Renormalization scheme dependence of the two-loop QCD corrections to the neutral Higgs-boson masses in the MSSM

Abstract

Reaching a theoretical accuracy in the prediction of the lightest MSSM Higgs-boson mass, $M_h$, at the level of the current experimental precision requires the inclusion of momentum-dependent contributions at the two-loop level. Recently two groups presented the two-loop QCD momentum-dependent corrections to $M_h$ (Borowka et al., Eur Phys J C 74(8):2994, 2014; Degrassi et al., Eur Phys J C 75(2):61, 2015), using a hybrid on-shell-$\overline{\mathrm{DR}}$ scheme, with apparently different results. We show that the differences can be traced back to a different renormalization of the top-quark mass, and that the claim in Ref. Degrassi et al. (Eur Phys J C 75(2):61, 2015) of an inconsistency in Ref. Borowka et al. (Eur Phys J C 74(8):2994, 2014) is incorrect. We furthermore compare consistently the results for $M_h$ obtained with the top-quark mass renormalized on-shell and $\overline{\mathrm{DR}}$. The latter calculation has been added to the FeynHiggs package and can be used to estimate missing higher-order corrections beyond the two-loop level.

Introduction

The particle discovered in the Higgs-boson searches by ATLAS [3] and CMS [4] at CERN shows, within experimental and theoretical uncertainties, properties compatible with the Higgs boson of the Standard Model (SM) [5–7]. It can also be interpreted as the Higgs boson of extended models, however, where the lightest Higgs boson of the Minimal Supersymmetric Standard Model (MSSM) [8–10] is a prime candidate.

The Higgs sector of the MSSM with two scalar doublets accommodates five physical Higgs bosons. In lowest order these are the light and heavy $\mathcal{CP}$-even h and H, the $\mathcal{CP}$-odd A, and the charged Higgs bosons $H^{\pm }$. At tree level, the Higgs sector can be parameterized in terms of the gauge couplings, the mass of the $\mathcal{CP}$-odd Higgs boson, $M_A$, and $tan\mathit{\beta }\equiv v_2/v_1$, the ratio of the two vacuum expectation values; all other masses and mixing angles follow as predictions.

Higher-order contributions can give large corrections to the tree-level relations [11–13], and in particular to the mass of the lightest Higgs boson, $M_h$. For the MSSM1 with real parameters the status of higher-order corrections to the masses and mixing angles in the neutral Higgs sector is quite advanced; see Refs. [19–26] for the calculations of the full one-loop level. At the two-loop level [18, 27–44] in particular the $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ and $\mathcal{O}({\mathit{\alpha }}_t^2)$ contributions (${\mathit{\alpha }}_t\equiv h_t^2/(4\mathit{\pi })$, $h_t$ being the top-quark Yukawa coupling) to the self-energies – evaluated in the Feynman-diagrammatic (FD) as well as in the effective potential (EP) method – as well as the $\mathcal{O}({\mathit{\alpha }}_b{\mathit{\alpha }}_s)$, $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_b)$ and $\mathcal{O}({\mathit{\alpha }}_b^2)$ contributions – evaluated in the EP approach – are known for vanishing external momenta. An evaluation of the momentum dependence at the two-loop level in a pure $\overline{\mathrm{DR}}$ calculation was presented in Ref. [45]. The latest status of the momentum-dependent two-loop corrections will be discussed below. A (nearly) full two-loop EP calculation, including even the leading three-loop corrections, has also been published [46–50, 52, 53]. Within the EP method all contributions are evaluated at zero external momentum, however, in contrast to the FD method which in principle allows for non-vanishing external momenta. Furthermore, the calculation presented in Refs. [46–50, 52, 53] is not publicly available as a computer code for Higgs-boson mass calculations. Subsequently, another leading three-loop calculation of $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s^2)$, depending on the various SUSY mass hierarchies, was completed [54–56], resulting in the code H3m which adds the three-loop corrections to the FeynHiggs [14, 29, 57–60] result. Most recently, a combination of the full one-loop result, supplemented with leading and subleading two-loop corrections evaluated in the FD/EP method and a resummation of the leading and subleading logarithmic corrections from the scalar-top sector has been published [60] in the latest version of the code FeynHiggs.

The measured mass value of the observed Higgs boson is currently known to about $250\mathrm{MeV}$ accuracy [5], reaching the level of a precision observable. At a future linear collider (ILC), the precise determination of the light Higgs-boson properties and/or heavier MSSM Higgs bosons within the kinematic reach will be possible [61]. In particular, a mass measurement of the light Higgs boson with an accuracy below $\sim 0.05\mathrm{GeV}$ is anticipated [62].

In Ref. [59] the remaining theoretical uncertainty in the calculation of $M_h$, from unknown higher-order corrections, was estimated to be up to $3\mathrm{GeV}$, depending on the parameter region; see also Refs. [60, 63] for updated results. As the accuracy of the $M_h$ prediction should at least match the one of the experimental result, higher-order corrections which do not dominate the size of the Higgs-boson mass values have to be included in the Higgs-boson mass predictions.

To better control the size of momentum-dependent contributions, we recently presented the calculation of the $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections to $M_h$ (the leading momentum-dependent two-loop QCD corrections). The calculation was performed in a hybrid on-shell/$\overline{\mathrm{DR}}$ scheme [1] at the two-loop level, where $M_A$ and the tadpoles are renormalized on-shell (OS), whereas the Higgs-boson fields and $tan\mathit{\beta }$ are renormalized $\overline{\mathrm{DR}}$. At the one-loop level the top/stop parameters are renormalized OS.2 Subsequently, in Ref. [2] this calculation was repeated with a different result (also, a calculation in a pure $\overline{\mathrm{DR}}$ scheme as well as the two-loop corrections of $\mathcal{O}(\mathit{\alpha }{\mathit{\alpha }}_s)$ were presented). Within Ref. [2] the discrepancy between Refs. [1, 2] was explained by an inconsistency in the renormalization scheme used for the Higgs-boson field renormalization in Ref. [1].

In this paper we demonstrate that this claim is incorrect. The renormalization scheme for the Higgs-boson fields used in Ref. [1] is (up to corrections beyond the two-loop level) identical to the one employed in Ref. [2]. We clarify that the differences between the two results originates in a difference of the top-quark-mass renormalization scheme. While in Ref. [1] a full OS renormalization was used, in Ref. [2] the contributions to the top-quark self-energy of $\mathcal{O}(\mathit{\epsilon })$ (with $4-D=2\mathit{\epsilon }$, D being the space-time dimension) were neglected, leading to the observed numerical differences. We also demonstrate how this difference in the treatment of the contributions from the top-quark mass can be linked to a difference in the two-loop field renormalization constant and explain why this difference should be regarded as a theoretical uncertainty at the two-loop level, which would be fixed only at three-loop order.

We further present a consistent calculation of the $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections to $M_h$ in a scheme where the top quark is renormalized $\overline{\mathrm{DR}}$, whereas the scalar tops continue to be renormalized OS. This new scheme is available from FeynHiggs version 2.11.1 on, allowing for an improved estimate of (some) unknown higher-order corrections beyond the two-loop level originating from the top/stop sector.

The paper is organized as follows. An overview of the relevant sectors and the renormalization employed in our calculation is given in Sect. 2. In Sect. 3 we compare analytically and numerically the results of Refs. [1, 2]. Results obtained using the $\overline{\mathrm{DR}}$ scheme for the top-quark mass are given in Sect. 4. Our conclusions are given in Sect. 5.

The relevant sectors and their renormalization

The Higgs-boson sector of the MSSM

The MSSM requires two scalar doublets, which are conventionally written in terms of their components as follows:

$$\begin{array}{cc}\hfill {\mathcal{H}}_1& =\left(\begin{array}{c}{\mathcal{H}}_1^0\\ {\mathcal{H}}_1^{-}\end{array}\right)=\left(\begin{array}{c}{v}_1+\frac{1}{\sqrt{2}}({\mathit{\varphi }}_1^0-i{\mathit{\chi }}_1^0)\\ -{\mathit{\varphi }}_1^{-}\end{array}\right),\hfill \\ \hfill {\mathcal{H}}_2& =\left(\begin{array}{c}{\mathcal{H}}_2^{+}\\ {\mathcal{H}}_2^0\end{array}\right)=\left(\begin{array}{c}{\mathit{\varphi }}_2^{+}\\ v_2+\frac{1}{\sqrt{2}}({\mathit{\varphi }}_2^0+i{\mathit{\chi }}_2^0)\end{array}\right).\hfill \end{array}$$

The bilinear part of the Higgs potential leads to the tree-level mass matrix for the neutral $\mathcal{CP}$-even Higgs bosons,

$$\begin{array}{cc}\hfill M_{\text{Higgs}}^{2,\text{tree}}& =\left(\begin{array}{cc}{m}_{{\mathit{\varphi }}_1}^2\hfill & m_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^2\hfill \\ m_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^2\hfill & m_{{\mathit{\varphi }}_2}^2\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}{M}_A^2{sin}^2\mathit{\beta }+M_Z^2{cos}^2\mathit{\beta }\hfill & -(M_A^2+M_Z^2)sin\mathit{\beta }cos\mathit{\beta }\hfill \\ -(M_A^2+M_Z^2)sin\mathit{\beta }cos\mathit{\beta }\hfill & M_A^2{cos}^2\mathit{\beta }+M_Z^2{sin}^2\mathit{\beta }\hfill \end{array}\right),\hfill \end{array}$$ 1

in the $({\mathit{\varphi }}_1,{\mathit{\varphi }}_2)$ basis, expressed in terms of the Z boson mass, $M_Z$, $M_A$, and the angle $\mathit{\beta }$. Diagonalization via the angle $\mathit{\alpha }$ yields the tree-level masses $m_{h,\mathrm{tree}}$ and $m_{H,\mathrm{tree}}$. Below we also use $M_W$, denoting the W boson mass and $s_{\mathrm{w}}$, the sine of the weak mixing angle, $s_{\mathrm{w}}=\sqrt{1-c_{\mathrm{w}}^2}=\sqrt{1-M_W^2/M_Z^2}$.

