On the two-loop corrections to the Higgs mass in trilinear R-parity violation

We study the impact of large trilinear R-parity violating couplings on the lightest CP-even Higgs boson mass in supersymmetric models. We use the publicly available computer codes SARAH and SPheno to compute the leading two-loop corrections. We use the effective potential approach. For not too heavy third generation squarks (<1 TeV) and couplings close to the unitarity bound we find positive corrections up to a few GeV in the Higgs mass.


I. INTRODUCTION
On July 4 th , 2012 the discovery of the Higgs boson was announced at CERN [1,2]. It is not yet established whether this is the Standard Model (SM) Higgs boson [3][4][5]. However, in the SM the Higgs sector suffers from the hierarchy problem [6], to which supersymmetry (SUSY) [7,8] is the most obvious solution. It predicts a wide range of observables at the Large Hadron Collider (LHC), for which the first run has finished; Run II is expected to start in the Spring, 2015.
Within SUSY the mass of the Higgs boson is restricted at tree-level to be less than the mass of the Z 0 -boson. However realised, the quantum corrections to the mass can be large [28,29]. The observed mass of the Higgs boson, m exp h ≈ 125.7 GeV [30][31][32], is well within the previous predicted allowed range for SUSY models [33]. Such large corrections however typically require very large mixing in the stop sector and/or a very heavy stop squark. This in turn is disfavoured by fine-tuning arguments [34,35].
When extending the MSSM these conclusions can be modified, e.g. in the NMSSM [36][37][38]. Here we consider the Higgs mass in supersymmetric models with RpV.  [40][41][42][43][44] and SPheno [45,46], as recently presented in [47]. This letter is organized as follows: we present in the next section our conventions for the models we consider, before we give details about the two-loop calculation in sec. III. The numerical results are presented in sec. IV, before we conclude in sec. V.

II. THE MSSM EXTENDED BY TRILINEAR
R-PARITY VIOLATION R-parity is a discrete multiplicative Z 2 symmetry of the MSSM, defined as [14-16, 18, 48] where s is the spin of the field and B, L are its baryon respectively lepton number. We consider the R-parity conserving superpotential of the MSSM and extend it by trilinear RpV operators [49,50] (3) We assume the bi-linear term has been rotated away [51]. Here i, j, k = 1, 2, 3 are generation indices, while SU (3) colour and SU (2) isospin indices are suppressed. Above L i ,Ē j , Q i ,Ū i ,D i , H d , H u denote the left chiral superfields of the MSSM in the standard notation [18]. We thus have for the total superpotential In the following we consider only the presence of one RpV operator at a time, including the anti-symmetric counter part, if it exists. This ensures the stability of the proton and avoids many constraints from flavour changing neutral currents and lepton flavour violation [51][52][53].
The corresponding standard soft supersymmetry breaking terms for the scalar fields L, E, Q, U , D, H d , H u and the gauginos B, W , g read withφ ∈ { Q, D, U , E, L}. The gaugino fields are two component fermions [54]. We have suppressed all generation indices in Eq. (5). The m 2 φ are 3×3 matrices and denote the squared soft masses of the scalar componentsφ of the corresponding chiral superfields Φ. The T u,d,e are 3×3 matrices of mass-dimension one. They can be written in terms of the standard A-terms [55], if no flavour violation is assumed, Similarly, for the baryon number violating term we have T λ,ijk = A ijk λ ijk .

III. TWO-LOOP CORRECTIONS FROM R-PARITY VIOLATING OPERATORS
In the presence of trilinear RpV there are new contributions to the Higgs mass at the two-loop level. We use the public codes SARAH and SPheno to compute them. These codes perform an effective potential calculation based on the generic results in Ref. [56] in the DR scheme. The precision of this calculation using SARAH and SPheno is the same for models beyond the MSSM as in many public computer tools for the MSSM, by using the results of Refs. [57][58][59][60][61]. For more general information about the calculation of two-loop Higgs masses in extensions of the MSSM with SARAH and SPheno we refer to Ref. [47].
The corrections to the effective potential at the twoloop level involving trilinear RpV couplings come from the diagrams shown in Fig. 1. From these, the tadpole contributions and self-energies are calculated by taking the first and second derivative of the two-loop effective potential V Here, h i are the real parts of the neutral Higgs scalar fields, There are two possibilites to take the derivatives: either calculate numerically the derivative of the entire potential as done in Ref. [62] for the MSSM, or take analytically the derivative of the potential with respect to the masses and numerically the derivative of the masses and couplings with respect to the VEVs (semi-analytical approach). The combination SARAH/SPheno has implemented both methods and we check their numerical agreement. Throughout we neglect the possibility of sneutrino vacuum expectation values for the LQD operators. These effects are very small since the bounds on neutrino masses restrict the sneutrino VEVs to be of order 10 MeV or smaller [18].
We use the results of Eqs. (7) and (8) together with the tree-level minimization conditions, T i , and the one-loop corrections to find the minimum of the effective potential by demanding and to calculate the loop corrected Higgs mass matrix squared [M

