see text).}\n\\label{fig:mbAb}\n\\end{center}\n\\end{figure}\n\n\nFor the different choice of MSSM parameters in S2\\ (without a\nhigher $A_b$ value) this divergences\ndoes not occur. However, for $\\tan \\beta \\lsim 7$ one finds \n$\\delta Y_b \\gsim \\mathop{\\mathrm{Re}}\\Sigma_{\\tilde{b}_{21}}(m_{\\tilde{b}_1}^2)$ \n(with $\\delta Y_b = \\mathop{\\mathrm{Re}}\\Sigma_{\\tilde{b}_{21}}(m_{\\tilde{b}_1}^2)$ for $\\tan \\beta \\approx 7.5$).\nIn this part of the parameter space we also find $m_{\\tilde{b}_1} \\approx m_{\\tilde{b}_2}$,\nyielding a relatively large value of $\\dZ{\\tilde{b}_{21}}$ according to\n\\refeq{dZSbot21}, as can be seen in the right plot of\n\\reffi{fig:mbAb}. This relatively large (negative) value of\n$\\dZ{\\tilde{b}_{21}}$ in \nturn induces relatively large corrections to $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$. \nHowever, the loop corrections do not exceed the tree-level value\nof $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ (for our choice of MSSM parameters).\nIn summary: while for S1\\ a divergence in $\\delta Y_b$ and thus in\n$\\dZ{\\tilde{b}_{12}}$ can appear for very large values of $|A_b|$\n(possibly in conflict with charge- or color-breaking minima),\ninvalidating the renormalization scheme~RS2 in this part of the \nparameter space, these kind of problems are not encountered in S2. Here\nonly moderate loop corrections to the respective tree-level values are\nfound, and RS2 can be applied safely.\n\n\n\n\n\\subsection{Problems of non-\\boldmath{$m_{b}$} renormalization schemes}\n\\label{sec:problem_non_mb}\n\nTwo of our schemes do not employ a renormalization condition for $m_{b}$:\nRS5 (``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS'') and \nRS6 (``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS''). \nSince $A_b$ and $Y_b$ are complex, we chose to renormalize $A_b$ and the\nreal part of $Y_b$. \n\nWe start with the discussion of the (simpler) ``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS''\nscheme. We will focus on the real case as a subclass of\nthe more general complex case. In this renormalization scheme the bottom\nquark mass counterterm has the following form for real parameters\n(compare to \\refeq{dmb_AbDRbar_ReYbOS}),\n\\begin{align}\n\\label{dmb_problem}\n\\dem_{b} &= - \\frac{m_{b}\\, \\deA_b + \\delta S}{(A_b - \\mu\\tan \\beta)}~.\n\\end{align}\nFor vanishing sbottom mixing one finds $(A_b - \\mu\\tan \\beta) \\to 0$.\nIn the ``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS'' scheme this yields a finite (and\nnegative) numerator in \\refeq{dmb_problem}, but a vanishing denominator.\n\nIn a numerical evaluation, starting out with a value for the bottom\nquark mass defined as \\ensuremath{\\overline{\\mathrm{DR}}}~parameter, the actual value of the bottom\nquark mass receives a shift with respect to the \\ensuremath{\\overline{\\mathrm{DR}}}~bottom quark\nmass according to \\refeq{eq:mbcorr}. This shift corresponds to the\nfinite part of $\\delta m_{b}$ in \\refeq{dmb_problem}.\nConsequently, large positive or negative contributions to the bottom quark \nmass can occur, yielding possibly \nnegative values for the bottom quark mass and thus invalidating the\nrenormalization scheme for \nthese parts of the parameter space.\nThis can be seen in the left plot of \\reffi{fig:mb.explanation}, \nwhere we show $m_{b}$ in RS5 (and RS6) for the two numerical scenarios \ngiven in \\refta{tab:para} as a function of $\\tan \\beta$. \n$m_{b}$ exhibits a strong upward\/downward shift around the pole reached\nfor $\\tan \\beta = A_b\/\\mu$ and consequently yields unreliable results in this\npart of the parameter space.\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RC.TB.mb.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RC.PhiAb.mb.eps} \n\\end{tabular}\n\\caption{\n Left plot: $m_{b}$ in RS5 and RS6 for S1, S2. \n For S1\\ the grey region is excluded and for S2\\ the dark grey region \n is excluded.\n Right plot: $m_{b}$ in RS6 for S1, S2\\ but both with $\\tan \\beta = 20$ and \n $\\varphi_{A_b}$ varied. In S2\\ we used also $|\\mu| = 120 \\,\\, \\mathrm{GeV}$.}\n\\label{fig:mb.explanation}\n\\end{center}\n\\end{figure}\n\n\n\\bigskip\nWe now turn to the RS6 (``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS'') scheme. \nFollowing the same analysis as for the\n``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS'' scheme an additional term in the\ndenominator of the bottom quark mass counterterm\n$\\sim U_m\/U_-$ appears,\n\\begin{align}\n\\dem_{b} &= - \\frac{\\delta S + F}{(A_b - \\mu\\tan \\beta) - U_m\/(\\tan \\beta\\,U_-)}~,\n\\end{align}\nwhere $F$ denotes other (relatively small) additional contributions. \nWith the help of \\refeq{def:Um} one finds for real parameters\n\\begin{align}\n\\frac{U_m}{\\tan \\beta\\,U_-} &= \\frac{U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}} (A_b \\tan \\beta + \\mu)\n -U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}} (A_b \\tan \\beta + \\mu)}\n {\\tan \\beta (U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}} - \n U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}})}\n = (A_b + \\mu\/\\tan \\beta)~,\n\\end{align}\nand therefore\n\\begin{align}\n\\label{dmb_problem2}\n\\dem_{b} &= \\frac{\\delta S + F}{\\mu\\, (\\tan \\beta + 1\/\\tan \\beta)}~.\n\\end{align}\nThe denominator of \\refeq{dmb_problem2} can go to zero only for \n$\\mu \\to 0$, which is experimentally already excluded. \nConsequently, the problem of (too) large contributions to $m_{b}$ is\navoided in this scheme. This can be seen in the left plot of\n\\reffi{fig:mb.explanation}, where RS6, contrary to RS5, does not\nexhibit any pole-like structure in $m_{b}$.\n\nIn the complex case the above argument is no longer valid, and larger\ncontributions to $\\dem_{b}$ can arise. In the limit of $\\tan \\beta \\gg 1$ and\n$\\mu$~real the denominator of $\\dem_{b}$ in \\refeq{dmb_AbOS_ReYbOS}\nreads\n\\begin{align}\n\\ed{\\dem_{b}} &\\sim 4\\,\\mu\\, \\tan^3\\beta\\, \\Big[\n \\mathop{\\mathrm{Re}} U_- \\left( |U_{\\tilde{b}_{11}}|^2 - |U_{\\tilde{b}_{12}}|^2 \\right) \n + \\mathop{\\mathrm{Im}} U_- \\frac{4\\, m_{b}}{m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2} \n \\mathop{\\mathrm{Im}} \\left( U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{12}} A_b \\right) \\Big]~. \n\\label{dmb_RS6_pole}\n\\end{align}\nDepending on $\\varphi_{\\Ab}$ this denominator can go to zero and thus yield\nunphysically large corrections to $m_{b}$ in RS6.\nIn the right plot of \\reffi{fig:mb.explanation} we show $m_{b}$ as \nfunction of $\\varphi_{\\Ab}$. At \n$\\varphi_{\\Ab} \\approx 41.5^\\circ$, $87.5^\\circ$, $272.5^\\circ$, $318.2^\\circ$ \nthe denominator in \\refeq{dmb_RS6_pole} goes to zero and changes its \nsign which explains the corresponding structures.\nThis divergence in $\\dem_{b}$ enters via \\refeq{eq:mbcorr} already\ninto the tree-level prediction.\nTo summarize: while in S1\\ the scheme RS6 is well-behaved and can\nbe safely applied (also for complex $A_b$), in S2\\ \n(with $|\\mu| = 120$ GeV) severe problems\n(divergences in the counterterms) arise once complex parameters are\ntaken into account. Consequently, for S2\\ the scheme RS6 cannot be applied.\n\nIt should be noted that \nthe ``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS'' (RS6) scheme is the complex version of the \nrenormalization scheme used in \\citeres{sbotrenold,mhiggsFDalbals} for the\n\\order{\\alpha_b\\alpha_s} corrections to the neutral Higgs boson self-energies\nand thus to the mass of the lightest MSSM Higgs boson, $M_h$. For\nreal parameters, no problems occured. \n Therefore,\nemploying this renormalization scheme in \\citeres{sbotrenold,mhiggsFDalbals} \nyields numerically stable results.\n\n\n\n\n\\subsection{Generic considerations for the \\boldmath{$b\/\\tilde{b}$} sector\n renormalization (II)} \n\\label{sec:genericII}\n\nIn the previous subsections we have analyzed analytically \n(and numerically) the\ndeficiences of the various renormalization schemes. We have shown that\ndespite of the variety of schemes, even concentrating on the two sets of\nparameters, S1\\ and S2, severe problems can be encountered\nin all schemes.\n\nFor the further numerical evaluation of the partial stop quark decay\nwidths we choose \nRS2 as our ``preferred scheme''. According to our analyses in the\nprevious subsections, RS2 shows the ``relatively most stable'' behavior,\nproblems only occur for maximal sbottom mixing, \n$|U_{\\tilde{b}_{11}}| = |U_{\\tilde{b}_{12}}|$, where a divergence in $\\delta Y_b$\nappears. \nHaving $\\delta Y_b$ as a dependent counterterm induces \neffects in the field renormalization constants $\\dZ{\\tilde{b}_{12}}$ and\n$\\dZ{\\tilde{b}_{21}}$ and in\n$\\delta M_{\\tilde{Q}_L}^2(\\tilde{b})$ entering the scalar bottom quark masses.\nIn a process with only internal scalar bottom quarks, no problems occur\ndue to the field renormalization, but \ncounterterms to propagators, which\ninduce a transition from a $\\tilde{b}_1$ squark to a $\\tilde{b}_2$ squark contain\nalso the term $\\delta Y_b$. However, $\\delta Y_b$ appearing in counterterms of\n{\\em internal} scalar bottom quarks does not exhibit a problem, since in\nthis case these ``dangerous'' contributions cancel (which we have checked\nanalytically).\nOn the other hand,\nother schemes with $\\dem_{b}$ or $\\deA_b$ as dependent counterterms \nmay exhibit problems in larger parts of the parameter\nspace and may induce large effects, since $m_{b}$ (or the bottom Yukawa \ncoupling) and $A_b$ enter prominently into the various couplings of the\nHiggs bosons to other particles.\n\nWe are not aware of any paper dealing with scalar quark decays (or\ndecays into scalar quarks) that has employed exactly RS2 \n(or its real version), see our discussion in the beginning of \\refse{sec:stop}.\nVery recently a calculation of the scalar top decay width in the rMSSM\nusing a pure \\ensuremath{\\overline{\\mathrm{DR}}}\\ scheme for all parameters was\nreported~\\cite{HelmutLL2010}.\n\n\n\n\\section{Numerical examples for our favorite scheme}\n\\label{sec:numex}\n\nFollowing the discussion in \\refse{sec:RSana} we pick the renormalization \nscheme that shows the ``most stable'' behavior over the MSSM parameter space.\nWe choose the ``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}''(RS2) scheme.\nTree-level values of the partial decay widths shown in this section have\nbeen obtained including a \nshift in $m_{b}$ according to \\refeq{eq:mbcorr}. \nWe will concentrate on the calculation of the partial $\\tilde{t}_2$ decay widths\nincluding one scalar bottom quark in the final state. A calculation of\nthe respective branching ratios \nrequires the evaluation of {\\em all} partial scalar top quark decay\nwidths, which in turn \nrequires the renormalization of the full cMSSM. This is beyond the scope of\nour paper and will be presented elsewhere~\\cite{Stop2decay}.\n \n\n\\subsection{Full one-loop results}\n\\label{sec:full1L}\n\nWe start our numerical analysis with the upper left plot of\n\\reffi{fig:st2sb1H}, where we show the partial decay width \n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ as a function of $\\tan \\beta$. ``tree'' denotes the tree-level value\nand ``full'' is the decay width including {\\em all} one-loop \ncorrections as described in \\refse{sec:calc}.\nAs one can see, the full one-loop corrections are negative and rather small \nover the full range of $\\tan \\beta$, the largest size of the loop corrections\nis found to be $\\sim 28\\%$ of the tree-level value for $\\tan \\beta = 50$ in S2.%\n\\footnote{\nIt is interesting to note that at $\\tan \\beta = |A_b|\/|\\mu| = 2\\, (5.33)$ in\nS1\\ (S2) we get $U_{\\tilde{b}_{11,22}} = 1$ and $U_{\\tilde{b}_{12,21}} = 0$,\nand consequently $\\tilde{b}_{L,R} = \\tilde{b}_{1,2}$, respectively.\n}%\n~In S1\\ the grey region and in S2\\ the dark grey region is excluded due \nto too small values of the mass of the lightest MSSM Higgs boson,\n$M_h$.\n\nIn the upper right plot of \\reffi{fig:st2sb1H} we show the partial\ndecay width varying $|A_b|$ for $\\tan \\beta = 20$. \nIn S1\\ and S2\\ the full one-loop corrections grow with \n$A_b$, but never exceed $\\sim 25\\%$ of the tree-level result. \nNote, that for S1 $|A_b| > 1130 \\,\\, \\mathrm{GeV}$\n(grey region) and S2 $|A_b| > 1800 \\,\\, \\mathrm{GeV}$ (dark grey region) \nis excluded due to the charge- or color-breaking minima.\nOver the full parameter space the loop corrections are smooth and\nsmall with respect to the tree-level results.\n\nIn the lower left plot of \\reffi{fig:st2sb1H} we analyze \nthe partial decay width varying $|\\mu|$ for $\\tan \\beta = 20$. \nValues for $|\\mu| \\lsim 120 \\,\\, \\mathrm{GeV}$ are excluded due to \n$\\mcha1 < 94 \\,\\, \\mathrm{GeV}$~\\cite{pdg}.\nThe loop corrected predictions for the partial decay width show several dips and\nspikes. In S1\\ \nthe first dip at $|\\mu| \\approx 285 \\,\\, \\mathrm{GeV}$ is due to \n$|U_{\\tilde{b}_{11}}| \\approx |U_{\\tilde{b}_{12}}|$, see the discussion in\n\\refse{sec:mbAb}.\nThe second peak\/dip (already present in the tree-level prediction) at\n$|\\mu| = 300 \\,\\, \\mathrm{GeV}$ is due to the \nrenormalization of $\\mu$~\\cite{dissTF} and will be discussed in more\ndetail in \\citere{Stop2decay}.%\n\\footnote{\nThe chosen renormalization exhibits a divergence for $\\mu = M_2$. \n$\\delta \\mu$ enters via $\\delta Y_b$ into $\\delta M_{\\tilde{Q}_L}^2(\\tilde{b})$ \nand thus into the values of $m_{\\tilde{b}_i}$. Consequently, the dip is already\npresent in \nthe tree-level result.}\n~The third dip at $|\\mu| \\approx 424 \\,\\, \\mathrm{GeV}$, which is hardly visible, is\ndue to the production threshold $m_t + \\mneu{3} = m_{\\tilde{t}_2}$.\nThe fourth dip at $|\\mu| \\approx 873 \\,\\, \\mathrm{GeV}$ is the threshold\n$m_{\\tilde{t}_1} + M_{H^\\pm} = m_{\\tilde{b}_1}$ of the self energy $\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1}^2)$ \nin the renormalization constants $\\dZ{\\tilde{b}_{11}}$ and $\\dem_{\\tilde{b}_1}^2$.\nThe fifth dip at $|\\mu| \\approx 1107 \\,\\, \\mathrm{GeV}$ is the production threshold \n$m_{\\tilde{b}_2} + M_W = m_{\\tilde{t}_2}$.\nFor $|\\mu| > 790 \\,\\, \\mathrm{GeV}$ the value of $M_h$ drops strongly, and the \nscenario S1 is excluded by LEP Higgs searches as indicated by the \ngray shading.\nApart from the dips analyzed above the loop corrections are very\nsmall and do not exceed $\\sim 7\\%$ of the tree-level result, \nthe prediction for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ is well under control.\nWe now turn to the scenario S2. Here, for growing $|\\mu|$, the squark mass\nsplitting in the $\\tilde{t}\/\\tilde{b}$ sector becomes very large, leading to\nlarge contributions to the electroweak precision observables. The dark\ngray region for $|\\mu| > 1060 \\,\\, \\mathrm{GeV}$ yields $W$~boson masses outside the\nexperimentally favored region at the $2\\,\\sigma$ level, \n$M_W \\gsim 80.445 \\,\\, \\mathrm{GeV}$~\\cite{lepewwgNEW}. \nSuch large $|\\mu|$ values are consequently disfavored.\nThe dip\/peak at $|\\mu| = 200 \\,\\, \\mathrm{GeV}$ in the tree and the loop contribution\nis due to $\\delta \\mu$, where $\\mu = M_2$ is reached, see above. \nThe second dip at $|\\mu| = 477 \\,\\, \\mathrm{GeV}$, which is hardly visible, \nis the threshold $m_t + \\mcha{2} = m_{\\tilde{b}_1}$ of the self energy \n$\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1}^2)$ in the renormalization constants \n$\\dZ{\\tilde{b}_{11}}$ and $\\dem_{\\tilde{b}_1}^2$.\nThe third dip at $|\\mu| = 725 \\,\\, \\mathrm{GeV}$ is the production threshold \n$m_t + \\mneu{3} = m_{\\tilde{t}_2}$.\nThe fourth dip at $|\\mu| = 850 \\,\\, \\mathrm{GeV}$ is again the threshold \n$m_{\\tilde{t}_1} + M_{H^\\pm} = m_{\\tilde{b}_1}$. \nIn S2\\ the one-loop corrections are negative and growing with $|\\mu|$.\nApart from the dips described above, also in this numerical evaluation\nthe loop corrections stay mostly relatively small \nwith respect to the tree-level result, \nreaching the largest relative contribution at the smallest \n$|\\mu|$ values, and are thus well under control.\n\n\nWe now turn to the case of complex parameters. \nAs discussed in \\refse{sec:numpar} we consider only $A_b$ as a complex\nparameter.\nIn the lower right plot of \\reffi{fig:st2sb1H} we show \nthe partial decay width depending on $\\varphi_{\\Ab}$ for $\\tan \\beta = 20$.\nIn S1, the tree-level values and the loop corrections are well-behaved. \nThe latter ones stay relatively small for the whole parameter space, \nnot exceeding $\\sim 18\\%$ of the tree-level result.\nIn S2, the largest corrections occur for real positive values of \n$A_b$ and reach $\\sim 12\\%$ of the tree-level values. \nFor negative $A_b$, the tree-level result becomes very small \n($< 0.01\\,\\, \\mathrm{GeV}$) and here the size of the loop corrections can be as \nlarge as the tree-level values. \nA small (and barely visible) asymmetry in the one-loop corrections \nappears in the lower right plot of \\reffi{fig:st2sb1H}, \ndue to terms $\\sim U_{\\tilde{b}_{ij}} \\times C_{0,1,2}$-function.\nThe peak\/dip at $\\varphi_{\\Ab} \\approx 117^\\circ, 243^\\circ$ are again due to \n$|U_{\\tilde{b}_{11}}| \\approx |U_{\\tilde{b}_{12}}|$, see \\refse{sec:mbAb}.\nIt can be seen that the peaks due to this divergence are relatively\nsharp, i.e.\\ the region of parameter space that is invalidated remains\nrelatively small.\n\n\\smallskip\nIn \\reffi{fig:st2sb2H} we show the results for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$ for \nthe same set and variation of parameters as above. \nConsequently, the same peak and dip structures are visible in \nthe lower plots of \\reffi{fig:st2sb2H}. \nIn the lower left plot of \\reffi{fig:st2sb2H} in S1\\ both lines \nend because the phase space closes, \n$m_{\\tilde{b}_2} + M_{H^\\pm} > m_{\\tilde{t}_2}$ for $|\\mu| > 300 \\,\\, \\mathrm{GeV}$.