The higher-order-corrected $\mathcal{CP}$-even Higgs-boson masses in the MSSM are obtained from the corresponding propagators dressed by their self-energies. The calculation of these and their renormalization is performed in the $({\mathit{\varphi }}_1,{\mathit{\varphi }}_2)$ basis, which has the advantage that the mixing angle $\mathit{\alpha }$ does not appear and expressions are in general simpler. The inverse propagator matrix in the $({\mathit{\varphi }}_1,{\mathit{\varphi }}_2)$ basis is given by

$$\begin{array}{cc}& {({\mathrm{\Delta }}_{\text{Higgs}})}^{-1}\hfill \\ \hfill & =-\text{i}\left(\begin{array}{cc}{p}^2-m_{{\mathit{\varphi }}_1}^2+{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1}(p^2)\hfill & -m_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^2+{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}(p^2)\hfill \\ -m_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^2+{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}(p^2)\hfill & p^2-m_{{\mathit{\varphi }}_2}^2+{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_2}(p^2)\hfill \end{array}\right),\hfill \end{array}$$ 2

where $\widehat{\mathrm{\Sigma }}(p^2)$ denote the renormalized Higgs-boson self-energies, p being the external momentum. The renormalized self-energies can be expressed through the unrenormalized self-energies, $\mathrm{\Sigma }(p^2)$, and counterterms involving renormalization constants $\mathit{\delta }{m}^2$ and $\mathit{\delta }Z$ from parameter and field renormalization. With the self-energies expanded up to two-loop order, $\widehat{\mathrm{\Sigma }}={\widehat{\mathrm{\Sigma }}}^{(1)}+{\widehat{\mathrm{\Sigma }}}^{(2)}$, one has for the $\mathcal{CP}$-even part at the i-loop level ($i=1,2$),

$$\begin{array}{cc}& {\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1}^{(i)}(p^2)={\mathrm{\Sigma }}_{{\mathit{\varphi }}_1}^{(i)}(p^2)+\mathit{\delta }{Z}_{{\mathit{\varphi }}_1}^{(i)}(p^2-m_{{\mathit{\varphi }}_1}^2)-\mathit{\delta }{m}_{{\mathit{\varphi }}_1}^{2(i)},\hfill \end{array}$$ 3a

$$\begin{array}{cc}& {\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^{(i)}(p^2)={\mathrm{\Sigma }}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^{(i)}(p^2)-\mathit{\delta }{Z}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^{(i)}{m}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^2-\mathit{\delta }{m}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^{2(i)},\hfill \end{array}$$ 3b

$$\begin{array}{cc}& {\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_2}^{(i)}(p^2)={\mathrm{\Sigma }}_{{\mathit{\varphi }}_2}^{(i)}(p^2)+\mathit{\delta }{Z}_{{\mathit{\varphi }}_2}^{(i)}(p^2-m_{{\mathit{\varphi }}_2}^2)-\mathit{\delta }{m}_{{\mathit{\varphi }}_2}^{2(i)}.\hfill \end{array}$$ 3c

At the two-loop level the expressions in Eqs. (3) do not contain contributions of the type (1-loop) $\times$ (1-loop); such terms do not appear at $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ and hence can be omitted in the context of this paper. For the general expressions see Ref. [18].

Beyond the one-loop level, unrenormalized self-energies contain sub-loop renormalizations. At the two-loop level, these are one-loop diagrams with counterterm insertions at the one-loop level.

Renormalization

The following section summarizes the renormalization worked out in Ref. [1], based on Ref. [29]. The field renormalization is carried out by assigning one renormalization constant to each doublet,

$$\begin{array}{c}\hfill {\mathcal{H}}_1\to (1+\frac{1}{2}\mathit{\delta }{Z}_{{\mathcal{H}}_1}){\mathcal{H}}_1,{\mathcal{H}}_2\to (1+\frac{1}{2}\mathit{\delta }{Z}_{{\mathcal{H}}_2}){\mathcal{H}}_2,\end{array}$$ 4

which can be expanded to one- and two-loop order according to

$$\begin{array}{cc}\hfill \mathit{\delta }{Z}_{{\mathcal{H}}_1}& =\mathit{\delta }{Z}_{{\mathcal{H}}_1}^{(1)}+\mathit{\delta }{Z}_{{\mathcal{H}}_1}^{(2)},\mathit{\delta }{Z}_{{\mathcal{H}}_2}=\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(1)}+\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(2)}.\hfill \end{array}$$ 5

The field renormalization constants appearing in (3) are then given by

$$\begin{array}{cc}& \mathit{\delta }{Z}_{{\mathit{\varphi }}_1}^{(i)}=\mathit{\delta }{Z}_{{\mathcal{H}}_1}^{(i)},\mathit{\delta }{Z}_{{\mathit{\varphi }}_2}^{(i)}=\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(i)},\hfill \\ \hfill & \mathit{\delta }{Z}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^{(i)}=\frac{1}{2}(\mathit{\delta }{Z}_{{\mathcal{H}}_1}^{(i)}+\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(i)}).\hfill \end{array}$$ 6

The mass counterterms $\mathit{\delta }{m}_{ab}^{2(i)}$ in Eq. (3) are derived from the Higgs potential, including the tadpoles, by the following parameter renormalization:

$$\begin{array}{cc}& M_A^2\to M_A^2+\mathit{\delta }{M}_A^{2(1)}+\mathit{\delta }{M}_A^{2(2)},\hfill \\ \hfill & T_1\to T_1+\mathit{\delta }{T}_1^{(1)}+\mathit{\delta }{T}_1^{(2)},\hfill \\ \hfill & M_Z^2\to M_Z^2+\mathit{\delta }{M}_Z^{2(1)}+\mathit{\delta }{M}_Z^{2(2)},\hfill \\ \hfill & T_2\to T_2+\mathit{\delta }{T}_2^{(1)}+\mathit{\delta }{T}_2^{(2)},\hfill \\ \hfill & tan\mathit{\beta }\to tan\mathit{\beta }\left(1+\mathit{\delta }tan{\mathit{\beta }}^{(1)}+\mathit{\delta }tan{\mathit{\beta }}^{(2)}\right).\hfill \end{array}$$ 7

The parameters $T_1$ and $T_2$ are the terms linear in ${\mathit{\varphi }}_1$ and ${\mathit{\varphi }}_2$ in the Higgs potential. The renormalization of the Z-mass $M_Z$ does not contribute to the $\mathcal{O}({\mathit{\alpha }}_s{\mathit{\alpha }}_t)$ corrections we are pursuing here; it is listed for completeness only.

The basic renormalization constants for parameters and fields have to be fixed by renormalization conditions according to a renormalization scheme. Here we choose the on-shell scheme for the parameters and the $\overline{\mathrm{DR}}$ scheme for field renormalization and give the expressions for the two-loop part. This is consistent with the renormalization scheme used at the one-loop level.

The tadpole coefficients are chosen to vanish at all orders; hence their two-loop counterterms follow from

$$\begin{array}{c}\hfill T_{1,2}^{(2)}+\mathit{\delta }{T}_{1,2}^{(2)}=0,\text{i.e.}\mathit{\delta }{T}_1^{(2)}=-T_1^{(2)},\mathit{\delta }{T}_2^{(2)}=-T_2^{(2)},\end{array}$$ 8

where $T_1^{(2)}$, $T_2^{(2)}$ are obtained from the two-loop tadpole diagrams. The two-loop renormalization constant of the A-boson mass reads

$$\begin{array}{c}\hfill \mathit{\delta }{M}_A^{2(2)}=\mathrm{Re}{\mathrm{\Sigma }}_{AA}^{(2)}(M_A^2),\end{array}$$ 9

in terms of the A-boson unrenormalized self-energy ${\mathrm{\Sigma }}_{AA}$. The appearance of a non-zero momentum in the self-energy goes beyond the $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections evaluated in Refs. [27–29, 35].

For the renormalization constants $\mathit{\delta }{Z}_{{\mathcal{H}}_1}$, $\mathit{\delta }{Z}_{{\mathcal{H}}_2}$, and $\mathit{\delta }tan\mathit{\beta }$ several choices are possible; see the discussion in [67–69]. As shown there, the most convenient choice is a $\overline{\mathrm{DR}}$ renormalization of $\mathit{\delta }tan\mathit{\beta }$, $\mathit{\delta }{Z}_{{\mathcal{H}}_1}$, and $\mathit{\delta }{Z}_{{\mathcal{H}}_2}$, which at the two-loop level reads

$$\begin{array}{cc}& \mathit{\delta }{Z}_{{\mathcal{H}}_1}^{(2)}=\mathit{\delta }{Z}_{{\mathcal{H}}_1}^{\overline{\mathrm{DR}}(2)}=-{\left[\mathrm{Re}{\mathrm{\Sigma }}_{{\mathit{\varphi }}_1}^{\prime (2)}\right]}_{|p^2=0}^{\text{div}},\hfill \end{array}$$ 10a

$$\begin{array}{cc}& \mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(2)}=\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\overline{\mathrm{DR}}(2)}=-{\left[\mathrm{Re}{\mathrm{\Sigma }}_{{\mathit{\varphi }}_2}^{\prime (2)}\right]}_{|p^2=0}^{\text{div}},\hfill \end{array}$$ 10b

$$\begin{array}{cc}& \mathit{\delta }tan{\mathit{\beta }}^{(2)}=\mathit{\delta }tan{\mathit{\beta }}^{\overline{\mathrm{DR}}(2)}=\frac{1}{2}\left(\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(2)}-\mathit{\delta }{Z}_{{\mathcal{H}}_1}^{(2)}\right).\hfill \end{array}$$ 10c

The term in Eq. (10c) is in general not the proper expression beyond one-loop order even in the $\overline{\mathrm{DR}}$ scheme. For our approximation, however, with only the top Yukawa coupling at the two-loop level, it is the correct $\overline{\mathrm{DR}}$ form [70, 71].

The two-loop mass counterterms in the renormalized self-energies (3) are now expressed in terms of the two-loop parameter renormalization constants, determined above, as follows:

$$\begin{array}{cc}\hfill \mathit{\delta }{m}_{{\mathit{\varphi }}_1}^{2(2)}& =\mathit{\delta }{M}_Z^{2(2)}{cos}^2\mathit{\beta }+\mathit{\delta }{M}_A^{2(2)}{sin}^2\mathit{\beta }\hfill \\ \hfill & -\mathit{\delta }{T}_1^{(2)}\frac{e}{2M_W{s}_{\mathrm{w}}}cos\mathit{\beta }(1+{sin}^2\mathit{\beta })\hfill \\ \hfill & +\mathit{\delta }{T}_2^{(2)}\frac{e}{2M_W{s}_{\mathrm{w}}}{cos}^2\mathit{\beta }sin\mathit{\beta }\hfill \\ \hfill & +2\mathit{\delta }tan{\mathit{\beta }}^{(2)}{cos}^2\mathit{\beta }{sin}^2\mathit{\beta }(M_A^2-M_Z^2),\hfill \end{array}$$ 11a

$$\begin{array}{cc}\hfill \mathit{\delta }{m}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^{2(2)}& =-(\mathit{\delta }{M}_Z^{2(2)}+\mathit{\delta }{M}_A^{2(2)})sin\mathit{\beta }cos\mathit{\beta }\hfill \\ \hfill & -\mathit{\delta }{T}_1^{(2)}\frac{e}{2M_W{s}_{\mathrm{w}}}{sin}^3\mathit{\beta }-\mathit{\delta }{T}_2^{(2)}\frac{e}{2M_W{s}_{\mathrm{w}}}{cos}^3\mathit{\beta }\hfill \\ \hfill & -\mathit{\delta }tan{\mathit{\beta }}^{(2)}cos\mathit{\beta }sin\mathit{\beta }cos2\mathit{\beta }(M_A^2+M_Z^2),\hfill \end{array}$$ 11b

$$\begin{array}{cc}\hfill \mathit{\delta }{m}_{{\mathit{\varphi }}_2}^{2(2)}& =\mathit{\delta }{M}_Z^{2(2)}{sin}^2\mathit{\beta }+\mathit{\delta }{M}_A^{2(2)}{cos}^2\mathit{\beta }\hfill \\ \hfill & +\mathit{\delta }{T}_1^{(2)}\frac{e}{2M_W{s}_{\mathrm{w}}}{sin}^2\mathit{\beta }cos\mathit{\beta }\hfill \\ \hfill & -\mathit{\delta }{T}_2^{(2)}\frac{e}{2M_W{s}_{\mathrm{w}}}sin\mathit{\beta }(1+{cos}^2\mathit{\beta })\hfill \\ \hfill & -2\mathit{\delta }tan{\mathit{\beta }}^{(2)}{cos}^2\mathit{\beta }{sin}^2\mathit{\beta }(M_A^2-M_Z^2).\hfill \end{array}$$ 11c

The Z-mass counterterm is again kept for completeness; it does not contribute in the approximation of $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ considered here.

