the accuracy of the measurement of the higgs - boson mass by the atlas and cms collaborations at the large hadron collider ( lhc ) has already reached the level of 300400  mev [ 1 , 2 ] and , being still dominated by statistics , is bound to improve further when the lhc restarts operations in 2015 . this puts new emphasis on the need for high - precision calculations in those extensions of the standard model ( sm ) , such as the minimal supersymmetric standard model ( mssm ) , in which the higgs - boson mass can be predicted as a function of other physical observables . the higgs sector of the mssm consists of two \documentclass[12pt]{minimal } 
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				 \begin{document}$$h_2$$\end{document}h2 , whose relative contribution to electroweak ( ew ) symmetry breaking is determined by the ratio of vacuum expectation values ( vevs ) of their neutral components , \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta \equiv v_2/v_1$$\end{document}tanv2/v1 . the spectrum of physical higgs bosons is richer than in the sm , consisting of two neutral scalars , \documentclass[12pt]{minimal } 
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				 \begin{document}$$h$$\end{document}h , one neutral pseudoscalar , \documentclass[12pt]{minimal } 
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				 \begin{document}$$h^\pm $ $ \end{document}h. at the tree level , the neutral scalar masses \documentclass[12pt]{minimal } 
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				 \begin{document}$$\alpha $ $ \end{document} can be computed in terms of the \documentclass[12pt]{minimal } 
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				 \begin{document}$$z$$\end{document}z - boson mass \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}$$\end{document}ma and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan , and the bound \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{h } } < |\cos 2\beta |\,{{m}_{{\scriptscriptstyle z}}}$$\end{document}mh<|cos2|mz applies . in a significant portion of the parameter space the lightest scalar \documentclass[12pt]{minimal } 
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				 \begin{document}$$h$$\end{document}h has sm - like couplings to fermions and gauge bosons , in which case the tree - level bound on \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{h}}$$\end{document}mh has long been disproved by the lep  [ 3 , 4 ] . however , radiative corrections can raise the prediction for the lightest - scalar mass up to the value \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{h}}\approx 125$$\end{document}mh125  gev observed at the lhc , and they bring along a dependence on all mssm parameters . among the latter , particularly relevant are the masses and mixing of the scalar partners of the third - generation quarks , the stop and sbottom squarks . due to the crucial role of radiative corrections in pushing the prediction for the lightest - scalar mass above the tree - level bound , an impressive theoretical effort has been devoted over more than 20 years to the precise determination of the higgs sector of the mssm.1 after the early realization  [ 59 ] of the importance of the one - loop \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t})$$\end{document}o(t ) corrections2 involving top and stop , full one - loop computations of the mssm higgs masses have been provided  [ 1013 ] , leading logarithmic effects at two loops have been included via renormalization - group methods  [ 1417 ] , and genuine two - loop corrections of \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts )  [ 1825 ] , \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}^2)$$\end{document}o(t2 )  [ 18 , 24 , 26 ] , \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { b}\alpha _ { s})$$\end{document}o(bs ) [ 27 , 28 ] and \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { b}+ \alpha _ { b}^2)$$\end{document}o(tb+b2 )   have been evaluated in the limit of vanishing external momentum in the higgs self - energies . all of these corrections have been implemented in widely used computer codes for the calculation of the mssm mass spectrum , such as feynhiggs  , softsusy  [ 31 , 32 ] , suspect   and spheno  [ 34 , 35 ] . furthermore , a complete two - loop calculation of the mssm higgs masses in the effective potential approach ( i.e. , at zero external momentum ) , including also two - loop corrections controlled by the ew gauge couplings , has been presented in refs . [ 36 , 37 ] . some of the dominant three - loop corrections to \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{h}}$$\end{document}mh have also been obtained , both via renormalization - group methods  [ 3840 ] and by explicit calculation of the higgs self - energy at zero external momentum  [ 41 , 42 ] . already at the two - loop level , going beyond the approximation of zero external momentum brings significant complications to the calculation of the higgs self - energies . different algorithms for expressing all two - loop self - energy integrals with arbitrary external momentum in terms of a minimal set of master integrals ( mis ) were developed in refs . however , explicit analytical formulas for the mis can be derived only for special values of the masses of the particles circulating in the loops , whereas in the general case a numerical calculation becomes unavoidable . a method to compute all the mis of ref .   by numerically solving a system of differential equations in the external momentum [ 4851 ] . a library of routines for the computation of the mis with the method of ref . a calculation of the two - loop contributions to the higgs self - energies involving the strong gauge coupling or the third - family yukawa couplings , based on the methods of refs . that calculation goes beyond the two - loop results implemented in public codes  [ 1921 , 2529 ] in that it includes external - momentum effects , as well as contributions involving the d - term - induced ew interactions between higgs bosons and sfermions . when combined with the effective - potential results of refs . provide an almost - complete two - loop calculation of the higgs masses in the mssm  the only missing contributions being external - momentum effects that involve only the ew gauge couplings . however , no code for the calculation of the mssm mass spectrum implementing the results of refs . [ 36 , 37 , 53 ] was ever made available , and the way those results are organized does not lend itself to a straightforward implementation in the existing public codes . on one hand , the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr renormalization scheme adopted in refs . [ 36 , 37 , 53 ] for the parameters of the mssm lagrangian does not match the mixed on - shell ( os)\documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr  scheme adopted in feynhiggs . on the other hand , implementation of the results of refs . [ 36 , 37 , 53 ] in softsusy , suspect and spheno , which also adopt the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme , is complicated by the fact that in refs .  [ 36 , 37 , 53 ] the running masses of the higgs bosons entering the loop corrections are defined by the second derivatives of the tree - level potential . while this choice amounts to a legitimate reshuffling of terms between different perturbative orders , it restricts the applicability of the calculation to rather specific ranges of renormalization scale where none of the running higgs masses  as defined in refs . [ 36 , 37 , 53 ]  is tachyonic . perhaps as a consequence of these complications , a full decade after the publication of ref . its results have yet to be included in phenomenological analyses of the mssm higgs sector . in this paper we present a new calculation of the momentum - dependent part of the two - loop corrections to the neutral higgs masses of \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) , i.e.  those involving both the top yukawa coupling and the strong gauge coupling . we also compute  mixed  two - loop corrections that we denote by \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) : they involve both the strong gauge coupling and the ew gauge couplings , under the approximation that the only non - vanishing yukawa coupling is the top one . it is natural to consider these two classes of corrections together , because in both of them the dominant terms affecting the lightest - scalar mass are expected to be suppressed by a factor of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{o}(m_z^2/{{m}_{t}}^2)$$\end{document}o(mz2/mt2 ) with respect to the zero - momentum \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections ( in practice , we find that both classes of corrections are considerably more suppressed than that , but still comparable to each other in size ) . in our calculation we rely on the integration - by - parts ( ibp ) technique of refs . [ 54 , 55 ] to express the momentum - dependent loop integrals in terms of the mis of ref .  , which we evaluate by means of the package tsil . we obtain results for both the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr renormalization schemes , organized in such a way that they can be directly implemented in the existing codes for the computation of the mssm mass spectrum . we verify that our results are in full agreement with the ones of ref .   where they overlap . after describing our calculation in some detail , we briefly discuss the numerical impact of the momentum - dependent \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections to the higgs masses in a set of representative points in the mssm parameter space . while our paper was in preparation , an independent calculation of the momentum - dependent \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections to the neutral higgs masses in the mssm appeared  , relying on the results of ref .   for the decomposition of two - loop integrals into mis and on the package secdec  [ 57 , 58 ] for the numerical evaluation of the latter . the results of that calculation are expressed in the os\documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme , and they have been implemented in the latest version of feynhiggs . although we have verified that our results for the contributions of genuine two - loop diagrams involving the strong - gauge and top - yukawa couplings agree numerically with those of ref . , we do not reproduce the overall values of the momentum - dependent \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections to the higgs masses . we trace the reason for the discrepancy to an inconsistency in ref .   concerning the definitions of the wave - function - renormalization ( wfr ) constants for the higgs fields and of the parameter \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan. we outline here our calculation of the two - loop corrections to the masses of the neutral higgs bosons in the mssm with real parameters ( we do not consider the possibility of cp violation in the higgs sector ) . we decompose the neutral components of the two higgs doublets into their vevs plus their cp - even and cp - odd fluctuations as follows:1\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned}&h_1 ^ 0 \equiv \frac{1}{\sqrt{2}}\,(v_1 + s_1 + i \ , p_1 ) , \nonumber \\&h_2 ^ 0 \equiv \frac{1}{\sqrt{2}}\,(v_2 + s_2 + i \ , p_2 ) . \end{aligned}$$\end{document}h1012(v1+s1+ip1),h2012(v2+s2+ip2).the cp - odd components \documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \mathrm{det } \left [ \gamma _ s(p^2 ) \right ] = 0 , \end{aligned}$$\end{document}dets(p2)=0,where \documentclass[12pt]{minimal } 