(T )
h ] 2 is the Higgs mass matrix squared at tree-level at the minimum of the effective potential. The two eigenvalues m 2 hi of M 2 h (p 2 = m 2 hi ), i = 1, 2, are the pole masses of the corresponding scalar fields. The smaller eigenvalue, m h ≡ m h1 , is the mass of the SM-like Higgs boson, which we are mainly interested in. In addition to the two-loop corrections to the Higgs potential due to trilinear RpV parameters, there are also one-loop corrections to the SM Yukawa couplings due to the trilinear RpV parameters, see for example [39]. In particular there are one-loop RpV contributions to the up and down quark self-energy matrices: Σ q L , Σ q R , Σ q S , q = u, d. These self-energies in turn contribute at oneloop to the Higgs potential, leading to an overall two-loop effect on the Higgs mass, i.e. of the same order as we are investigating. These self-energies enter the calculation of the Yukawas couplings as [63] v which has to be solved iteratively. The dots stand for two-loop corrections important for the top quark, U L , U R are the matrices which diagonalize the Yukawa matrix Y q . m q,pole is a diagonal matrix with the pole masses as entries.

IV. RESULTS
We now discuss the numerical impact of the RpV operators on the Higgs mass at the two-loop level. To be specific, we consider the supersymmetric parameter point fixed by tan β = 10, M 1 = M 2 = 1 2 M 3 = 1 TeV, µ = 0.5 TeV, and M A = 1 TeV. All slepton soft masses as well as all squark soft masses of the first two generations are set to 1.5 TeV. For the third generation squarks soft masses we distinguish two exemplary mass hierarchies (i) mQ ,33 = 1.5 TeV, mŨ ,33 = mD ,33 = 0.5 TeV , (ii) mQ ,33 = mŨ ,33 = mD ,33 = 2.5 TeV .
In (i) the third generation is lighter than the other sfermions, in (ii) it is heavier. The two hierarchies are assumed in the two plots shown in Fig. 2. We choose the Rparity conserving trilinear parameters as T t = −2.5 TeV, resulting in large mixing in the stop sector; all other R-parity conserving trilinear parameters vanish. In the RpV sector we choose where the Higgs mass in the R-parity conserving case, m h (0), for the two hierarchies is given by Since we just wish to demonstrate an effect, we have not attempted to tune our parameters to get the correct Higgs mass in all scenarios. We restrict ourselves to the couplings λ 313 , λ 312 , λ 213 , λ 313 , λ 331 , and λ 333 . However through the radiative corrections we dynamically generate further couplings. As mentioned, since the operators corresponding to λ ijk do not couple to squarks, the associated corrections to the Higgs mass are negligible. For the green line in the two plots of Fig. 2, this is also the case, corresponding to squark contributions not involving stops: λ 213 , λ 313 . In general, we find that for light third generation squarks, hierarchy (i), shown in the top plot in Fig. 2, there can be large positive contributions of several GeV to the Higgs mass, if stops are involved in the RpV operator. If the third generation squarks are heavier (hierarchy (ii)) shown in the bottom plot in Fig. 2, the effects are significantly smaller.
To get large effects, the RpV couplings have to be very large. An enhancement of several GeV is only found for couplings which are close to or even above the perturbativity limit, which is approximately 1 at the weak scale [64,65] 3 . In order to avoid the Landau pole the large coupling scenarios must have a low cut-off similar to the λ-SUSY setup [66].
The couplings involving stops are hardly constrained by flavor physics, especially if the non-stop masses are in the TeV range [67]. Furthermore, we have checked that for example for λ 312 if we choose instead tan β = 25, the resulting shift in the Higgs mass changes by less than 5%. We have to note that very small soft masses together with large trilinear couplings often suffer from an unstable electroweak vacuum and have to be considered carefully [68][69][70]. We used the public code Vevacious [71] to check that hierarchy (i) is meta-stable with a lifetime longer than the age of the universe. We show in Fig. 3 the change in the top-Yukawa coupling from including the RpV loop corrections to all quarks.
Here Y t (0) 0.85, for tan β = 10. The effect is very small. The dependence of the Higgs mass on the mass of the involved squarks is depicted in Fig. 4, where we kept λ 313 = 1, respectively λ 333 = 1, fixed and varied mQ ,33 , mŨ ,33 , and mD ,33 , separately. The soft masses not being varied are fixed at 1.5 TeV.
The largest corrections appear in the case of light right-handed squarks together with largeŪDD operators. For LQD operators the strongest dependence is on the left-squark soft mass. The value of mD ,33 plays always a subdominant role. We finally consider the dependence on A 0 . For this purpose we show in Fig. 5 the light Higgs mass as function of A 0 with and without RpV operators. Here, we have chosen light right-handed stops, mŨ ,33 = 0.5 TeV, while all other scalar soft masses are set to 1.5 TeV. Once again the RpV couplings can easily shift the light Higgs mass by a few GeV. In the case of λ 313 the shift shows a clear dependence on A 0 while it is rather insensitive to A 0 if λ couplings are considered. That is consistent with our choice of small mŨ ,33 . For small mQ ,33 the λ would show a stronger dependence on A 0 .

V. CONCLUSION
We have discussed the impact of large trilinear RpV couplings on the light CP-even Higgs mass at the twoloop level. We have shown that in particular for light stops these corrections can be very important, increasing