Diagram evaluation

Our calculation is performed in the Feynman-diagrammatic (FD) approach. To arrive at expressions for the unrenormalized self-energies and tadpoles at $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$, the evaluation of genuine two-loop diagrams and one-loop graphs with counterterm insertions is required. For the counterterm insertions, described in Sect. 2.4, one-loop diagrams with external top quarks/squarks have to be evaluated as well, as displayed in Fig. 1. The calculation is performed in dimensional reduction [72, 73].

Fig. 1.

Generic one-loop diagrams for subrenormalization counterterms for the top quark (upper row) and for the scalar tops (lower row) ($i,j,k=1,2$)

The complete set of contributing Feynman diagrams was generated with the program FeynArts [74–76] (using the model file including counterterms from Ref. [77]), tensor reduction and the evaluation of traces was done with support from the programs FormCalc [78] and TwoCalc [79, 80], yielding algebraic expressions in terms of the scalar one-loop functions $A_0$, $B_0$ [81], the massive vacuum two-loop functions [82], and two-loop integrals which depend on the external momentum. These integrals were evaluated with the program SecDec [64–66], where up to four different masses in 34 different mass configurations needed to be considered, with differences in the kinematic invariants of several orders of magnitude.

The scalar-top sector of the MSSM

The bilinear part of the top-squark Lagrangian,

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\tilde{t},\text{mass}}& =-\left(\begin{array}{c}{{\tilde{t}}_L}^{†},{{\tilde{t}}_R}^{†}\end{array}\right){\mathbf{M}}_{\tilde{t}}\left(\begin{array}{c}{\tilde{t}}_L\\ {\tilde{t}}_R\end{array}\right),\hfill \end{array}$$ 12

contains the stop-mass matrix

$$\begin{array}{c}\hfill {\mathbf{M}}_{\tilde{t}}=\left(\begin{array}{cc}{M}_{{\tilde{t}}_L}^2+m_t^2+M_Z^{2}cos2\mathit{\beta }(T_t^3-Q_t{s}_{\mathrm{w}}^2)\hfill & m_t{X}_t\hfill \\ m_t{X}_t\hfill & M_{{\tilde{t}}_R}^2+m_t^2+M_Z^{2}cos2\mathit{\beta }{Q}_t{s}_{\mathrm{w}}^2\hfill \end{array}\right),\end{array}$$ 13

with

$$\begin{array}{c}\hfill X_t=A_t-\mathit{\mu }cot\mathit{\beta }\end{array}$$ 14

where $Q_t$ and $T_t^3$ denote the charge and isospin of the top quark, $A_t$ the trilinear coupling between the Higgs bosons and the scalar tops, and $\mathit{\mu }$ the Higgsino mass parameter. Below we use $M_{\mathrm{SUSY}}:=M_{{\tilde{t}}_L}=M_{{\tilde{t}}_R}$ for our numerical evaluation. The analytical calculation was performed for arbitrary $M_{{\tilde{t}}_L}$ and $M_{{\tilde{t}}_R}$, however. ${\mathbf{M}}_{\tilde{t}}$ can be diagonalized with the help of a unitary transformation matrix ${\mathbf{U}}_{\tilde{t}}$, parameterized by a mixing angle ${\mathit{\theta }}_{\tilde{t}}$, to provide the eigenvalues $m_{{\tilde{t}}_1}^2$ and $m_{{\tilde{t}}_2}^2$ as the squares of the two on-shell top-squark masses.

For the evaluation of the $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ two-loop contributions to the self-energies and tadpoles of the Higgs sector, renormalization of the top/stop sector at $\mathcal{O}({\mathit{\alpha }}_s)$ is required, giving rise to the counterterms for sub-loop renormalization.

We follow the renormalization at the one-loop level given in Refs. [31, 83–85], where details can be found. In particular, in the context of this paper, an OS renormalization is performed for the top-quark mass as well as for the scalar-top masses. This is different from the approach pursued, for example, in Ref. [45], where a $\overline{\mathrm{DR}}$ renormalization was employed, or similarly in the pure $\overline{\mathrm{DR}}$ renormalization presented in Ref. [2]. Using the OS scheme allows us to consistently combine our new correction terms with the hitherto available self-energies included in FeynHiggs.

Besides employing a pure OS renormalization for the top/stop masses in our calculation, we also obtain a result in which the top-quark mass is renormalized $\overline{\mathrm{DR}}$. This new top-quark mass renormalization is included as a new option in the code FeynHiggs. The comparison of the results using the $\overline{\mathrm{DR}}$ and the OS renormalization allows one to estimate (some) missing three-loop corrections in the top/stop sector.

Finally, at $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$, gluinos appear as virtual particles only at the two-loop level (hence, no renormalization for the gluinos is needed). The corresponding soft-breaking gluino mass parameter $M_3$ determines the gluino mass, $m_{\tilde{g}}=M_3$.

Evaluation and implementation in the program FeynHiggs

The resulting new contributions to the neutral $\mathcal{CP}$-even Higgs-boson self-energies, containing all momentum-dependent and additional constant terms, are assigned to the differences

$$\begin{array}{c}\hfill \mathrm{\Delta }{\widehat{\mathrm{\Sigma }}}_{ab}(p^2)={\widehat{\mathrm{\Sigma }}}_{ab}^{(2)}(p^2)-{\tilde{\mathrm{\Sigma }}}_{ab}^{(2)}(0),ab=\{HH,hH,hh\}.\\ \hfill \end{array}$$ 15

These are the new terms evaluated in Ref. [1], included in FeynHiggs. Note the tilde (not hat) on ${\tilde{\mathrm{\Sigma }}}^{(2)}(0)$, which signifies that not only the self-energies are evaluated at zero external momentum but also the corresponding counterterms, following Refs. [27–29]. A finite shift $\mathrm{\Delta }\widehat{\mathrm{\Sigma }}(0)$ therefore remains in the limit $p^2\to 0$ due to $\mathit{\delta }{M}_A^{2(2)}=\mathrm{Re}{\mathrm{\Sigma }}_{AA}^{(2)}(M_A^2)$ being computed at $p^2=M_A^2$ in ${\widehat{\mathrm{\Sigma }}}^{(2)}$, but at $p^2=0$ in ${\tilde{\mathrm{\Sigma }}}^{(2)}$; for details see Eqs. (9) and (11). For the sake of simplicity we will refer to these terms as $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ despite the $M_A^2$ dependence.

Discussion of renormalization schemes

In this section we compare our results for the $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ contributions to the MSSM Higgs-boson self-energies, as given in Ref. [1] to the ones presented subsequently in Ref. [2]. We first show analytically the agreement in the Higgs field renormalization in the two calculations and discuss the differences in the $m_t$ renormalizations. We also present some numerical results in both schemes, demonstrating agreement with Ref. [2] once the $\mathcal{O}(\mathit{\epsilon })$ terms are dropped from the top-quark mass counterterm.

Using an OS renormalization for the top-quark mass, the counterterm is determined from the components of the $\mathcal{O}({\mathit{\alpha }}_s)$ top-quark self-energy (Fig. 1) as follows:

$$\begin{array}{cc}\hfill \frac{\mathit{\delta }{m}_t^{\mathrm{OS}}}{m_t}& =\frac{1}{2}\mathrm{Re}\left\{\left[{\mathrm{\Sigma }}_t^L(m_t^2)+{\mathrm{\Sigma }}_t^R(m_t^2)\right]\right.\hfill \\ \hfill & \left.+\left[{\mathrm{\Sigma }}_t^{SL}(m_t^2)+{\mathrm{\Sigma }}_t^{SR}(m_t^2)\right]\right\},\hfill \end{array}$$ 16

where the top-quark self-energy is decomposed according to

$$\begin{array}{cc}\hfill {\mathrm{\Sigma }}_t(p)& =p/{\mathit{\omega }}_{-}{\mathrm{\Sigma }}_t^L(p^2)+p/{\mathit{\omega }}_{+}{\mathrm{\Sigma }}_t^R(p^2)\hfill \\ \hfill & +m_t{\mathit{\omega }}_{-}{\mathrm{\Sigma }}_t^{SL}(p^2)+m_t{\mathit{\omega }}_{+}{\mathrm{\Sigma }}_t^{SR}(p^2).\hfill \end{array}$$ 17

with the projectors ${\mathit{\omega }}_{\pm }=\frac{1}{2}(1\mathrm{l}\pm {\mathit{\gamma }}_5)$.

Analytical comparison

In the $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ calculation of the Higgs-boson self-energies the renormalization of the top-quark mass at $\mathcal{O}({\mathit{\alpha }}_s)$ is required. The contributing diagrams are shown in the top row of Fig. 1. The top-quark mass counterterm is inserted into the sub-loop renormalization of the two-loop contributions to the Higgs-boson self-energies, where two sample diagrams are shown in Fig. 2. The left diagram contributes to the momentum-dependent two-loop self-energies, while the right one contributes only to the momentum-independent part.

Fig. 2.