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				 \begin{document}$$\gamma _ s(p^2)$$\end{document}s(p2 ) denotes the \documentclass[12pt]{minimal } 
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				 \begin{document}$$(s_1,s_2)$$\end{document}(s1,s2 ) basis , \documentclass[12pt]{minimal } 
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				 \begin{document}$$p$$\end{document}p being the external momentum flowing into the scalar self - energies . we can decompose \documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \gamma _ s(p^2 ) = p^2 - \mathcal{m}_0 ^ 2 - \delta \mathcal{m}^2(p^2 ) , \end{aligned}$$\end{document}s(p2)=p2-m02-m2(p2),where \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{m}_0 ^ 2$$\end{document}m02 denotes the tree - level mass matrix written in terms of renormalized parameters , and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \mathcal{m}^2(p^2)$$\end{document}m2(p2 ) collectively denotes the radiative corrections . at each order in the perturbative expansion , the latter include both the contributions of one - particle - irreducible ( 1pi ) diagrams and non-1pi counterterm contributions arising from the renormalization of parameters that enter the lower - order parts of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\gamma _ s(p^2)$$\end{document}s(p2 ) . we express \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{m}_0 ^ 2$$\end{document}m02 in terms of the parameter \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan renormalized in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme , and of the physical masses of the pseudoscalar and of the \documentclass[12pt]{minimal } 
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				 \begin{document}$$z$$\end{document}z boson4\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \mathcal{m}_0 ^ 2 = \left ( \begin{array}{cc } c^2_\beta \,m_z^2 + s^2_\beta \,m_a^2 & { } \quad - { { s}_{\beta } } \ , { { c}_{\beta } } \left ( m_z^2 + m_a^2 \right ) \\ - { { s}_{\beta } } \ , { { c}_{\beta } } \left ( m_z^2 + m_a^2\right ) & { } \quad s^2_\beta \ , m_z^2 + c^2_\beta \ , m_a^2 \end{array}\right ) , \end{aligned}$$\end{document}m02=c2mz2+s2ma2-scmz2+ma2-scmz2+ma2s2mz2+c2ma2,using ( here and thereafter ) the shortcuts \documentclass[12pt]{minimal } 
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				 \begin{document}$$c_\phi \equiv \cos \phi $ $ \end{document}ccos and \documentclass[12pt]{minimal } 
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				 \begin{document}$$s_\phi \equiv \sin \phi $ $ \end{document}ssin for a generic angle \documentclass[12pt]{minimal } 
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				 \begin{document}$$\phi $ $ \end{document}. neglecting terms that do not contribute at \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) or \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) , our choices for the parameters entering \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{m}_0 ^ 2$$\end{document}m02 lead to the following expressions for the two - loop parts of the individual entries of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \mathcal{m}^2(p^2)$$\end{document}m2(p2):5\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \left [ \delta \mathcal{m}^2(p^2)\right ] ^{(2)}_{11}&= s_\beta ^2\ , \mathrm{re}\ , \pi _ { { \scriptscriptstyle a}{\scriptscriptstyle a}}^{(2)}(m_a^2 ) + c_\beta ^2\,\mathrm{re}\ , \pi _ { { \scriptscriptstyle z}{\scriptscriptstyle z}}^{(2)}(m_z^2)\nonumber \\&-\pi _ { 11}^{(2)}(p^2 ) -\delta \mathcal { z}_1^{(2 ) } \left ( p^2- c^2_\beta \ , m_z^2 - s^2_\beta \,m_a^2\right ) \nonumber \\&+ \left ( 1- { { s}_{\beta } } ^4\right ) \frac{~\,t_1^{(2)}}{v_1 } - s_\beta ^2 c_\beta ^2 \frac{~\,t_2^{(2)}}{v_2}\nonumber \\&-2\,s_\beta ^2 c_\beta ^2\ , \left ( m_z^2-m_a^2\right ) \frac{~\delta \tan \beta ^{(2)}}{\tan \beta } , \end{aligned}$$\end{document}m2(p2)11(2)=s2reaa(2)(ma2)+c2rezz(2)(mz2)-11(2)(p2)-z1(2)p2-c2mz2-s2ma2 + 1-s4t1(2)v1-s2c2t2(2)v2 - 2s2c2mz2-ma2tan(2)tan,6\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \left [ \delta \mathcal{m}^2(p^2)\right ] ^{(2)}_{12}&= - { { s}_{\beta } } { { c}_{\beta } } \left [ \mathrm{re}\ , \pi _ { { \scriptscriptstyle a}{\scriptscriptstyle a}}^{(2)}(m_a^2 ) + \mathrm{re}\ , \pi _ { { \scriptscriptstyle z}{\scriptscriptstyle z}}^{(2)}(m_z^2)\right . s_\beta ^2 \frac{~\,t_1^{(2)}}{v_1 } - c_\beta ^2 \frac{~\,t_2^{(2)}}{v_2 } \right ] -\pi _ { 12}^{(2)}(p^2)\nonumber \\&-\frac{1}{2}{{s}_{\beta } } { { c}_{\beta } } \left ( m_z^2 + m_a^2\right ) \nonumber \\&\times \left [ 2\ , ( c_\beta ^2-s_\beta ^2)\frac{~\delta \tan \beta ^{(2)}}{\tan \beta } \!+\ ! \delta \mathcal { z}_1^{(2)}\!+\!\delta \mathcal { z}_2^{(2)}\!\right ] , \nonumber \\ \end{aligned}$$\end{document}m2(p2)12(2)=-screaa(2)(ma2)+rezz(2)(mz2)-s2t1(2)v1-c2t2(2)v2-12(2)(p2)-12scmz2+ma22(c2-s2)tan(2)tan+z1(2)+z2(2),7\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \left [ \delta \mathcal{m}^2(p^2)\right ] ^{(2)}_{22}&= c_\beta ^2\ , \mathrm{re}\ , \pi _ { { \scriptscriptstyle a}{\scriptscriptstyle a}}^{(2)}(m_a^2 ) + s_\beta ^2 \ , \mathrm{re}\ , \pi _ { { \scriptscriptstyle z}{\scriptscriptstyle z}}^{(2)}(m_z^2)\nonumber \\&-\pi _ { 22}^{(2)}(p^2 ) -\delta \mathcal { z}_2^{(2 ) } \left ( p^2- s^2_\beta \,m_z^2 - c^2_\beta \ , s_\beta ^2 c_\beta ^2 \frac{~\,t_1^{(2)}}{v_1 } + \left ( 1- { { c}_{\beta } } ^4\right ) \frac{~\,t_2^{(2)}}{v_2}\nonumber \\&+2\,s_\beta ^2c_\beta ^2\ , \left ( m_z^2-m_a^2\right ) \frac{~\delta \tan \beta ^{(2)}}{\tan \beta } . \end{aligned}$$\end{document}m2(p2)22(2)=c2reaa(2)(ma2)+s2rezz(2)(mz2)-22(2)(p2)-z2(2)p2-s2mz2-c2ma2-s2c2t1(2)v1 + 1-c4t2(2)v2 + 2s2c2mz2-ma2tan(2)tan.in the equations above , \documentclass[12pt]{minimal } 
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				 \begin{document}$$t^{(2)}_i$$\end{document}ti(2 ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$$i , j=1,2$$\end{document}i , j=1,2 ) denote the two - loop parts of tadpoles and self - energies , respectively , for the scalars \documentclass[12pt]{minimal } 
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				 \begin{document}$$s_i$$\end{document}si , while \documentclass[12pt]{minimal } 
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				 \begin{document}$$\pi ^{(2)}_{{\scriptscriptstyle z}{\scriptscriptstyle z}}$$\end{document}zz(2 ) denote the two - loop parts of the pseudoscalar and \documentclass[12pt]{minimal } 
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				 \begin{document}$$z$$\end{document}z - boson self - energies . in addition , \documentclass[12pt]{minimal } 
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				 \begin{document}$$i=1,2$$\end{document}i=1,2 ) in eqs . ( 5)(7 ) denote the two - loop parts of the wfr counterterms for the higgs fields \documentclass[12pt]{minimal } 
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				 \begin{document}$$h^0_i$$\end{document}hi0 , which we renormalize as follows:8\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } h^0_i ~\longrightarrow ~\sqrt{\mathcal{z}_i}\,h^0_i ~\simeq ~ \left ( 1 + \frac{1}{2}\,\delta \mathcal{z}^{(1)}_i + \frac{1}{2}\,\delta \mathcal{z}^{(2)}_i \right ) \,h^0_i , \end{aligned}$$\end{document}hi0zihi01 + 12zi(1)+12zi(2)hi0,where in the expansion of the square root we have again neglected terms that do not contribute at \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) or \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) . we adopt a \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr definition for the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{z}_i$$\end{document}zi , which can then be determined from the anomalous dimensions of the higgs fields and from the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\beta $ $ \end{document} functions of the couplings entering the anomalous dimensions . taking from the general formulas of refs . [ 59 , 60 ] only the terms relevant to our approximation , we get9\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \delta \mathcal{z}^{(1)}_1&= \delta \mathcal{z}^{(2)}_1= 0,\quad \delta \mathcal{z}^{(1)}_2 = - \frac{\alpha _ { t}}{4\pi } \,n_c\,\cdot \,\frac{1}{\epsilon } , \nonumber \\ \delta \mathcal{z}^{(2)}_2&= \frac{\alpha _ { t}\alpha _ { s}}{(4\pi ) ^2}\,2\,n_c\,c_f\cdot \left ( \frac{1}{\epsilon ^2}-\frac{1}{\epsilon } \right ) , \end{aligned}$$\end{document}z1(1)=z1(2)=0,z2(1)=-t4nc1,z2(2)=ts(4)22nccf12 - 1,where \documentclass[12pt]{minimal } 
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				 \begin{document}$$c_f=4/3$$\end{document}cf=4/3 are color factors , \documentclass[12pt]{minimal } 