\nOverall the partial decay width is much smaller than for\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$, which can \npartially be attributed to the smaller phase space, see for instance\nthe results within S2\\ in\nthe upper left plot of \\reffi{fig:st2sb2H}, and partially to the \nsmallness of the tree-level coupling.\nOnly in S2\\ for $\\tan \\beta \\gsim 35$ we find $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+) \\gsim 1 \\,\\, \\mathrm{GeV}$.\nThe relative corrections become very large for $|A_b| \\gsim 1200 \\,\\, \\mathrm{GeV}$\nas shown in the upper right plot of \\reffi{fig:st2sb2H}, \nhowever these values are disfavored by the constraints from charge and \ncolor breaking minima as discussed above.\nThe smallness of $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$ at the tree-level can lead sometimes to \na ``negative value at the loop level''. \nIn this case of (accidental) smallness of the\ntree-level partial decay width also $|{\\cal M}_{\\rm loop}|^2$ would have to\nbe taken into \naccount, yielding a positive value for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$.\nOverall, because of the smallness of the tree-level result due to the tree-level\ncoupling the {\\em relative} size of the loop corrections are a bit larger than\nfor $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$. Nevertheless, apart from the peaks visible in\nthe lower plots of \\reffis{fig:st2sb2H}, the loop corrections are\nwell under control also for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$ using the renormalization\nscheme RS2. \nAgain a small asymmetry in the one-loop corrections in\nthe lower right plot of \\reffi{fig:st2sb2H} can be observed, \nwhich is due to terms $\\sim U_{\\tilde{b}_{ij}} \\times C_{0,1,2}$-function.\n\n\\smallskip\nFinally we evaluate the partial decay width of a scalar top quark to a\nscalar bottom quark\nand a $W$~boson, $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 W^+)$ and $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 W^+)$. Since the\n$W$~boson is relatively light, also the latter channel is open.\nIn \\reffi{fig:st2sb1W} the results for $\\tilde{t}_2 \\to \\tilde{b}_1 W^+$ are shown, \nin \\reffi{fig:st2sb2W} the ones for $\\tilde{t}_2 \\to \\tilde{b}_2 W^+$.\nThe divergences visible in the various plots are the same ones as found\nin the respective plot for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$. \nAn additional (finite) dip is visible in the lower left plot of \n\\reffi{fig:st2sb1W} in S2\\ for $|\\mu| \\approx 521 \\,\\, \\mathrm{GeV}$, \ndue to an interplay of $t\/\\cha{2}$ contributions to \n$\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1})$, similar to the structure discussed for \n\\reffi{fig:dAb}. In this part of the parameter space the results\n calculated within the\n renormalization scheme RS2 have to be discarded.\n\nOverall, the loop corrections to $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 W^+)$ calculated within the\nrenormalization scheme RS2 behave similar\nto the ones to $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$. The size is relatively small, i.e.\n$\\lsim 20\\%$ and $\\lsim 30\\%$ of the tree-level results in the upper left\n and \n in the upper right plot of \\reffi{fig:st2sb1W}, respectively, \nfor the regions which are not in conflict with charge- or color\nbreaking minima \n(for $|A_b| = 2000 \\,\\, \\mathrm{GeV}$ a correction of $\\sim 70\\%$ of the tree-level\nresult can be observed in \nS1\\ due to the smallness of the tree-level value). \nWe find loop corrections of the size of $\\lsim 20\\%$ of the tree-level\nresults in the lower left plot of \\reffi{fig:st2sb1W} \nexcept for very small values of $|\\mu|$ and in the lower right plot of\n\\reffi{fig:st2sb1W}. \nIn the latter plot for S2\\ the known divergences appear at \n$\\varphi_{\\Ab} \\approx 117^\\circ, 243^\\circ$, leading to larger loop\ncorrections for intermediate values of $\\varphi_{\\Ab}$.\nApart from the latter case the full one-loop corrections to\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 W^+)$ are well under control employing the renormalization\nscheme RS2.\n\nSimilar observations hold for the decay $\\tilde{t}_2 \\to \\tilde{b}_2 W^+$, as shown in\n\\reffi{fig:st2sb2W}.\nIn the upper left plot of \\reffi{fig:st2sb2W} in the scenario S2\\ for \n$\\tan \\beta = |A_b|\/|\\mu| \\approx 5.3$, the tree-level partial decay width vanishes, \nleading to a ``negative value at the loop level''. \nAs discussed above, in this case also $|{\\cal M}_{\\rm loop}|^2$ would have to\nbe taken into account, yielding a positive value for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 W^+)$.\n(A similar situation is found in the lower left plot of\n\\reffi{fig:st2sb2W} for $|\\mu| \\approx 200 \\,\\, \\mathrm{GeV}$.) For somewhat\nlarger $\\tan \\beta$ values, loop corrections of $\\sim 50\\%$ of the tree-level\nvalues are reached, while\nin S1\\ they stay below $\\sim 23\\%$ of the tree-level results.\nIn the upper right plot of \\reffi{fig:st2sb2W} the loop corrections \nare smaller than $\\sim 40\\%$ of the tree-level values, depending on the\nsize of $|A_b|$, see above.\n The loop corrections shown in the lower left plot of \\reffi{fig:st2sb2W}\nyield maximal $\\sim 9 (37) \\%$ of the tree-level results in S1\\\n(S2), apart from very small $\\mu$ \nvalues, where the tree-level partial decay width can become accidentally small.\n\nFinally, looking at the dependence on $\\varphi_{\\Ab}$ in\nthe lower right plot of \\reffi{fig:st2sb2W}, apart from \nthe known divergences in S2\\ around $\\varphi_{\\Ab} \\approx 117^\\circ, 243^\\circ$, \nthe loop corrections do not exceed $\\sim 6\\%$ and $\\sim 35\\%$ of\nthe tree-level values\nin S1\\ and in S2, respectively. Overall, except for the small\nparameter regions around $\\varphi_{\\Ab} \\approx 117^\\circ, 243^\\circ$, the full\none-loop corrections to $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 W^+)$ are well under control employing the\nrenormalization scheme RS2.\n\n\\newpage\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{TB.st2sb1H.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{Ab.st2sb1H.eps} \n\\\\[4em]\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{MUE.st2sb1H.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{PhiAb.st2sb1H.eps}\n\\end{tabular}\n\\vspace{2em}\n\\caption{\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$. Tree-level and full one-loop corrected partial decay widths \n for the renormalization scheme RS2. The parameters are chosen according to\n the scenarios S1\\ and S2\\ (see \\refta{tab:para}). \n For S1\\ the grey region is excluded and for S2\\ the dark grey region \n is excluded.\n Upper left plot: $\\tan \\beta$ varied.\n Upper right plot: $\\tan \\beta = 20$ and $|A_b|$ varied.\n Lower left plot: $\\tan \\beta = 20$ and $|\\mu|$ varied.\n Lower right plot: $\\tan \\beta = 20$ and $\\varphi_{A_b}$ varied.\n}\n\\label{fig:st2sb1H}\n\\end{center}\n\\end{figure}\n\n\n\\newpage\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{TB.st2sb2H.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{Ab.st2sb2H.eps} \n\\\\[4em]\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{MUE.st2sb2H.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{PhiAb.st2sb2H.eps}\n\\end{tabular}\n\\vspace{2em}\n\\caption{\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$. Tree-level and full one-loop corrected partial decay widths \n for the renormalization scheme RS2. The parameters are chosen\n according to the scenarios S1\\ and S2\\ \n (see \\refta{tab:para}). \n For S1\\ the grey region is excluded and for S2\\ the dark grey region \n is excluded.\n Upper left plot: $\\tan \\beta$ varied.\n Upper right plot: $\\tan \\beta = 20$ and $|A_b|$ varied.\n Lower left plot: $\\tan \\beta = 20$ and $|\\mu|$ varied.\n Lower right plot: $\\tan \\beta = 20$ and $\\varphi_{A_b}$ varied.\n}\n\\label{fig:st2sb2H}\n\\end{center}\n\\end{figure}\n\n\n\\newpage\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{TB.st2sb1W.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{Ab.st2sb1W.eps} \n\\\\[4em]\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{MUE.st2sb1W.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{PhiAb.st2sb1W.eps}\n\\end{tabular}\n\\vspace{2em}\n\\caption{\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 W^+)$. Tree-level and full one-loop corrected partial decay widths \n for the renormalization scheme RS2. The parameters are chosen\n according to the scenarios S1\\ and S2\\ \n (see \\refta{tab:para}). \n For S1\\ the grey region is excluded and for S2\\ the dark grey region \n is excluded.\n Upper left plot: $\\tan \\beta$ varied.\n Upper right plot: $\\tan \\beta = 20$ and $|A_b|$ varied.\n Lower left plot: $\\tan \\beta = 20$ and $|\\mu|$ varied.\n Lower right plot: $\\tan \\beta = 20$ and $\\varphi_{A_b}$ varied.\n}\n\\label{fig:st2sb1W}\n\\end{center}\n\\end{figure}\n\n\n\\newpage\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{TB.st2sb2W.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{Ab.st2sb2W.eps} \n\\\\[4em]\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{MUE.st2sb2W.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{PhiAb.st2sb2W.eps}\n\\end{tabular}\n\\vspace{2em}\n\\caption{\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 W^+)$. Tree-level and full one-loop corrected partial decay widths \n for the renormalization scheme RS2. The parameters are chosen\n according to the scenarios S1\\ and S2\\ \n (see \\refta{tab:para}). \n For S1\\ the grey region is excluded and for S2\\ the dark grey region \n is excluded.\n Upper left plot: $\\tan \\beta$ varied.\n Upper right plot: $\\tan \\beta = 20$ and $|A_b|$ varied.\n Lower left plot: $\\tan \\beta = 20$ and $|\\mu|$ varied.\n Lower right plot: $\\tan \\beta = 20$ and $\\varphi_{A_b}$ varied.\n}\n\\label{fig:st2sb2W}\n\\end{center}\n\\end{figure}\n\n\n\n \n\n\\subsection{Comparison with SQCD calculation}\n\\label{sec:sqcd}\n\nOften QCD corrections to SM or MSSM processes are considered as the\nleading higher-order contributions. However, it has also been observed\nfor SM processes (e.g.\\ in the case of $WH$ and $ZH$ production at the\nTevatron and LHC~\\cite{WHZH}, for $H + 2$\\,jet production at the\nLHC~\\cite{H2j}, or for the Higgs decay to four fermions in the\nSM~\\cite{HVV4f}) that the electroweak (EW) corrections can be of similar\nsize as the QCD corrections. \nTherefore, in the last step of our numerical evaluation, we show the size\nof the one-loop effects based on SUSY QCD (SQCD) only. The size of the\nSQCD corrections \ncan then be compared to the full calculation presented in the previous\nsubsection. It should be kept in mind that, following\n\\refeq{MSbotshift}, also the masses of the scalar bottom quarks depend\non the order of the calculation. Consequently, we do not explicitly compare\nSQCD with the full one-loop calculation, but analyze only the size and\nthe sign of the pure SQCD corrections.\n\nIn \\reffi{fig:SQCD} we show the tree-level values and SQCD one-loop \ncorrected partial decay widths for \n$\\tilde{t}_2 \\to \\tilde{b}_1 H^+$, $\\tilde{t}_2 \\to \\tilde{b}_1 W^+$, $\\tilde{t}_2 \\to \\tilde{b}_2 H^+$, $\\tilde{t}_2 \\to \\tilde{b}_2 W^+$,\nrespectively. The renormalization scheme RS2 is used, \nand hard gluon radiation is taken into account.\nThe parameters are chosen according to S1\\ and S2\\ with $\\tan \\beta$ varied. \nFor S1\\ and S2\\ the grey and the dark grey region \nis excluded via LEP Higgs searches, respectively.\nIn the lower left plot of \\reffi{fig:SQCD} the curves in S1\\ \nend at $\\tan \\beta \\approx 27$ due to the closing of the phase space.\nThe size of the SQCD one-loop corrections reaches the highest values for\nlarge $\\tan \\beta$ \nin the case of $\\tilde{t}_2 \\to \\tilde{b}_{1,2} H^+$ and for intermediate $\\tan \\beta$\nin the case of $\\tilde{t}_2 \\to \\tilde{b}_{1,2} W^+$. \nThe relative size in percent of the tree-level values do not exceed \n$-8\\%$ in $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$, \n$+18\\%$ in $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 W^+)$,\n$-24\\%$ in $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$ and\n$-6\\%$ in $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 W^+)$. \nThe absolute size of the SQCD corrections can be compared with\nthe upper left plots of \\reffis{fig:st2sb1H}--\\ref{fig:st2sb2W}, where\nthe full one-loop corrections are shown. It becomes obvious, especially\nin S2, that restricting an evaluation to the pure SQCD corrections would\nstrongly underestimate the full one-loop corrections. (Hard photon radiation\ncan be as relevant as hard gluon radiation.) Consequently, \nthe full set of one-loop corrections must be taken into account to yield\na reliable prediction of the scalar top quark decay width.\n\n\n\n\\begin{figure}[htb!]\n\\vspace{2em}\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{SQCD.TB.st2sb1H.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{SQCD.TB.st2sb1W.eps} \n\\\\[4em]\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{SQCD.TB.st2sb2H.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{SQCD.TB.st2sb2W.eps}\n\\end{tabular}\n\\vspace{2em}\n\\caption{\n Tree-level and SQCD corrected partial decay widths for the\n renormalization scheme RS2 with $\\tan \\beta$ varied. \n The parameters are chosen according to the scenarios S1\\ and S2\\\n (see \\refta{tab:para}). \n For S1\\ the grey region is excluded and for S2\\ the dark grey region \n is excluded.\n Upper left plot: $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$.\n Upper right plot: $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 W^+)$. \n Lower left plot: $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$. \n Lower right plot: $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 W^+)$. \n}\n\\label{fig:SQCD}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\section{Conclusions}\n\nA scalar top quark can decay into a scalar bottom quark \nand a charged Higgs boson or a $W$~boson\nif the process is kinematically allowed.\nThese decay modes can comprise a large part of\nthe total stop decay width. The decay channels with a charged Higgs\nboson in the final state form a potentially important subprocess\nof cascade decays \nwhich are interesting for the search of charged Higgs bosons at the LHC.\nIn order to arrive at a precise prediction of these scalar top quark\npartial decay widths\nat least a (full) one-loop calculation has to be performed.\nIn such a calculation a renormalization procedure has to be applied\nthat takes into account the top quark\/squark as well as the bottom\nquark\/squark sector in the MSSM. These two sectors are connected via the soft\nSUSY-breaking mass parameter $M_{\\tilde{Q}_L}$ of the superpartners of the\nleft-handed quarks, \nwhich is the same in both sectors due to the $SU(2)_L$ invariance.\n\nWithin the MSSM with complex parameters (cMSSM)\nwe defined six different renormalization schemes for the bottom\nquark\/squark sector, while \nin the top quark\/squark sector we applied a commonly used on-shell\nrenormalization scheme, which is well suited for processes with external\ntop and stop quarks. \nIn our analysis we focused on the problem that, for\ncertain parameter sets, an applied renormalization scheme might fail\nand cause large counterterm contributions that enhance the loop\ncorrections to unphysically large values. \nWe have analyzed analytically the drawbacks and shortcomings of each of the\nsix renormalization schemes. Because of the relations between the\nparameters that have to be respected also at the one-loop level\n we did not find \nany renormalization scheme that results in reasonably small counterterm\ncontributions over {\\em all}\nthe cMSSM parameter space we have analyzed (we did not consider\na pure \\ensuremath{\\overline{\\mathrm{DR}}}~scheme which is not well suited to describe external particles). \nSome renormalization schemes (for instance, the ``on-shell'' scheme\nwhich is defined analogously to the one applied in the\ntop quark\/squark sector) fail over large parts of the parameter\nspace. Others fail only in relatively small parts where, \nfor instance, a divergence due to a vanishing denominator occurs.\nThe most robust schemes turn out to be the \n``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}''(RS2) scheme and the\n``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS''(RS6) scheme.\nThese renormalization schemes appear to be most suitable for\nhigher-order corrections \ninvolving scalar top and bottom quarks.\n\nWe performed a detailed numerical analysis for the full one-loop result \nof the partial decay widths corresponding to the four processes $\\tilde{t}_2\n\\to \\tilde{b}_j H^+\/W^+$ ($j = 1,2$) in our ``preferred'' scheme,\n``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}''. \nThe higher-order corrections, besides the full set of one-loop diagrams, also\ncontain soft and hard QED and QCD radiation. \nWe evaluated the higher-order predictions of the four partial decay\nwidths as a function of $\\tan \\beta$, $\\mu$, $A_b$ and $\\varphi_{\\Ab}$. \nWe found mainly modest corrections at the one-loop level.\nLarger corrections are mostly found in regions of the parameter space that are\ndisfavored by experimental constraints and\/or charge and color breaking\nminima. \nA comparison of the full one-loop calculation with a pure SQCD\ncalculation showed that the latter one can result in a very poor\napproximation of the full result and cannot be used for a reliable\nprediction.\n\nA full one-loop calculation of the corresponding branching ratios requires the\ncalculation of all possible partial decay widths of the scalar top quark\n(and consequently a\nrenormalization of the full cMSSM) and will be presented\nelsewhere~\\cite{Stop2decay}. \n\n\n\n\\subsection*{Acknowledgements}\n\nWe thank for helpful discussions:\nF.~Campanario, \nS.~Dittmaier,\nT.~Fritzsche, \nJ.~Guasch, \nT.~Hahn, \nW.~Hollik,\nL.~Mihaila, \nF.