One-loop subrenormalization diagram contributing to ${\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}(p^2)$ and ${\mathit{\delta }}_A(p^2)$, with the counterterm insertion denoted by a cross. The right diagram only contributes to ${\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}(0)$ and ${\mathit{\delta }}_A(0)$

Evaluating the expression in Eq. (16) in $4-2\mathit{\epsilon }$ dimensions yields the OS top-quark mass counterterm at the one-loop level, which can be written as a Laurent expansion in $\mathit{\epsilon }$,

$$\begin{array}{c}\hfill \mathit{\delta }{m}_t^{\mathrm{OS}}=\frac{1}{\mathit{\epsilon }}\mathit{\delta }{m}_t^{\mathrm{div}}+\mathit{\delta }{m}_t^{\mathrm{fin}}+\mathit{\epsilon }\mathit{\delta }{m}_t^{\mathit{\epsilon }}+\cdots ;\end{array}$$ 18

higher powers in $\mathit{\epsilon }$, indicated by the ellipses, do not contribute at the two-loop level for $\mathit{\epsilon }\to 0$ after renormalization. Accordingly, the $\overline{\mathrm{DR}}$ top-quark mass counterterm is given by the singular part of Eq. (18),

$$\begin{array}{c}\hfill \mathit{\delta }{m}_t^{\overline{\mathrm{DR}}}=\frac{1}{\mathit{\epsilon }}\mathit{\delta }{m}_t^{\mathrm{div}}.\end{array}$$ 19

For further use we define the quantity

$$\begin{array}{c}\hfill \mathit{\delta }{m}_t^{\mathrm{FIN}}=\frac{1}{\mathit{\epsilon }}\mathit{\delta }{m}_t^{\mathrm{div}}+\mathit{\delta }{m}_t^{\mathrm{fin}}.\end{array}$$ 20

At $\mathcal{O}({\mathit{\alpha }}_s)$ the OS counterterm is given as

$$\begin{array}{cc}\hfill \frac{\mathit{\delta }{m}_t^{\mathrm{OS}}}{m_t}& =\frac{{\mathit{\alpha }}_s}{6\mathit{\pi }}\left\{-2\frac{A_0(m_t^2)}{m_t^2}-4B_0(m_t^2,0,m_t^2)\right.\hfill \\ \hfill & -2\frac{A_0(m_{\tilde{g}}^2)}{m_t^2}+\frac{A_0(m_{{\tilde{t}}_1}^2)}{m_t^2}+\frac{A_0(m_{{\tilde{t}}_2}^2)}{m_t^2}\hfill \\ \hfill & +\frac{m_{\tilde{g}}^2+m_t^2-m_{{\tilde{t}}_1}^2-4sin{\mathit{\theta }}_{\tilde{t}}cos{\mathit{\theta }}_{\tilde{t}}{m}_{\tilde{g}}{m}_t}{m_t^2}\hfill \\ \hfill & \times \mathrm{Re}\left[B_0(m_t^2,m_{\tilde{g}}^2,m_{{\tilde{t}}_1}^2)\right]\hfill \\ \hfill & +\frac{m_{\tilde{g}}^2+m_t^2-m_{{\tilde{t}}_2}^2+4sin{\mathit{\theta }}_{\tilde{t}}cos{\mathit{\theta }}_{\tilde{t}}{m}_{\tilde{g}}{m}_t}{m_t^2}\hfill \\ \hfill & \left.\times \mathrm{Re}\left[B_0(m_t^2,m_{\tilde{g}}^2,m_{{\tilde{t}}_2}^2)\right]\right\}.\hfill \end{array}$$ 21

The one- and two-point functions $A_0(m^2)$ and $B_0(p^2,m_1^2,m_2^2)$ are expanded in $\mathit{\epsilon }$ as follows:

$$\begin{array}{cc}\hfill A_0(m^2)& =\frac{1}{\mathit{\epsilon }}{A}_0^{\text{div}}(m^2)+A_0^{\text{fin}}(m^2)+\mathit{\epsilon }{A}_0^{\mathit{\epsilon }}(m^2),\hfill \\ \hfill B_0(p^2,m_1^2,m_2^2)& =\frac{1}{\mathit{\epsilon }}{B}_0^{\text{div}}(p^2,m_1^2,m_2^2)+B_0^{\text{fin}}(p^2,m_1^2,m_2^2)\hfill \\ \hfill & +\mathit{\epsilon }{B}_0^{\mathit{\epsilon }}(p^2,m_1^2,m_2^2).\hfill \end{array}$$ 22

Consequently, the term at $\mathcal{O}(\mathit{\epsilon })$, $\mathit{\delta }{m}_t^{\mathit{\epsilon }}/m_t$, is given by Eq. (21), but taking only into account the pieces $\propto A_0^{\mathit{\epsilon }},B_0^{\mathit{\epsilon }}$. The special cases of $A_0^{\mathit{\epsilon }}(m^2)$ and $B_0^{\mathit{\epsilon }}(m^2,0,m^2)$ are given by

$$\begin{array}{cc}& A_0^{\mathit{\epsilon }}(m^2)=m^2\left\{1-log(m^2/{\mathit{\mu }}^2)+\frac{1}{2}{log}^2(m^2/{\mathit{\mu }}^2)+\frac{{\mathit{\pi }}^2}{12}\right\},\hfill \\ \hfill & B_0^{\mathit{\epsilon }}(m^2,0,m^2)=4-2log(m^2/{\mathit{\mu }}^2)+\frac{1}{2}{log}^2(m^2/{\mathit{\mu }}^2)+\frac{{\mathit{\pi }}^2}{12},\hfill \end{array}$$ 23

where the factor $4\mathit{\pi }{\mathrm{e}}^{-{\mathit{\gamma }}_E}$ is absorbed into the renormalization scale. The expression for $B_0^{\mathit{\epsilon }}$ depending on three mass scales can be found e.g. in Ref. [86].

In our calculation in Ref. [1] we include terms up to $\mathcal{O}(\mathit{\epsilon })$, originating from the top-quark self-energy, in the top-mass counterterm,3 i.e.

$$\begin{array}{c}\hfill \mathit{\delta }{m}_t^{[1]}=\mathit{\delta }{m}_t^{\mathrm{OS}}.\end{array}$$ 24

The derivation in Ref. [2] proceeds differently. The renormalized Higgs-boson self-energies are first calculated in a pure $\overline{\mathrm{DR}}$ scheme. This concerns the top mass, the scalar-top masses, the Higgs field renormalization, and $tan\mathit{\beta }$. In this way it is ensured that in particular the Higgs fields are renormalized using $\overline{\mathrm{DR}}$, $\mathit{\delta }{Z}_{{\mathcal{H}}_i}=\mathit{\delta }{Z}_{{\mathcal{H}}_i}^{\overline{\mathrm{DR}}}$, where this quantity contains the contribution from the one- and two-loop level. Using this pure $\overline{\mathrm{DR}}$ scheme a finite result is obtained in which all poles in $1/\mathit{\epsilon }$ and $1/{\mathit{\epsilon }}^2$ cancel, such that the limit $\mathit{\epsilon }\to 0$ can be taken. Subsequently, the $\overline{\mathrm{DR}}$ top-quark mass counterterm, $\mathit{\delta }{m}_t^{\overline{\mathrm{DR}}}$, is replaced by an on-shell counterterm, and the top-quark mass definition is changed accordingly. The same procedure is applied for the scalar-top masses. Since these finite expressions for the renormalized Higgs-boson self-energies do not contain any term of $\mathcal{O}(1/\mathit{\epsilon })$, the $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ part of the OS top-quark mass counterterm does not contribute, i.e.

$$\begin{array}{c}\hfill \mathit{\delta }{m}_t^{[2]}=\mathit{\delta }{m}_t^{\mathrm{FIN}}.\end{array}$$ 25

The numerical results for the renormalized Higgs-boson self-energies obtained this way differ significantly from the ones obtained in Ref. [1], as pointed out in Ref. [2].

In the following we discuss the different Higgs-boson field renormalizations, where we use the notation of $\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\text{X}}}$ for the field renormalization derived using $\mathit{\delta }{m}_t^{\text{X}}$, with $\text{X}=\overline{\mathrm{DR}}$, FIN, $\mathrm{OS}$. The field renormalization can be decomposed into one-loop, two-loop, ...parts as

$$\begin{array}{c}\hfill \mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\text{X}}}=\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\text{X}}(1)}+\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\text{X}}(2)}+\cdots \end{array}$$ 26

In Ref. [2] it was claimed that using an OS top-quark mass renormalization from the start results in a non-$\overline{\mathrm{DR}}$ renormalization of $\mathit{\delta }{Z}_{{\mathcal{H}}_2}$. While it is correct that an OS value for $m_t$ yields different results in the one- and two-loop part,

$$\begin{array}{c}\hfill \mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\mathrm{OS}}(1)}\ne \mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\overline{\mathrm{DR}}}(1)},\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\mathrm{OS}}(2)}\ne \mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\overline{\mathrm{DR}}}(2)},\end{array}$$ 27

the sum of the one- and two-loop parts are identical, independently of the choice of the top-quark mass renormalization (see e.g. Eqs. (3.60)–(3.62) in Ref. [90]),

$$\begin{array}{cc}\hfill \left(\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{[1]}=\right)\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\mathrm{OS}}}{|}_{\text{div}}& =\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\mathrm{FIN}}}=\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\overline{\mathrm{DR}}}}\left(=\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{[2]}\right),\hfill \end{array}$$ 28

provided that also in $\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\mathrm{OS}}}$ all finite pieces are dropped, as done in Ref. [1]. Differences between $\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{[1]}$ and $\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{[2]}$ arise only at the three-loop level. Consequently, the claim in Ref. [2] that using $\mathit{\delta }{m}_t^{\mathrm{OS}}$ leads to an inconsistency in the Higgs field renormalization in Ref. [1] is not correct. The field renormalizations thus cannot be responsible for the observed differences between Refs. [1, 2].

More explicitly, the difference between the two calculations results from non-vanishing $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ terms in the renormalized Higgs-boson self-energies. Those terms naturally appear when performing a full expansion in the dimensional regulator $\mathit{\epsilon }$. The latter corresponds to choosing $\mathit{\delta }{m}_t^{\mathrm{OS}}$ (as done in Ref. [1]) instead of $\mathit{\delta }{m}_t^{\mathrm{FIN}}$ (as done in Ref. [2]).