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				 \begin{document}$$\epsilon = ( 4-d)/2\,$$\end{document}=(4-d)/2 in dimensional reduction , and the coupling \documentclass[12pt]{minimal } 
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				 \begin{document}$$\alpha _ { t}$$\end{document}t entering the one - loop counterterm \documentclass[12pt]{minimal } 
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				 \setlength{\oddsidemargin}{-69pt } \begin{document}$$\delta \mathcal{z}^{(1)}_2$$\end{document}z2(1 ) is in turn renormalized in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \tan \beta ^{(2)}$$\end{document}tan(2 ) in eqs . ( 5)(7 ) denotes the two - loop part of the counterterm for the parameter \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan. the choice of a \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan implies that , in our approximation , its counterterm can be expressed via the wfr counterterms:10\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \frac{~\delta \tan \beta ^{(\ell ) } } { \tan \beta } = \frac{1}{2}\,\left ( \delta \mathcal{z}^{(\ell ) } _ 2 -\delta \mathcal{z}^{(\ell ) } _ ( 5)(7 ) include both 1pi two - loop contributions and non-1pi contributions arising from the renormalization of the parameters entering their one - loop counterparts . since we are focusing on the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections to the higgs masses , we need to introduce counterterms only for the parameters that are subject to \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { s})$$\end{document}o(s ) corrections , namely the top mass \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_t$$\end{document}at and the masses \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_{\tilde{q}_i}$$\end{document}mq~i of all squarks other than the stops . the latter enter the one - loop tadpoles and self - energies of the higgs bosons via d - term - induced ew couplings , and the one - loop self - energy of the \documentclass[12pt]{minimal } 
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				 \begin{document}$$z$$\end{document}z boson via the gauge interaction . in our calculation we neglect all yukawa couplings ( and hence quark masses ) other than the top one,3 therefore none of the other squarks mix . we obtained results for the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) contributions to tadpoles and self - energies assuming that the relevant quark / squark parameters are renormalized either in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr or in the os scheme . formulas for the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dros shift of the parameters in the top / stop sector can be found , e.g. , in appendix b of ref .  , while the shifts for the remaining squark masses can be obtained by setting \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{t}}=\theta _ t=0$$\end{document}mt=t=0 in the corresponding formulas for the stop masses . ( 5)(7 ) are constructed to give finite entries in the inverse - propagator matrix of the scalars . the \documentclass[12pt]{minimal } 
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				 \begin{document}$$1/\epsilon $ $ \end{document}1/ poles in the right - hand sides of eqs . ( 5)(7 ) cancel out . in principle , the two - loop contributions to the higgs inverse propagator given in eqs . ( 5)(7 ) must be combined with a full calculation of the corresponding one - loop contributions , and used to determine the physical higgs masses by solving directly eq . however , as will be discussed in the next section , the computing times required for the evaluation of momentum - dependent two - loop integrals are not negligible . a numerical search for the solutions of eq . ( 2 ) could significantly slow down the codes for the calculation of the higgs masses , making them unsuitable for extensive phenomenological analyses of the mssm parameter space . it is therefore convenient to compute the higgs masses in two steps , with a procedure similar to the one discussed in refs . ( 2 ) including in \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \mathcal{m}^2(p^2)$$\end{document}m2(p2 ) the full one - loop corrections plus the dominant two - loop corrections of \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) , \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { b}\alpha _ { s})$$\end{document}o(bs ) , and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{o}(\alpha _ { t}^2+\alpha _ { t}\alpha _ { b}+\alpha _ { b}^2)$$\end{document}o(t2+tb+b2 ) computed in the approximation of vanishing external momentum . from feynhiggs we obtain the scalar masses \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{m}_h^2$$\end{document}mh2 and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{m}_{{\scriptscriptstyle h}}^2$$\end{document}mh2 , and an effective mixing angle \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\alpha } $ $ \end{document} which diagonalizes the loop - corrected scalar mass matrix at vanishing external momentum . our full results for the scalar masses are then obtained by adding to the results of feynhiggs the momentum - dependent parts of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections and the whole \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections:11\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } m^2_{h,{\scriptscriptstyle h } } = \overline{m}^2_{h,{\scriptscriptstyle h } } ~+~ ( \delta m^2_{h,{\scriptscriptstyle h}})^{\alpha _ { t}\alpha _ { s},\,p^2 } ~+~ ( \delta m^2_{h,{\scriptscriptstyle h}})^{\alpha \alpha _ { s}}. \end{aligned}$$\end{document}mh , h2=mh , h2+(mh , h2)ts , p2+(mh , h2)s.concerning the former , we have12\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \left ( \delta m^2_{h}\right ) ^{\alpha _ { t}\alpha _ { s},\,p^2}&= c^2_{\beta -\overline{\alpha } } \,\delta \pi _ { { \scriptscriptstyle a}{\scriptscriptstyle a}}^{(2)}\left ( m_a^2\right ) - s^2_{\overline{\alpha } } \ , \delta \pi _ { 11}^{(2)}\left ( \overline{m}_h^2\right ) \nonumber \\&\quad + s_{2\overline{\alpha } } \,\delta \pi _ { 12}^{(2)}\left ( \overline{m}_h^2\right ) - c^2_{\overline{\alpha } } \,\delta \pi _ { 22}^{(2)}\left ( \overline{m}_h^2\right ) , \end{aligned}$$\end{document}mh2ts , p2=c-2aa(2)ma2-s211(2)mh2+s212(2)mh2-c222(2)mh2,13\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \left ( \delta m^2_{{\scriptscriptstyle h}}\right ) ^{\alpha _ { t}\alpha _ { s},\,p^2}&= s^2_{\beta -\overline{\alpha } } \,\delta \pi _ { { \scriptscriptstyle a}{\scriptscriptstyle a}}^{(2)}\left ( m_a^2\right ) - c^2_{\overline{\alpha } } \ , \delta \pi _ { 11}^{(2)}\left ( \overline{m}_{{\scriptscriptstyle h}}^2\right ) \nonumber \\&\quad - s_{2\overline{\alpha } } \,\delta \pi _ { 12}^{(2)}\left ( \overline{m}_{{\scriptscriptstyle h}}^2\right ) - s^2_{\overline{\alpha } } \,\delta \pi _ { 22}^{(2)}\left ( \overline{m}_{{\scriptscriptstyle h}}^2\right ) , \end{aligned}$$\end{document}mh2ts , p2=s-2aa(2)ma2-c211(2)mh2-s212(2)mh2-s222(2)mh2,where we define \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \pi ( p^2 ) \,\equiv \ , \pi ( p^2)-\pi ( 0)$$\end{document}(p2)(p2)-(0 ) , and retain only the real and finite part of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) contributions to the two - loop self - energies . for what concerns the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) ( 5)(7):14\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \left ( \delta m^2_{h}\right ) ^{\alpha \alpha _ { s}}&= s^2_{\overline{\alpha } } \,\left [ \delta \mathcal{m}^2\left ( \overline{m}_h^2\right ) \right ] ^{\alpha \alpha _ { s}}_{11}\nonumber \\&\quad - s_{2\overline{\alpha } } \,\left [ \delta \mathcal{m}^2\left ( \overline{m}_h^2\right ) \right ] ^{\alpha \alpha _ { s}}_{12 } + c^2_{\overline{\alpha } } \,\left [ \delta \mathcal{m}^2\left ( \overline{m}_h^2\right ) \right ] ^{\alpha \alpha _ { s}}_{22},\end{aligned}$$\end{document}mh2s = s2m2mh211s - s2m2mh212s+c2m2mh222s,15\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \left ( \delta m^2_{{\scriptscriptstyle h}}\right ) ^{\alpha \alpha _ { s}}&= c^2_{\overline{\alpha } } \,\left [ \delta \mathcal{m}^2\left ( \overline{m}_{{\scriptscriptstyle h}}^2\right ) \right ] ^{\alpha \alpha _ { s}}_{11}\nonumber \\&\quad + \ , s_{2\overline{\alpha } } \,\left [ \delta \mathcal{m}^2\left ( \overline{m}_{{\scriptscriptstyle h}}^2\right ) \right ] ^{\alpha \alpha _ { s}}_{12 } + s^2_{\overline{\alpha } } \,\left [ \delta \mathcal{m}^2\left ( \overline{m}_{{\scriptscriptstyle h}}^2\right ) \right ] ^{\alpha \alpha _ { s}}_{22 } , \end{aligned}$$\end{document}mh2s = c2m2mh211s+s2m2mh212s+s2m2mh222s , where again we take the real part of all two - loop self - energies . .   proposes an alternative two - step procedure to include the momentum - dependent parts of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections in feynhiggs , differing from the one outlined above only by higher - order effects . finally , a comment is in order about the dependence of the corrections to the higgs masses on the wfr constants.4 in principle , the predictions for the physical higgs masses at a given order in the perturbative expansion should not depend directly on the wfr constants ( although they could still depend indirectly on them via the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan counterterm ) . indeed , if the two - loop contributions to the inverse - propagator matrix are computed with \documentclass[12pt]{minimal } 