~von der Pahlen, \nT.~Plehn, \nM.~Spira,\nD.~St\\\"ockinger\nand \nG.~Weiglein.\nThe work of S.H.\\ was partially supported by CICYT (grant FPA 2007--66387).\nWork supported in part by the European Community's Marie-Curie Research\nTraining Network under contract MRTN-CT-2006-035505\n`Tools and Precision Calculations for Physics Discoveries at Colliders'.\nH.R.\\ acknowledges support by the Deutsche\nForschungsgemeinschaft via the Sonderforschungsbereich\/Transregio\nSFB\/TR-9 ``Computational Particle Physics'' and the Initiative and\nNetworking Fund of the Helmholtz Association, contract HA-101 ``Physics\nat the Terascale''.\n\n\n\\newpage\n\n\\begin{appendix}\n\\section*{Appendix: \\boldmath{$C$}-functions}\n\nAs explained in \\refse{sec:AbOS_ReYbOS}, \nin RS6 we have to deal with infrared divergent $C$-functions \n(appearing in $\\Lambda(p_1^2 = 0,p^2,p^2)$) with vanishing \nGram-determinants. This case is not implemented in \n{\\em LoopTools}~\\cite{formcalc}. \nTherefore we follow \\citere{cfunc} (and references therein) and \nreplace the corresponding $C$-functions by well behaving linear \ncombinations of $B$-functions \\footnote{FormCalc \\cite{formcalc} \nsorts the loop integrals with help of the masses. \nConsequently, any momentum can become zero, \nnot only $p_1$. Furthermore {\\em LoopTools} uses a different convention \nthan \\cite{cfunc}: $C_1 = C_{11} - C_{12}$, $C_2 = C_{12}$.}.\nFor sake of completeness we briefly review our implementation.\nThe class of $C$-functions with only one external momentum zero, can be\ncompletely reduced to $B$-functions. Having three different masses we\ncan use partial fraction decomposition:\n\\begin{align}\\label{C0}\nC_0(0,p,p,m_1,m_2,m_3) = \n \\frac{B_0(p, m_1, m_3) - B_0(p, m_2, m_3)}{m_1^2 - m_2^2}~.\n\\end{align}\nWith only two different masses applying partial differentiation \n(l'Hospital) yields\n\\begin{align}\n\\label{dB0}\nC_0(0,p,p,m_1,m_1,m_3) &= \n \\frac{\\partial\\, B_0(p,m_1,m_3)}{\\partial\\,(m_1^2)}~.\n\\end{align}\nWe also used symmetry relations and decompositions which can be found \nin \\cite{cfunc} and the following short hand notation:\n\\begin{align}\nD_{ij} &= p^4 + m_i^4 + m_j^4 - 2 (p^2 m_i^2 + p^2 m_j^2 + m_i^2 m_j^2)~.\n\\end{align}\nWe included the following replacements of $C_i$\nfunctions with $p_1 = 0$:\n\\begin{align}\nC_0(0, p, p, m_1, m_2, m_3) &\\to \n \\frac{B_0(p, m_1, m_3) - B_0(p, m_2, m_3)}{m_1^2 - m_2^2}~, \\\\\nC_0(0, p, p, m_1, m_1, m_3) &\\to \n \\frac{1}{D_{13}} \\Big[ (p^2 + m_3^2 - m_1^2) (2 - B_0(p, m_1, m_3)) \\nonumber \\\\\n &\\qquad + (p^2 - m_3^2 - m_1^2) B_0(0, m_1, m_1)\n + 2 m_3^2 B_0(0, m_3, m_3)\\Big]~, \\\\\nC_1(0, p, p, m_1, m_2, m_3) &\\to \n \\frac{1}{3 (m_1^2 - m_2^2)^2} \\Big[ 2 m_1^2 (B_0(p, m_2, m_3) \n - B_0(p, m_1, m_3)) \\nonumber \\\\\n &\\qquad + (m_1^2 - m_2^2) B_0(p, m_2, m_3) - m_1^2 + m_2^2 \\nonumber \\\\\n &\\qquad + (3 m_1^2 - 2 m_2^2 - m_3^2 + p^2) B_1(p, m_2, m_3) \\nonumber \\\\\n &\\qquad + (m_3^2 - m_1^2 - p^2) B_1(p, m_1, m_3) \\Big]~, \\\\\nC_2(0, p, p, m_1, m_2, m_3) &\\to \n \\frac{B_1(p, m_1, m_3) - B_1(p, m_2, m_3)}{m_1^2 - m_2^2}~, \\\\\nC_2(0, p, p, m_1, m_1, m_3) &\\to \n \\frac{1}{2 p^2} \\bigg\\{B_0(0, m_1, m_1) - B_0(p, m_1, m_3) \\nonumber \\\\\n &\\qquad - \\frac{p^2 + m_1^2 - m_3^2}{D_{13}} \\Big[ \n (p^2 - m_1^2 + m_3^2) (2 - B_0(p, m_1, m_3)) \\nonumber \\\\\n &\\qquad + (p^2 - m_3^2 - m_1^2) B_0(0, m_1, m_1) \n + 2 m_3^2 B_0(0, m_3, m_3)\\Big] \\bigg\\}~.\n\\end{align}\nIn the case of $p_2 = 0$, we used the following replacements:\n\\begin{align}\nC_0(p, 0, p, m_1, m_2, m_3) &\\to \n \\frac{B_0(p, m_1, m_2) - B_0(p, m_1, m_3)}{m_2^2 - m_3^2}~, \\\\\nC_0(p, 0, p, m_1, m_2, m_2) &\\to\n \\frac{1}{D_{12}} \\Big[ (p^2 - m_2^2 + m_1^2) (2 - B_0(p, m_1, m_2)) \\nonumber \\\\\n &\\qquad + (p^2 - m_1^2 - m_2^2) B_0(0, m_2, m_2) \n + 2 m_1^2 B_0(0, m_1, m_1)\\Big]~, \\\\\nC_1(p, 0, p, m_1, m_2, m_3) &\\to \n \\frac{1}{3 (m_2^2 - m_3^2)^2} \\Big[ 2 m_1^2 (B_0(p, m_1, m_2) \n - B_0(p, m_1, m_3)) \\nonumber \\\\\n &\\qquad + (m_2^2 - m_3^2) B_0(0, m_2, m_3) - m_3^2 + m_2^2 \\nonumber \\\\\n &\\qquad - (3 m_3^2 - 2 m_2^2 - m_1^2 - p^2) B_1(p, m_1, m_2) \\nonumber \\\\\n &\\qquad + (m_3^2 - m_1^2 - p^2) B_1(p, m_1, m_3) \\Big]~, \\\\\nC_2(p, 0, p, m_1, m_2, m_3) &\\to \n \\frac{1}{3 (m_2^2 - m_3^2)^2} \\Big[ 2 m_1^2 (B_0(p, m_1, m_3) \n - B_0(p, m_1, m_2)) \\nonumber \\\\\n &\\qquad - (m_2^2 - m_3^2) B_0(0, m_2, m_3) + m_3^2 - m_2^2 \\nonumber \\\\\n &\\qquad - (3 m_2^2 - 2 m_3^2 - m_1^2 - p^2) B_1(p, m_1, m_3) \\nonumber \\\\\n &\\qquad + (m_2^2 - m_1^2 - p^2) B_1(p, m_1, m_2) \\Big]~.\n\\end{align}\nFinally, for $p_3 = (p_1 + p_2) = 0$ we employed:\n\\begin{align}\nC_0(p, p, 0, m_1, m_2, m_3) &\\to \n \\frac{B_0(p, m_1, m_2) - B_0(p, m_2, m_3)}{m_1^2 - m_3^2}~, \\\\\nC_0(p, p, 0, m_1, m_2, m_1) &\\to\n \\frac{1}{D_{12}} \\Big[ (p^2 - m_1^2 + m_2^2) (2 - B_0(p, m_1, m_2)) \\nonumber \\\\\n &\\qquad + (p^2 - m_2^2 - m_1^2) B_0(0, m_1, m_1) \n + 2 m_2^2 B_0(0, m_2, m_2)\\Big]~, \\\\\nC_1(p, p, 0, m_1, m_2, m_3) &\\to \n \\frac{B_0(p, m_2, m_3) + B_1(p, m_1, m_2) \n + B_1(p, m_2, m_3)}{m_1^2 - m_3^2}~, \\\\\nC_1(p, p, 0, m_1, m_2, m_1) &\\to \n \\frac{1}{2 p^2} \\bigg\\{ B_0(0, m_1, m_1) - B_0(p, m_1, m_2) \\nonumber \\\\\n &\\qquad - \\frac{p^2 + m_1^2 - m_2^2}{D_{12}} \\Big[ \n (p^2 - m_1^2 + m_2^2) (2 - B_0(p, m_1, m_2)) \\nonumber \\\\\n &\\qquad + (p^2 - m_2^2 - m_1^2) B_0(0, m_1, m_1) \n + 2 m_2^2 B_0(0, m_2, m_2) \\Big] \\bigg\\}~, \\\\\nC_2(p, p, 0, m_1, m_2, m_3) &\\to \n \\frac{1}{3 (m_1^2 - m_3^2)^2} \\Big[ 2 m_1^2 (B_0(p, m_2, m_3) \n - B_0(p, m_1, m_2)) \\nonumber \\\\\n &\\qquad - (2 m_1^2 - m_2^2 - m_3^2 + p^2) B_0(p, m_2, m_3) \n + m_3^2 - m_1^2 \\nonumber \\\\\n &\\qquad - (3 m_1^2 - 2 m_3^2 - m_2^2 + p^2) B_1(p, m_2, m_3) \\nonumber \\\\\n &\\qquad + (m_2^2 - m_1^2 - p^2) B_1(p, m_1, m_2) \\Big]~.\n\\end{align}\n\n\n\\end{appendix}\n\n\n\n\n\\newpage\n\\pagebreak\n\\clearpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsubsection*{Introduction}\n\\label{submission}\nThe Census Organisation in India has been publishing the tabulated results of Census since the initiation of modern Census in 1872 in various book forms. As the information given by any respondent is treated as confidential, this micro-data is not publicly available. \\cite{isi-census} However Agent-based models, such as infectious disease modelling for Covid-19 require individual level information \\cite{Bonabeau02} to model and simulate the behavior of the system's constituent units (the agents) and their interactions. Though, within the 12th Five Year Plan (2012-17) Indian Census set-up 18 Workstations country-wide for research on micro-data from Census, it lacked the flexibility which researchers need in using such huge datasets. Thus it is crucial to have the ability to model population data and generate synthetic population at various administrative levels like, the Country, State, District, Sub-District, Town, Village or Ward in Town as the case may be. \n\nIn this paper, we use a variety of data sources to generate a population of individuals and households with demographic attributes that are statistically identical to real data. This population is generated using a hybrid \\footnote{Our model is open-sourced at \\url{https:\/\/github.com\/bhaveshneekhra\/synthpop} } of statistical methods and machine learning algorithms that are flexible enough to generate data at various administrative levels, ranging from small communities to states. The primary sources of data for these algorithms include the Census of India \\cite{census-2011}, the India Human Development Survey (IHDS) (\\cite{Desai18}, the National Sample Survey (NSS) \\cite{nss18}, and the Gridded Population of the World (GPW) \\cite{ciesin2016}.\n\nWhile the synthetic population should faithfully reproduce demographic statistics, it must also incorporate other realistic network structures, such as those appropriate to households and workplaces. \\footnote{Otherwise, we could end up, for example, with ``families'' composed entirely of toddlers, or workplaces with strange mixes of professions.} Because different kinds of data respond well to different techniques, a hybrid process is used to scale up these datasets. First, the data is cleaned to remove obvious inconsistencies. Next, a customized hybrid of Iterative Proportional Fitting (IPF) \\cite{beckman-1996, Deming1940}, Iterative Proportional Updating (IPU) \\cite{ipu2009}, and a specialized variant of a neural network, called Conditional-Tabular Generative Adversarial Network (CTGAN) \\cite{xu2019}, is used to generate new data. \n\nBriefly, Iterative Proportional Fitting finds a joint distribution that matches the marginals, while trying to stay as close to the sample distribution as possible. Iterative Proportional Updating is a heuristic iterative approach which can simultaneously match or fit to multiple distributions (constraints). Finally, Conditional-Tabular Generative Adversarial Networks is a method to model the tabular data distribution and sample rows from the distribution. A Generative Adversarial Network (GAN) \\cite{goodfellow14} uses two ``competing'' neural networks, the generator and the discriminator. The generator creates realistic samples with the goal that the discriminator should be unable to differentiate between a real sample and a generated sample. In this zero-sum game, capabilities of both the networks are enhanced iteratively. Critically, our techniques are designed to work seamlessly across data-scarce and data-rich areas; even if a particular area has error-prone or missing data, a synthetic population can still be generated, albeit of a lower quality.\n\n\n\n\n\\subsubsection*{The Population Generation Process}\n \nWe use IPF to generate a base population, using census data for the demographics and the IHDS survey dataset for personal and household attributes. The base population thus consists of individual data and household data. We assign each household to an administrative unit within a district. \n\nWe also experimented with CTGAN to generate a base population. The major advantage of IPU over CTGAN is that IPU is capable of matching individual level and household level characteristics of an individual while making sure that members of the household have a realistic age and gender joint distribution.\n\nTo assign job labels to individuals, the relevant data from the IHDS dataset is used. For the time-being, we classify individuals below the age of 18 as students, but could easily relax this assumption. A subset of the population is also assumed be home-bound. This subset consists of unemployed individuals, homemakers, infants and children under the age of 3 and elderly people over the retirement age. We use data from the NSS survey to determine the percentage of adult males and females in a city who are home-bound. A random independent sample is drawn from a Bernoulli distribution with this gender-based marginal value as a parameter in order to decide if an individual will be home-bound or not.\n\n\nEach student in the population is assigned a school. Similarly, each working individual is assigned a workplace based upon their job label. We generate a synthetic latitude and longitude pair for each home, school and workplace in our dataset using GADM grid population density data \\cite{j-hijmans-2018}. We select a subset of grid points that lie within a given geographical boundary and sample grid-points with replacement grid points from the subset, weighing each point by the population density in the associated grid. We add independent random noise drawn from a uniform distribution to the latitudes and longitudes, rejecting those samples which fall outside the given geographical boundary. We follow this process to generate synthetic geolocation data for households, schools and workplaces.\n\nTo assign an individual a school, we sample from the list of schools within that geographical boundary, weighing each school by the inverse of the euclidean distance between it and the individual's home. This weighting factor increases the probability of assigning an individual a school that is closer to their home \\cite{rte-2019}. We follow a similar method to assign workplaces to adults. Additionally, based on every individual's job label, a workplace is assigned at random from a suitable subset of allowed workplaces.\n\n\nA number of additional attributes are included in our synthetic population, including whether an individual uses public transport or whether an individual is an essential worker. These values are assigned using the individual's job label.\n\n\\begin{figure}[htb]\n\\vskip 0.2in\n\\begin{center}\n\n\\centerline{\\includegraphics[width=\\columnwidth]{histogram}}\n\\caption{Histogram: Comparing source population (left) with synthetic population for the city of Mumbai in India}\n\\label{hist-plot-comparison}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\\begin{figure}[htb]\n\\vskip 0.2in\n\\begin{center}\n\n\\centerline{\\includegraphics[width=\\columnwidth]{plot_3}}\n\\caption{Scatter plot: Comparing source population (left) with synthetic population for the city of Mumbai in India}\n\\label{scatter-plot-comparison}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\\subsubsection*{Population Verification Metrics}\nTo compare and verify the the generated synthetic population with the survey data, we used several methods. We used the Bhattacharya\ndistance to quantify the similarity of the joint age-height and age-weight distributions. In addition, apart from comparing the two\npopulations visually as seen in \\cref{hist-plot-comparison} and \\cref{scatter-plot-comparison}, we have also used a number of\nother metrics such as statistical likelihood techniques (CS-test, KS-test). We also visualise the geographical spread of the households, schools and the workplaces in the population as in \\cref{geo-plots}. The visual comparison shows the synthetic population resembles the real population.\n\n\n\n\\begin{figure}[H]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{gpd_1.png}}\n\\caption{Geographical distribution of Households, Schools and Workplaces, respectively, in the Synthetic Population}\n\\label{geo-plots}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection*{Conclusion}\nLack of data due to access and privacy issues results in poor model design. To tackle this issue, we propose a hybrid method to generate a synthetic population. We also provide a combination of metrics to verify the generated data. In ongoing work, we are generating the synthetic population for the entire country of India. As future work, we want to explore the possibility of modelling more nuanced and complex features for the synthetic population. \n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nThis work was funded by the Mphasis foundation.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\n\nModels with softly broken supersymmetry\n(SUSY)~\\cite{Ramond,Golfand,Volkov,Wess,Fayet} predict superpartners\nof the standard model (SM) particles. Experimental limits from the\nTevatron and LEP showed that superpartner particles, if they exist,\nare significantly heavier than their SM counterparts. Proposed\nexperimental searches for $R$-parity conserving SUSY at the Large\nHadron Collider (LHC) have therefore focused on a combination of two\nSUSY signatures: multiple energetic jets and\/or leptons from the\ndecays of pair-produced squarks and gluinos, and large missing\ntransverse energy (\\ETm) from the two weakly interacting\nlightest superpartners (LSP) produced in separate decay chains.\n\nIn this article a new approach is presented that is inclusive not only\nfor SUSY but also in the larger context of physics beyond the standard\nmodel. The focal point for this novel \\textit{razor}\nanalysis~\\cite{rogan} is the production of pairs of heavy particles\n(of which squarks and gluinos are examples), whose masses are\nsignificantly larger than those of any SM particle. The analysis is\ndesigned to kinematically discriminate the pair production of heavy\nparticles from SM backgrounds, without making strong assumptions about\nthe \\ETm spectrum or details of the decay chains of these\nparticles. The baseline selection requires two or more reconstructed\nobjects, which can be calorimetric jets, isolated electrons or\nisolated muons. These objects are grouped into two\n \\textit{megajets}. The razor analysis tests the consistency, event by\nevent, of the hypothesis that the two megajets represent the visible\nportion of the decays of two heavy particles. This strategy is\ncomplementary to traditional searches for signals in the tails of the\n\\ETm distribution ~\\cite{:2007ww, Aaltonen:2008rv, Collaboration:2011xk,\n daCosta:2011qk, Aad:2011hh, Aad:2011xm, RA2, alphaT, :2011wb, Chatrchyan:2011bz}\nand is applied to data collected with the Compact Muon Solenoid (CMS)\ndetector from pp collisions at $\\sqrt{s}=7\\TeV$ corresponding to an\nintegrated luminosity of 35\\pbinv.\n\n\\section{The CMS Apparatus}\n\n\nA description of the CMS detector can be found\nelsewhere~\\cite{:2008zzk}.\nA characteristic feature of the CMS detector is its superconducting\nsolenoid magnet, of 6~m internal diameter, providing a field of\n3.8~T. The silicon pixel and strip tracker, the crystal\nelectromagnetic calorimeter (ECAL) and the brass\/scintillator hadron\ncalorimeter (HCAL) are contained within the solenoid. Muons are\ndetected in gas-ionization chambers embedded in the steel return\nyoke. The ECAL has an energy resolution of better than 0.5\\,\\% above\n100\\GeV. The HCAL combined with the ECAL, measures the jet energy with\na resolution $\\Delta E\/E \\approx 100\\,\\% \/ \\sqrt{E\/\\GeV} \\oplus\n5\\,\\%$.\n\nCMS uses a coordinate system with the origin located at the nominal\ncollision point, the $x$-axis pointing towards the center of the LHC,\nthe $y$-axis pointing up (perpendicular to the LHC plane), and the\n$z$-axis along the counterclockwise beam direction. The azimuthal\nangle $\\phi$ is measured with respect to the $x$-axis in the $xy$\nplane and the polar angle $\\theta$ is defined with respect to the\n$z$-axis. The pseudorapidity is $\\eta = -\\ln [\\tan(\\theta \/ 2)]$.