In order to isolate the contributions coming from $\mathcal{O}(\mathit{\epsilon })$ terms $\times$$1/\mathit{\epsilon }$ poles we define the following quantities, where superscripts $\mathrm{OS}$, $\mathrm{FIN}$ refer to the respective use of $\mathit{\delta }{m}_t^{\mathrm{OS}}$, $\mathit{\delta }{m}_t^{\mathrm{FIN}}$:

$$\begin{array}{cc}& \mathit{\delta }{T}_i^{(2)\mathrm{OS}}=\mathit{\delta }{T}_i^{(2)\mathrm{FIN}}+{\mathit{\delta }}_{T_i},\hfill \end{array}$$ 29a

$$\begin{array}{cc}& {\mathrm{\Sigma }}_{{\mathit{\varphi }}_{ij}}^{(2)\mathrm{OS}}(p^2)={\mathrm{\Sigma }}_{{\mathit{\varphi }}_{ij}}^{(2)\mathrm{FIN}}(p^2)+{\mathit{\delta }}_{{\mathrm{\Sigma }}_{ij}}(p^2),\hfill \end{array}$$ 29b

$$\begin{array}{cc}& {\mathrm{\Sigma }}_{AA}^{(2)\mathrm{OS}}(p^2)={\mathrm{\Sigma }}_{AA}^{(2)\mathrm{FIN}}(p^2)+{\mathit{\delta }}_A(p^2),\hfill \end{array}$$ 29c

where the last equation yields a shift for the A-boson mass counterterm in Eq. (7),

$$\begin{array}{c}\hfill \mathit{\delta }{M}_A^{2(2)\mathrm{OS}}=\mathit{\delta }{M}_A^{2(2)\mathrm{FIN}}+{\mathit{\delta }}_A(M_A^2).\end{array}$$ 30

The $\mathit{\delta }$-terms are defined as the finite contributions stemming from $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$-dependent parts in the counterterms (see the left diagram in Fig. 2 for an example). The $\overline{\mathrm{DR}}$-renormalized quantities do not contain a finite $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$-dependent part by definition. Furthermore, since ${\mathit{\varphi }}_1$ has no coupling to the top quark, there are no terms proportional to $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ in ${\mathrm{\Sigma }}_{{\mathit{\varphi }}_1}^{(2)}$, ${\mathrm{\Sigma }}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^{(2)}$, and $\mathit{\delta }{T}_1^{(2)}$, and it is sufficient to consider ${\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}$, ${\mathit{\delta }}_A$, and ${\mathit{\delta }}_{T_2}$ only. While ${\mathit{\delta }}_{T_2}$ is $p^2$-independent, we find

$$\begin{array}{cc}\hfill {\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}(p^2)& =\frac{3{\mathit{\alpha }}_t}{2\mathit{\pi }}{p}^2\frac{\mathit{\delta }{m}_t^{\mathit{\epsilon }}}{m_t}+{\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}(0),\hfill \end{array}$$ 31

$$\begin{array}{cc}\hfill {\mathit{\delta }}_A(p^2)& =\frac{3{\mathit{\alpha }}_t}{2\mathit{\pi }}{p}^2{cos}^2\mathit{\beta }\frac{\mathit{\delta }{m}_t^{\mathit{\epsilon }}}{m_t}+{\mathit{\delta }}_A(0).\hfill \end{array}$$ 32

Using Eqs. (3), (11) we find that the following relations hold for the renormalized Higgs-boson self-energies:

$$\begin{array}{cc}& -{sin}^2\mathit{\beta }{\mathit{\delta }}_A(0)-\frac{e}{2M_W{s}_{\mathrm{w}}}{cos}^2\mathit{\beta }sin\mathit{\beta }{\mathit{\delta }}_{T_2}=0(\text{for}{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1}^{(2)}),\hfill \\ \hfill & sin\mathit{\beta }cos\mathit{\beta }{\mathit{\delta }}_A(0)+\frac{e}{2M_W{s}_{\mathrm{w}}}{cos}^3\mathit{\beta }{\mathit{\delta }}_{T_2}=0(\text{for}{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^{(2)}),\hfill \\ \hfill & {\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}(0)-{cos}^2\mathit{\beta }{\mathit{\delta }}_A(0)+\frac{e}{2M_W{s}_{\mathrm{w}}}sin\mathit{\beta }(1+{cos}^2\mathit{\beta }){\mathit{\delta }}_{T_2}\hfill \\ \hfill & =0(\text{for}{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_2}^{(2)}).\hfill \end{array}$$ 33

This is in agreement with the observation that in the renormalized Higgs-boson self-energies at zero external momentum at $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$, the terms containing $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ drop out in the final (finite) result. Such a cancellation is to be expected as the same combination of one-loop self-energies that potentially contributes to this finite contribution also appears in the $\mathcal{O}(1/\mathit{\epsilon })$ term, where they must cancel. This argument in principle still holds when the momentum-dependent $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections are calculated and all counterterms are evaluated with a full expansion in $\mathit{\epsilon }$. Since the counterterm ${\mathit{\delta }}_A$ is evaluated at $p^2=M_A^2$, and the Higgs-boson fields are renormalized in the $\overline{\mathrm{DR}}$ scheme, however, one finds, using Eqs. (3) and (11) for the three renormalized Higgs-boson self-energies,

$$\begin{array}{cc}& -{sin}^2\mathit{\beta }\left({\mathit{\delta }}_A(M_A^2)-{\mathit{\delta }}_A(0)\right)\hfill \\ \hfill & =\frac{3{\mathit{\alpha }}_t}{2\mathit{\pi }}\left(-{cos}^2\mathit{\beta }{sin}^2\mathit{\beta }{M}_A^2\right)\frac{\mathit{\delta }{m}_t^{\mathit{\epsilon }}}{m_t}\left(\text{for}{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1}^{(2)}\right),\hfill \\ \hfill & sin\mathit{\beta }cos\mathit{\beta }\left({\mathit{\delta }}_A(M_A^2)-{\mathit{\delta }}_A(0)\right)\hfill \\ \hfill & =\frac{3{\mathit{\alpha }}_t}{2\mathit{\pi }}\left(+{cos}^3\mathit{\beta }sin\mathit{\beta }{M}_A^2\right)\frac{\mathit{\delta }{m}_t^{\mathit{\epsilon }}}{m_t}\left(\text{for}{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^{(2)}\right),\hfill \\ \hfill & \left({\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}(p^2)-{\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}(0)\right)-{cos}^2\mathit{\beta }\left({\mathit{\delta }}_A(M_A^2)-{\mathit{\delta }}_A(0)\right)\hfill \\ \hfill & =\frac{3{\mathit{\alpha }}_t}{2\mathit{\pi }}\left(p^2-{cos}^4\mathit{\beta }{M}_A^2\right)\frac{\mathit{\delta }{m}_t^{\mathit{\epsilon }}}{m_t}\left(\text{for}{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_2}^{(2)}\right),\hfill \end{array}$$ 34

i.e. the $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ terms contribute in the newly evaluated $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections. They are $p^2$-independent in ${\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1}^{(2)}$ and ${\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}^{(2)}$, while they do depend on $p^2$ in ${\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_2}^{(2)}$.

The $p^2$-dependent terms coming from the expansion of terms like ${(-p^2)}^{-\mathit{\epsilon }}$ multiplying a $1/{\mathit{\epsilon }}^2$ divergence must certainly cancel after inclusion of the counterterms, because non-local terms cannot appear in a renormalizable theory. However, the cancellation of the $\mathit{\epsilon }$-dependent terms stemming from the mass renormalization is not necessarily fulfilled once the two-loop amplitude carries full momentum dependence. Similarly, the truncation of the field renormalization to the divergent part cuts away terms involving $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$, leading to further non-cancellations. The explicit $\overline{\mathrm{DR}}$ renormalization of the Higgs-boson fields drops the corresponding finite contributions, such that no $\mathit{\delta }{m}_t^{\mathrm{fin}}$, $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ terms are taken into account. The different dependence on the external momentum and the $\overline{\mathrm{DR}}$ prescription for the Higgs field renormalization leads to Eqs. (34).

Equivalent momentum-dependent terms of $\mathcal{O}(\mathit{\epsilon })$ of the scalar-top mass counterterms, evaluated from the diagrams in the lower row of Fig. 1, do not contribute. The diagrams with top-squark counterterm insertions are depicted in Fig. 3. The first diagram is momentum independent. In the second diagram, the corresponding loop integral is a massive scalar three-point function ($C_0$) with only scalar particles running in the loop, and thus is UV finite. Consequently, the top-squark mass counterterm insertions of $\mathcal{O}(\mathit{\epsilon })$ do not contribute. In the third diagram the stop mass counterterm can enter via the (dependent) counterterm for $A_t$ [29, 84]. This diagram does not possess a momentum-dependent divergence, however, and thus the $\mathcal{O}(\mathit{\epsilon })$ term of the scalar-top mass counterterm again does not contribute.

Fig. 3.

One-loop subrenormalization diagrams containing top-squark loops with counterterm insertions

Physics content and interpretation

In the following we give another view on the finite $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ term from the top mass renormalization and on the interpretation of the different results for the Higgs-boson masses with and without this term.

In the approximation with $p^2=0$ for the two-loop self-energies, the results are the same for either dropping or including the $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ term, provided that this is done everywhere in the contributions from the top–stop sector in the renormalized two-loop self-energies.

As explained above, abandoning the $p^2=0$ approximation yields an additional $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ in the $p^2$-coefficient of the self-energy ${\mathrm{\Sigma }}_{{\mathit{\varphi }}_2}^{(2)}(p^2)$ when the on-shell top-quark mass counterterm, see Eq. (18), is used, as well as in the A-boson self-energy ${\mathrm{\Sigma }}_{AA}(p^2)$ from which it induces an additive term $\sim M_A^2\mathit{\delta }{m}_t^{\mathit{\epsilon }}/m_t$ to the mass counterterm $\mathit{\delta }{M}_A^2$.

In the renormalized self-energy ${\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_2}^{(2)}(p^2)$, Eq. (3c), this extra $p^2$-dependent term survives when $\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(2)}$ is defined in the minimal way containing only the $1/\mathit{\epsilon }$ and $1/{\mathit{\epsilon }}^2$ singular parts; however, it disappears in ${\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_2}^{(2)}(p^2)$ when the minimal $\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(2)}=\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{\mathit{\delta }{m}_t^{\mathrm{OS}}(2)}{|}_{\mathrm{div}}$ is replaced by

$$\begin{array}{c}\hfill \mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(2)}\to \mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(2)}-\frac{3{\mathit{\alpha }}_t}{2\mathit{\pi }}\frac{\mathit{\delta }{m}_t^{\mathit{\epsilon }}}{m_t},\end{array}$$ 35

which now accommodates also a finite part of two-loop order.

This shift in $\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(2)}$ by a finite term has also an impact on the counterterm for $tan\mathit{\beta }$ via $\mathit{\delta }tan\mathit{\beta }=\frac{1}{2}\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(2)}$. This has the consequence that the extra $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ term in $\mathit{\delta }{M}_A^2$ drops out in the constant counterterms for the renormalized self-energies ${\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_{ij}}^{(2)}(p^2)$ in Eq. (3) because of cancellations with the $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ term in $\mathit{\delta }tan\mathit{\beta }$ and $\mathit{\delta }{Z}_{{\mathcal{H}}_2}^{(2)}$ [this can be seen from the explicit expressions given in Eqs. (6) and (11)].

Accordingly, keeping or dropping the finite $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ part is thus equivalent to a finite shift in the field-renormalization constant $\mathit{\delta }{Z}_{{\mathcal{H}}_2}$ at the two-loop level, which corresponds to a finite shift in $tan\mathit{\beta }$ as input quantity. Numerically, the shift in $tan\mathit{\beta }$ is small, and cannot explain the differences in the $M_h$ predictions from the two schemes. Hence, these differences originate from the different $p^2$ coefficients in ${\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_2}^{(2)}(p^2)$.