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				 \begin{document}$$p^2$$\end{document}p2 equal to the tree - level scalar masses and then rotated to the mass - eigenstate basis via the tree - level mixing angle , so that the computation is performed strictly at the two - loop level , the terms in eqs . ( 5)(7 ) that depend explicitly on \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \mathcal{z}^{(2)}_i$$\end{document}zi(2 ) drop out of the mass corrections \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta m^2_{h,{\scriptscriptstyle h}}\,$$\end{document}mh , h2 . on the other hand , if the loop - corrected scalar masses \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{m}_h^2$$\end{document}mh2 and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{m}_{{\scriptscriptstyle h}}^2$$\end{document}mh2 and the effective mixing angle \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\alpha } $ $ \end{document} are used , as in eqs . ( 12)(15 ) above , or if the zeroes of the inverse - propagator matrix are determined numerically , the corrections to the scalar masses retain a dependence on the wfr counterterms . in our calculation we adopt a \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr definition for the wfr ; therefore the terms involving \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \mathcal{z}^{(2)}_i$$\end{document}zi(2 ) are purely divergent and cancel out against other divergent terms in the individual entries of the inverse - propagator matrix , hence they do not show up in eqs . if , however , one adopts a non - minimal definition of the wfr , higgs - mass corrections computed as in eqs . ( 14 ) and ( 15 ) will contain non - vanishing terms that depend on the finite part of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \mathcal{z}^{(2)}_i$$\end{document}zi(2 ) . albeit formally of higher order in the perturbative expansion , these terms can be numerically relevant when the loop - corrected scalar masses differ significantly from their tree - level values ( as is the case for a sm - like scalar \documentclass[12pt]{minimal } 
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				 \begin{document}$$h$$\end{document}h with mass around 125  gev ) . the computation of the two - loop corrections to the neutral mssm higgs masses considered in this paper requires the knowledge of the tadpole and self - energy diagrams entering eqs . while the strategy for the computation in the zero - momentum approximation is well known , the evaluation of the self - energies with arbitrary external momentum is more involved . we illustrate in this section the details of our calculation , which we performed in a fully automated way . the relevant diagrams are generated with feynarts , using a modified version of the original mssm model file   that implements the qcd interactions in the background field gauge . the diagrams contributing to the vacuum polarization of the \documentclass[12pt]{minimal } 
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				 \begin{document}$$z$$\end{document}z boson are contracted with a suitable projector in order to single out their transverse part . the color factors are simplified with a private package and the dirac algebra is handled by form  . the computation is performed in dimensional reduction , which we can implement in this case by generating the diagrams in dimensional regularization and replacing , in each diagram involving an internal \documentclass[12pt]{minimal } 
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				 \begin{document}$$d$$\end{document}d - dimensional gluon , \documentclass[12pt]{minimal } 
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				 \begin{document}$$g^{\mu \nu } \rightarrow g^{\mu \nu } + g^{\hat{\mu } \hat{\nu } } $ $ \end{document}gg+g^^ ( where \documentclass[12pt]{minimal } 
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				 \begin{document}$$g^{\hat{\mu } \hat{\nu } } $ $ \end{document}g^^ is the \documentclass[12pt]{minimal } 
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				 \begin{document}$$2\epsilon $ $ \end{document}2-dimensional metric tensor ) in order to include the corresponding \documentclass[12pt]{minimal } 
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				 \begin{document}$$\epsilon $ $ \end{document}-scalar contribution . we are then left with feynman integrals of the form16\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \int d^dk_1 d^dk_2 \frac{\left ( k_1 ^ 2\right ) ^\alpha \left ( k_2 ^ 2\right ) ^\beta ( k_1\cdot p)^\gamma ( k_2\cdot p)^\delta ( k_1\cdot k_2)^\eta } { d_1^{a_1 } d_2^{a_2 } d_3^{a_3 } d_4^{a_4 } d_5^{a_5 } } , \end{aligned}$$\end{document}ddk1ddk2k12k22(k1p)(k2p)(k1k2)d1a1d2a2d3a3d4a4d5a5,where \documentclass[12pt]{minimal } 
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				 \begin{document}$$\alpha , \ldots , \eta , \ , a_1 , \ldots , a_5$$\end{document},  ,,a1,  ,a5 are positive ( or zero ) integer exponents and the \documentclass[12pt]{minimal } 
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				 \begin{document}$$d_i$$\end{document}di s are defined as\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } d_1&= k_1 ^ 2 - m_1 ^ 2 , \ , d_2 = ( k_1-p)^2 - m_2 ^ 2 , \ , d_3 = k_2 ^ 2 - m_3 ^ 2,\\ d_4&= ( k_2-p)^2 - m_4 ^ 2 , \ , d_5 = ( k_1-k_2)^2 - m_5 ^ 2 . \end{aligned}$$\end{document}d1=k12-m12,d2=(k1-p)2-m22,d3=k22-m32,d4=(k2-p)2-m42,d5=(k1-k2)2-m52.integrals belonging to the class above are in general not linearly independent of each other . when the scalar products in the numerator are expressed in terms of the denominators , powers of a \documentclass[12pt]{minimal } 
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				 \begin{document}$$d_i$$\end{document}di present in the original integral might cancel against a \documentclass[12pt]{minimal } 
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				 \begin{document}$$d_i$$\end{document}di in the numerator , possibly generating a feynman integral in which \documentclass[12pt]{minimal } 
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				 \begin{document}$$d_i$$\end{document}di does not appear , i.e. in which the corresponding line has been shrunk to a point . for given \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_i$$\end{document}ai s and high enough \documentclass[12pt]{minimal } 
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				 \begin{document}$$\alpha , \ldots , \eta $ $ \end{document}, , , some \documentclass[12pt]{minimal } 
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				 \begin{document}$$d_i$$\end{document}di s may acquire negative exponents . ( 16 ) therefore reduces to the evaluation of a number of integrals of the form17\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \int \frac{d^dk_1 d^dk_2}{d_1^{n_1 } d_2^{n_2 } d_3^{n_3 } d_4^{n_4 } d_5^{n_5 } } , \end{aligned}$$\end{document}ddk1ddk2d1n1d2n2d3n3d4n4d5n5,where the exponents \documentclass[12pt]{minimal } 
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				 \begin{document}$$n_i\in \mathbb { z}$$\end{document}niz . in the present case , one has to evaluate \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal { o}(300)$$\end{document}o(300 ) different feynman integrals . there exists a convenient procedure for dealing with such large numbers of different feynman integrals in a more efficient way than their direct evaluation . dimensionally regularized integrals , at arbitrary loop order and with arbitrary number of external legs , satisfy identities of the ibp type  [ 54 , 55 ] . these identities are linear relations that connect integrals with different sets of exponents \documentclass[12pt]{minimal } 
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				 \begin{document}$$\{n_1,\ldots , n_5\}$$\end{document}{n1,  ,n5}. after a set of independent integrals , the mis , has been identified , all the remaining integrals can then be reduced to linear combinations of the mis , the coefficients being rational functions of the masses , the kinematic invariants and the space - time dimension \documentclass[12pt]{minimal } 
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				 \begin{document}$$d$$\end{document}d . one practical advantage of such a procedure is its  divide and conquer  spirit . on the one hand few mis encode the analyticity properties ( singularities , thresholds , branch cuts ) of the problem under consideration . on the other hand , the evaluation of the large number of different feynman integrals entering a computation is reduced to a problem of linear algebra , which can easily be automated , if the mis are known . we perform the reduction to mis with the public code reduze  [ 70 , 71 ] , which implements the laporta algorithm   and produces the ibp identities relevant to our case . the evaluation of the mis is in general a complicated problem and can proceed via different techniques , like the integration in the feynman , schwinger or mellin a remarkable consequence of the aforementioned ibp relations is that the mis obey linear systems of first - order differential equations ( des ) in the kinematic invariants , which provide an alternative means for their computation  [ 73 , 74 ] . finding the analytic solution of the des for arbitrary \documentclass[12pt]{minimal } 
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				 \begin{document}$$d$$\end{document}d is possible only in some simple cases . in more general cases the mis are expanded in powers of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\epsilon = ( 4-d)/2$$\end{document}=(4-d)/2 , giving rise to a ( generally coupled ) system of des for the expansion coefficients . in the limit of vanishing external momentum , two - loop self - energies become two - loop vacuum diagrams , which reduce via ibp to linear combinations of only one genuine two - loop mi and products of the one - loop one - propagator mi  . two - loop self - energies with arbitrary external momentum , in the general case with five different masses in the loops , reduce via ibp to linear combinations of 30 mis  . the finite part of such mis can be expressed in terms of four functions , in addition to the well - known one - loop mis.5 as already mentioned , analytic solutions for such functions have been derived only for special patterns of up to two internal masses ( only one function is known in a particular case with three different masses ) . on the other hand , the diagrams that in our approximation contribute to the self - energies entering eqs . ( 5)(7 ) require the knowledge of mis with up to four different masses , the most complicated ones being those involving simultaneously \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_t^2,{m}_{\tilde{t}_1}^{2},{m}_{\tilde{t}_2}^{2}$$\end{document}mt2,mt~12,mt~22 , and \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_{\tilde{g}}^{2}$$\end{document}mg~2 . in our computation we rely on the package tsil  , which implements ( besides all the analytically known cases ) the numerical solution of the des for the two - loop self - energy mis . [ 4751 ] is based on the fact that the value at \documentclass[12pt]{minimal } 
