Two-loop correction to the Higgs boson mass in the MRSSM

We present the impact of two-loop corrections on the mass of the lightest Higgs boson in the Minimal R-symmetric Supersymmetric Standard Model (MRSSM). These shift the Higgs boson mass up by typically 5 GeV or more. The dominant corrections arise from strong interactions, from the gluon and its N=2 superpartners, the sgluon and Dirac gluino, and these corrections further increase with large Dirac gluino mass. The two-loop contributions governed purely by Yukawa couplings and the MRSSM $\lambda,\Lambda$ parameters are smaller. We also update an earlier analysis [Diessner:2014ksa], which showed that the MRSSM can accommodate the measured Higgs and W boson masses. Including the two-loop corrections increases the parameter space where the theory prediction agrees with the measurement.


Introduction
The recent discovery at the LHC of a particle consistent with the long sought Higgs boson seemingly completes the Standard Model (SM). The mass of the particle is measured with an astonishingly high accuracy of m H = 125.09 ± 0.24 GeV [2]. The precise determination of this mass is of paramount importance not only within the context of the Standard Model, but also for finding the path beyond it. In fact, a number of experimental observations suggest that the SM cannot be the ultimate theory and many theoretical scenarios for the beyond the SM physics (BSM) have been proposed in past decades. In some models of BSM, in particular in supersymmetric extensions of the SM, the Higgs boson mass can be predicted. However, the current experimental accuracy is far better than theoretical predictions for Higgs boson mass in any given model of BSM physics. From the point of view of theory, the best accuracy has been achieved in the minimal supersymmetric extension of the SM (MSSM), in which the discovery of the Higgs boson and the determination of its mass have given a new impetus to the theoretical efforts. The most recent improvements comprise the inclusion of leading three-loop corrections [3,4], resummations of leading logarithms beyond the two-loop level [5,6], inclusion of the external momenta of two-loop self-energies [7,8], and the evaluation of the O(α 2 t )-contributions in the complex MSSM [9,10]. The MSSM two-loop corrections controlled by Yukawa couplings and α s have been known for quite some time for the real MSSM (see the above references for an overview of the literature).
The absence of any direct signal of supersymmetric particle production at the LHC, and the observed Higgs boson mass of ∼125 GeV being rather close to the upper value of ∼135 GeV achievable in the MSSM, are a strong motivation to consider non-minimal SUSY scenarios. In fact, non-minimal SUSY models can lift the Higgs boson mass (at the tree level by new F -or D-term contributions or at the loop level from additional new states), which makes these models more natural by reducing fine-tuning. They can also weaken SUSY limits either by predicting compressed spectra, or by reducing the expected missing transverse energy, or by reducing production cross-sections. The comparison of the measured Higgs boson mass with the theoretically predicted values in any given model is therefore highly desirable. Although the theoretical calculations for the SM-like Higgs boson mass in such models are less advanced, progress is being made in the development of highly automated tools which greatly facilitate the computations in non-minimal SUSY models: SARAH [11][12][13] automatically