The impact of a modification of the two-loop field-renormalization constant on the mass $M_h$ can best be studied in terms of the self-energy ${\mathrm{\Sigma }}_{hh}$ in the h, H basis, which is composed of the ${\mathrm{\Sigma }}_{{\mathit{\varphi }}_{ij}}$ in the following way:

$$\begin{array}{c}\hfill {\mathrm{\Sigma }}_{hh}={cos}^2\mathit{\alpha }{\mathrm{\Sigma }}_{{\mathit{\varphi }}_2}+{sin}^2\mathit{\alpha }{\mathrm{\Sigma }}_{{\mathit{\varphi }}_1}-2sin\mathit{\alpha }cos\mathit{\alpha }{\mathrm{\Sigma }}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2},\end{array}$$ 36

where only ${\mathrm{\Sigma }}_{{\mathit{\varphi }}_2}$ contains the $p^2$-dependent $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ contribution. In order to simplify the discussion and to point to the main features, we assume sufficiently large values of $tan\mathit{\beta }$, such that we can write ${\widehat{\mathrm{\Sigma }}}_{hh}\simeq {\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_2}$, and h, H mixing effects play only a marginal role (both simplifications apply to the numerical discussions in the subsequent section). Moreover, to simplify the notation, we drop the indices and define

$$\begin{array}{c}\hfill {\mathrm{\Sigma }}_{hh}\equiv {\mathrm{\Sigma }}_{},{\widehat{\mathrm{\Sigma }}}_{hh}\equiv {\widehat{\mathrm{\Sigma }}}_{},\mathit{\delta }{Z}_{hh}\equiv \mathit{\delta }Z,\end{array}$$ 37

where $\mathit{\delta }{Z}_{hh}={cos}^2\mathit{\alpha }\mathit{\delta }{Z}_{{\mathcal{H}}_2}+{sin}^2\mathit{\alpha }\mathit{\delta }{Z}_{{\mathcal{H}}_1}\simeq \mathit{\delta }{Z}_{{\mathcal{H}}_2}$. Starting from the tree-level mass $m_h$ and the renormalized h self-energy up to the two-loop level,

$$\begin{array}{c}\hfill {\widehat{\mathrm{\Sigma }}}_{}(p^2)={\mathrm{\Sigma }}_{}(p^2)-\mathit{\delta }{m}_h^2+\mathit{\delta }Z(p^2-m_h^2),\end{array}$$ 38

we obtain the higher-order corrected mass $M_h$ from the pole of the propagator, i.e.

$$\begin{array}{c}\hfill M_h^2-m_h^2+{\widehat{\mathrm{\Sigma }}}_{}(M_h^2)=0.\end{array}$$ 39

The Taylor-expansion of the unrenormalized self-energy around $p^2=0$,

$$\begin{array}{c}\hfill {\mathrm{\Sigma }}_{}(p^2)={\mathrm{\Sigma }}_{}(0)+p^2{\mathrm{\Sigma }}_{}^{\prime }(0)+\widetilde{{\mathrm{\Sigma }}_{}}(p^2),\end{array}$$ 40

yields the first two terms containing the singularities in $1/\mathit{\epsilon }$ and $1/{\mathit{\epsilon }}^2$, and the residual fully finite and scheme-independent part denoted by $\widetilde{{\mathrm{\Sigma }}_{}}(p^2)$. With this expansion inserted into Eq. (38) one obtains from the pole condition Eq. (39) the relation

$$\begin{array}{cc}& (M_h^2-m_h^2)\left[1+\mathit{\delta }Z+{\mathrm{\Sigma }}_{}^{\prime }(0)\right]+\left[{\mathrm{\Sigma }}_{}(0)-\mathit{\delta }{m}_h^2+m_h^2{\mathrm{\Sigma }}_{}^{\prime }(0)\right]\hfill \\ \hfill & +\widetilde{{\mathrm{\Sigma }}_{}}(M_h^2)=0,\hfill \end{array}$$ 41

where the expressions in the square brackets are each finite, irrespective of a possible finite term in the definition of $\mathit{\delta }Z$.

Taking into account that $M_h^2$ differs from $m_h^2$ by a higher-order shift, we can replace

$$\begin{array}{c}\hfill \widetilde{{\mathrm{\Sigma }}_{}}(M_h^2)=\widetilde{{\mathrm{\Sigma }}_{}}(m_h^2)+(M_h^2-m_h^2){\widetilde{{\mathrm{\Sigma }}_{}}}^{\prime }(m_h^2)+\cdots \end{array}$$ 42

and obtain

$$\begin{array}{cc}\hfill M_h^2-m_h^2& =-\frac{{\mathrm{\Sigma }}_{}(0)-\mathit{\delta }{m}_h^2+m_h^2{\mathrm{\Sigma }}_{}^{\prime }(0)+\widetilde{{\mathrm{\Sigma }}_{}}(m_h^2)}{1+\mathit{\delta }Z+{\mathrm{\Sigma }}_{}^{\prime }(0)+{\widetilde{{\mathrm{\Sigma }}_{}}}^{\prime }(m_h^2)}\hfill \\ \hfill & =-[{\mathrm{\Sigma }}_{}(0)-\mathit{\delta }{m}_h^2+m_h^2{\mathrm{\Sigma }}_{}^{\prime }(0)+\widetilde{{\mathrm{\Sigma }}_{}}(m_h^2){]}_{1\mathrm{loop}+2\mathrm{loop}}\hfill \\ \hfill & +[{\mathrm{\Sigma }}_{}(0)-\mathit{\delta }{m}_h^2+m_h^2{\mathrm{\Sigma }}_{}^{\prime }(0)+\widetilde{{\mathrm{\Sigma }}_{}}(m_h^2){]}_{1\mathrm{loop}}\hfill \\ \hfill & ·[\mathit{\delta }Z+{\mathrm{\Sigma }}_{}^{\prime }(0)+{\widetilde{{\mathrm{\Sigma }}_{}}}^{\prime }(m_h^2){]}_{1\mathrm{loop}}+\cdots \hfill \end{array}$$ 43

showing explicitly all terms up to two-loop order. It does not contain the two-loop part of the field-renormalization constant, which indeed would show up at the three-loop level. Hence, effects resulting from different conventions for $\mathit{\delta }{Z}^{(2\mathrm{loop})}$ in the finite part have to be considered in the current situation as part of the theoretical uncertainty.

Numerical comparison

In this section the renormalized momentum-dependent $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ self-energy contributions $\mathrm{\Delta }{\widehat{\mathrm{\Sigma }}}_{hh}$, $\mathrm{\Delta }{\widehat{\mathrm{\Sigma }}}_{hH}$, $\mathrm{\Delta }{\widehat{\mathrm{\Sigma }}}_{HH}$ of Eq. (15) and the mass shifts

$$\begin{array}{c}\hfill \mathrm{\Delta }{M}_h=M_h-M_{h,0},\mathrm{\Delta }{M}_H=M_H-M_{H,0}\end{array}$$ 44

are compared using either $\mathit{\delta }{m}_t^{\mathrm{OS}}$ or $\mathit{\delta }{m}_t^{\mathrm{FIN}}$, as discussed above. $M_{h,0}$ and $M_{H,0}$ denote the Higgs-boson mass predictions without the newly obtained $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections.

The results are obtained for two different scenarios. Scenario 1 is adopted from the $m_h^{\max }$ scenario described in Ref. [91]. We use the following parameters:

$$\begin{array}{cc}\hfill m_t& =173.2\mathrm{GeV},M_{\mathrm{SUSY}}=1\mathrm{TeV},X_t=2M_{\mathrm{SUSY}},\hfill \\ \hfill m_{\tilde{g}}& =1500\mathrm{GeV},\mathit{\mu }=M_2=200\mathrm{GeV}.\hfill \end{array}$$ 45

Here $M_2$ denotes the SU(2) soft SUSY-breaking parameter, where the U(1) parameter is derived via the GUT relation $M_1=(5/3)(s_{\mathrm{w}}^2/c_{\mathrm{w}}^2)M_2$. Scenario 2 is an updated version of the “light-stop scenario” of Refs. [91, 92]

$$\begin{array}{cc}\hfill m_t& =173.2\mathrm{GeV},M_{\mathrm{SUSY}}=0.5\mathrm{TeV},X_t=2M_{\mathrm{SUSY}},\hfill \\ \hfill m_{\tilde{g}}& =1500\mathrm{GeV},\mathit{\mu }=M_2=400\mathrm{GeV}{M}_1=340\mathrm{GeV},\hfill \end{array}$$ 46

leading to stop mass values of

$$\begin{array}{c}\hfill m_{{\tilde{t}}_1}=326.8\mathrm{GeV},m_{{\tilde{t}}_2}=673.2\mathrm{GeV}.\end{array}$$ 47

A renormalization scale of $\mathit{\mu }=m_t$ is set in all numerical evaluations.

Self-energies

In Fig. 4 we present the results for the ${\mathit{\delta }}_A$ (upper plot) and ${\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}$ (lower plot) contributions for $tan\mathit{\beta }=5(20)$ in red (blue) in Scenario 1, where ${\mathit{\delta }}_A,{\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}$ are defined in Eqs. (31) and (32). In the upper plot ${\mathit{\delta }}_A(M_A^2)$ (${\mathit{\delta }}_A(0)$) is shown as solid (dashed) line; correspondingly, in the lower plot ${\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}(p^2)$ (${\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}(0)$) is depicted as solid (dashed) line. The contribution is seen to decrease quadratically with $M_A$ or $p(:=\sqrt{p^2})$ when including the momentum-dependent terms; see Eq. (34). For ${\mathit{\delta }}_A$ it is suppressed with ${tan}^2\mathit{\beta }$. For high values of $M_A$ and low $tan\mathit{\beta }$, the ${\mathit{\delta }}_A$ contribution becomes sizable. Similarly, for large p the ${\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}$ term becomes sizable, showing the relevance of the $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ contribution.

Fig. 4.

${\mathit{\delta }}_A(M_A^2)$ and ${\mathit{\delta }}_A(0)$ varying $M_A$ shown in the upper plot, ${\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}(p^2)$ and ${\mathit{\delta }}_{{\mathrm{\Sigma }}_{22}}(0)$ in the lower plot, both within Scenario 1

The behavior of the real parts of the two-loop contributions to the self-energies $\mathrm{\Delta }{\widehat{\mathrm{\Sigma }}}_{ab}$ is analyzed in Fig. 5. Solid lines show the result evaluated with $\mathit{\delta }{m}_t^{\mathrm{OS}}$, as obtained in Ref. [1] [i.e. the new contribution added to the previous FeynHiggs result in Ref. [1]; see Eq. (15)]. Dashed lines show the result evaluated with $\mathit{\delta }{m}_t^{\mathrm{FIN}}$, as obtained in Ref. [2]. We show $M_A=250\mathrm{GeV}$ and $tan\mathit{\beta }=5(20)$ as red (blue) lines. The difference between the $\mathit{\delta }{m}_t^{\mathrm{FIN}}$ and $\mathit{\delta }{m}_t^{\mathrm{OS}}$ calculations for $\mathrm{\Delta }{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1}$ and $\mathrm{\Delta }{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_1{\mathit{\varphi }}_2}$ is p-independent, as discussed below Eq. (34), and the difference between the two schemes is numerically small. For $\mathrm{\Delta }{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_2}$, on the other hand, the difference becomes large for large values of p. This self-energy contribution is mostly relevant for the light $\mathcal{CP}$-even Higgs boson, however, i.e. for $p\sim M_h$, and thus the relevant numerical difference remains relatively small (but non-zero) compared to the larger differences at large p.