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				 \begin{document}$$p^2=0$$\end{document}p2=0 ( or the expansion for small \documentclass[12pt]{minimal } 
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				 \begin{document}$$p^2$$\end{document}p2 in the case of logarithmically divergent integrals ) is known for each function and can be used to build the set of initial conditions needed for the solution of the des . in the computation of the self - energies entering eqs . ( 5)(7 ) we need to evaluate the corresponding mis at \documentclass[12pt]{minimal } 
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				 \begin{document}$$p^2=m_z^2,m_a^2,m_h^2,m_h^2$$\end{document}p2=mz2,ma2,mh2,mh2 . given that we include the contribution of light quarks to \documentclass[12pt]{minimal } 
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				 \begin{document}$$\pi _ { { \scriptscriptstyle z}{\scriptscriptstyle z}}$$\end{document}zz ( in the approximation \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_q=0$$\end{document}mq=0 ) and that \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}$$\end{document}ma is a free parameter , it is clear that the way thresholds are handled in the numerical evaluation of the mis is of particular relevance . in the des approach , the physical two- and three - particle thresholds show up , together with the pseudothresholds , as poles in the coefficients of the des . tsil overcomes the numerical instabilities related to such poles by displacing the \documentclass[12pt]{minimal } 
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				 \begin{document}$$p^2$$\end{document}p2-integration contour in the upper half - plane when the momentum is above the smallest ( pseudo)threshold . the evaluation at ( or very close to ) the ( pseudo)thresholds is performed through a variant of the algorithm , which is slightly less efficient but ensures reliable results in such critical cases . as an example , the time needed on an intel core i7 - 4650u cpu for the evaluation of the complete set of mis , for any of the mass patterns entering the self - energies , ranges between \documentclass[12pt]{minimal } 
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				 \begin{document}$$5 \times 10^{-4}\,s$$\end{document}510 - 4s and \documentclass[12pt]{minimal } 
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				 \begin{document}$$8 \times 10^{-2}\,s$$\end{document}810 - 2s , the latter being the typical time for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\sqrt{p^2}$$\end{document}p2 close or equal to the heavy stop pair threshold and to the three - particle ( pseudo)thresholds \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_{\tilde{t}_i}+m_{\tilde{g}}\pm { { m}_{t}}$$\end{document}mt~i+mg~mt . in ref .   the relative accuracy of tsil is claimed to be better than \documentclass[12pt]{minimal } 
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				 \begin{document}$$10^{-10}$$\end{document}10 - 10 for generic cases , or worse in cases with large mass hierarchies . being tsil a package dedicated to the evaluation of the mis for two - loop self - energy diagrams , it is not surprising that its speed and accuracy prove much better than those quoted in ref .  , where the general - purpose package secdec is used and , in the most complicated case , \documentclass[12pt]{minimal } 
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				 \begin{document}$$100\,s$$\end{document}100s are needed in order to reach a relative accuracy of at least \documentclass[12pt]{minimal } 
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				 \begin{document}$$10^{-5}$$\end{document}10 - 5 close to a threshold . in this section we assess the numerical impact of the momentum - dependent part of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections on the predictions for the neutral higgs - boson masses in the mssm . we focus on six benchmark scenarios introduced in ref .  , which identify regions in the mssm parameter space compatible with the current bounds from susy - particle searches and with the requirement that the predicted value of \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{h}}$$\end{document}mh agrees , within the theoretical uncertainty of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\pm 3~\mathrm{gev}$$\end{document}3gev estimated in refs . [ 78 , 79 ] , with the mass of the sm - like higgs boson discovered at the lhc.6 in our numerical examples we adopt the mixed os\documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme described in sect .  2 . the sm input parameters are chosen as the pole top mass \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_t = 173.2~\mathrm{gev}$$\end{document}mt=173.2gev , the running bottom mass \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_b(m_b ) = 4.2~\mathrm{gev}$$\end{document}mb(mb)=4.2gev , the fermi constant \documentclass[12pt]{minimal } 
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				 \begin{document}$$g_f = 1.16639 \times 10^{-5}~\mathrm{gev}^{-2}$$\end{document}gf=1.1663910 - 5gev-2 , the strong gauge coupling \documentclass[12pt]{minimal } 
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				 \begin{document}$$\alpha _ s({{m}_{{\scriptscriptstyle z } } } ) = 0.118$$\end{document}s(mz)=0.118 , and the pole gauge - boson masses \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle z}}}= 91.1876~\mathrm{gev}$$\end{document}mz=91.1876gev and \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle w}}}= 80.385~\mathrm{gev}$$\end{document}mw=80.385gev . to compute the scalar masses \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{m}^2_{h,{\scriptscriptstyle h}}$$\end{document}mh , h2 and the effective mixing angle \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\alpha } $ $ \end{document} entering the corrections in eqs . we use default values for all settings with the exception of runningmt = 0 , i.e.  the top mass in the radiative corrections is identified with the pole mass ( to match the renormalization conditions imposed both in our os\documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr calculation and in the one of ref .  ) . by default , the renormalization scale associated to the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr definition of the higgs wfr and of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan is fixed as \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu _ { { \scriptscriptstyle r}}=m_t$$\end{document}r = mt . in fig .  1 we present our predictions for the lightest - scalar mass \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_h$$\end{document}mh in the six benchmark scenarios . we choose \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}= 500~\mathrm{gev}$$\end{document}ma=500gev and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta = 20$$\end{document}tan=20 , so that the lightest scalar \documentclass[12pt]{minimal } 
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				 \begin{document}$$h$$\end{document}h is sm - like , the bound on its tree - level mass is saturated , and the corrections controlled by the bottom yukawa coupling , which we do not compute beyond the approximations of feynhiggs , are not expected to be particularly relevant . for each scenario mass \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{m}_{h}$$\end{document}mh , obtained from feynhiggs without additional corrections ; the middle bar includes the effect of the momentum - dependent part of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections , i.e.  the \documentclass[12pt]{minimal } 
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				 \begin{document}$$(\delta m_h^2)^{\alpha _ ( 12 ) ; finally , the lower bar represents our final result for \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_h$$\end{document}mh , and includes the effects of both the momentum - dependent part of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections and the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections , i.e.  the \documentclass[12pt]{minimal } 
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				 \begin{document}$$(\delta m_h^2)^{\alpha \alpha _ { s}}$$\end{document}(mh2)s defined in eq . 1predictions for the mass of the lightest scalar \documentclass[12pt]{minimal } 
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				 \begin{document}$$h$$\end{document}h in the six benchmark scenarios of ref .  , for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{m}_h$$\end{document}mh computed with feynhiggs  2.10.2 ( upper ) , the inclusion of the momentum - dependent part of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections ( middle ) and the additional inclusion of the whole \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections ( lower ) . from top to bottom , the considered scenarios are \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{h}}^\mathrm{mod-}$$\end{document}mhmod- ( green ) , light stop ( turquoise ) , light stau ( purple ) , tau - phobic ( orange ) predictions for the mass of the lightest scalar \documentclass[12pt]{minimal } 
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				 \begin{document}$$h$$\end{document}h in the six benchmark scenarios of ref .  , for \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{h}}^\mathrm{mod-}$$\end{document}mhmod- ( green ) , light stop ( turquoise ) , light stau ( purple ) , tau - phobic ( orange ) figure  1 shows that , in all considered scenarios , the momentum - dependent part of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections can shift the prediction for \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_h$$\end{document}mh by several hundred mev each ( the largest shifts , of about \documentclass[12pt]{minimal } 