\n\n\n\\section{The Razor Analysis\\label{intro}}\n\nThe pair production of two heavy particles, each decaying to an unseen\nLSP plus jets, gives rise to a generic SUSY-like signal. Events in\nthis analysis are forced into a dijet topology by combining all jets\nin the event into two megajets.\nWhen an isolated lepton is present, it can be included\nin the megajets or not, as described in Sections~\\ref{sec:es}\nand~\\ref{sec:be}. To the extent that the pair of megajets accurately\nreconstructs the visible portion of the underlying parent particle\ndecays, the kinematic properties of the signal are equivalent to the pair production of,\nfor example, two heavy squarks $\\PSq_1$, $\\PSq_2$, with\n$\\PSq_i\\rightarrow {{\\rm{j}}_i} \\PSGcz_i $, for $i=1,~2$,\nwhere ${\\rm{j}}_i$ and $\\PSGcz_i$ denote the visible and invisible\nproducts of the decays, respectively. In the approximation that the\nheavy squarks are produced at threshold and their visible decay\nproducts are massless, the center of mass (CM) frame four-momenta are\n\n\\begin{eqnarray}\n&&\\hspace*{-50pt}\np_{\\rm j_1} = \\frac{M_\\Delta}{2}(1,\\hat{\\boldmath{u}}_1) \\;,\\quad\np_{\\rm j_2} = \\frac{M_\\Delta}{2}(1,\\hat{\\boldmath{u}}_2)\\; , \\\\\n&&\\hspace*{-50pt}\np_{\\PSGc_{1}} = \\frac{M_\\Delta}{2} \\left( \\frac{2M_{\\PSq}}{M_\\Delta}-1,-\\hat{\\boldmath{u}}_1\\right) \\; ,\\quad\np_{\\PSGc_{2}} = \\frac{M_\\Delta}{2} \\left(\n \\frac{2M_{\\PSq}}{M_\\Delta}-1,-\\hat{\\boldmath{u}}_2 \\right) \\; ,\n\\end{eqnarray}\n\nwhere $\\hat{\\boldmath{u}}_i$ is the unit vector in the direction of $\\rm{j}_i$, and\n\n\\begin{eqnarray}\n&&\\hspace*{-50pt}\nM_\\Delta \\equiv \\frac{M_{\\PSq}^{2}-M_{\\PSGc}^{2}}{M_{\\PSq}}~,~ \\;\n\\end{eqnarray}\n\nwhere $M_{\\PSq}$ and $M_{\\PSGc}$ are the squark and LSP\nmasses, respectively.\n\nIn events with two undetected particles in the partonic final state, it\nis not possible to reconstruct the actual CM frame. Instead, an\napproximate event-by-event reconstruction is made assuming the dijet signal\ntopology, replacing the CM frame with the \\textit{$R$ frame} \\cite{rogan},\ndefined as the longitudinally boosted frame that equalizes the magnitude\nof the two megajets' three-momenta. The $R$ frame would be the CM frame for\nsignal events, if the squarks were produced at threshold and if the CM\nsystem had no overall transverse momentum from initial-state\nradiation. The longitudinal Lorentz boost factor is defined by\n\\begin{equation}\n\\beta_{R}\\equiv\\frac{E^{\\rm j_1}-E^{\\rm j_2}}{p^{\\rm j_1}_{z}-p^{\\rm j_2}_{z}}\\; ,\n\\end{equation}\nwhere $E^{\\rm j_1}$, $E^{\\rm j_2}$ and $p^{\\rm j_1}_{z}$, $p^{\\rm\n j_2}_{z}$ are the megajet energies and longitudinal momenta ,\nrespectively. To the extent that the $R$ frame matches the true CM\nframe, the maximum value of the scalar sum of the megajets'\ntransverse momenta ($\\PT^1,~\\PT^2$) is $M_\\Delta$ for signal\nevents. The maximum value of the \\ETm is also $M_\\Delta$. A\ntransverse mass $M_T^{R}$ is defined whose maximum value for signal events is\nalso $M_\\Delta$ in the limit that the $R$ and CM frames coincide:\n\n\\begin{equation}\nM_T^{R}\\equiv \\sqrt{ \\frac{\\ETm(\\PT^{\\rm j_1}+\\PT^{\\rm j_2}) -\n \\VEtmiss {\\mathbf \\cdot}\n (\\vec{p}_T^{\\,\\rm j_1}+\\vec{p}_T^{\\,\\rm j_2})}{2}} \\; .\n\\end{equation}\n\nThe event-by-event estimator of $M_\\Delta$ is\n\\begin{equation}\nM_R\\equiv 2|\\vec{p}^{R}_{{\\rm j_1}}| = 2|\\vec{p}^{R}_{{\\rm j_2}}|\\; ,\n\\end{equation}\nwhere $\\vec{p}^{R}_{{\\rm j_1}}$ and $\\vec{p}^{R}_{{\\rm j_2}}$ are the\n3-momenta of the megajets in the $R$ frame. For signal events in the\nlimit where the $R$ frame and the true CM frame coincide, $M_R$ equals\n$M_\\Delta$, and more generally $M_R$ is expected to peak around\n$M_\\Delta$ for signal events.\nFor QCD dijet and multijet events the only relevant scale is\n$\\sqrt{\\hat{s}}$, the CM energy of the partonic subprocess. The\nsearch for an excess of signal events in a tail of a distribution is\nthus recast as a search for a peak on top of a steeply falling SM\nresidual tail in the $M_R$ distribution. To extract the peaking\nsignal, the QCD multijet background needs to be reduced to manageable\nlevels. This is achieved using the \\textit{razor} variable defined as:\n\\begin{equation}\n R\\equiv \\frac{M_T^{R}}{M_R} \\; .\\end{equation}\n\nSince for signal events $M_T^R$ has a maximum value of $M_\\Delta$\n(i.e., a kinematic edge), $R$ has a maximum value of approximately 1\nand the distribution of $R$ for signal events peaks around 0.5. These\nproperties motivate the appropriate kinematic requirements for the\nsignal selection and background reduction. It is noted that, while $M_T^R$\nand $M_R$ measure the same scale (one as an end-point, the other as a\npeak), they are largely uncorrelated for signal events, as shown in\nFig.~\\ref{fig:MR_v_R}. In this figure, the $\\PW$+jets and $\\ttbar$+jets\nbackgrounds peak at $M_R$ values partially determined by the $\\PW$ and top quark masses,\nrespectively.\n\\begin{figure*}[ht!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{R-MR-QCD.pdf}\n\\includegraphics[width=0.49\\textwidth]{R-MR-Wjets.pdf}\n\\includegraphics[width=0.49\\textwidth]{R-MR-top.pdf}\n\\includegraphics[width=0.49\\textwidth]{R-MR-LM1.pdf}\n\\caption{ Scatter plot in the ($M_R$, $R$) plane for simulated events:\n (top left) QCD multijet, (top right) $\\PW$+jets, (bottom left)\n $\\ttbar$+jets, and (bottom right) the SUSY benchmark model LM1\n \\cite{PTDR2} with $M_\\Delta = 597\\GeV$. The yields are normalized to\n an integrated luminosity of 35\\pbinv. The bin size is (20\\GeV\n $\\times$ 0.015). \\label{fig:MR_v_R}}\n\\end{center}\n\\end{figure*}\n\nIn this analysis the SM background shapes and normalizations are\nobtained from data. The backgrounds are extracted from control regions\nin the $R$ and $M_R$ distributions dominated by SM processes.\nInitial estimates of the background distributions in these regions are\nobtained from the individual simulated background components, but\ntheir shapes and normalizations are then corrected using data.\nThe analysis flow is as follows:\n\n\\begin{enumerate}\n\\item The inclusive data sets are collected using\n the electron, muon, and hadronic-jet triggers.\n\\item These data sets are examined for the presence of a\n well-identified isolated electron or muon, irrespective of the\n trigger path. Based on the presence or absence of such a lepton,\n each event is assigned to one of three disjoint event samples,\n referred to as the electron, muon, and hadronic \\textit{boxes}. These\n boxes serve as controls of processes in the SM with leptons, jets,\n and neutrinos, e.g. QCD multijet, $\\PW$+jets or $\\cPZ$+jets, and\n $\\cPqt$+X. The diboson background is found to be negligible. Exclusive\n multilepton boxes are also defined but are not sufficiently\n populated to be used in this analysis.\n\\item Megajets are constructed for events passing a baseline kinematic\n selection, and the $R$ and $M_R$ event variables are computed. In\n the electron box, electrons are clustered with jets in the\n definition of the megajets. Jets matched to these electrons are\n removed to avoid double-counting. In the muon box, muons are\n included in the megajet clustering.\n\\item In order to characterize the distribution of the SM background\n events in the ($M_R$, $R$) plane, a kinematic region is identified\n in the lepton boxes that is dominated by $\\PW(\\ell \\nu)+$jets. Another\n region is found that is dominated by the sum of the non-QCD\n backgrounds.\n\\item Events remaining in the hadronic box primarily consist of QCD\n multijet, $\\cPZ(\\nu\\bar{\\nu})$+jets, $\\PW(\\ell\\nu)$+jets, and $\\cPqt$+X\n events that produce $\\ell$+jets+$\\ETm$ final states with\n charged leptons that do not satisfy the electron or muon selections.\n The shapes and normalizations of these non-QCD background processes\n in the hadronic box are estimated using the results from the lepton\n boxes in appropriate regions in the ($M_R$, $R$) plane.\n\\item The QCD background shape and normalization in each of the lepton\n boxes is extracted by reversing the lepton isolation requirements to\n obtain control samples dominated by QCD background.\n\\item The QCD background in the hadronic box is estimated using QCD\n control samples collected with prescaled jet triggers.\n\\item The large-$R$ and high-$M_R$ regions of all boxes are signal candidate\n regions not used for the background estimates. Above a given\n $R$ threshold, the $M_R$ distribution of the backgrounds observed in the\n data is well modeled by simple exponential functions. Having determined the $R$\n and $M_R$ shape and normalization of the backgrounds in the control\n regions, the SM yields are extrapolated to the large-$R$\n and high-$M_R$ signal candidate regions for each box.\n\\end{enumerate}\n\n\\section{Event Selection\\label{sec:es}}\n\nThe analysis uses data sets recorded with triggers based on the\npresence of an electron, a muon, or on $H_T$, the uncorrected scalar\nsum of the transverse energy of jets reconstructed at the trigger\nlevel. Prescaled jet triggers with low thresholds are used for the\nQCD multijet background estimation in the hadronic box.\n\nThe analysis is guided by studies of Monte Carlo (MC) event\nsamples generated with the {\\sc Pythia} \\cite{Sjostrand:2006za} and\n{\\sc Madgraph} \\cite{Maltoni:2002qb} programs, simulated using the CMS\n{\\sc Geant}-based \\cite{G4} detector simulation, and then processed by\nthe same software used to reconstruct real collision data. Events\nwith QCD multijet, top quarks, and electroweak bosons were generated\nwith {\\sc Madgraph} interfaced with {\\sc Pythia} for parton showering,\nhadronization, and underlying event description. To generate Monte\nCarlo samples for SUSY, the mass spectrum was first calculated with\n{\\sc {Softsusy}} \\cite{softsusy} and the decays with {\\sc {Susyhit}}\n\\cite{Susyhit}. The {\\sc {Pythia}} program was used with the {\\sc SLHA}\ninterface \\cite{SLHA} to generate the events. The generator level\ncross section and the K factors for the next-to-leading order (NLO) cross\nsection calculation were computed using {\\sc Prospino} \\cite{prospino}.\n\n\nEvents are required to have at least one good reconstructed\ninteraction vertex \\cite{TRK-10-005}. When multiple vertices are\nfound, the one with the highest associated $\\sum_{\\rm track}\\PT$ is\nused. Jets are reconstructed offline from calorimeter energy deposits\nusing the infrared-safe anti-k$_{\\rm{T}}$~\\cite{antikt} algorithm with\nradius parameter $0.5$. Jets are corrected for the nonuniformity of\nthe calorimeter response in energy and $\\eta$ using corrections\nderived with the simulation and are required to have $\\PT> 30\\GeV$\nand $|\\eta| < 3.0$. The jet energy scale uncertainty for these\ncorrected jets is $5\\%$ \\cite{JES}. The \\ETm is\nreconstructed using the particle flow algorithm \\cite{PFMET}.\n\nThe electron and muon reconstruction and identification criteria are\ndescribed in ~\\cite{EWK-PAS}. Isolated electrons and muons are\nrequired to have $\\PT>20\\GeV$ and $\\vert\\eta|<$ 2.5 and 2.1,\nrespectively, and to satisfy the selection requirements from\n\\cite{EWK-PAS}. The typical lepton trigger and reconstruction\nefficiencies are 98\\% and 99\\%, respectively, for electrons and 95\\%\nand 98\\% for muons.\n\n\nThe reconstructed hadronic jets, isolated electrons, and isolated\nmuons are grouped into two megajets, when at least two such objects\nare present in the event. The megajets are constructed as a sum of\nthe four-momenta of their constituent objects. After considering all\npossible partitions of the objects into two megajets, the combination\nminimizing the invariant masses summed in quadrature of the resulting\nmegajets is selected among all combinations for which the $R$ frame is\nwell defined.\n\n\nAfter the construction of the two megajets the boost variable\n$|\\beta_{R}|$ is computed; due to the approximations mentioned above,\n$|\\beta_{R}|$ can fall in an unphysical region (${\\ge}1$) for signal or\nbackground events; these events are removed. The additional\nrequirement $|\\beta_{R}|\\le 0.99$ is imposed to remove events for\nwhich the razor variables become singular. This requirement is typically\n85\\% efficient for simulated SUSY events. The azimuthal angular\ndifference between the megajets is required to be less than $2.8$ radians;\nthis requirement suppresses nearly back-to-back QCD dijet\nevents. These requirements define the inclusive baseline\nselection. After this selection, the signal efficiency in the\nconstrained minimal supersymmetric standard model\n(CMSSM)~\\cite{Chamseddine,Barbieri,Hall,Kane} parameter space for a gluino\nmass of ${\\sim} 600\\GeV$ is over 50\\%.\n\n\\section{Background Estimation\\label{sec:be}}\n\nIn traditional searches for SUSY based on missing transverse energy, it\nis difficult to model the tails of the \\ETm\ndistribution and the contribution from events with spurious\ninstrumental effects. The QCD multijet production is an especially\ndaunting background because of its very high cross section and complicated\nmodeling of its high-\\PT and \\ETm tails. In this analysis\na cut on $R$ makes it possible to isolate the QCD multijet\nbackground in the low-$M_R$ region.\n\nApart from QCD multijet backgrounds, the remaining backgrounds in the\nlepton and hadronic boxes are processes with genuine \\ETm due to\nenergetic neutrinos and leptons from massive vector boson decays\n(including $\\PW$ bosons from top quark decays). After applying an $R$ threshold,\nthe $M_{R}$ distributions in the lepton and hadronic boxes are\nvery similar for these backgrounds; this similarity is exploited in\nthe modeling and normalization of these backgrounds.\n\n\\subsection{QCD multijet background\\label{sec:qcd}}\n\nThe QCD multijet control sample for the hadronic box is defined from\nevent samples recorded with prescaled jet triggers and passing the\nbaseline analysis selection for events without a well-identified\nisolated electron or muon. The trigger requires at least two jets with\nan average uncorrected $\\PT > 15\\GeV$. Because of the low jet\nthreshold, the QCD multijet background dominates this sample for low\n$M_R$, thus allowing the extraction of the $M_R$ shapes with different\n$R$ thresholds for QCD multijet events. These shapes are corrected\nfor the $H_T$ trigger turn-on efficiency.\n\nThe $M_{R}$ distributions for events satisfying the QCD control box\nselection, for different values of the $R$ threshold, are shown in\nFig.~\\ref{fig:DATA_QCD_calo} (left). The $M_{R}$ distribution is\nexponentially falling, after a turn-on at low $M_{R}$ resulting from\nthe \\PT threshold requirement on the jets entering the megajet\ncalculation. After the turn-on which is fitted with an asymmetric\nGaussian, the exponential region of these distributions is fitted for\neach value of $R$ to extract the exponential slope, denoted by $S$.\nThe value of $S$ that maximizes the likelihood in the exponential fit\nis found to be a linear function of $R^{2}$, as shown in\nFig.~\\ref{fig:DATA_QCD_calo} (right); fitting $S$ to the form $S = a +\nbR^{2}$ determines the values of $a$ and $b$.\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth]{DATA_QCD_calo_slope_log.pdf}\n\\includegraphics[width=0.47\\textwidth]{DATA_QCD_calo_slopefit.pdf}\n\\caption{(Left) $M_{R}$ distributions for different values of the $R$\n threshold for data events in the QCD control box. Fits of the\n $M_{R}$ distribution to an exponential function and an asymmetric\n Gaussian at low $M_R$, are shown as dotted black curves. (Right)\n The exponential slope $S$ from fits to the $M_{R}$ distribution, as\n a function of the square of the $R$ threshold for data events in the\n QCD control box.}\n\\label{fig:DATA_QCD_calo}\n\\end{center}\n\\end{figure*}\n\nWhen measuring the exponential slopes of the $M_{R}$ distributions as\na function of the $R$ threshold, the correlations due to events\nsatisfying multiple $R$ threshold requirements are neglected. The\neffect of these correlations on the measurement of the slopes is\nstudied by using pseudo-experiments and is found to be negligible.\n\nTo measure the shape of the QCD background component in the lepton boxes, the corresponding\nlepton trigger data sets are used with the baseline selection and\nreversed lepton isolation criteria. The QCD background component in the lepton\nboxes is found to be negligible.\n\nThe $R$ threshold shapes the $M_R$ distribution in a simple therefore\npredictable way. Event selections with combined $R$ and $M_R$\nthresholds are found to suppress jet mismeasurements, including\nsevere mismeasurements of the electromagnetic or hadronic\ncomponent of the jet energy, or other anomalous calorimetric noise\nsignals such as the ones described in~\\cite{hcalnoise,ecalnoise}.\n\n\n\\subsection{\\texorpdfstring{\\PW+jets, \\cPZ+jets, and \\cPqt+X backgrounds}{W+jets, Z+jets, and t+X backgrounds}\\label{bg-prop}}\n\nUsing the muon (MU) and electron (ELE) control boxes defined in\nSection~\\ref{intro}, $M_R$ intervals dominated by $\\PW(\\ell\\nu)$+jets\nevents are identified for different $R$ thresholds. In both simulated\nand data events, the $M_{R}$ distribution is well described by two\nindependent exponential components. The first component of\n$W(\\ell\\nu)$+jets corresponds to events where the highest \\PT\nobject in one of the megajets is the isolated electron or muon;\nthe second component consists of events where the leading object\nin both megajets is a jet, as is typical also for the $t$+X background\nevents. The first component of $W(\\ell\\nu)$+jets can be measured\ndirectly in data, because it dominates over all other backgrounds in a control region\nof lower $M_R$ set by the $R$ threshold.