Refinements of the Bottom and Strange MSSM Higgs Yukawa Couplings at NNLO

We extend the already existing two-loop calculation of the effective bottom-Yukawa coupling in the MSSM. In addition to the resummation of the dominant corrections for large values of tg$\beta$, we include the subleading terms induced by the trilinear Higgs coupling $A_b$. This calculation has been extended to the NNLO corrections to the MSSM strange-Yukawa coupling. Our analysis leads to residual theoretical uncertainties at the per-cent level.


Introduction
The discovery of a Standard-Model-like Higgs boson at the LHC [1] completed the theory of electroweak and strong interactions. The existence of an elementary Higgs boson [2] is a necessary ingredient of a weakly interacting renormalizable theory with spontaneous symmetry breaking [3]. The measured Higgs mass of (125.09 ± 0.24) GeV [4] ranges at the order of the weak scale. However, if embedded in a Grand Unified Theory (GUT), radiative corrections tend to push the Higgs mass towards the GUT scale, if the Higgs couples to particles at this large scale. This problem is known as the hierarchy problem [5]. A possible solution to this problem is provided by supersymmetry (SUSY) at the TeV scale [6,7].
The minimal supersymmetric extension of the Standard Model (MSSM) requires the introduction of two Higgs doublets implying the existence of five elementary Higgs bosons, two neutral CP-even (scalar) bosons h, H, one neutral CP-odd (pseudoscalar) boson A and two charged bosons H ± . At lowest order the MSSM Higgs sector is entirely fixed by two independent input parameters, which are generally chosen as tgβ = v 2 /v 1 , the ratio of the two vacuum expectation values v 1,2 , and the pseudoscalar Higgs mass M A if all SUSY parameters are real. Including the one-loop and dominant two-loop corrections the upper bound on the light scalar Higgs mass is lifted to M h 135 GeV [8]. More recent first three-loop results confirm this upper bound within less than 1 GeV [9]. The various Higgs couplings to fermions and gauge bosons depend on mixing angles α and β, which are defined by diagonalizing the neutral and charged Higgs mass matrices. They are collected in Table 1 relative to the SM Higgs couplings. For large values of tgβ the down-type Yukawa couplings are strongly enhanced, while the up-type H 1 1 1 MSSM h cos α/ sin β − sin α/ cos β sin(β − α) H sin α/ sin β cos α/ cos β cos(β − α) A 1/tgβ tgβ 0 The soft SUSY-breaking terms in the MSSM induce mixing of the current sfermion eigenstatesf L andf R . The sfermion mass matrix in the current eigenstate basis is given by 1 [10] to the terms induced by the soft SUSY-breaking trilinear coupling A b and to the SUSY-QCD corrections to the strange-Yukawa couplings. The results will play a role in all processes to which the bottom-and strange-Yukawa couplings contribute, i.e. in particular the neutral and charged Higgs decay widths and Higgs radiation off bottom quarks at hadron colliders which constitutes the dominant Higgs boson production channel for large tgβ at the LHC [11].

Effective Bottom-and Strange-Yukawa Couplings
The leading parts of the SUSY-QCD (and SUSY-electroweak) corrections to bottom-and strange-Yukawa-coupling induced processes can be absorbed in effective bottom-and strange-Yukawa couplings. These contributions arise in the limit of heavy supersymmetric particle masses compared to the energy scale of the particular process. The accuracy of this heavy mass approximation has been investigated for neutral MSSM Higgs decays into bottom quarks h/H/A → bb [12], charged Higgs decays to top and bottom quarks H ± → tb [13] and Higgs radiation off bottom quarks at e + e − colliders [14] and hadron colliders [15,16] by comparing it to the full NLO results. For large values of tgβ the approximation turns out to agree with the next-to-leading-order (NLO) results to better than one per cent.