The MRSSM
The MRSSM has been constructed in Ref. [20] as a minimal supersymmetric model with unbroken continuous R-symmetry. The superpotential of the model reads as whereĤ u,d are the MSSM-like Higgs weak iso-doublets, andŜ,T ,R u,d are the singlet, weak iso-triplet andR-Higgs weak iso-doublets, respectively. The usual MSSM µ-term is forbidden; instead the µ u,dterms involving R-Higgs fields are allowed. The Λ, λ-terms are similar to the usual Yukawa terms, where theR-Higgs andŜ orT play the role of the quark/lepton doublets and singlets. The usual soft mass terms of the MSSM scalar fields are allowed just like in the MSSM. In contrast, A-terms and soft Majorana gaugino masses are forbidden by R-symmetry. The fermionic components of the chiral adjoints,Φ i =Ô,T ,Ŝ for each standard model gauge group i = SU (3), SU (2), U (1) respectively, are paired with standard gauginosg,W ,B to build Dirac fermions and the corresponding mass terms. The Dirac gaugino masses generated by D-type spurions produce additional terms with the auxiliary D-fields in the Lagrangian, which after being eliminated through their equations of motion, lead to the appearance of Dirac masses in the scalar sector as well. The soft-breaking scalar mass terms read The electroweak symmetry breaking (EWSB) is triggered by non-zero vacuum expectation values of the R = 0 neutral EW scalars, which are parameterized as R-Higgs bosons carry R-charge 2 and therefore do not develop vacuum expectation values. We stress that in general the mixing of φ T , φ S with φ u and φ d leads to a reduction of the lightest Higgs boson mass at the tree-level compared to the MSSM.
3 Higgs mass dependence on the λ, Λ superpotential parameters We now present the MRSSM Higgs boson mass prediction at the two-loop level. We use the same renormalization scheme as in Ref. [1], where all SUSY parameters are defined in the DR scheme and m 2 H d , m 2 Hu , v S and v T are determined by minimizing the effective potential at the two-loop order. The discussion is divided into two parts. In the present section we begin with the one-loop contributions, which are dominated by terms of O(α t,b,λ ), where α λ collectively denotes squares of the superpotential couplings λ u,d and Λ u,d . We then discuss the two-loop contributions of O(α 2 t,b,λ ), i.e. ones which depend on parameters which already play a role at the one-loop level. In the subsequent section we then discuss those two-loop corrections which involve new parameters.
In the usual MSSM, the one-loop contributions to the Higgs boson mass are dominated by top/stop contributions. In the MRSSM, these contributions are also important, but they are simpler since stop mixing is forbidden by R-symmetry (corresponding to the MSSM parameter X t ≡ A t − µ/ tan β = 0). This implies that the top/stop contributions cannot reach values as high as in the MSSM for a given stop mass scale. However, as mentioned above, the MRSSM superpotential contains new terms governed by λ u,d and Λ u,d which have a Yukawa-like structure. References [1,34] have given a useful analytical approximation for these contributions. In the the limit λ = λ u = −λ d ,  large tan β, we get This result shows a behavior proportional to λ 4 , Λ 4 and log m 2 soft . This is similar to the top/stop contributions as the λ's and Y t appear in a similar fashion in superpotential.
We expect therefore that the two-loop result will depend on these model parameters (which already entered at the one-loop level) in a manner similar to the pure top quark/squarks two-loop contributions, i.e. similar to the MSSM O(α 2 t ) contributions without stop mixing. In Figs. 1 and 2 the dependence of the lightest Higgs boson mass calculated at tree-, one-and two-loop levels for two benchmarks BMP1 and BMP3 on different model parameters is shown. All parameters except the ones shown on the horizontal axes are set to the values of the benchmark points defined in Ref. [1] (see Tab. 2). Indeed the λ, Λ behavior of the two-loop corrections is very similar to the one of the corresponding one-loop corrections. The numerical impact of the two-loop λ, Λcontributions is rather small, typically less than 1 GeV, except for very large |λ u |, |Λ u | > 1, where they can reach several GeV. Particularly, the strong λ u dependence for large λ u is already manifest for the tree-level mass; this is due to the mixing with the singlet state already present in the tree-level mass matrix.
One should remember that very large one-loop contributions are required to bring the predicted Higgs boson mass close to the experimental one. In the preferred parameter regions, the λ, Λ are large but still moderate enough not to blow up the two-loop contributions.
Overall, the total two-loop contributions (including the ones to be discussed in the subsequent section) are in the range between 4 and 5 GeV, except in the very large λ, Λ regions. This is in agreement with the estimate given in Ref. [1], and it confirms the validity of the perturbative expansion in spite of the large one-loop corrections.

QCD corrections and the two-loop corrected Higgs boson mass
At two-loop level the strongly interacting sector and the strong coupling α s appear directly in the Higgs boson mass predictions. These two-loop corrections involve not only the gluon but also the Dirac gluino and the sgluon, the scalar component of the octet superfieldÔ. They can be expected to be sizable, and they depend on the gluino Dirac mass and sgluon soft mass parameters. 1 The gluino Dirac mass parameter M D O appears not only directly as the gluino mass but, via Eq. (2), also in couplings and mass terms of sgluons, inducing the mass splitting 2 between the real and imaginary parts of the sgluon field, O = 1