Fig. 5.

$\mathrm{\Delta }{\widehat{\mathrm{\Sigma }}}_{{\mathit{\varphi }}_{ij}}$ in Scenario 1 (with $M_A=250\mathrm{GeV}$) for $ij=11,12,22$ in the upper, the middle and the lower plot, respectively. The solid (dashed) lines show the result obtained with $\mathit{\delta }{m}_t^{\mathrm{OS}}$ ($\mathit{\delta }{m}_t^{\mathrm{FIN}}$); the red (blue) lines correspond to $tan\mathit{\beta }=5(20)$

For completeness it should be mentioned that the imaginary part is not affected by the variation of the top-quark renormalization, as only the real parts of the counterterm insertions enter the calculation.

Scenario 2 was omitted as the relevant aspects for the analysis of the self-energies using $\mathit{\delta }{m}_t^{\mathrm{OS}}$ vs. $\mathit{\delta }{m}_t^{\mathrm{FIN}}$ have become sufficiently apparent within Scenario 1.

Mass shifts

We now turn to the effects on the neutral $\mathcal{CP}$-even Higgs-boson masses themselves. The numerical effects on the two-loop corrections to the Higgs-boson masses $M_{h,H}$ are investigated by analyzing the mass shifts $\mathrm{\Delta }{M}_h$ and $\mathrm{\Delta }{M}_H$ of Eq. (44). The results are shown for the two renormalization schemes for the top-quark mass, i.e. using $\mathit{\delta }{m}_t^{\mathrm{OS}}$ or $\mathit{\delta }{m}_t^{\mathrm{FIN}}$. The color coding is as in Fig. 5. The results for Scenario 1 are shown in Fig. 6 and are in agreement with Figs. 2 and 3 (left) in Ref. [2], i.e. we reproduce the results of Ref. [2] using $\mathit{\delta }{m}_t^{\mathrm{FIN}}$.

Fig. 6.

Variation of the mass shifts $\mathrm{\Delta }{M}_h,\mathrm{\Delta }{M}_H$ with the A-boson mass $M_A$ within Scenario 1, for $tan\mathit{\beta }=5$ (red) and $tan\mathit{\beta }=20$ (blue) including or excluding some $\mathit{\delta }$ terms. The peak in $\mathrm{\Delta }{M}_H$ originates from a threshold at $2m_t$

The results for Scenario 2 are shown in Fig. 7. The results are again in agreement with Figs. 2 and 3 (right) in Ref. [2]. This agreement confirms the use of $\mathit{\delta }{m}_t^{\mathrm{FIN}}$ in Ref. [2], in comparison with $\mathit{\delta }{m}_t^{\mathrm{OS}}$ used in the evaluation of our results.

Fig. 7.

Variation of the mass shifts $\mathrm{\Delta }{M}_h,\mathrm{\Delta }{M}_H$ with the A-boson mass $M_A$ within Scenario 2, for $tan\mathit{\beta }=5$ (red) and $tan\mathit{\beta }=20$ (blue) including or excluding some $\mathit{\delta }$ terms. The peaks in $\mathrm{\Delta }{M}_H$ originate from thresholds at $2m_t$, $2m_{{\tilde{t}}_1}$, $m_{{\tilde{t}}_1}+m_{{\tilde{t}}_2}$, and $2m_{{\tilde{t}}_2}$, where the threshold at $2m_t$ is suppressed by $1/{tan}^2\mathit{\beta }$

For the contribution to $M_H$, peaks can be observed at $M_A=2m_{{\tilde{t}}_1}$, $m_{{\tilde{t}}_1}+m_{{\tilde{t}}_2}$, $2m_{{\tilde{t}}_2}$; see also Ref. [1] and the discussion of Fig. 9 below.

Fig. 9.

$\overline{\mathrm{\Delta }}{M}_{\mathit{\varphi }}=M_{\mathit{\varphi }}(m_t^{\mathrm{OS}})-M_{\mathit{\varphi }}(m_t^{\overline{\mathrm{DR}}})$ for $\mathit{\varphi }=h$ (upper plot) and $\mathit{\varphi }=H$ (lower plot) as a function of $M_A$ within Scenario 2, with the same line/color coding as in Fig. 8. The peak in the lower plot originates from a threshold at $2m_{{\tilde{t}}_1}$. The threshold at $2m_t$ is suppressed by $1/{tan}^2\mathit{\beta }$

Since the results using $\mathit{\delta }{m}_t^{\mathrm{OS}}$ and $\mathit{\delta }{m}_t^{\mathrm{FIN}}$ correspond to two different renormalization schemes, their difference should be regarded as an indication of missing higher-order momentum-dependent corrections.

Comparison with the ${\mathit{m}}_{\mathit{t}}$$\overline{\mathrm{DR}}$ renormalization

Having examined the renormalization of the top-quark mass, we will now analyze the numerical differences between an $m_t^{\overline{\mathrm{DR}}}$ and an $m_t^{\mathrm{OS}}$ calculation. This has been realized by employing a $\overline{\mathrm{DR}}$ renormalization of the top-quark mass in all steps of the calculation. The top-squark masses are kept renormalized on-shell. This can be seen as an intermediate step toward a full $\overline{\mathrm{DR}}$ analysis.

Implementation in the program FeynHiggs

In the $\overline{\mathrm{DR}}$ scheme the top-quark mass parameter entering the calculation is the MSSM $\overline{\mathrm{DR}}$ top-quark mass, which at one-loop order is related to the pole mass $m_t$ (given in the user input) in the following way:

$$\begin{array}{c}\hfill m_t^{\overline{\mathrm{DR}}}(\mathit{\mu })=m_t·\left[1+\frac{\mathit{\delta }{m}_t^{\mathrm{fin}}}{m_t}+\mathcal{O}\left(({\mathit{\alpha }}_s^{\overline{\mathrm{DR}}}{)}^2\right)\right].\end{array}$$ 48

The term $\mathit{\delta }{m}_t^{\mathrm{fin}}$ can be obtained from Eq. (18), with the formal replacement ${\mathit{\alpha }}_s\to {\mathit{\alpha }}_s^{\overline{\mathrm{DR}}}(\mathit{\mu })$, yielding

$$\begin{array}{cc}\hfill \frac{\mathit{\delta }{m}_t^{\mathrm{fin}}}{m_t}& ={\mathit{\alpha }}_s^{\overline{\mathrm{DR}}}(\mathit{\mu })\left(-\frac{5}{3\mathit{\pi }}+\frac{1}{\mathit{\pi }}log(m_t^2/{\mathit{\mu }}^2)\right.\hfill \\ \hfill & +\frac{m_{\tilde{g}}^2}{3m_t^2\mathit{\pi }}\left(-1+log(m_{\tilde{g}}^2/{\mathit{\mu }}^2)\right)\hfill \\ \hfill & +\frac{1}{6m_t^2\mathit{\pi }}\left(m_{{\tilde{t}}_1}^2(1-log(m_{{\tilde{t}}_1}^2/{\mathit{\mu }}^2))\right.\hfill \\ \hfill & +m_{{\tilde{t}}_2}^2(1-log(m_{{\tilde{t}}_2}^2/{\mathit{\mu }}^2))\hfill \\ \hfill & +(m_{\tilde{g}}^2+m_t^2-m_{{\tilde{t}}_1}^2-2m_{\tilde{g}}{m}_{t}sin(2{\mathit{\theta }}_t))\hfill \\ \hfill & \times \mathrm{Re}[B_0^{\mathrm{fin}}(m_t^2,m_{\tilde{g}}^2,m_{{\tilde{t}}_1}^2)]\hfill \\ \hfill & +(m_{\tilde{g}}^2+m_t^2-m_{{\tilde{t}}_2}^2+2m_{\tilde{g}}{m}_{t}sin(2{\mathit{\theta }}_t))\hfill \\ \hfill & \left.\left.\times \mathrm{Re}[B_0^{\mathrm{fin}}(m_t^2,m_{\tilde{g}}^2,m_{{\tilde{t}}_2}^2)]\right)\right).\hfill \end{array}$$ 49

At zeroth order, ${\mathit{\alpha }}_s^{\overline{\mathrm{DR}}}(\mathit{\mu })={\mathit{\alpha }}_s^{\overline{\mathrm{MS}}}(\mathit{\mu })$.

As on-shell renormalized quantities the stop masses $m_{{\tilde{t}}_1}$ and $m_{{\tilde{t}}_2}$ should have fixed values, independently of the renormalization chosen for the top-quark mass. We compensate for the changes induced by $\mathit{\delta }{m}_t^{\mathrm{fin}}$ in the stop mass matrix, Eq. (13), by shifting the SUSY-breaking parameters as follows:

$$\begin{array}{cc}& M_{{\tilde{t}}_L}^2\to M_{{\tilde{t}}_L}^{\prime 2}=M_{{\tilde{t}}_L}^2+{(m_t^{\mathrm{OS}})}^2-{(m_t^{\overline{\mathrm{DR}}})}^2,\hfill \end{array}$$ 50a

$$\begin{array}{cc}& 7M_{{\tilde{t}}_R}^2\to M_{{\tilde{t}}_R}^{\prime 2}=M_{{\tilde{t}}_R}^2+{(m_t^{\mathrm{OS}})}^2-{(m_t^{\overline{\mathrm{DR}}})}^2,\hfill \end{array}$$ 50b

$$\begin{array}{cc}& A_t\to A_t^{\prime }=\frac{m_t^{\mathrm{OS}}}{m_t^{\overline{\mathrm{DR}}}}\left(A_t-\frac{\mathit{\mu }}{tan\mathit{\beta }}\right)+\frac{\mathit{\mu }}{tan\mathit{\beta }}.\hfill \end{array}$$ 50c

(Except for $A_t$, which actually appears in the Feynman rules, FeynHiggs only pretends to perform these shifts but computes the sfermion masses using $m_t^{\mathrm{OS}}$.)

This procedure is available in FeynHiggs from version 2.11.1 on and is activated by setting the new value 2 for the runningMT flag. The comparison of the results with $\overline{\mathrm{DR}}$ and with OS renormalization admits an improved estimate of (some) of the missing three-loop corrections in the top/stop sector.

Numerical analysis

In the following plots we show the difference

$$\begin{array}{c}\hfill \overline{\mathrm{\Delta }}{M}_{\mathit{\varphi }}:=M_{\mathit{\varphi }}(m_t^{\mathrm{OS}})-M_{\mathit{\varphi }}(m_t^{\overline{\mathrm{DR}}}),\mathit{\varphi }=h,H,\end{array}$$ 51

between $M_{\mathit{\varphi }}$ evaluated in the OS scheme, i.e. using $m_t^{\mathrm{OS}}$ (not$m_t^{\mathrm{FIN}}$), and in the $\overline{\mathrm{DR}}$ scheme, i.e. using $m_t^{\overline{\mathrm{DR}}}$.