\nAt higher values of $M_R$, the\nfirst component of $W(\\ell\\nu)$+jets falls off rapidly,\nand the remaining background is instead dominated by the\nsum of $t$+X and the second component of $W(\\ell\\nu)$+jets;\nthis defines a second control region of intermediate $M_R$\nset by the $R$ threshold.\n\nUsing these two control regions in a given box,\na simultaneous fit determines both exponential slopes along\nwith the absolute normalization of the first component of $W(\\ell\\nu)$+jets\nand the relative normalization of the sum of the second\ncomponent of $W(\\ell\\nu)$+jets with the other backgrounds.\nThe $M_R$ distributions as a function of $R$ are shown in\nFig.~\\ref{fig:DATA_MU_slopes} (left). The slope parameters\ncharacterizing the exponential behavior of the first\n$W(\\ell\\nu)$+jets component are shown in Fig.~\\ref{fig:DATA_MU_slopes}\n(right); they are consistent within uncertainties between the electron\nand muon channels. The values of the parameters $a$ and $b$ that\ndescribe the $R^{2}$ dependence of the slope are in good agreement\nwith the values extracted from simulated $W(\\ell\\nu)$+jets events.\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{DATA_MUBOX_slopes2.pdf}\n\\includegraphics[width=0.49\\textwidth]{DATA_MUBOX_slopefit.pdf}\\\\\n\\includegraphics[width=0.49\\textwidth]{DATA_EGBOX_slopes.pdf}\n\\includegraphics[width=0.49\\textwidth]{DATA_EGBOX_slopefit.pdf}\n\\caption{(Left) $M_{R}$ distributions for different values of the $R$\n threshold from data events selected in the MU (upper) and ELE\n (lower) boxes. Dotted curves show the results of fits using two\n independent exponential functions and an asymmetric Gaussian at low\n $M_R$. (Right) The slope $S$ of the first exponential component\n as a function of the square of the $R$ threshold in the MU (upper)\n and ELE (lower) boxes. The dotted lines show the results of the fits\n to the form $S = a + bR^{2}$.}\n\\label{fig:DATA_MU_slopes}\n\\end{center}\n\\end{figure*}\n\nThe data\/MC ratios $\\rho(a)_1^{\\mathrm{data\/MC}}$, $\\rho(b)_1^{\\mathrm{data\/MC}}$\nof the first component slope parameters $a$, $b$ measured in the MU and ELE boxes are thus combined\nyielding\n\n\\begin{eqnarray}\n\\rho(a)_{1}^{\\mathrm{data\/MC}} = 0.97 \\pm 0.02 ~~;~~\n\\rho(b)_{1}^{\\mathrm{data\/MC}} = 0.97 \\pm 0.02~~,\n\\end{eqnarray}\nwhere the quoted uncertainties are determined from the fits.\n\nThe ratios\n$\\rho^{\\mathrm{data\/MC}}$\nare taken as correction factors for the shapes of the\n$Z$+jets and $t+$X backgrounds as extracted from simulated samples\nfor the MU and ELE boxes; the same corrections are used\nfor the shape of the first component of $W(\\ell\\nu)$+jets as\nextracted from simulated samples for the\nhadronic (HAD) box.\n\nThe data\/MC correction factors for\nthe $\\cPZ(\\nu\\bar\\nu)$+jets and $\\cPqt$+X backgrounds in the HAD box, as well as\nthe second component of $\\PW(\\ell\\nu)$+jets in the MU, ELE, and HAD boxes,\nare measured in the MU and ELE boxes\nusing a \\textit{lepton-as-neutrino} treatment of leptonic events. Here\nthe electron or muon is excluded from the megajet reconstruction,\nkinematically mimicking the presence of an additional neutrino. With\nthe lepton-as-neutrino treatment in the MU and ELE boxes only one\nexponential component is observed both in data and in\n$\\PW(\\ell\\nu)$+jets simulated events. In the simulation, the value of\nthis single exponential component slope is found to agree with the\nvalue for the second component of $\\PW(\\ell\\nu)$+jets obtained in\nthe default treatment.\n\nThe combined data\/MC correction factors measured using this\nlepton-as-neutrino treatment are\n\\begin{eqnarray}\n\\rho(a)_{2}^{\\mathrm{data\/MC}} = 1.01 \\pm 0.02 ~~;~~\n\\rho(b)_{2}^{\\mathrm{data\/MC}} = 0.94 \\pm 0.07.\n\\end{eqnarray}\nFor the final background prediction the magnitude of the relative\nnormalization between the two $\\PW(\\ell\\nu)$+jets components, denoted\n$f^{\\PW}$, is determined from a binned maximum likelihood fit\nin the region 200 $< M_{R} < $ 400\\GeV.\n\n\\section{Results}\n\\subsection{Lepton box background predictions \\label{sec:LEPBOX}}\n\nHaving extracted the $M_{R}$ shape of the $\\PW$+jets and $\\cPZ$+jets\nbackgrounds, their relative normalization is set from the $\\PW$ and $\\cPZ$\ncross sections measured by CMS in electron and muon final\nstates~\\cite{EWK-PAS}.\nSimilarly, the normalization of the \\ccbar\nbackground relative to $\\PW$+jets is taken from the \\ttbar cross section measured by CMS in the dilepton channel~\\cite{top}. The measured values of these cross sections are summarized below:\n\\begin{eqnarray}\n\\sigma(pp \\to {\\PW}X) \\times \\mathrm{B}(\\PW \\to \\ell\\nu) &=& 9.951 \\pm\n0.073~(\\mathrm{stat}) \\pm 0.280~(\\mathrm{syst}) \\pm\n1.095~(\\mathrm{lum})~ \\mathrm{nb}~,~ \\nonumber \\\\\n\\sigma(\\Pp\\Pp \\to {\\cPZ}X) \\times \\mathrm{B}(\\cPZ \\to \\ell\\ell) &=& 0.931 \\pm 0.026~(\\mathrm{stat}) \\pm 0.023~(\\mathrm{syst}) \\pm 0.102~(\\mathrm{lum})~ \\mathrm{nb}~,~ \\\\\n\\sigma(\\Pp\\Pp \\to \\ttbar) &=& 194 \\pm 72~(\\mathrm{stat}) \\pm 24~(\\mathrm{syst}) \\pm 21~(\\mathrm{lum})~ \\mathrm{pb}~.~\\nonumber\n\\end{eqnarray}\n\nFor an $R > 0.45$ threshold the QCD background is virtually\neliminated. The region 125~$< M_{R}< 175\\GeV$ where the QCD contribution is\nnegligible and the $\\PW(\\ell\\nu)$+jets component is dominant is used to\nfix the overall normalization of the total background prediction.\nThe final background prediction in the ELE and MU boxes for $R > 0.45$ is shown in Fig.~\\ref{fig:ELEMUBOX}.\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{ELEBOX_45.pdf}\n\\includegraphics[width=0.49\\textwidth]{MUBOX_R45.pdf}\n\\caption{The $M_R$ distributions with $R > 0.45$ in the\n ELE (left) and MU (right) boxes for data (points) and backgrounds\n (curves). The bands show the uncertainties of the background predictions.}\n\\label{fig:ELEMUBOX}\n\\end{center}\n\\end{figure*}\nThe number of events with $M_{R}>500\\GeV$ observed in data and the\ncorresponding number of predicted background events are given in\nTable~\\ref{tab:ELEMUBOX} for the ELE and MU boxes.\nAgreement between the predicted and observed yields is found. The\n$p$-value of the measurement in the MU box is 0.1, given the predicted\nbackground (with its statistical and systematic uncertainties) and the\nobserved number of\nevents.\nA summary of the uncertainties entering the background measurements is\npresented in Table~\\ref{tab:LEPSYS}.\n\\begin{table}[ht!]\n\\caption{The number of predicted background events in the ELE and MU\n boxes for $R>$0.45 and $M_{R}>500\\GeV$ and the number of events\n observed in data. \\label{tab:ELEMUBOX}}\n\\smallskip\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n & Predicted & Observed \\\\\n\\hline\n\\hline\nELE box & 0.63 $\\pm$ 0.23 & 0 \\\\\n\\hline\nMU box & 0.51 $\\pm$ 0.20 & 3 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table*}[ht!]\n \\caption{Summary of the uncertainties on the background predictions\n for the ELE and MU boxes and their relative magnitudes. The range\n in the Monte Carlo uncertainties is owing to the different\n statistical precisions of the simulated background samples.\\label{tab:LEPSYS}}\n\\smallskip\n\\centering\n\\begin{tabular}{|c||c|c|c|}\n\\hline\nParameter & Description & Relative magnitude \\\\\n\\hline\n\\hline\nSlope parameter $a$ & systematic bias from correlations in fits & 5\\% \\\\\n\\hline\nSlope parameter $b$ & systematic bias from correlations in fits & 10\\% \\\\\n\\hline\nSlope parameter $a$ & uncertainty from Monte Carlo & 1--10\\% \\\\\n\\hline\nSlope parameter $b$ & uncertainty from Monte Carlo & 1--10\\% \\\\\n\\hline\n$\\rho(a)^{\\mathrm{data\/MC}}$ & data fit & 3\\% \\\\\n\\hline\n$\\rho(b)^{\\mathrm{data\/MC}}$ & data fit & 3\\% \\\\\n\\hline\nNormalization & systematic+statistical component & 3--8\\% \\\\\n\\hline\n$f^{\\PW}$ & extracted from fit ($\\PW$ only) & 30\\% \\\\\n\\hline\n$PW\/\\ttbar$ cross section ratio & CMS measurements (top only) & 40\\% \\\\\n\\hline\n$\\PW\/\\cPZ$ cross section ratio & CMS measurements ($\\cPZ$ only) & 19\\% \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Hadronic box background predictions\\label{sec:yesyoucan}}\n\nThe procedure for estimating the total background predictions in the\nhadronic box is summarized as follows:\n\\begin{itemize}\n\\item Construct the non-QCD background shapes in $M_{R}$ using measured values\n of $a$ and $b$ from simulated events, applying correction\n factors derived from data control samples, and taking into account the $H_T$ trigger turn-on\n efficiency.\n\\item Set the relative normalizations of the $\\PW$+jets, $\\cPZ$+jets, and $\\cPqt$+X\n backgrounds using the relevant inclusive cross section measurements from\n CMS (Eq. 10).\n\\item Set the overall normalization by measuring the event yields in\n the lepton boxes, corrected for lepton reconstruction and\n identification efficiencies. The shapes and normalizations of all the\n non-QCD backgrounds are now fixed.\n\\item The shape of the QCD background is extracted, as described in\n Section ~\\ref{sec:qcd}, and its normalization in the HAD box is\n determined from a fit to the low-$M_{R}$ region, as described below.\n\\end{itemize}\n\n\nThe final hadronic box background prediction is calculated from a binned\nlikelihood fit of the total background shape to the data in the\ninterval 80~$ 0.5$. The observed $M_{R}$ distribution is consistent with\nthe predicted one over the entire $M_{R}$ range. The predicted and\nobserved background yields in the high-$M_{R}$ region are summarized\nin Table~\\ref{table:PRED}. A summary of the uncertainties entering\nthese background predictions is listed in Table~\\ref{tab:HADSYS}. A\nlarger $R$ requirement is used in the HAD box analysis due to the\nlarger background.\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{HADBOX_R50_lin.pdf}\n\\includegraphics[width=0.85\\columnwidth]{HADBOX_R50_log.pdf}\n\\caption{The $M_R$ distributions with $R > 0.5$ in the HAD box for\n data (points) and backgrounds (curves) on (top) linear and (bottom)\n logarithmic scales. The bands show the uncertainties of the\n background predictions. The corresponding distributions for SUSY\n benchmark models LM1 \\cite{PTDR2} with $M_\\Delta = 597\\GeV$ and LM0\n \\cite{alphaT} with $M_\\Delta = 400\\GeV$ are overlaid. }\n\\label{fig:HADBOX}\n\\end{center}\n\\end{figure}\n\n\\begin{table}[ht!]\n\\caption{Predicted and observed yields for $M_{R }$$>500\\GeV$ with $R > 0.5$ in the HAD box.\n\\label{table:PRED}}\n\\smallskip\n\\centering\n\\begin{tabular}{|c||c|\nc|}\n\\hline\n$M_{R}$ & Predicted & Observed \\\\\n\\hline\n\\hline\n\\hline\n$M_{R} > 500\\GeV$ & 5.5 $\\pm$ 1.4 & 7 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table*}[ht!]\n\\caption{Summary of uncertainties entering the background predictions for the HAD box.\n\\label{tab:HADSYS}}\n\\smallskip\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nParameter & Description & Relative magnitude \\\\\n\\hline\n\\hline\nSlope parameter $a$ & systematic bias from correlations in fits & 5\\% \\\\\n\\hline\nSlope parameter $b$ & systematic bias from correlations in fits & 10\\% \\\\\n\\hline\nSlope parameter $a$ & uncertainty from Monte Carlo & 1--10\\% \\\\\n\\hline\nSlope parameter $b$ & uncertainty from Monte Carlo & 1--10\\% \\\\\n\\hline\n$\\rho(a)^{\\mathrm{data\/MC}}$ & data fit & 3\\% \\\\\n\\hline\n$\\rho(b)^{\\mathrm{data\/MC}}$ & data fit & 3\\% \\\\\n\\hline\nNormalization & systematic+statistical component & 8\\% \\\\\n\\hline\nTrigger parameters & systematic from fit pseudo-experiments & 2\\% \\\\\n\\hline\n$f^{\\PW}$ & extracted from fit ($\\PW$ only) & 13\\% \\\\\n\\hline\n$W\/\\ttbar$ cross section ratio & CMS measurements (top only) & 40\\% \\\\\n\\hline\n$\\PW\/\\cPZ$ cross section ratio & CMS measurements ($\\cPZ$ only) & 19\\% \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\\section{Limits in the CMSSM Parameter Space}\nHaving observed no significant excess of events beyond the SM\nexpectations, we extract a model-independent 95\\% confidence level (CL) limit on\nthe number of signal events. This limit is then\ninterpreted in the parameter spaces of SUSY models.\n\nThe likelihood for the number of observed events $n$ is modeled as a\nPoisson function, given the sum of the number of signal events ($s$)\nand the number of background events. A posterior probability\ndensity function $P(s)$ for the signal yield is derived using Bayes theorem,\nassuming a flat prior for the signal and a log-normal prior for the\nbackground.\n\nThe model-independent upper limit is derived by integrating the\nposterior probability density function between 0 and $s^*$ so that\n$\\int_0^{s^*}P(s)ds=0.95$. The observed upper limit in the hadronic\nbox is $s^{*}=8.4$ (expected limit 7.2 $\\pm$ 2.7); in the muon box\n$s^{*}=6.3$ (expected limit 3.5 $\\pm$ 1.1); and in the electron box\n$s^{*}=2.9$ (expected limit 3.6 $\\pm$ 1.1). For 10\\% of the\npseudo-experiments in the muon box the expected limit is higher than\nthe observed. The stability of the result was studied with different\nchoices of the signal prior. In particular, using the reference priors\nderived with the methods described in Ref.~\\cite{refprior}, the\nobserved upper limits in the hadronic, muon, and electron boxes\nare 8.0, 5.3, and 2.9, respectively.\n\nThe results can be interpreted in the context of the CMSSM, which is a\ntruncation of the full SUSY parameter space motivated by the minimal\nsupergravity framework for spontaneous soft breaking of\nsupersymmetry. In the CMSSM the soft breaking parameters are reduced\nto five: three mass parameters $m_0$, $m_{1\/2}$, and $A_0$ being,\nrespectively, a universal scalar mass, a universal gaugino mass, and a\nuniversal trilinear scalar coupling, as well as $\\tan\\beta$, the\nratio of the up-type and down-type Higgs vacuum expectation values,\nand the sign of the supersymmetric Higgs mass parameter\n$\\mu$. Scanning over these parameters yields models which, while not\nentirely representative of the complete SUSY parameter space, vary\nwidely in their superpartner spectra and thus in the dominant\nproduction channels and decay chains.\n\nThe upper limits are projected onto the ($m_0$, $m_{1\/2}$) plane by\ncomparing them with the predicted yields, and excluding any model if\n$s(m_{0},m_{1\/2})>s^{*}$. The systematic uncertainty on the signal\nyield (coming from the uncertainty on the luminosity, the selection\nefficiency, and the theoretical uncertainty associated with the cross\nsection calculation) is modeled according to a log-normal prior. The\nuncertainty on the selection efficiency includes the effect of\njet energy scale (JES) corrections, parton distribution function (PDF)\nuncertainties~\\cite{Bourilkov:2006cj}, and the description of\ninitial-state radiation (ISR). All the effects are summed in\nquadrature as shown in Table~\\ref{tab:syst}. The JES, ISR, and PDF\nuncertainties are relatively small owing to the insensitivity of the\nsignal $R$ and $M_R$ distributions to these effects.\n\n\\begin{table}[ht!]\n \\caption{Summary of the systematic uncertainties on the signal yield\n and totals for each of the event boxes. For the CMSSM scan the NLO signal cross section uncertainty is included. \\label{tab:syst}}\n\\centering\n\\smallskip\n\\begin{tabular}{|lccc|}\n\\hline\nbox & MU & ELE & HAD \\\\\\hline\n\\multicolumn{4}{|c|}{Experiment}\\\\\\hline\nJES & 1\\% & 1\\% & 1\\% \\\\\\hline\nData\/MC $\\epsilon$& 6\\% & 6\\% & 6\\% \\\\\\hline\n$\\mathcal{L}$\\cite{lumi-moriond} & 4\\% & 4\\% & 4\\% \\\\\n\\hline\\hline\n\\multicolumn{4}{|c|}{Theory}\\\\\\hline\nISR & 1\\% & 1\\% & 0.5\\% \\\\\\hline\nPDF & 3--6\\% & 3--6\\% & 3--6\\% \\\\ \\hline\nSubtotal & 8--9\\% & 8--9\\% & 8--9\\% \\\\\n\\hline\\hline\n\\multicolumn{4}{|c|}{CMSSM}\\\\\\hline\nNLO & 16--18\\% & 16--18\\% & 16--18\\% \\\\\\hline\nTotal & 17--19\\% & 17--19\\% & 17--19\\% \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table}\n\nThe observed limits from the ELE, MU, and HAD boxes are shown in\nFigs.~\\ref{fig:ELE_LIMIT},~\\ref{fig:test}, and~\\ref{fig:HAD_LIMIT},\nrespectively, in the CMSSM\n($m_{0}$, $m_{1\/2}$) plane for the values $\\tan\\beta = 3$, $A_{0} = 0$,\n$\\operatorname{sgn}(\\mu) = +1$,\n together with the 68\\% probability band for the\nexpected limits, obtained by applying the same procedure to an ensemble of\nbackground-only pseudo-experiments. The band is computed\naround the median of the limit distribution. Observed limits are also shown\nin Figs.~\\ref{fig:ELE_LIMIT10}\n--\\ref{fig:HAD_LIMIT10} in the CMSSM ($m_{0}$, $m_{1\/2}$) plane for the\nvalues $\\tan\\beta = 10$, $A_{0} = 0$,\n$\\operatorname{sgn}(\\mu) = +1$, and in\nFigs.~\\ref{fig:ELE_LIMIT50}--\\ref{fig:HAD_LIMIT50} for the values\n$\\tan\\beta = 50$, $A_{0} = 0$,\n$\\operatorname{sgn}(\\mu) = +1$.\n\nFigure ~\\ref{fig:SMS} shows the same result in terms of 95\\% CL upper\nlimits on the cross section as a function of the physical masses for two benchmark\nsimplified models~\\cite{Alwall-1,Alwall-2,Sanjay,RA2}: four-flavor\nsquark pair production and gluino pair production. In\nthe former, each squark decays to one quark and the LSP, resulting in\nfinal states with two jets and missing transverse energy, while in the\nlatter each gluino decays directly to two light quarks and the LSP,\ngiving events with four jets and missing transverse energy.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_ELEBOX_tanB3.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=3$,\n $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from the ELE box selection ($R >\n 0.45$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n equivalent variations in the uncertainties are shown as a band\n around the expected limits. The area labeled $\\tilde{\\tau}$=LSP\n is the region of the parameter space where the LSP is a\n $\\tilde{\\tau}$ and not the lightest neutralino.