Analytic formulas
As in the previous section, we begin with an analytic approximation for the leading contributions of O(α t α s ), i.e. two-loop strong corrections proportional to Y 2 t . This provides us with qualitative insight and serves as a check of the code. Generally, in the gaugeless limit the two-loop corrections from gluinos and sgluons contribute only to the diagonal part of the {φ d , φ u } submatrix of the scalar Higgs boson mass matrix. In the MRSSM the O(α t α s ) terms contribute only to the φ u φ u element. This already constitutes a difference to the MSSM, where the µ-term violates R-symmetry and Peccei-Quinn symmetry leading to couplings of stops to φ d . Figure 4 shows two-loop diagrams contributing to the Higgs boson mass at O(α t α s ) that explicitly depend on m O and/or M D O . These diagrams provide the following contribution to the effective potential: where the functions f SSS and f F F S are defined in [39]. The effective potential V ef f depends on v u through stop masses, which in the gaugeless limit approach Equation (5) can be obtained from Ref. [39] by applying translation rules from real fields to complex ones. Many such rules can be found in Ref. [35]; an additional rule needed here for the case of a ). An important difference to the MSSM is that contributions with fermion mass insertions, corresponding to F F S-type contributions in Ref. [39], are not present in the MRSSM. Such contributions vanish due to the lack of L-R mixing between squarks. Hence the gluino mass appears in a simpler way than in the MSSM. Likewise, the sgluon only enters via the SSS-type diagram of Fig. 4. An SS-type diagram vanishes due to the color structure.
The corresponding two-loop contribution to the φ u φ u Higgs boson mass matrix element in zeromomentum approximation is then given by 3 For large tan β, corrections of order O(α b α s ) cannot be neglected any more. But since they contribute only to φ d φ d matrix element, their impact on mass of the lightest Higgs, which stems mainly from the φ u φ u element, is small. Results of Eq. (5) where compared with the results of two-loop routines from the SARAH-generated SPheno module.

Numerical analysis
We now turn to the numerical analysis of the complete two-loop corrections to the SM-like Higgs boson mass, using the full evaluation within the framework of SARAH and SPheno. The first two panels of Fig. 5 focus on the gluino and sgluon mass dependence, which arises mainly from the O(α t α s ) corrections; they show the two-loop corrections as a function of the gluino mass parameter for two different values of the soft sgluon mass, m O = 2 and 10 TeV for two benchmarks BMP1 and BMP3; other parameters are fixed at benchmark values. For comparison, the two-loop result without the sgluon contribution is shown as well (i.e. without the first diagram of Fig. 4). We also plot the MSSM prediction with strong stop mixing and without any sfermion mixing the at tree-level. The first two panels show that the dependence in the MRSSM without sgluon contributions is very similar to the one in the MSSM without stop mixing. The corresponding thin solid red and thin dashed light blue curves in Fig. 5 show a characteristic drop for large gluino masses. This is understandable as in the MSSM without sfermion mixing the gluino contribution is precisely the same as in the MRSSM and given by the two corresponding diagrams in Fig. 4. The Dirac or Majorana nature of the gluino does not matter since the Dirac partner, the octet superfieldÔ has no direct couplings to quark superfields. A few TeV gluino masses slightly increase the Higgs boson mass, but for larger values of M D O the f F F S function becomes negative and drives the correction downwards. In the full MRSSM calculations, including the sgluon diagrams strongly changes the behavior. Surprisingly, the full MRSSM two-loop contributions resemble the MSSM contributions with large stop mixing. In both cases, large gluino masses strongly enhance the Higgs boson mass, however for different reasons. In the MSSM the increase can be traced back to the additional F F S-type diagram which is directly proportional to M D O and which vanishes in the limit of no stop-mixing. In the MRSSM, on the other hand, the sgluon diagram grows with M D O both due to the sgluon-stop-stop coupling, which scales like M D O , and to an increase in the scalar (but not pseudoscalar) sgluon mass. Due to the sgluon contributions the total two-loop contributions to the Higgs boson mass in the MRSSM are larger than the ones in the MSSM. They are further increased by heavy sgluons.
The third panel of Figure 5 compares the numerical impact of individual contributions by successively switching off contributions. It allows to read off the contributions from sgluon, gluino and gluon, of O(α 2 t , α t α b ), and the remaining two-loop contributions (particularly the λ, Λ contributions). The gluon diagrams alone contribute approximately +4 GeV. The negative gluino and the positive sgluon corrections together amount to an additional upward shift of the Higgs boson mass, which can reach several GeV for large Dirac gluino masses. The remaining contributions are far smaller and amount to around −1 GeV for the O(α 2 t , α t α b ) contributions and +0.5 GeV for the remaining contributions. In this section we present an update of the analysis of Ref. [1], using the more precise evaluation of the Higgs boson mass. Ref. [1] studied the mass predictions of the W and lightest Higgs bosons in the MRSSM and showed that agreement with experimental data is possible, in spite of tree-level shifts from violations of custodial symmetry and from mixing with other Higgs states, respectively. Table 2 shows benchmark parameter points defined in that reference. They exemplify parameter regions in which m W and m H 1 agree with experiment. They are characterized by large |Λ| ≈ 1, rather light Dirac higgsinos and gauginos, and they have tan β = 3, 10, 40, respectively.