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				 \begin{document}$$\pm 1$$\end{document}1  gev , occur in the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections and of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections on the prediction for the lightest - scalar mass as a function of \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}$$\end{document}ma , and in fig .  3 we do the same for the heaviest - scalar mass . in each figure , the mssm parameters for the left plot are chosen as in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_h^\mathrm{max}$$\end{document}mhmax benchmark scenario of ref .  , while for the right plot they are chosen as in the modified light - stop scenario . in each plot , the dashed lines represent the contribution of the sole momentum - dependent part of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections , while the solid lines include both the momentum - dependent \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections and the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections . the red lines were obtained with \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta = 5$$\end{document}tan=5 while the blue lines were obtained with \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta = 20$$\end{document}tan=20 . 2corrections to the lightest - scalar mass as a function of \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}$$\end{document}ma , for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta = 5$$\end{document}tan=5 ( red ) and for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta = 20$$\end{document}tan=20 ( blue ) . the other mssm parameters are chosen as in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_h^\mathrm{max}$$\end{document}mhmax scenario ( left ) or as in the light - stop scenario ( right ) . the dashed lines represent the contribution of the sole momentum - dependent part of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections , the solid lines include both the momentum - dependent \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections and the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) correctionsfig . 3same as fig .  2 for the corrections to the heaviest - scalar mass corrections to the lightest - scalar mass as a function of \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}$$\end{document}ma , for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta = 5$$\end{document}tan=5 ( red ) and for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta = 20$$\end{document}tan=20 ( blue ) . the other mssm parameters are chosen as in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_h^\mathrm{max}$$\end{document}mhmax scenario ( left ) or as in the light - stop scenario ( right ) . the dashed lines represent the contribution of the sole momentum - dependent part of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections , the solid lines include both the momentum - dependent \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections and the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) 2 for the corrections to the heaviest - scalar mass figure  2 shows that the corrections to the lightest - scalar mass are negligible at low values of \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}$$\end{document}ma , but they become larger and essentially independent of \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}$$\end{document}ma as the latter increases . the transition to this  decoupling  regime  where the lightest scalar has sm - like couplings and its mass is insensitive to the value of \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}$$\end{document}ma  is sharper for larger values of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan. in both the \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_h^\mathrm{max}$$\end{document}mhmax and light - stop scenarios , the momentum - dependent \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) effects decrease \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{h}}$$\end{document}mh by 300400  mev at large \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}$$\end{document}ma . however , as already seen in fig .  1 , the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) effects enter the prediction for \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{h}}$$\end{document}mh with comparable magnitude but opposite sign , significantly reducing the total size of the correction . figure  3 shows that for low values of \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}$$\end{document}ma , where the heaviest scalar is the one with sm - like couplings , the corrections to its mass are comparable to the ones that affect the lightest - scalar mass in the decoupling region . on the other hand , for larger values of the pseudoscalar mass  where \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle h}}}\approx { { m}_{{\scriptscriptstyle a}}}$$\end{document}mhma  the corrections to the heaviest - scalar mass show a series of spikes , related to the opening of real - particle thresholds in diagrams that involve a virtual gluon . the first spike is visible in correspondence with \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}= 2\,{{m}_{t}}$$\end{document}ma=2mt in the plot on the left for the \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_h^\mathrm{max}$$\end{document}mhmax scenario . more - pronounced spikes ( note the different scale on the \documentclass[12pt]{minimal } 
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				 \begin{document}$$y$$\end{document}y axis ) are visible in correspondence with \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle h}}}= 2\,m_{\tilde{t}_1}$$\end{document}mh=2mt~1 , \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle h}}}= m_{\tilde{t}_1}+m_{\tilde{t}_2}\,$$\end{document}mh = mt~1+mt~2 , and \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle h}}}= 2\,m_{\tilde{t}_2}\,$$\end{document}mh=2mt~2 in the plot on the right for the light - stop scenario . analogous spikes would appear at larger values of \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}$$\end{document}ma in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_h^\mathrm{max}$$\end{document}mhmax scenario , where the stops are heavier . we stress that our results are not reliable in the vicinity of these thresholds : the two - loop correction to the heaviest - scalar mass is actually divergent there , and the height of the spikes in the plots carries no physical meaning . a more sophisticated analysis , taking into account the widths of the virtual particles in the loops as well as non - perturbative qcd effects , would be necessary around the thresholds , but it is beyond the scope of our calculation . figure  3 also shows that , in the decoupling region and away from the thresholds , the corrections to the heaviest - scalar mass amount at most to a few hundred mev , and they decrease in size with increasing \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan. moreover , the effect of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections is negligible ( the dashed and solid lines are practically overlapping in the plots ) . inspection of our analytic formulae shows that , in the decoupling limit , the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections to \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle h}}}$$\end{document}mh are suppressed by one or two powers of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan , whereas the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections contain unsuppressed contributions proportional to the square of the superpotential higgs - mass parameter \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu $ $ \end{document}. while corrections of this size might be considered negligible in comparison with the value of \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle h}}}$$\end{document}mh itself , they are not entirely irrelevant when compared to the difference \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle h}}}-{{m}_{{\scriptscriptstyle a}}}$$\end{document}mh - ma , which can be of the order of a few gev and is the quantity of interest when a large physical mass for the pseudoscalar is taken as input in the calculation . the way we compute the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections to the entries of the inverse - propagator matrix for the neutral scalars allows for a relatively easy comparison with earlier calculations . we first renormalize all the relevant parameters in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme , i.e.  we introduce minimal counterterms that , by definition , subtract only powers of \documentclass[12pt]{minimal } 
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				 \begin{document}$$1/\epsilon , $ $ \end{document}1/ , multiplied by coefficients that should be polynomial in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr-renormalized masses and couplings . in a second step , we convert our results to the mixed os\documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme adopted in feynhiggs , replacing the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr top / stop parameters entering the one - loop part of the corrections with the corresponding os parameters plus the finite one - loop shifts given in ref .  . as a first obvious check , we took the limit of vanishing external momentum in the scalar self - energies entering the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections and we compared our results with those in ref .  , finding full agreement . we also successfully compared the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections at vanishing external momentum with the results of an independent calculation based on the effective - potential techniques of ref .  . note , however , that this comparison does not cover the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) contributions to the \documentclass[12pt]{minimal } 