}\n\\label{fig:ELE_LIMIT}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_MUBOX_tanB3.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=3$,\n $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from the MU box selection ($R >\n 0.45$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n equivalent variations in the uncertainties are shown as a band\n around the expected limits. }\n\\label{fig:test}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_HADBOX_tanB3.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=3$,\n $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from the HAD box selection ($R >\n 0.5$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n equivalent variations in the uncertainties are shown as a band\n around the expected limits. }\n\\label{fig:HAD_LIMIT}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_ELEBOX_tanB10.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=10$,\n $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from the ELE box selection ($R >\n 0.45$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n equivalent variations in the uncertainties are shown as a band\n around the expected limits. }\n\\label{fig:ELE_LIMIT10}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_MUBOX_tanB10.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=10$,\n $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from the MU box selection ($R >\n 0.45$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n equivalent variations in the uncertainties are shown as a band\n around the expected limits. }\n\\label{fig:MU_LIMIT10}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_HADBOX_tanB10.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=10$,\n $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from the HAD box selection ($R >\n 0.5$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n equivalent variations in the uncertainties are shown as a band\n around the expected limits. }\n\\label{fig:HAD_LIMIT10}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_ELEBOX_tanB50.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=50$,\n $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from the ELE box selection ($R >\n 0.45$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n equivalent variations in the uncertainties are shown as a band\n around the expected limits. }\n\\label{fig:ELE_LIMIT50}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_HADBOX_tanB50.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% L\n limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=50$,\n $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from the HAD box selection ($R >\n 0.5$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n equivalent variations in the uncertainties are shown as a band\n around the expected limits. }\n\\label{fig:HAD_LIMIT50}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.7\\columnwidth]{T2_limit.pdf}\n\\includegraphics[width=0.7\\columnwidth]{T1_limit.pdf}\n\\caption{Upper limits on two simplified models: di-squark production (top)\n resulting in a 2-jet + \\ETm final state and di-gluino (lower)\n production resulting in a 4-jet + \\ETm final state. The\n shade scale indicates the value of the cross section excluded at 95\\%\n CL for each value of $m_{\\text{LSP}}$ and $m_{\\text{gluino}}$\n or $m_{\\text{squark}}$. The solid and dashed contours indicate the\n 95\\% CL limits assuming the NLO cross section and its variations\n up and down by a factor of three.}\n\\label{fig:SMS}\n\\end{center}\n\\end{figure}\n\\section{Summary}\nWe performed a search for squarks and gluinos using a data\nsample of 35\\pbinv integrated luminosity from pp collisions at\n$\\sqrt{s} = 7\\TeV$, recorded by the CMS detector at the LHC. The\nkinematic consistency of the selected events was tested against the\nhypothesis of heavy particle pair production using the dimensionless\nrazor variable $R$ related to the missing transverse energy \\ETm,\nand $M_R$, an event-by-event indicator of the heavy particle\nmass scale. We used events with large $R$ and high $M_R$ in inclusive\ntopologies.\n\nThe search relied on predictions of the SM backgrounds determined from\ndata samples dominated by SM processes. No significant excess over\nthe background expectations was observed, and model-independent upper\nlimits on the numbers of signal events were calculated.\nThe results were presented in the ($m_0$, $m_{1\/2}$) CMSSM parameter\nspace. For simplified models the results were given as limits on the\nproduction cross sections as a function of the squark, gluino, and LSP\nmasses.\n\nThese results demonstrate the strengths of the razor analysis\napproach; the simple exponential behavior of the various SM\nbackgrounds when described in terms of the razor variables is useful\nin suppressing these backgrounds and in making reliable estimates from\ndata of the background residuals in the signal regions. Hence, the\nrazor method provides an additional powerful probe in searching for\nphysics beyond the SM at the LHC.\n\n\n\\section*{Acknowledgments}\n\n\n\n\\hyphenation{Bundes-ministerium Forschungs-gemeinschaft Forschungs-zentren} We wish to congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC machine. We thank the technical and administrative staff at CERN and other CMS institutes. This work was supported by the Austrian Federal Ministry of Science and Research; the Belgian Fonds de la Recherche Scientifique, and Fonds voor Wetenschappelijk Onderzoek; the Brazilian Funding Agencies (CNPq, CAPES, FAPERJ, and FAPESP); the Bulgarian Ministry of Education and Science; CERN; the Chinese Academy of Sciences, Ministry of Science and Technology, and National Natural Science Foundation of China; the Colombian Funding Agency (COLCIENCIAS); the Croatian Ministry of Science, Education and Sport; the Research Promotion Foundation, Cyprus; the Estonian Academy of Sciences and NICPB; the Academy of Finland, Finnish Ministry of Education and Culture, and Helsinki Institute of Physics; the Institut National de Physique Nucl\\'eaire et de Physique des Particules~\/~CNRS, and Commissariat \\`a l'\\'Energie Atomique et aux \\'Energies Alternatives~\/~CEA, France; the Bundesministerium f\\\"ur Bildung und Forschung, Deutsche Forschungsgemeinschaft, and Helmholtz-Gemeinschaft Deutscher Forschungszentren, Germany; the General Secretariat for Research and Technology, Greece; the National Scientific Research Foundation, and National Office for Research and Technology, Hungary; the Department of Atomic Energy and the Department of Science and Technology, India; the Institute for Studies in Theoretical Physics and Mathematics, Iran; the Science Foundation, Ireland; the Istituto Nazionale di Fisica Nucleare, Italy; the Korean Ministry of Education, Science and Technology and the World Class University program of NRF, Korea; the Lithuanian Academy of Sciences; the Mexican Funding Agencies (CINVESTAV, CONACYT, SEP, and UASLP-FAI); the Ministry of Science and Innovation, New Zealand; the Pakistan Atomic Energy Commission; the State Commission for Scientific Research, Poland; the Funda\\c{c}\\~ao para a Ci\\^encia e a Tecnologia, Portugal; JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan); the Ministry of Science and Technologies of the Russian Federation, and Russian Ministry of Atomic Energy; the Ministry of Science and Technological Development of Serbia; the Ministerio de Ciencia e Innovaci\\'on, and Programa Consolider-Ingenio 2010, Spain; the Swiss Funding Agencies (ETH Board, ETH Zurich, PSI, SNF, UniZH, Canton Zurich, and SER); the National Science Council, Taipei; the Scientific and Technical Research Council of Turkey, and Turkish Atomic Energy Authority; the Science and Technology Facilities Council, UK; the US Department of Energy, and the US National Science Foundation.\n\nIndividuals have received support from the Marie-Curie programme and the European Research Council (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Associazione per lo Sviluppo Scientifico e Tecnologico del Piemonte (Italy); the Belgian Federal Science Policy Office; the Fonds pour la Formation \\`a la Recherche dans l'Industrie et dans l'Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); and the Council of Science and Industrial Research, India.\n\\cleardoublepage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nThere are several indications by now from angle-resolved photoemission spectroscopy (ARPES) that in underdoped cuprates a gap opens at the Fermi surface in the diagonal (nodal) direction~\\cite{Drachuck,AHarter,vishik2012,razzoli2013evolution,peng2013disappearance}. In La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) this nodal gap (NG) extends to $x=8\\%$. At doping around $x=12.5\\%$ samples develop a charge-density-wave (CDW) below $T\\approx100$~K \\cite{croft14}. Traces of antiferromagnetism (AFM) in the form of spin-density-waves (SDW) \\cite{matsuda2002} or spin-glass \\cite{Niedermayer98} appear at doping up to $x=12.5\\%$ and temperatures $T\\approx10$~K. It is therefore natural to speculate that one of these symmetry breaking phenomena is responsible for the opening of a nodal gap. In this work, we would like to clarify which one is the most likely. Our strategy is to carefully examine a sample which is known to at least have both AFM and SDW order, and opens a nodal gap at low temperatures. The sample is LSCO with $x=1.92\\%$~\\cite{Drachuck}.\n\nPrevious neutron diffraction measurements on LSCO $x=1.92\\%$ \\cite{Drachuck} showed a magnetic Bragg peak at the AFM wave vector $\\textbf{Q}_{AF}$ below $T=140$~K, and two satellites that stand for static SDW order (on top of the AFM one). The satellites appear below $T=30$~K. Like in Matsuda et al. ~\\cite{matsuda2002}, there are two domains in the sample. We focus on one of them, in which the AFM peak is observed when scanning near (1,0,0), with no contribution from SDW. In contrast, the SDW peaks are observed when scanning near (0,1,0), with no contribution from the AFM peak. Neutron scattering detects the component of spin fluctuations perpendicular to the momentum transfer $\\textbf{q}$ ~\\cite{squires2012introduction}. Hence, the SDW fluctuations are perpendicular to the AFM order. ARPES measurements on the same sample found that a nodal gap opens below $T_{NG}=45$~K~\\cite{Drachuck}. Even though there is a temperature mismatch between the NG and SDW appearance, the two phenomena might be related. Moreover, CDW in LSCO $x=1.92\\%$ is expected to be very weak \\cite{capati2015}, and indeed this sample is out of the CDW dome ~\\cite{croft14,Hucker14}. Therefore, \\textit{a priori}, CDW is not expected to generate the nodal gap.\n\nHere we add to the available ARPES and neutron diffraction data, inelastic neutron scattering (INS) and x-ray diffraction data on the same piece of LSCO $x=1.92\\%$. We show that the fluctuating SDW amplitude of the frequency where it is the strongest, decreases at a temperature equal to $T_{NG}$ within experimental error. In addition, we could not find any indications for CDW in our sample. We argue that these findings explain the previously measured $15$~K discrepancy between the SDW freezing and the opening of a NG, and tie the latter to fluctuating SDW.\n\n\n\\begin{figure*}[h!t]\n\t\\begin{center}\n\t\t\\includegraphics[trim=0cm 0cm 0.5cm 0cm ,clip=true,width=18cm]{dispersion.png}\n\t\\end{center}\n\t\\caption{\\textbf{SDW dispersion.}\n\t\tFalse color map of normalized intensity as a function of neutron energy transfer $\\hbar \\omega$ and momentum transfer $\\textbf{q}$ at $T=50$~K (a) and $T=2$~K (b). The raw data is interpolated. The black horizontal line in panel (b) demonstrates a constant energy cut along which the intensity is integrated and plotted in Fig.~\\ref{fig:edep}. Dashed white lines in panel (b) represent cuts along which the background is determined. The black symbols indicate the center of the $\\hbar \\omega =0$ and $2$~meV peaks demonstrated in Fig.~\\ref{fig:qdep}, and make the bottom part of an hourglass.}\n\t\\label{fig:dispersion}\n\\end{figure*}\n\n\nThe neutron experiment was performed at Rita-II, the cold neutrons triple axis spectrometer at the Paul Scherrer Institut. Throughout this paper, we work in orthorhombic notation, with cell parameters $a=5.344$~\\AA, $b=5.421$~\\AA\\ and $c=13.14$~\\AA\\ at $T=2$~K. In this notation, the tetragonal 2D $\\textbf{Q}_{AF}=\\left(1\/2,1\/2,0\\right)$ is equivalent to $\\left( 0,1,0\\right)$ in reciprocal lattice units (r.l.u) of $2\\pi\/a$. More information is available in the Methods section. In Fig.~\\ref{fig:dispersion} we present a false color map of neutron counts versus energy transfer $\\hbar \\omega$ and momentum transfer $\\textbf{q}$. The raw data, in this figure alone, is interpolated for presentation purpose. Data is presented at two temperatures, $2$~K and $50$~K, which are below and above the freezing temperature of the incommensurate magnetic order of $30$~K ~\\cite{Drachuck}. In both cases, strong intensity is observed at $\\hbar \\omega=0$. This is due to high order contamination of the incoming beam scattering from a nuclear Bragg peak at $(0,2,0)$, despite the use of Br filter. Around $(0,1,0)$, the intensity extends to energy transfers as high as 8 meV for both temperatures, in a cone shape, which is in fact a poorly-resolved bottom part of an hourglass. This will be demonstrated subsequently. The scattering intensity is stronger at elevated temperatures. Interestingly, at $T=2$~K spectral weight is missing at low energies, suggesting the presence of a soft gap for spin excitations. A similar spectrum, including the gap, was observed at the fully developed hourglass dispersion of La$_{1.875}$Ba$_{0.125}$CuO$_4$~\\cite{tranquada2004}, La$_{1.88}$Sr$_{0.12}$CuO$_4$ ~\\cite{matsuda2008,Romer13}, and La$_{1.6}$Sr$_{0.4}$CoO$_4$ ~\\cite{drees2013hour}. \n\nq-scans at specific constant energies at $T=2$~K are presented in Fig.~\\ref{fig:qdep}, showing the evolution of the SDW peaks with energy transfer. The intensities are shifted vertically for clarity. At $\\hbar \\omega=0.6$~meV, some intensity is detected around $(0,1,0)$ above the background. However, this could stem from the tail of the high order contamination. At $\\hbar \\omega=2$~meV two clear peaks appear \n\nFor fitting, the instrument was modeled using Popovici ResCal5 ~\\cite{popovici1975resolution}, and the resolution was calculated. Black horizontal lines in Fig.~\\ref{fig:qdep} represent the q-resolution at each energy. This was taken into account as a constant width Gaussian at each energy, which was convoluted with a Lorentzian (Voigt function). The fit with two Voigt functions is demonstrated in Fig.~\\ref{fig:qdep} by solid lines. The fit to the $\\hbar \\omega=2$~meV data indicates a peak separation of 0.04 r.l.u. The same separation is found in the elastic peaks ~\\cite{Drachuck}, as demonstrated in the inset. The peaks centers are illustrated in Fig.~\\ref{fig:dispersion}(b) by the solid points. The static and dynamic SDW correlation lengths, determined from the peaks width, are $85\\pm12~\\AA$ and $44\\pm5~\\AA$ respectively. With increasing energy to $4$~meV and then to $6$~meV, the two peaks are no longer resolved. However, the measured peak is asymmetric because of the two underlying incommensurate peaks coming closer together. At $8$~meV the intensity diminishes. This behavior reminds two ``legs\" dispersing downwards from some crossing energy as in the hourglass.\n\n\\begin{figure}[tbph]\n\t\t\\includegraphics[trim=1cm 2cm 1cm 2.5cm ,clip=true,width=\\columnwidth]{qdep.pdf}\n\n\t\\caption{\\textbf{Evolution of the SDW peaks with energy at $T=2$~K.}\n\t\tMomentum scan along k centered at (0,1,0) for different energy transfers at $T=2$~K. Scans are shifted consecutively by $2.5\\times10^{-5}$ counts\/monitor for clarity. Inset: SDW elastic peaks for the same $\\textbf{q}$ scan also at $T=2$~K. Background from higher temperature was subtracted. For energies of $\\hbar \\omega=0$ and $2$~meV, a sum of two Voigt functions is fitted to the data (solid black lines). The peak separation for $\\hbar \\omega=2$~meV is 0.04 r.l.u, as in the $\\hbar \\omega=0$ case (see inset). Black horizontal lines represents the instrumental resolution. \n\t}\n\t\\label{fig:qdep}\n\\end{figure}\n\nTo further investigate the inelastic behavior, we sum the intensity over $\\textbf{q}$ at constant energy cuts. The horizontal line in Fig.~\\ref{fig:dispersion}(b) presents one such cut. Background contribution is estimated from the data along the dashed diagonal lines in Fig.~\\ref{fig:dispersion}(b), and subtracted. Fig.~\\ref{fig:edep} presents the background subtracted \\textbf{q}-integrated intensity versus energy transfer $ \\langle I \\rangle (\\omega)=\\sum_\\textbf{q} I(\\textbf{q},\\omega)$, starting from $\\hbar\\omega = 0.15$ meV to avoid the high intensity elastic peak. At $T=50$~K, $\\langle I \\rangle (\\omega)$ monotonically grows as the frequency decreases. In contrast, at $T=2$~K, $\\langle I \\rangle (\\omega)$ reaches a maximum at some $\\hbar\\omega_{max}$ between 2 and 3~meV, and drops towards $\\hbar\\omega=0$, although residual elastic scattering intensity is observed near $\\hbar\\omega=0$. Measurements on La$_{2-x}$Ba$_x$CuO$_4$ with $0.0125\\le x \\le 0.035$ which were limited to energies below 1meV agree with our results ~\\cite{Wagman2013}. This plot demonstrates more clearly the aforementioned soft gap in spin excitations which develops at low temperatures.\n\n\\begin{figure}[tbph]\n\n\t\t\\includegraphics[trim=1cm 1cm 1cm 1.2cm ,clip=true,width=\\columnwidth]{E_dependence.pdf}\n\n\t\\caption{\\textbf{q-integrated intensity vs. neutron energy transfer at low (2 K) and high (50 K) temperatures.}\n\t\tIntegrated intensity is calculated for each energy as sum of the counts over $\\textbf{q}$ along horizontal lines like the one shown in Fig.