Update of benchmarks
For all three benchmark points the two-loop correction to m H 1 is around +5 GeV. As discussed in the previous sections, the largest part of this is due to the O(α t α s ) corrections. The MRSSM-specific  Table 2: Benchmark points of Ref. [1]. Dimensionful parameters are given in GeV or GeV 2 , as appropriate. The first two parts define input parameters. The third part shows parameters derived from electroweak symmetry breaking after solving the tadpole equations at two loops. The last part gives the theory predictions for the Higgs boson mass at the two-loop level and further quantities relevant for comparison with experiment.
corrections of O(α 2 Λ ) are small since the values of Λ u , though large, are still not as large as needed to make these corrections dominate, see Figs. 1, 2 for two out of three benchmarks. The magnitude of the total two-loop correction is consistent with the theory error estimate given in Ref. [1].
The upward shift of m H 1 implies that it is easier to obtain agreement with the measured value, i.e. smaller values of |Λ u | are sufficient. In Tab. 3 we provide new, slightly modified benchmark points, whose definitions differ only in the values of Λ u . The two-loop Higgs boson mass prediction agrees well with experiment, and the good agreement of m W with experiment is unchanged. Likewise, both the old and the new set of benchmark points pass checks against HiggsBounds [40][41][42] and HiggsSignals [43,44].
In Fig. 6 we give an update to some of the subfigures from Figs. 4 and 5 of Ref. [1]. These show the predictions of m W and m H 1 as contour lines in several two-dimensional parameter spaces. The Higgs boson mass is evaluated at the two-loop level. As discussed before, with the exception of the regions of very large Λ, there is a general positive contribution to the lightest Higgs boson mass between 4 and 5 GeV. Accordingly, the contour lines, in particular the central green region in which the Higgs boson mass agrees with experiment, shift to slightly lower values of Λ. Also, the overlap region, where Higgs and W boson masses agree with experiment, is enlarged.

Conclusions
In this work we have presented the impact of two-loop corrections on the mass of the lightest Higgs boson in the MRSSM. The calculation has been performed using the framework of SARAH in the approximation of the vanishing electroweak gauge couplings and external momenta of the Higgs self energies. The code has been cross-checked with an analytic calculation of the most important new corrections. We have separately analyzed the impact of contributions involving the λ, Λ-couplings, which already appear in the one-loop corrections, and of the strong corrections involving gluon, Dirac gluino, and sgluon exchange.
In the previous work [1] and the present paper we have found that the lightest Higgs boson mass in the MRSSM differs from the one in the usual MSSM in several respects. At tree-level the additional mixing with additional scalar states reduces the MRSSM Higgs mass below the MSSM value. At the one-loop level, the top/stop contributions cannot be as large as in the MSSM, because stop mixing is forbidden by R-symmetry. However, the new contributions from the superpotential λ, Λ-terms have a similar structure as the top/stop contributions. If the λ, Λ-couplings are similar in magnitude to the top Yukawa coupling, the lightest Higgs boson mass can easily be in the ballpark of the experimentally allowed range.
The two-loop corrections governed by these λ, Λ-couplings, however, amount to only 1 GeV or less in parameter regions in which the Higgs boson mass agrees with experiment. The most important two-loop contributions are the strong corrections of O(α t α s ). As we have shown the Dirac gluino and gluon contributions alone are very similar to the MSSM strong contributions for vanishing stop mixing. The inclusion of the sgluons changes the picture. The sgluon contributions are positive and rise with the Dirac gluino mass, such that the total O(α t α s ) corrections of the MRSSM are larger than the ones of the MSSM, independently of the magnitude of stop mixing.
Overall, the MRSSM two-loop corrections to the lightest Higgs boson mass are typically positive. E.g. for the benchmark parameter points proposed in Ref. [1], the two-loop corrections to the Higgs boson mass amount to approximately +5 GeV, within the error estimate of that reference. Since perturbation theory shows a converging behavior and since the λ, Λ-corrections are subdominant (for |λ|, |Λ| less than around 1.2), we estimate the remaining theory uncertainty to be not larger than the one of the MSSM.
The positive two-loop corrections make it easier to achieve agreement between the theory prediction for the lightest Higgs boson mass and the measured value. We have provided an update of the analysis of Ref. [1], showing parameter regions of simultaneous agreement of the Higgs and W boson mass predictions with experiment. Compared to Ref. [1], the allowed parameter regions are slightly larger and located at smaller values of the λ, Λ-couplings.