Dependence on ${\mathit{M}}_{\mathit{A}}$

In the upper half of Fig. 8, $\overline{\mathrm{\Delta }}{M}_h$ is plotted in Scenario 1 as a function of $M_A$ with $tan\mathit{\beta }=5(20)$ in red (blue). The solid (dashed) lines show the difference evaluated at the full one-loop level (including the $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections). The dotted lines include the newly calculated $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections. For $M_A\gtrsim 200\mathrm{GeV}$ one observes large differences of $\mathcal{O}(10\mathrm{GeV})$ at the one-loop level, indicating the size of missing higher-order corrections from the top/stop sector beyond one loop. This difference is strongly reduced at the two-loop level, to about $\sim 3\mathrm{GeV}$, now corresponding to missing higher orders beyond two loops from the top/stop sector. The dotted lines are barely visible below the dashed lines, indicating the relatively small effect of the $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections as derived in Ref. [1].

Fig. 8.

$\overline{\mathrm{\Delta }}{M}_{\mathit{\varphi }}=M_{\mathit{\varphi }}(m_t^{\mathrm{OS}})-M_{\mathit{\varphi }}(m_t^{\overline{\mathrm{DR}}})$ for $\mathit{\varphi }=h$ (upper plot) and $\mathit{\varphi }=H$ (lower plot). The difference is shown as solid (dashed/dotted) line at the one-loop ($\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$/$\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$) level as a function of $M_A$ for $tan\mathit{\beta }=5(20)$ in red (blue) within Scenario 1

The lower plot of Fig. 8 shows the corresponding results for $\overline{\mathrm{\Delta }}{M}_H$ with the same color/line coding. Here large effects are only visible for low $M_A$, where the higher-order corrections to $M_H$ are sizable (and the light Higgs boson receives only very small higher-order corrections). In this part of the parameter space the same reduction of $\overline{\mathrm{\Delta }}{M}_H$ going from one loop to two loops can be observed.

The behavior is similar for Scenario 2, shown in Fig. 9 (with the same line/color coding as in Fig. 8), only the size of the difference $\overline{\mathrm{\Delta }}{M}_h$ is $\sim 20\%$ smaller at the one-loop level, and $\sim 50\%$ smaller at the two-loop level compared to Scenario 1. The same peak structure due to thresholds as in Fig. 7 is visible.

Dependence on ${\mathit{m}}_{\stackrel{\mathbf{~}}{\mathit{g}}}$

In Figs. 10 and 11 we analyze $\overline{\mathrm{\Delta }}{M}_{\mathit{\varphi }}$ as a function of $m_{\tilde{g}}$ in Scenario 1 and 2, respectively. We fix $M_A=250\mathrm{GeV}$ and use the same line/color coding as in Fig. 8. Due to the choice of an MSSM $\overline{\mathrm{DR}}$ top-quark mass definition, $m_t^{\overline{\mathrm{DR}}}$ varies with $m_{\tilde{g}}$ already at the one-loop level.

Fig. 10.

$\overline{\mathrm{\Delta }}{M}_{\mathit{\varphi }}=M_{\mathit{\varphi }}(m_t^{\mathrm{OS}})-M_{\mathit{\varphi }}(m_t^{\overline{\mathrm{DR}}})$ for $\mathit{\varphi }=h$ (upper plot) and $\mathit{\varphi }=H$ (lower plot) as a function of $m_{\tilde{g}}$ within Scenario 1, for $M_A=250\mathrm{GeV}$ and with the same line/color coding as in Fig. 8

Fig. 11.

$\overline{\mathrm{\Delta }}{M}_{\mathit{\varphi }}=M_{\mathit{\varphi }}(m_t^{\mathrm{OS}})-M_{\mathit{\varphi }}(m_t^{\overline{\mathrm{DR}}})$ for $\mathit{\varphi }=h$ (upper plot) and $\mathit{\varphi }=H$ (lower plot) as a function of $m_{\tilde{g}}$ within Scenario 2, for $M_A=250\mathrm{GeV}$ and with the same line/color coding as in Fig. 8

In the upper plots we show the light $\mathcal{CP}$-even Higgs-boson case, where it can be observed that the scheme dependence is strongly reduced at the two-loop level. It reaches 2–3$\mathrm{GeV}$ in Scenario 1 and $\sim 1\mathrm{GeV}$ in Scenario 2, largely independently of $tan\mathit{\beta }$. At the one-loop level the scheme dependence grows with $m_{\tilde{g}}$, whereas the dependence is much milder at the two-loop level. The effects of the $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections become visible at larger $m_{\tilde{g}}$, in agreement with Ref. [1].

The heavy $\mathcal{CP}$-even Higgs-boson case is shown in the lower plots. At small $tan\mathit{\beta }$ scheme differences of $\mathcal{O}(600\mathrm{MeV}(150\mathrm{MeV}))$ can be observed at the one- (two-) loop level. For large $tan\mathit{\beta }$ the differences always stay below $\mathcal{O}(50\mathrm{MeV})$, in agreement with Fig. 8. The dependence on $m_{\tilde{g}}$ is similar to the light Higgs boson, but again somewhat weaker.

Dependence on ${\mathit{X}}_{\mathit{t}}$

Finally, in Figs. 12 and 13 we analyze $\overline{\mathrm{\Delta }}{M}_{\mathit{\varphi }}$ as a function of $X_t=X_t^{\mathrm{OS}}$ in Scenario 1 and 2, respectively. We again fix $M_A=250\mathrm{GeV}$ and use the same line/color coding as in Fig. 8.

Fig. 12.

$\overline{\mathrm{\Delta }}{M}_{\mathit{\varphi }}=M_{\mathit{\varphi }}(m_t^{\mathrm{OS}})-M_{\mathit{\varphi }}(m_t^{\overline{\mathrm{DR}}})$ for $\mathit{\varphi }=h$ (upper plot) and $\mathit{\varphi }=H$ (lower plot) as a function of $X_t=X_t^{\mathrm{OS}}$ within Scenario 1, with the same line/color coding as in Fig. 8

Fig. 13.

$\overline{\mathrm{\Delta }}{M}_{\mathit{\varphi }}=M_{\mathit{\varphi }}(m_t^{\mathrm{OS}})-M_{\mathit{\varphi }}(m_t^{\overline{\mathrm{DR}}})$ for $\mathit{\varphi }=h$ (upper plot) and $\mathit{\varphi }=H$ (lower plot) as a function of $X_t=X_t^{\mathrm{OS}}$ within Scenario 2, with the same line/color coding as in Fig. 8

In the upper plots we show the light $\mathcal{CP}$-even Higgs-boson case. As before the scheme dependence is strongly reduced when going from the one-loop to the two-loop case. In general a smaller scheme dependence is found from small $X_t$, while it increases for larger $|X_t|$ values, in agreement with Ref. [93]. For most parts of the parameter space, when the two-loop corrections are included, it is found to be below $\sim 3\mathrm{GeV}$. The contribution of $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ remains small for all $X_t$ values.

In the heavy $\mathcal{CP}$-even Higgs-boson case, shown in the lower plots, the dependence of the size of the effects is slightly more involved, though the general picture of a strongly reduced scheme dependence can be observed here, too. In both scenarios, for large negative $X_t$ and $tan\mathit{\beta }=5$ the $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ contributions can become sizable with respect to the $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections.

In conclusion, the scheme dependence is found to be reduced substantially when going from the pure one-loop calculation to the two-loop $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections. This indicates that corrections at the three-loop level and beyond, stemming from the top/stop sector are expected at the order of the observed scheme dependence, i.e. at the level of $\sim 3\mathrm{GeV}$. This is in agreement with existing calculations beyond two loops [54–56, 60].

A further reduction of the scheme dependence might be expected by adding the $\mathcal{O}({\mathit{\alpha }}_t^2)$ contributions. The $m_t^{\overline{\mathrm{DR}}}$ value calculated at $\mathcal{O}({\mathit{\alpha }}_s+{\mathit{\alpha }}_t)$ is substantially closer to $m_t^{\mathrm{OS}}$, reducing already strongly the scheme dependence at the one-loop level. This extended analysis is beyond the scope of our paper, however.

Conclusions

In this paper we analyzed the scheme dependence of the $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections to the neutral $\mathcal{CP}$-even Higgs-boson masses in the MSSM. In a first step we investigated the differences in the $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections as obtained in Refs. [1, 2]. We have shown that the difference can be attributed to different renormalizations of the top-quark mass. In both calculations an “on-shell” top-quark mass was employed. The evaluation in Ref. [1] includes the $\mathcal{O}(\mathit{\epsilon })$ terms of the top-quark mass counterterm, $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$, however, whereas this contribution was omitted in Ref. [2]. We have shown analytically that the terms involving $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ do not cancel in the $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections to the renormalized Higgs-boson self-energies (an effect that was already observed in the $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections in the NMSSM Higgs sector [90]). Numerical agreement between Refs. [1, 2] is found as soon as the $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ terms are dropped from the calculation in Ref. [1]. Moreover, as an alternative interpretation, we have shown that omitting the $\mathit{\delta }{m}_t^{\mathit{\epsilon }}$ terms is equivalent to a redefinition of the finite part of the two-loop field-renormalization constant which affects the Higgs-boson mass prediction at the three-loop order (apart from a numerically insignificant shift in $tan\mathit{\beta }$ as an input parameter). The differences between the two calculations can thus be regarded as an indication of the size of the missing momentum-dependent corrections beyond the two-loop level, and they reach up to several hundred MeV in the case of the light $\mathcal{CP}$-even Higgs boson.

In a second step we performed a calculation of the $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ and $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections employing a $\overline{\mathrm{DR}}$ top-quark mass counterterm. We analyzed the numerical difference of the Higgs-boson masses evaluated with $\mathit{\delta }{m}_t^{\mathrm{OS}}$ and with $\mathit{\delta }{m}_t^{\overline{\mathrm{DR}}}$. By varying the $\mathcal{CP}$-odd Higgs-boson mass, $M_A$, the gluino mass, $m_{\tilde{g}}$ and the off-diagonal entry in the scalar-top mass matrix, $X_t$, we found that in all cases the scheme dependence, in particular of the light $\mathcal{CP}$-even Higgs-boson mass, is strongly reduced by going from the full one-loop result to the two-loop result including the $\mathcal{O}({\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ corrections. The further inclusion of the $\mathcal{O}(p^2{\mathit{\alpha }}_t{\mathit{\alpha }}_s)$ contributions had a numerically small effect. The differences found at the two-loop level indicate that corrections at the three-loop level and beyond, stemming from the top/stop sector, are expected at the level of $\sim 3\mathrm{GeV}$. This is in agreement with existing calculations beyond two loops [54–56, 60]. The possibility to use $m_t^{\overline{\mathrm{DR}}}$ instead of $m_t^{\mathrm{OS}}$ has been added to the FeynHiggs package and allows an improved estimate of the size of missing corrections beyond the two-loop order.