~\\ref{fig:dispersion}. Background is estimated from the counts along the two dashed lines shown in Fig.~\\ref{fig:dispersion}(a) and subtracted from the raw data.\n\t}\n\t\\label{fig:edep}\n\\end{figure}\n\n\nWe summarize the available data on LSCO $x=1.92\\%$ in Fig.~\\ref{fig:tdep}(a). In this figure we show the temperature dependence of the $\\textbf{q}$-integrated scattering intensity at three different energies. The data at $\\hbar \\omega=0$ is taken from Ref.~\\cite{Drachuck} and multiplied by $2\\times 10^{-3}$ for clarity. It shows that a long range static SDW appears at a temperature of $30$~K. The intensity at $\\hbar \\omega=0.6$~meV increases as the temperature is lowered, peaks at $38$~K, and then decreases. This result demonstrates that dynamically fluctuating SDW at $\\hbar\\omega >0$ diminishes upon cooling before long range static incommensurate order develops. The same effect, although less sharp, is observed for $\\hbar \\omega=2$~meV at $45$~K. \n\nFigure ~\\ref{fig:tdep}(b) depicts the temperature dependence of the nodal gap from Ref.~\\cite{Drachuck} as measured by ARPES. This gap opens at $T_{NG}=45$~K, which is the same temperature where the spectral density at $\\hbar\\omega_{max}$ begins to diminish. The maximum electronic gap value $\\Delta$ agrees with isolated dopant-hole bound state calculations~\\cite{sushkov2005}. We note that $\\hbar\\omega_{max}$ and $k_B T_{NG}$ are of the same order of magnitude. Our result indicates a strong link between the dynamically fluctuating SDW and the nodal gap. \n\n\n\\begin{figure}[tbph]\n\t\\begin{center}\n\t\t\\includegraphics[trim=0cm 3cm 0cm 1cm ,clip=true,width=\\columnwidth]{Tdep.pdf}\n\t\\end{center}\n\t\\caption{\\textbf{Temperature dependence of all experimental parameters.}\n\t\t(a) Elastic and inelastic incommensurate SDW intensities at different energies from neutron scattering. (b) ARPES measurement of the nodal gap at $k_F$ \\cite{Drachuck}. The dashed vertical line emphasizes the fact that the nodal gap opens when the amplitude of dynamic spin fluctuations at $\\hbar\\omega\\approx~2$meV decreases.\n\t}\n\t\\label{fig:tdep}\n\\end{figure}\n\nIn order to investigate whether CDW plays a role in the nodal gap \\cite{Berg08}, we conducted a search for CDW in this sample by two different methods: off resonance x-ray diffraction (XRD) and resonance elastic x-ray scattering (REXS). The experiments were done at PETRA III on the P09 beam-line and at BESSY on the UE46-PGM1 beam-line, respectively. In REXS, the background subtraction is not trivial, so we only present here our XRD data. Nonetheless, the final conclusion from both methods is the same.\n\n\nIn Fig.~\\ref{fig:cdw} we show results from LSCO samples with $x=1.92 \\%$, $x=6.0\\%$, and La$_{2-x}$Ba$_x$CuO$_4$ (LBCO) $x=12.5\\%$. The data sets are shifted vertically for clarity. The LBCO sample is used as a test case, since it has well established CDW and presents strong diffraction peaks. The measurements were taken at $7$~K and at $70$~K, which are below and above the CDW critical temperature of LBCO~\\cite{Tranquada2008cdw}. We performed two types of scans: a ``stripes\" scan along $(0,\\delta q,8.5)$ direction and a ``checkerboard\" scan along $(\\delta q,\\delta q,8.5)$ direction. We chose to work at $l=8.5$ to minimize contribution from a Bragg peak at $l=8$ or $l=9$. For LBCO at $T=7$~K, there is a clear CDW peak at $\\delta q=\\pm 0.24$ in the ``checkerboard\" scan, which is absent at high temperatures. In contrast, for the LSCO samples there is no difference between the signal at high and low temperatures. Since $\\delta q$ of the CDW peak depends on doping, in our sample it is expected to be close to $\\delta q=0$, where a tail of the Bragg peak could potentially obscure the CDW peak. Arrows in Fig.~\\ref{fig:cdw} show where we might expect the CDW peaks, should they appear, based on linear scaling with doping. These positions are out of the $\\delta q=0$ peak tail, and not obscured. Thus, although we are in experimental conditions appropriate to find a CDW, it is not observed within our sensitivity. In fact, CDW is even absent at higher doping as demonstrated by our experiment with LSCO $x=6\\%$ sample. We observed the same null-result with the REXS experiment. It is important to mention that hourglass excitations with no stripe-like CDW were observed previously \\cite{drees2013hour}.\n\nOur main results are as follows: we find the bottom part of an hourglass dispersion inside the AFM phase of LSCO. The hourglass does not start from zero energy, but has a soft gap from the static SDW order. A CDW order seems to be absent in our sample. Upon cooling the system, a nodal gap in electronic excitations opens just when the strongest spin excitations start to diminish. It is therefore sufficient for the SDW fluctuations to slow down without completely freezing out in order to modify the band structure.\n\n\n\\begin{figure}[tbph]\n\t\\begin{center}\n\t\t\\includegraphics[trim=1.5cm 1cm 3cm 0cm ,clip=true,width=\\columnwidth]{NoCDWL.pdf}\n\t\\end{center}\n\t\\caption{\\textbf{Hard x-ray diffraction on three different samples: }LSCO with $x=1.92\\%$ and $x=6\\%$, and LBCO $x=12.5\\%$. Scans are done in two different orientations and two different temperatures. CDW is detected only in LBCO. }\n\t\\label{fig:cdw}\n\\end{figure}\n\n\\section*{Methods}\n\nFor the Neutron scattering experiment, the sample was mounted on aluminum holder covered with Cd foils, and oriented in the (h,k,0) scattering plane. A Be filter was used to minimize contamination from high order monochromator Bragg reflections. The scattered neutrons are recorded with a nine bladed graphite analyzer. All the blades are set to scatter neutrons at the same final energy of 5 meV, and direct the scattered neutrons through an adjustable radial collimator to different predefined areas on a position sensitive detector ~\\cite{bahl2006inelastic,lefmann2006realizing}. This monochromatic q dispersive mode allows for an efficient mapping of magnetic excitations with an excellent q resolution.\n\nTwo types of scans were used: I) energy scan, in which the incoming neutrons energy is swept, and the $\\textbf{q}$ information is embedded in the position of each blade. II) momentum scan, in which the incoming neutrons energy is fixed, the nine blades cover a small window in $\\textbf{q}$, and the entire window is scanned. The contribution to a given $\\textbf{q}$ is a weighted sum from the different blades.\n\nDespite the Be filter, some contribution from the nuclear structure is unavoidable. For elastic scattering, this contribution survives to higher temperature than does the magnetic part, and therefore can be easily subtracted. For inelastic scattering, the contribution from phonons could not be subtracted, but it is expected to vary slowly with temperature close to the magnetic phase transitions. Therefore, all features in this scattering experiment which show abrupt temperature dependence around and below $T=50$~K are associated with the electronic (magnetic) system.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\\section{Introduction}\n\\subsection{The model}\nThe aim of the present paper is to study the ergodicity of a piecewise\ndeterministic Markov process (PDMP) linked to Lotka-Volterra type dynamics.\nThese lines can be seen as a companion paper to~\\cite{lotka} since we go one\nstep further in the description of different regimes of the process and the support of the\ninvariant measures. Let us first provide an overview of the \nmain results in~\\cite{lotka} before stating our contribution.\n\nFor a given set of positive parameters $\\mathcal{E}=(a,b,c,d,\\alpha,\\beta)$,\nconsider the Lotka-Volterra differential system in $\\dR_+^2$, given by \n\\begin{equation}\n \\label{eq:lv}\n \\begin{cases}\n \\dot{x}=\\alpha x (1-ax-by) \\\\\n \\dot{y}=\\beta y (1-cx-dy) \\\\\n (x_0,y_0)\\in\\dR_+^2. \n \\end{cases}\n\\end{equation}\nThis system modelizes the evolution of the populations of two species ($x$ of\ntype~$\\textbf{x}$ and $y$ of type~$\\textbf{y}$). The populations\ngrow logistically --- as encoded by the $\\alpha x(1-ax)$ and $\\beta y(1-dy)$ terms --- and\ncompete with each other, which gives rise to the cross terms $\\alpha b xy$\nand $\\beta c xy$. We denote by $F_\\mathcal{E}$ the associated vector\nfield: $(\\dot x,\\dot y)=F_{\\mathcal{E}}(x,y)$. In the sequel, the variable $z$\nstands for $(x,y)$ and we will sometimes write $F_{\\mathcal{E}}(z)$ instead of\n$F_{\\mathcal{E}}(x,y)$. This ODE system, taken alone, is easy to analyze. \nIn particular it has only a finite number of equilibrium points, towards which \nthe dynamics converges. These equilibria may be on the coordinate axes --- meaning\nthat one of the species gets extinct --- or inside the positive quadrant. The \nposition and nature of the equilibria turn out to depend only on the signs of\n$c-a$ and $d-b$. A complete picture will be given in Section~\\ref{sec:notation};\nlet us note already that when $a0$ as soon as\n $x_0>0$. \n\\end{ass}\n\nFinally, introduce the random process obtained by switching between \nthese two deterministic dynamics, at rates $(\\lambda_i)_{i=0,1}$. More precisely, \nwe consider the process $(Z,I)$ on $\\dR^2\\times\\BRA{0,1}$ driven by the \ninfinitesimal generator \n\\[\nLf(z,i)=F_{\\mathcal{E}_i}(z)\\cdot\\nabla_z f(z,i) +\\lambda_i (f(z,1-i)-f(z,i)). \n\\]\nIn other words, $I$ jumps from $i$ to $1-i$ after a random time with an exponential distribution \nof parameter $\\lambda_i$, and while $I_t$ is equal to $i$,\n$Z$ evolves deterministically by $\\dot Z_t=F_{\\mathcal{E}_i}(Z_t)$. The coordinates \nof $Z_t$ are denoted by $X_t$ and $Y_t$. We refer to~\\cite{lotka} for a detailed biological \nmotivation. \n\nIt is shown in \\cite{lotka} that, depending on the environments $\\cE_0$, $\\cE_1$ \nand the jump rates $\\lambda_0$, $\\lambda_1$, \none of the following four things occur almost surely: \n\\begin{itemize}\n\\item extinction of species $\\textbf{x}$: $X_t \\to 0$ and $\\limsup Y_t>0$,\n\\item extinction of species \\textbf{y}: $Y_t\\to 0$ and $\\limsup X_t>0$, \n\\item extinction of one of the two species, chosen randomly,\n\\item persistence: the empirical occupation measure (and, in many cases, the\n distribution) of ${(X_t,Y_t)}_{t\\geq 0}$ \nconverges to a probability measure on $(0,+\\infty)^2$ that is absolutely continuous with \nrespect to the Lebesgue measure. \n\\end{itemize}\nMoreover, one or more of these regimes may occur when the jump \nrates $(\\lambda_0,\\lambda_1)$ vary, the environments $(\\mathcal{E}_0,\\mathcal{E}_1)$ \nbeing fixed. Similar surprising behaviors for switched processes have\nbeen studied for linear ODEs in \\cite{BLMZ1,mattingly}.\n\n\n\\subsection{The frequent jumps asymptotics and the averaged vector field}\nRecall that $\\lambda_0$, $\\lambda_1$ are the jump rates from one environment\nto the other. \nNote that the index process $(I_t)_{t\\geq 0}$ is Markov by itself, \nand its invariant measure is a Bernoulli distribution with parameter \n$\\lambda_0\/(\\lambda_0+\\lambda_1)$. As a consequence, \nit will be convenient to choose the alternative\nparametrization\n\\begin{equation}\\label{eq:para_st}\n(s,t)\\in [0,1]\\times(0,+\\infty)\\mapsto (st,(1-s)t) \n\\end{equation}\nfor the jump rates, that is, let $t$ be the sum $\\lambda_0+\\lambda_1$ \nand $s$ be the ratio $\\lambda_0\/(\\lambda_0+\\lambda_1)$. \n\\begin{rem}[Length of interjump times I]\\label{rem:st}\nNotice that the expectations of the interjump times are given by $(st)^{-1}$ and $((1-s)t)^{-1}$. \nIf $t$ is small, the jumps are rare and the jump times are large in average; as \n$t$ grows the jumps become more frequent and the jump times shorter on average. \n\\end{rem}\nAs the parameter $t$ goes to infinity --- the frequent jumps asymptotics --- it\ncan be shown that the stochastic process ${(Z_t)}_{t\\geq 0}$ converges to the \nsolution of \n\\[\n\\dot{z}_t=F_s(z_t) \\quad\\text{where}\\quad \nF_s=(1-s)F_{\\mathcal{E}_0}+sF_{\\mathcal{E}_1}. \n\\]\nAs noticed in \\cite{lotka}, for any $s\\in [0,1]$, the vector field \n$F_s$ is the Lotka-Volterra system associated to the \"averaged\" environment \n$\\mathcal{E}_s=(a_s,b_s,c_s,d_s,\\alpha_s,\\beta_s)$ with \n\\begin{align}\n\\alpha_s &= (1-s)\\alpha_0+s\\alpha_1, \n&\na_s &= \\frac{(1-s)\\alpha_0a_0+s\\alpha_1a_1}{\\alpha_s},\n&\nb_s &= \\frac{(1-s)\\alpha_0b_0+s\\alpha_1b_1}{\\alpha_s},\\label{eq:def_alphas}\n\\\\\n\\beta_s &= (1-s)\\beta_0+s\\beta_1,\n&\nc_s &= \\frac{(1-s)\\beta_0c_0+s\\beta_1c_1}{\\beta_s},\n&\nd_s &= \\frac{(1-s)\\beta_0d_0+s\\beta_1d_1}{\\beta_s}.\\label{eq:def_betas}\n\\end{align}\nRecall that by our standing assumption, $a_i c_s} \n\\quad\\text{and}\\quad \nJ=(s_3,s_4)=\\BRA{ s\\in[0,1], b_s > d_s}. \n \\end{equation}\n\\end{defi}\n The fact that $I$ and $J$ are indeed intervals is obvious from the definition\n of $a_s$, $b_s$, $c_s$ and $d_s$ by~\\eqref{eq:def_alphas}\n and~\\eqref{eq:def_betas}. As we have seen, the relevance of these\n intervals stems from the fact that they correspond to different types\n for the averaged environment $\\cE_s$. For example, \n the vector field $F_s$ always has two stationary points on the coordinate axes, \n but their nature vary: \n\\begin{itemize}\n\\item the stationary point $(1\/a_s,0)$ is a well if $s\\notin I$ and a saddle point if $s\\in I$, \n\\item the stationary point $(0,1\/d_s)$ is a saddle point if $s\\notin J$ and a well if $s\\in J$. \n \\end{itemize}\n\\begin{rem}\\label{rem:adiff}\nNotice that if $a_0=a_1$ then the interval $I$ is empty. In the sequel we will focus on the \ncase when $a_0\\neq a_1$ and without loss of generality we will assume that $a_00$ & $\\Lambda_{\\textbf{y}}<0 $ \\\\\n\\hline \n$\\Lambda_\\textbf{x}>0$& persistence of the two species & extinction of species $\\textbf{y}$\\\\\n\\hline\n$\\Lambda_\\textbf{x}<0$ & extinction of species $\\textbf{x}$ & random extinction of one of the two species \\\\\n\\bottomrule\n\\end{tabular}\n\\label{tab:signs_of_lambda12}\n\\end{center}\n\n\\subsection{Our contribution}\nIn view of the previous result, the study of the model is reduced to \nfinding the sign of the invasion rates, depending on the parameters \nof the environment and on the jump rates. To state our results, \nwe need to introduce a second parametrization for the jump rates \n$(\\lambda_0,\\lambda_1)\\in (0,+\\infty)^2$ slightly different from~\\eqref{eq:para_st}:\n\\[\n(u,v)\\in [0,1]\\times(0,+\\infty) \\mapsto (\\alpha_0 uv, \\alpha_1 (1-u)v)\n\\]\nin such a way that \n\\begin{equation}\\label{eq:def_u_v}\nu = \\gamma_0\/(\\gamma_0+\\gamma_1)\n\\quad\\text{and}\\quad \nv=\\gamma_0+\\gamma_1\n\\quad\\text{where }\\gamma_i = \\lambda_i\/\\alpha_i\n\\quad\\text{for } i=0,1. \n\\end{equation}\n\n\nThe change of parameters $(u,v)=\\xi(s,t)$ is triangular in the sense\nthat $u$ only depends on $s$: \n\\[\n(u,v)=\\xi(s,t)=\\PAR{\n \\frac{s\\alpha_1}{(1-s)\\alpha_0 + s\\alpha_1},\n \\frac{t}{\\alpha_0\\alpha_1}( (1-s)\\alpha_0 + s\\alpha_1)\n}.\n\\]\n\\begin{rem}[Length of interjump times II]\nNotice that the new parameter $v$ is proportional to $t$ when $u$ (or $s$) is fixed. \nAs a consequence, as in Remark \\ref{rem:st}, the interjump times are short when $v$ \nis large and large when $v$ is small. \n\\end{rem}\n\n\\begin{defi}[Reparametrized invasion rates]\nThe invasion rates in the $(u,v)$ coordinates are denoted by\n\\[\n\\tilde \\Lambda_\\emph{\\textbf{x}}(u,v)=\\Lambda_\\emph{\\textbf{x}}(\\xi^{-1}(u,v))\n\\quad\\text{and}\\quad \n\\tilde \\Lambda_\\emph{\\textbf{y}}(u,v)=\\Lambda_\\emph{\\textbf{y}}(\\xi^{-1}(u,v)). \n\\]\nSimilarly, $\\tilde{I}$ (resp. $\\tilde{J}$) is the image of $I$ (resp. $J$) \nfor the other parametrization. \n\\end{defi}\nNote that $\\tilde{I}$ and $\\tilde{J}$ still are \n(possibly empty) intervals. \n\n\n\n\\begin{rem}\n The parameter $u$ is already implicitly considered in~\\cite{lotka}, where \n it appears in the computations leading to the explicit conditions for the \n non-emptyness of $I$ (which are equivalent to the positivity of a second \n degree polynomial). \n\\end{rem}\n\nOur first result is an explicit formula for~$\\tilde \\Lambda_\\textbf{y}$, \nsuited both to fast numerical computations and theoretical study. \n\\begin{lem}[Expression of $\\tilde\\Lambda_\\textbf{y}$]\n \\label{lem:exprLambda2}\n Assume that $a_00$ \n when $v>v_\\emph{\\textbf{y}}(u)$. \n\n Moreover $v_\\emph{\\textbf{y}}$ is quasi-convex, continuous on its domain $\\tilde{I}$, and \n tends to $+\\infty$ on the endpoints of $\\tilde{I}$. \n\n Similarly, there exists a function $s\\mapsto t_\\emph{\\textbf{y}}(s)\\in[0,\\infty]$, with\n domain $I$, going to infinity at the endpoints of $I$, such that: \n \\begin{itemize}\n\t\\item $\\Lambda_\\emph{\\textbf{y}}(s,t) < 0$ if $t 0 $ if $t>t_\\emph{\\textbf{y}(s)}$. \n \\end{itemize}\n\n The same statement holds in the parameters $(s,t)$ for the function\n $(-\\Lambda_\\emph{\\textbf{x}})$ with~$I$ replaced by~$J$ and\n with a critical function $t_\\emph{\\textbf{x}}(s)$. \n\\end{thm}\n\n\\begin{rem}\\label{rem:convex}\n Numerical computations suggest that both $v_\\emph{\\textbf{y}}$ \n and~$t_\\emph{\\textbf{y}}$ are in fact smooth and convex on $\\tilde I$ and $I$ \n respectively. \n\\end{rem}\n\n\\begin{rem}\n This result is cited in \\cite[Proposition 2.5]{lotka}, since it\n answers a conjecture that appeared in a preprint version of~\\cite{lotka}. \n\\end{rem}\nFor an illustration of Theorem~\\ref{thm:mainResult} and Remark~\\ref{rem:convex}, \nsee Figure~\\ref{fig:lambda_12}. \n\n\\begin{figure}\n \n {\\centering\n \\input{lambda1Et2.tex}\n\n \\bigskip\n\n \\input{lambda1Et2_bis.tex}\n\n}\n\n \n {\\small\n These plots represent the \"critical\" functions $t_\\textbf{y}$ and \n $t_\\textbf{x}$ for different choices of the environments. \n Denoting environments by the couple \n $\\PAR{\\begin{smallmatrix} \\alpha \\\\ \\beta\\end{smallmatrix}}$; \n $\\PAR{\\begin{smallmatrix} a & b \\\\ c & d\\end{smallmatrix}}$,\n the functions are plotted