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				 \begin{document}$$z$$\end{document}z - boson self - energy . concerning the latter , we checked that we can reproduce the result of ref . for the subset of two - loop diagrams that involve only quarks and a gluon , taking into account the fact that ref . we then compared our results for the momentum - dependent corrections with those of ref .  , where the two - loop calculation of the higgs masses was performed entirely in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme . as mentioned in sect .  1 , the higgs - mass corrections in ref are organized in a different way with respect to our calculation , therefore we could compare only at the level of individual two - loop self - energies for scalars and pseudoscalars ( the two - loop self - energy for the \documentclass[12pt]{minimal } 
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				 \begin{document}$$z$$\end{document}z boson was not computed in ref .  ) . rotating our scalar self - energies from the \documentclass[12pt]{minimal } 
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				 \begin{document}$$(s_1,s_2)$$\end{document}(s1,s2 ) basis to the \documentclass[12pt]{minimal } 
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				 \begin{document}$$(h , h)$$\end{document}(h , h ) basis with the tree - level mixing angle defined in ref . , we reproduce perfectly the results for the  top / gluon  and  top / stop / gluino  contributions to \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \pi _ { hh}$$\end{document}hh shown in figure  2 of that paper . this provides a full cross - check of the momentum - dependent \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) contribution to the self - energy , as well as a partial check of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) contribution , restricted to diagrams involving the stop squarks ( the diagrams involving the other squarks are included in the  others  line in the above - mentioned figure ) . we also checked the analogous contributions to \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \pi _ { h{\scriptscriptstyle h}}$$\end{document}hh , \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \pi _ { { \scriptscriptstyle h}{\scriptscriptstyle h}}$$\end{document}hh , and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \pi _ { { \scriptscriptstyle a}{\scriptscriptstyle a}}$$\end{document}aa against results provided by the author of ref .  , finding again perfect numerical agreement . although our calculation and the one in ref .   both use tsil to compute the mis , and thus can not be considered entirely independent , the agreement in the results for the self - energies gives us confidence that the computation of two - loop feynman diagrams in terms of mis and the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr subtraction of their divergences are correct in both papers . our results for the momentum - dependent \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections in the mixed os\documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme can in turn be compared with those of ref .  . to start with , we compared our two - loop 1pi contributions to the scalar and pseudoscalar self - energies with analogous results provided by the authors of ref .  , and we found agreement within the accuracy of the sector - decomposition procedure used to compute the loop integrals in that paper . the successful comparison between two sets of self - energies in which the loop integrals were evaluated with tsil and secdec , respectively , validates the results for the two - loop mis , thus reinforcing our cross - check of ref .  . on the other hand , our results for the momentum - dependent \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections to the higgs masses , obtained by combining the 1pi diagrams with all the necessary counterterm contributions , differ significantly from the ones in ref .  . considering for example the \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_h^{\mathrm{max}}$$\end{document}mhmax scenario discussed in the previous section , we find that for large \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle a}}}$$\end{document}ma the lightest - scalar mass is subject to a negative correction of about 350400  mev ( depending on \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan ; see the left plot in fig . is also negative but quite smaller , about 5060  mev ( see the upper plot in figure 7 of that paper ) . , related to the renormalization conditions for the higgs fields and for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan. in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme , the wfr counterterm for each field \documentclass[12pt]{minimal } 
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				 \begin{document}$$h^0_i$$\end{document}hi0 can be related to the divergent part of the derivative of the scalar self - energy with respect to the external momentum:18\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \delta \mathcal{z}^{(\ell ) } _ i = - \left [ \frac{d\ , \mathrm{re}\,\pi ^{(\ell ) } _ { ii}(p^2)}{dp^2 } \right ] ^\mathrm{div}\quad ( \ell = 1,2 ) . \end{aligned}$$\end{document}zi()=-dreii()(p2)dp2div(=1,2).indeed , when all parameters entering the one - loop part of the scalar self - energies are renormalized in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme , eq . ( 18 ) leads to the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr wfr counterterms given in eq . ( 9 ) , in accordance with the anomalous dimensions of the higgs fields given in refs . however , in the mixed os\documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme of ref . the top / stop parameters in the one - loop self - energies are renormalized os . in that case , the use of eq . ( 18 ) to determine the wfr counterterms leads to19\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \mathcal{z}_2^{}&= 1 - \frac{~\alpha _ { t}^{{\mathrm{os}}}}{4\pi } \,n_c\,\cdot \,\frac{1}{\epsilon } ~+~ \frac{\alpha _ { t}\alpha _ { s}}{(4\pi ) ^2}\,2\,n_c\,c_f\nonumber \\&\cdot \left ( \frac{1}{\epsilon ^2}-\frac{1}{\epsilon } \right ) - \frac{\alpha _ { t}}{2\pi } \,n_c\ , \frac{\delta m_t}{m_t}\,\cdot \,\frac{1}{\epsilon } , \end{aligned}$$\end{document}z2=1-tos4nc1+ts(4)22nccf12 - 1-t2ncmtmt1,where \documentclass[12pt]{minimal } 
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				 \begin{document}$$\alpha _ { t}^{{\mathrm{os}}}$$\end{document}tos is a scale - independent coupling extracted from the pole top mass , and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta m_t$$\end{document}mt is the finite one - loop shift for the top mass given in eq .  ( b2 ) of ref .  . by converting the coupling \documentclass[12pt]{minimal } 
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				 \begin{document}$$\alpha _ { t}^{{\mathrm{os}}}$$\end{document}tos in the one - loop term of eq . ( 19 ) into the corresponding \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr coupling , it is easy to see that \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{z}_2^{}$$\end{document}z2 differs from the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr wfr constant in eq . ( 9 ) by a finite two - loop term:20\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \mathcal{z}_2^ { } = \mathcal{z}_2^{\,{\overline{\mathrm{dr } } } } ~+~ \frac{\alpha _ { t}}{2\pi } \,n_c\ , \frac{\delta _ \epsilon m_t}{m_t } , \end{aligned}$$\end{document}z2=z2dr+t2ncmtmt , where \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta _ \epsilon m_t$$\end{document}mt denotes the part proportional to \documentclass[12pt]{minimal } 
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				 \begin{document}$$\epsilon $ $ \end{document} in the top self - energy regularized with dimensional reduction:21\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \frac{\delta _ s}{4 \pi } \,c_f\ , \left\ { -\frac{3}{2}\,\ln ^2 \frac{{{m}_{t}}^2}{\mu _ { { \scriptscriptstyle r}}^2 } + 5\,\ln \frac{{{m}_{t}}^2}{\mu _ { { \scriptscriptstyle r}}^2 } - \frac{~\pi ^2}{4 } - 9 - \frac{m_{\tilde{g}}^{2}}{{{m}_{t}}^2}\right . \left ( \frac{1}{2 } \ , \ln ^2 \frac{m_{\tilde{g}}^{2}}{\mu _ { { \scriptscriptstyle r}}^2 } - \ln \frac{m_{\tilde{g}}^{2}}{\mu _ { { \scriptscriptstyle r}}^2 } + \frac{~\pi ^2}{12 } + 1 \right ) \right . \nonumber \\&+ \frac{1}{2}\left [ ~ \frac{m_{\tilde{g}}^{2}+{{m}_{t}}^2 - { m}_{\tilde{t}_1}^{2}- 2 \,{s}_{2\theta _ { t}}\,m_{\tilde{g}}\ , { { m}_{t}}}{{{m}_{t}}^2 } \,b_\epsilon ( { { m}_{t}}^2,m_{\tilde{g}}^{2},{m}_{\tilde{t}_1}^{2 } ) \right . \nonumber \\&+\left . \left . \,\frac{{m}_{\tilde{t}_1}^{2}}{{{m}_{t}}^2 } \left ( \frac{1}{2 } \ , \ln ^2 \frac{{m}_{\tilde{t}_1}^{2}}{\mu _ { { \scriptscriptstyle r}}^2 } - \ln \frac{{m}_{\tilde{t}_1}^{2}}{\mu _ { { \scriptscriptstyle r}}^2 } + \frac{~\pi ^2}{12 } + 1 \right ) \right . \right . ( \tilde{t}_1 \rightarrow \tilde{t}_2 , ~~ { s}_{2\theta _ { t}}\rightarrow -{s}_{2\theta _ { t } } ) \right ] \right\ } . \end{aligned}$$\end{document}mtmt=s4cf-32ln2mt2r2 + 5lnmt2r2-24 - 9-mg~2mt212ln2mg~2r2-lnmg~2r2+212 + 1 + 12mg~2+mt2-mt~12 - 2s2tmg~mtmt2b(mt2,mg~2,mt~12)+mt~12mt212ln2mt~12r2-lnmt~12r2+212 + 1+(t~1t~2,s2t-s2t).in the equation above \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu _ { { \scriptscriptstyle r}}$$\end{document}r is the renormalization scale associated to the higgs wfr and to \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan , while \documentclass[12pt]{minimal } 
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				 \begin{document}$$b_\epsilon ( s , x , y)$$\end{document}b(s , x , y ) denotes the coefficient of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\epsilon $ $ \end{document} in the expansion of the passarino veltman function \documentclass[12pt]{minimal } 
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				 \begin{document}$$b_0$$\end{document}b0 . an explicit expression for \documentclass[12pt]{minimal } 
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				 \begin{document}$$b_\epsilon $ $ \end{document}b can be found , e.g. , in eq . ( 2.31 ) of the tsil manual  . in the calculation of ref .  , where the top / stop parameters entering the one - loop part of the corrections are directly renormalized os instead of being first renormalized in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme and then converted to the os scheme via a finite shift , the two - loop self - energies and tadpoles contain terms proportional to \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta _ \epsilon m_t$$\end{document}mt . such terms would drop out of the final result for the renormalized inverse - propagator matrix if eq . ( 20 ) was used to obtain the correct \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr definition for the wfr constant \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{z}_2^{\,{\overline{\mathrm{dr}}}}$$\end{document}z2dr , and consequently for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \tan \beta $ $ \end{document}tan , but they survive if the wfr constant is defined as in eq . ( 19 ) . to prove that these terms are indeed at the root of the observed discrepancies , we modified our own calculation , using \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{z}_2^{}$$\end{document}z2  as obtained from eq . ( 20 )  instead of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{z}_2^{\,{\overline{\mathrm{dr}}}}$$\end{document}z2dr and then computing a non - minimal counterterm for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan via eq . we checked that , with this modification , we reproduce exactly the corrections to the renormalized inverse propagator shown in figures  5 and  10 of ref .  . we also reproduce the corrections to the scalar masses shown in figures  7 , 8 , 12 and  13 of that paper , although small discrepancies persist in the case of the heaviest scalar when its mass is above the threshold \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle h}}}=2\,m_t$$\end{document}mh=2mt . these residual discrepancies are formally of higher order in the perturbative expansion , and they result from different approximations in the two - step procedure for the determination of the poles of the propagator ( namely , we drop the imaginary parts of the two - loop self - energies , while ref the two - loop renormalization of the higgs fields and of the parameter \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan is not performed in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme as claimed in the paper , but rather in some non - minimal scheme where the wfr counterterms and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \tan \beta $ $ \end{document}tan differ from their \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr counterparts by finite , non - polynomial terms , and neither the higgs fields nor \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan obey their usual renormalization - group equations ( because of the explicit scale dependence of the additional terms ) . this inconsistency should be taken into account when comparing the results of ref .   with those of calculations that actually employ \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr definitions for the wfr and for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan. first of all , to account for the difference in \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \tan \beta $ $ \end{document}tan , the input value for the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr-renormalized parameter \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan should be converted to the corresponding value in the non - minimal scheme of ref .  , according to22\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \tan \beta ^ { } = \tan \beta ^{\,{\overline{\mathrm{dr } } } } - \frac{\alpha _ { t}}{4\pi } \,n_c\,\tan \beta ~ \frac{\delta _ ( 5)(7 ) show that the contributions of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \tan \beta ^{(2)}$$\end{document}tan(2 ) to the entries of the higgs mass matrix are suppressed by powers of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\cos \beta $ $ \end{document}cos. consequently , the effect on the higgs masses arising from a two - loop difference in \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta \tan \beta $ $ \end{document}tan is very small already for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta = 5$$\end{document}tan=5 . in fact , the bulk of the numerical discrepancy between our results and those of ref .   is due to higher - order effects that are directly related to the finite shift in the wfr . as discussed at the end of sect .  2 , such effects are included in the higgs - mass corrections when the latter are computed in terms of loop - corrected higgs masses and mixing , and can become numerically relevant when the loop - corrected masses differ significantly from their tree - level values . we computed the two - loop corrections to the neutral mssm higgs masses of \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) i.e. , all two - loop corrections that involve the strong gauge coupling when the only non - vanishing yukawa coupling is \documentclass[12pt]{minimal } 
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				 \begin{document}$$h_t$$\end{document}ht  including the effect of non - vanishing external momenta in the self - energies . we relied on an ibps technique to express the momentum - dependent loop integrals in terms of a minimal set of master integrals , and we used the public code tsil to evaluate the latter . we obtained results for the higgs - mass corrections valid when all parameters in the one - loop part of the corrections are renormalized in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme , as well as results valid in a mixed os\documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme where the top / stop parameters are renormalized on - shell . our results for the scalar and pseudoscalar self - energies in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme confirm the results of an earlier calculation , ref . in addition , we obtained new results for the two - loop contributions to the \documentclass[12pt]{minimal } 
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				 \begin{document}$$z$$\end{document}z - boson self - energy that involve the strong gauge coupling . , enter the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections to the higgs masses when the tree - level mass matrix of the scalars is expressed in terms of the physical \documentclass[12pt]{minimal } 
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				 \begin{document}$$z$$\end{document}z - boson mass . we also compared our results for the momentum - dependent \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections in the mixed os\documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme with those of a recent calculation of the same corrections , ref .  , and found disagreement . we traced the reason for the discrepancy to the fact that , contrary to what stated in ref . , in that calculation the higgs fields and the parameter \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tan \beta $ $ \end{document}tan are renormalized in a non - minimal scheme instead of the usual \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{dr}}$$\end{document}dr scheme . when this difference is taken into account , we reproduce the results of ref .  , providing in passing a cross - check of the codes used to evaluate the loop integrals in the two calculations ( i.e. , tsil and secdec , respectively ) . however , we noticed that tsil , which implements dedicated algorithms for two - loop self - energy integrals , can be a thousand times faster than a multi - purpose code such as secdec in the computation of the higgs - mass corrections . this should be taken into consideration when including the momentum - dependent corrections in phenomenological analyses of the mssm parameter space . as to the numerical impact of the corrections computed in this paper , it could at best be described as moderate . we considered six benchmark scenarios compatible with the results of higgs and susy searches at the lhc , and we found that both the momentum - dependent part of the \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha _ { t}\alpha _ { s})$$\end{document}o(ts ) corrections and the whole \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(\alpha \alpha _ { s})$$\end{document}o(s ) corrections can shift the prediction for the lightest - scalar mass \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{h}}$$\end{document}mh by several hundred mev . however , we noticed that  at least in the considered scenarios  the two classes of corrections enter the prediction for \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{h}}$$\end{document}mh with opposite sign , and they compensate each other to a good extent . for what concerns the heaviest - scalar mass \documentclass[12pt]{minimal } 
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				 \begin{document}$${{m}_{{\scriptscriptstyle h}}}$$\end{document}mh , the impact of the new corrections is also modest , with the exception of regions around real - particle thresholds where a fixed - order calculation is not reliable anyway . the predictions for the lightest - scalar mass , as obtained from popular codes for the determination of the mssm mass spectrum , carry a theoretical uncertainty that has been estimated to be ( at least ) of the order of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\pm 3$$\end{document}3  gev  see , e.g. , refs .  [ 78 , 79 ] and the more recent discussion in ref .  . against this backdrop , the corrections presented in this paper can be considered sub - dominant , and their inclusion in public codes might seem less urgent than , e.g. , the inclusion of the dominant three - loop effects  [ 41 , 42 ] or the proper resummation of large logarithms in scenarios with multi - tev stop masses  [ 39 , 40 , 82 ] , both of which can shift the prediction for the lightest - scalar mass by several gev . nevertheless , one should not forget that the accuracy of the measurement of the higgs mass at the lhc has already reached the level of a few hundred mev i.e. , comparable to our sub - dominant corrections  and will improve further when more data become available . if susy shows up at last when the lhc operates at 1314  tev , the higgs mass will serve as a precision observable to constrain mssm parameters that might not be directly accessible by experiment . to this purpose , the accuracy of the theoretical prediction will have to match the experimental one , making a full inclusion of the two - loop corrections to the higgs masses unavoidable .