with\n \\[\n \\cE_0 = \n \\begin{pmatrix} 1 \\\\ 5 \\end{pmatrix} ; \n \\begin{pmatrix} 1 & 1 \\\\ 2 & 2\\end{pmatrix} \n \\text{ (top plot);}\n \\qquad\n \\cE_0 = \n \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix} ; \n \\begin{pmatrix} 1 & 2\/3 \\\\ 2 & 4\/3\\end{pmatrix} \n \\text{ (bottom plot);}\n \\qquad \\cE_1 = \n \\begin{pmatrix} 5 \\\\ 1 \\end{pmatrix} ; \n \\begin{pmatrix} 3 & 3 \\\\ 4 & \\rho\\end{pmatrix}\n \\]\n for various values of the parameter $\\rho$ appearing in the definition of the environment $\\cE_1$. \n The black curve in both plots is $t_\\textbf{y}$, \n and does not depend on the value of $\\rho$. The colored curves are $t_\\textbf{x}$. \nThe respective domains of these curves are the intervals~$I$ and~$J$. All configurations\nare possible: $I\\cap J$ may be empty (bottom plot, $\\rho=6.8$), a strict subset of\n$I$ and $J$ (bottom plot, $\\rho=6.2$) or may be $I$ or $J$ itself (top plot). \n\nThanks to the results of \\cite{lotka} summarized in the table page\n\\pageref{tab:signs_of_lambda12}, these plots\ndescribe exactly what regimes are possible when the jump rates (parametrized by \n$s$ and $t$) vary, for a given choice of the environments. \n\nFor example the top plot for $\\rho=5.5$ has three regimes: extinction of $\\textbf{x}$ \n(above the red curve), persistence (between the red and the black curves) \nand extinction of $\\textbf{y}$ (below the black curve). For $\\rho=4.5$ there is an \nadditonal zone (above the yellow curve and below the black one) of extinction of\na random species. In particular, the knowledge of the relative positions\nof $I$ and $J$ is not enough to determine the possible regimes. \n\nAll these plots are computed by finding, for a fixed $s$, the zero of the function\n$t\\mapsto\\Lambda(s,t)$; this is done by a simple root finding \nalgorithm, using the explicit formula given in Lemma~\\ref{lem:exprLambda2}\nto evaluate $\\Lambda(s,t)$. \n}\n\n \\caption{Shape of positivity regions for $\\Lambda_\\textbf{x}$ and $\\Lambda_{\\textbf{y}}$ }\n \\label{fig:lambda_12}\n\\end{figure}\n\nFinally, our last results are dedicated to the support of the non-trivial invariant probability \nmeasure in the persistence regime. In~\\cite{lotka}, it is shown that this measure has a \ndensity with respect to the Lebesgue measure on the quadrant. Theorem~\\ref{thm:support}\nprovides a full description of its support when the set $I\\cap J$ is not empty. Since a precise \nstatement requires several notations introduced in Section~\\ref{sec:notation}, we postpone \nit to the last section of the document. \n\nThe remainder of the paper is organized as follows. In\nSection~\\ref{sec:notation} we describe the various phase portraits for\nLotka-Volterra vector fields, and narrow down the choices of $\\cE_0$, $\\cE_1$\nthat lead to interesting behaviour. In Section~\\ref{sec:proofLemma} we prove\nLemma~\\ref{lem:exprLambda2}; the main result is proved in\nSection~\\ref{sec:proofTheorem}. The final section is dedicated \nto the description of the support of invariant measures in the persistence \nregime. \n\n\n\\section{Deterministic picture}\\label{sec:notation}\n\\subsection{Phase portraits of Lotka-Volterra vector field}\n\n\nWe consider here the ODE \\eqref{eq:lv} in an environment $\\mathcal{E}=(a,b,c,d,\\alpha,\\beta)$\nand describe its possible qualitative behaviours. Much of this description\ncan be found in~\\cite{lotka}, we give it here for the sake of clarity. \n\nBarring limit cases that we will not consider, there are essentially\nfour different phase portraits for the system, that are depicted in Figure~\\ref{fi:cases}. \nThese four regimes are obtained as follows. \n\nNotice first that the vector $F_{\\mathcal{E}}(x,y)$ is horizontal if $y=0$ or\n$cx+dy=1$: we call the line $cx+dy=1$ the horizontal isocline. Similarly\n$F_\\cE(x,y)$ is vertical if $x=0$ or if $(x,y)$ is on the vertical isocline\n$ax+by=1$. These isoclines are the bold straight lines in\nFigure~\\ref{fi:cases}. \n\nEach axis is invariant that why in the sequel we are only interested in initial \nconditions with positive coordinates. The three points $(0,0)$, $(0,1\/d)$ and \n$(1\/a,0)$ are stationary for $F_{\\mathcal{E}}$. The origin is always a source. \nThe nature of the other points and the existence of a fourth stationary point \ndepends on the parameters; this gives rise to the four types announced above. \n\n\\emph{Type 1.} If $ac$ and $b>d$, species $\\textbf{x}$ gets extinct. \n$(0,1\/d)$ is the unique sink and $(1\/a,0)$ is a saddle point. This is the same\nas Type 1 except that the two species $\\textbf{x}$ and $\\textbf{y}$ are\nswapped. \n\n\\emph{Type 3.} If $a>c$ and $b>d$, both species survive. The points\n$(1\/a,0)$ and $(0,1\/d)$ are saddle points. The isoclines meet at\nthe sink $(\\bar x,\\bar y) = (ad-bc)^{-1} (a-c,b-d)$, which \nis the unique global attractor. \n\n\\emph{Type 4.} If $a>c$ and $b] (m-2-1)-- node[auto] {$s\\mapsto a_s$} (m-1-2);\n \\draw[->] (m-2-3)-- node[auto,swap] {$u\\mapsto \\tilde{a}(u)$} (m-1-2);\n \\draw[->] (m-2-1)-- node[auto,swap] {$s\\mapsto p_s$} (m-3-2);\n \\draw[->] (m-2-3)-- node[auto] {$u\\mapsto \\tilde{p}(u)$} (m-3-2);\n \\draw[->] (m-1-2)-- node[auto] {$x\\mapsto 1\/x$} (m-3-2);\n\\end{tikzpicture}\n\\end{center}\n\nThis parameter $u$ is the one given in the introduction\nand\ncorresponds\nto a ratio of the $\\gamma$, when $s$ corresponds to a ratio of $\\lambda$, \nin the sense that:\n\\[ \n \\tilde{p}\\PAR{\\frac{\\gamma_0}{\\gamma_0 + \\gamma_1}} = p\\PAR{\\frac{\\lambda_0}{\\lambda_0+\n\\lambda_1}}.\n\\]\n\\begin{rem}\n As already mentioned above, the parameter $u$ and the interval $\\tilde{I}$\n are used implicitly in~\\cite{lotka}: $u$ appears in Remark 1, and the map $S$\n defined at the beginning of Section $4$ is given in our notation by $S(u) =\n p^{-1}(\\tilde{p}(u))$. \n\\end{rem}\n\n\n\nLet us study the integral $\\int_{p_1}^{p_0} P(x)\\theta(x)dx$. \nSet $y=\\tilde{p}^{-1}(x)$, so that:\n\\begin{align*}\n x&= \\tilde{p}(y) = \\frac{1}{\\tilde{a}(y)} = \\frac{1}{a_0+\\delta y}, &\n dx &= -\\delta \\tilde{p}(y)^2 dy \\\\\n p_0 - x &= \\delta p_0 y\\tilde{p}(y), &\n x - p_1 &= \\delta p_1 (1-y)\\tilde{p}(y).\n\\end{align*}\nChanging variables in the integral yields:\n\\begin{align*}\n \\int_{p_1}^{p_0} P(x) \\theta(x) dx \n &= \\int_0^1 P(\\tilde{p}(y))\\PAR{\\delta p_0 y \\tilde{p}(y)}^{\\gamma_0 - 1}\n \\PAR{\\delta p_1 (1-y) \\tilde{p}(y)}^{\\gamma_1 - 1} \n \\tilde{p}(y)^{-\\gamma_0 - \\gamma_1 - 1}\n \\delta \\tilde{p}(y)^2 dy \\\\\n &= \n \\delta^{\\gamma_0+\\gamma_1 - 1} p_0^{\\gamma_0-1}p_1^{\\gamma_1 - 1}\n \\int_0^1 P(\\tilde{p}(y)) \\frac{1}{\\tilde{p}(y)} y^{\\gamma_0 -1} (1-y)^{\\gamma_1 - 1} dy \\\\\n &= \n \\delta^{\\gamma_0+\\gamma_1 - 1} p_0^{\\gamma_0-1}p_1^{\\gamma_1 - 1}\n \\int_0^1 \\phi(y) y^{\\gamma_0 -1} (1-y)^{\\gamma_1 - 1} dy \\\\\n &= \n \\delta^{\\gamma_0+\\gamma_1 - 1} p_0^{\\gamma_0-1}p_1^{\\gamma_1 - 1}\n B(uv,(1-u)v) \\esp{\\phi(U_{u,v})}.\n\\end{align*}\nsince $\\phi(y) = \\frac{1}{\\tilde{p}(y)} P (\\tilde{p}(y))$. \nA similar computation gives the exact formula\n\\[ C^{-1} = \\PAR{ \\delta^{\\gamma_0+\\gamma_1 - 1 } p_0^{\\gamma_0 - 1} p_1^{\\gamma_1 - 1}\n B(uv,(1-u)v) \n}p_0p_1\\delta \\PAR{ \\frac{1}{\\alpha_0}(1- u) + \\frac{1}{\\alpha_1} u} \n\\]\nfor the normalization constant $C$. Therefore\n\\begin{align*}\n \\Lambda_\\textbf{y}(\\gamma_0,\\gamma_1) \n &= \\frac{1}{\\delta\\PAR{\\frac{1}{\\alpha_0} (1-u) + \\frac{1}{\\alpha_1} u}}\n \\esp{\\phi(U_{u,v})}.\n\\end{align*}\n\n\nLet us study $\\phi$ more precisely. Since $P$ is a second-degree polynomial, \nlet us write it down as $P(x)=A_2x^2 + A_1x + A_0$. Then \n\\[ \\phi(y) = \\frac{A_2}{a_0 + \\delta y} + A_1 + A_0(a_0 + \\delta y).\\]\nThe second derivative is \n\\[ \\phi''(y) = \\frac{2A_2\\delta^2}{(a_0 + \\delta y)^3},\\]\nwhich has the sign of $A_2$ on $[0,1]$, so $\\phi$ is either strictly\nconvex or strictly concave. However, the proof of the first item\nof Proposition 2.2 in~\\cite{lotka} shows that (still in the case $a_00$ such that for all $t\\in[0,1]$, \n \\[\n \\psi(X+tZ) = \\psi( (1-t)X + t(X+Z))\\leq (1-t)\\psi(X) + t\\psi(X+Z) - \\frac{mt(1-t)}{2}Z^2. \n \\]\n Taking expectations we get\n \\[\n \\esp{\\psi(X)} \\leq \\esp{\\psi(X+tZ)} \\leq (1-t)\\esp{\\psi(X)} + t\\esp{\\psi(Y)} \n - \\frac{mt(1-t)}{2} \\esp{Z^2},\n \\]\n where the first inequality comes from Jensen's inequality and $\\esp{Z|X}=0$. \n Since $\\esp{\\psi(Y)} = \\esp{\\psi(X)}$, $Z$ must be zero almost surely, \n so $X$ and $Y$ have the same distribution. \n\\end{proof}\n\n\n\n\\begin{thm}[Orderings between Beta r.v.]\n Let $X\\sim \\mathrm{Beta}(\\alpha,\\beta)$ and $X'\\sim\\mathrm{Beta}(\\alpha',\\beta')$. \n\n If $\\alpha< \\alpha'$, $\\beta<\\beta'$ and $\\alpha\/(\\alpha+\\beta) = \\alpha'\/(\\alpha'+\\beta')$, \n then $X' \\leq_{\\mathrm{cvx}} X$. \n\n \n\\end{thm}\n\\begin{proof}\n Call $f_{\\alpha,\\beta}$, $f_{\\alpha',\\beta'}$ the densities of the distributions. \n Compute their ratio:\n \\[ \\frac{f_{\\alpha',\\beta'}(x)}{f_{\\alpha,\\beta}(x)} = C_{\\alpha,\\beta,\\alpha',\\beta'} x^{\\alpha'-\\alpha} (1-\nx)^{\\beta'-\\beta}.\\]\n In the first case, \n this ratio starts and ends in zero, is strictly increasing on $[0,x_0]$ and\n strictly decreasing on $[x_0,1]$. Since the\n two functions are densities, the ratio must cross $1$ exactly twice, say in $x_1$, $x_2$. Therefore\n \\[\n d(x) = f_{\\alpha',\\beta'} - f_{\\alpha,\\beta}\n \\]\n is positive on $(x_1,x_2)$ and negative on $(0,x_1)$ and $(x_2,1)$. \n Therefore\n \\[ D(x) = F_{X'}(x) - F_{X}(x)\\]\n starts at zero, decreases on $[0,x_1]$, increases on $[x_1,x_2]$ and decreases on $[x_2,1]$, so\n $D(x)$ is negative on $[0,x_3]$ and positive on $[x_3,1]$ (since it ends at zero). \n Integrating once more, \n \\[ \\int_0^x D(t) dt\\]\n starts and ends at zero (since $\\esp{X} = \\esp{X'}$) and is decreasing-increasing, therefore it is non-\npositive. Thanks to Theorem~\\ref{thm:convex_order_cdf}, this implies $X'\\leq_{\\mathrm{cvx}} X$. \n\\end{proof}\n\n\\begin{proof}[Proof of the monotonicity in $v$]\n Suppose $v\\max(v_c(a), v_c(c))$. Since $\\tilde \\Lambda_\\textbf{y}(u,\\cdot)$ \nis increasing, $\\tilde \\Lambda_\\textbf{y}(a,M)$ and $\\tilde \\Lambda_\\textbf{y}(c,M)$ are positive. \nSince $u\\mapsto \\dE\\phi(U_{u,v})$ is concave, $\\tilde \\Lambda_\\textbf{y}(b,M)$ is positive. \nTherefore $v_c(c) \\leq M$. Sending $M$ to $\\max(v_c(a), v_c(b))$ yields the \nquasi-convexity of $v_c$. \n\nLet us now show the regularity properties. Let $u_n$ be an increasing \nsequence in $\\tilde{I}$, converging to some $u\\in(0,1)$. Since\n$v_c$ is quasi-convex, $v_n = v_c(u_n)$ is eventually monotone, \nso it converges to some $v\\in [0,\\infty]$. If $v$ is finite, since \nthe zero set of $\\Lambda_\\textbf{y}$ is closed, by continuity, $v = v_c(u)$, so \n$u$ must be in $\\tilde{I}$. Conversely, \nif $u\\in \\tilde{I}$, $v_c$ is bounded on\na neighborhood of $u$ by quasi-convexity, so $v$ is finite. This shows\nthat $v_c$ is continuous on $\\tilde{I}$ and converges to $\\infty$ \nat the endpoints. \n\nThe properties of the change of variables $(s,t)\\leftrightarrow(u,v)$ \nshow that $v_c$ is well-defined and continuous with the correct limits. \n\n\\section{Support of the invariant measure} \\label{se:support}\n\nNote that the stochastic Lotka-Volterra process has at least two invariant \nprobability measures, supported on the coordinate axes. In the persistence regime, \nwe are interested in the third invariant measure, whose support \n$\\Gamma\\times \\BRA{0,1}$ is such that $\\Gamma$ \nhas non empty interior. Several properties of $\\Gamma$ are \nestablished in~\\cite{lotka} (see below). In this section, we aim at providing \na full description of $\\Gamma$. Its shape essentially depends \non the fact that $I\\cap J$ is empty or not. \n\n\\subsection{Persistence with \"full support\"}\n\nIn this subsection, we assume that $I\\cap J$ is not empty. According to \nLemma~\\ref{lem:parameters}, the vector fields $F_{\\mathcal{E}_0}$ and $F_{\\mathcal{E}_1}$\nare such that \\eqref{eq:parameters} holds.\nLet us denote by $\\Sigma_i$ the intersection of $[0,\\infty)^2$ and the\nunstable manifold of $(0,1\/d_i)$ and $\\Gamma'$ the bounded subset of\n$[0,+\\infty)^2$ with border \n\\[\n\\Sigma_1\\cup\\BRA{(x,0)\\,:\\,1\/a_1\\leq x\\leq 1\/a_0}\n\\cup\\Sigma_0\\cup\\BRA{(0,y)\\,:\\,1\/d_1\\leq x\\leq 1\/d_0}.\n\\]\n\n\\begin{thm}\\label{thm:support}\nSuppose that $I\\cap J\\neq \\emptyset$. Then, for any $(s,t)\\in [0,1]\\times\n(0,\\infty)$ such that $\\Lambda_\\emph{\\textbf{x}}(s,t)>0$ and\n$\\Lambda_\\emph{\\textbf{y}}(s,t)>0$, then $\\Gamma'=\\Gamma$. \n\\end{thm}\n\n\\begin{figure}\n {\n \\centering\n \\input{proof_support.tex}\n \n \\par\n}\n\\small\nThe isoclines (straight lines) and unstable manifolds (curved lines)\nfor the three environments $\\cE_0$ (bottom left, in blue), $\\cE_s$ (middle,\nin purple) and $\\cE_1$ (upper right, in red). Note how the isoclines \nare \"swapped\" for $\\cE_s$, a Type 2 environment. \n\\caption{Full support case: isoclines and unstable manifolds}\n \\label{fig:IinterJ_non_vide}\n\\end{figure}\n\n\\begin{figure}\n {\\centering\n \n \\includegraphics[width=10cm]{support_away.png}\n \\par\n }\n\n \\small\n The outer curves are $\\Sigma_0$ and $\\Sigma_1$. The region \n between these curves is positively invariant. The\n inner curves are the two trajectories coming from the \n unique point $z\\in T$: they form the boundary of the support. \n The sample trajectory shows that the invariant measure is \n in practice often concentrated on a smaller subset. \n \\caption{Support away from the $y$ axis}\n \\label{fig:away}\n\\end{figure}\n\n\\begin{proof}\nFirstly, notice that the set $\\Gamma'$ is positively invariant for each flow \nsince both vector fields $F_{\\mathcal{E}_0}$ and $F_{\\mathcal{E}_1}$ \npoint inside $\\Gamma'$. \n\nPick an $s\\in I\\cap J$. The isoclines and the unstable manifold of the \nsaddle point for the three environments $\\cE_0$, \n$\\cE_1$ and $\\cE_s$ are necessarily in the position depicted in\nFigure~\\ref{fig:IinterJ_non_vide}. Denote by $\\Sigma_s$ the \nintersection of the unstable manifold of $(1\/a_s,0)$ with the\nupper right quadrant. \n\nFirst step: the set $\\Sigma_s$ is contained in the support. \nIndeed, pick a point $(x,y)$ in the interior of the support (such\na point exists by \\cite[Remark 6]{lotka}). The loop formed \nby the trajectories starting from $(x,y)$ with both flows (converging\nto $A_1:(1\/a_1,0)$ and $A_0:(1\/a_0,0)$ and the line segment $[A_0,A_1]$\nis included in the support (by positive invariance). As a consequence, \nthe support must contain a closed half ball centered on $A_s$ --- \nlet us call it $\\cB$. Now pick a point $(x,y)\\in \\Sigma_s$: by definition its $\\cE_s$ \nflow converges for $t\\to-\\infty$ to $1\/a_s$. By continuity there exists a \npoint in the past of $(x,y)$ which is in $\\cB$. Running the time forward again, \nthe point $(x,y)$ must be in the support. \n\nSecond step: any point (strictly) between $\\Sigma_1$ and $\\Sigma_s$ is in $\\Gamma$. \nStarting from such a point $(x,y)$, run the $\\cE_1$ flow in reverse time. \nThe trajectory must cross $\\Sigma_s$. So $(x,y)$ is in the \nfuture of a point in $\\Sigma_s\\subset\\Gamma$, and $(x,y)\\in\\Gamma$ by positive invariance. \n\nThird step: any point between $\\Sigma_0$ and $\\Sigma_s$ is in $\\Gamma$. \nThis step is similar to the previous one and is omitted. \n\nSimilarly, any point between $\\Sigma_1$ and $\\Sigma_s$ is in $\\Gamma$.\n\\end{proof}\n\n\\subsection{Support away from the $y$ axis}\n\nWe suppose in the sequel that $I\\cap J$ is empty. Let us introduce the \nset where the two vector fields $F_0$ and $F_1$ are collinear:\n\\[\nC=\\BRA{z\\in\\dR_+^2\\, : \\, \\det(F_0(z),F_1(z))=0}.\n\\]\nThis set is the union of $\\BRA{(0,y)\\,:\\, y\\geq 0}$, $\\BRA{(x,0)\\,:\\, x\\geq 0}$, and \n\\[\n\\tilde C=\\BRA{(x,y)\\in\\dR_+^2\\, : \\,G(x,y)=0}\n\\]\nwhere $G$ is a polynomial of degree 2. As a consequence, the set~$\\tilde C$ is \na subset of a conic. It is easy to see that $\\tilde{C}$ is also the \nset of non-degenerate equilibrium points for the vector field $F_{\\cE_s}$, as\n$s$ varies from $0$ to $1$. When $s\\in I$, $\\cE_s$ is of Type~$3$ so the\nequilibrium point is stable and globally attractive. Therefore the \npart of $\\tilde{C}$ that corresponds to $s\\in I$ must be included in $\\Gamma$, as\nwell as all trajectories (for both flows) starting from it. \n\nNumerical experiments suggest that there is a unique \"extremal point\" \non this part of $\\tilde{C}$, such that the trajectories starting from \nthis point form the boundary of $\\Gamma$. See Figure~\\ref{fig:away}. \n\nTo describe it more precisely, consider the subset of $\\tilde C$ made of the points \nwhere $F_0$ (or $F_1$) is tangent to the curve $\\tilde C$. This set is given by \n\\[\nT=\\BRA{(x,y)\\in \\dR_+^2\\,:\\, G(x,y)=0\\text{ and } (F_0\\cdot \\nabla G)(x,y)=0}.\n\\]\nSince $G$ and $F_0\\cdot\\nabla G$ are polynomials with respective degrees 2 and 3, $T$ \nis made of at most six points according to Bezout's Theorem. \n\n\nFor any $z\\in T$ let us define $C(z)$ the bounded region enclosed\nby the Jordan curve\n\\[\n\\BRA{\\varphi^{0}_z(t)\\, :\\, t\\in [0,\\infty)}\\cup \\BRA{\\varphi^{1}_z(t)\\, :\\, t\\in [0,\\infty)} \n\\cup [1\/a_1,1\/a_0]\\times \\BRA{0}, \n\\]\nwhere $t\\mapsto \\varphi^{i}_z(t)$ is the flow associated to the vector field $F_i$ for $i=0,1$. \n\n\\begin{conj}\nThe set $T$ is a singleton $\\BRA{z_0}$ and the support of the invariant measure \nwhich is not supported by one of the two axes is $C(z_0)\\times\\BRA{0,1}$. \n\\end{conj}\n\n\n\n\\paragraph*{Acknowledgements.}\nWe thank an anonymous referee for constructive remarks.\nWe acknowledge financial support from the French ANR project ANR-12-JS01-0006-PIECE.\nNumerical computations were done in Julia and graphics in TikZ. \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}