Effective Lagrangian
The leading corrections to the MSSM bottom-and strange-Yukawa couplings can be obtained from the effective Lagrangian [12,13] with the individual leading one-loop expressions (C F = 4/3) [17] and the final contribution in the mass-eigenstate-basis The generic function I is given by The fields φ 0 1 and φ 0 2 are the neutral components of the Higgs doublets coupling to down-and up-type quarks, respectively. They are related to the mass eigenstates h, H, A by The two vacuum expectation values are related to the Fermi constant The would-be Goldstone field G 0 is absorbed by the Z boson and generates its longitudinal component. The top-Yukawa coupling λ t is related to the top mass by m t = λ t v 2 / √ 2 at lowest order. The soft SUSY-breaking trilinear couplings of the top, bottom and strange squarks are denoted by A t , A b and A s , the higgsino mass parameter by µ and the strong coupling constant by α s . The renormalization scale is depicted as µ R . The corrections ∆ b,s induce a modification of the relation between the bottom (strange) quark mass m b (m s ) and the bottom (strange) Yukawa coupling λ b (λ s ), The effective Lagrangian of Eq. (5) can be expressed as (omitting the mass and Goldstone terms) with the effective (resummed) couplings Although the SUSY corrections ∆ q are loop suppressed, they are significant for large values of tgβ. In these cases they constitute the dominant supersymmetric radiative corrections to the bottom-and strange-Yukawa couplings. The effective Lagrangian in Eq. (5) has been derived by integrating out the heavy SUSY particles so that it is not restricted to large values of tgβ only. In order to improve the perturbative result it has been shown by power counting that the Lagrangian of Eq. (5) [12,13] (including mixed contributions).

Low Energy Theorems
The derivation of higher-order corrections to the effective bottom-and strange-Yukawa couplings would require the calculation of the corresponding three-point functions. This, however, can be reduced to the evaluation of self-energy diagrams by the use of low energy theorems [18]. These are based on the idea that any matrix element with an external Higgs boson can be related to the analogous matrix element without the external Higgs particle in the limit of vanishing Higgs momentum by the simple replacements v 1 → √ 2φ 0 1 and v 2 → √ 2φ 0 * 2 in the latter. Thus we only need to calculate the corresponding pieces of the bottom and strange quark self-energies. The leading pieces ∆ q,1/2 (q = b, s) emerge from the scalar part Σ S (m 2 q ) of the self-energy 2 giving rise to the following relation between the mass m q of the bottom (strange) quark and the bottom (strange) Yukawa coupling λ b (λ s ), where the leading terms of the self-energy Σ S (m 2 q ) for heavy SUSY particles are given by The NLO-QCD parts of ∆ b and ∆ s in Eq. (5) can be derived from off-diagonal mass insertions of the type λ q (A q v 1 −µv 2 ) (up to a factor 1/ √ 2) in the virtual squark propagators, as illustrated in Fig. 1 at one-loop level. The result of these diagrams is given by the finite expressions of Eq. (6) (supplemented by the SUSY-electroweak corrections originating from charged higgsino exchange to the bottom-Yukawa couplings) after rotation of the fields in the current-eigenstate basis to the mass eigenstates. These expressions are not renormalized since there is no tree-level coupling of bottom and strange quarks proportional to A q or µ.

NNLO Corrections
The determination of the NNLO corrections to the effective bottom-and strange-Yukawa couplings requires the calculation of the leading NNLO corrections to the bottom and strange self-energies. The NNLO results of the bottom-Yukawa couplings have been obtained in Refs. [10,19]. We will extend these results to the non-tgβ-enhanced A b terms and to the strange Yukawa couplings.

Bottom-Yukawa Couplings
Due to the fact that the effective insertions according to Fig. 1 are always proportional to A b − µ tgβ, the contributions of all two-loop diagrams for the bottom-Yukawa coupling is the same for the A b and the µ tgβ contributions (up to the relative overall sign). The renormalization proceeds along the lines of Ref. [10] so that the SUSY-QCD corrections to the ∆ b,1 terms are the same as for the ∆ b,2 contributions after renormalization (including the SUSY-restoring counter terms [20]). Denoting the (renormalized) NNLO-corrected SUSY-QCD part ∆ QCD b,2 of Ref. [10] as with the NNLO correction δ 1 and the effective correction to the bottom-Yukawa couplings of Eq. (7) acquires the form where A 0 b denotes the bare trilinear coupling that is renormalized in SUSY-QCD and δ 2 the SUSY-QCD corrections to ∆ elw b,2 [10]. However, the renormalization of A b emerges from a nonleading order in our context: for the MS-renormalized trilinear coupling within dimensional regularization in n = 4 − 2ǫ dimensions we obtain so that A b is not renormalized at O(α s A b ). We have explicitly checked that the divergence corresponding to the counter term of A b is generated by the diagram of Fig. 2 with an insertion λ b v 1 (up to a factor 1/ √ 2) at the virtual bottom-quark line. Thus the final expression including the O(A b ) terms is given by Eq. (17) Figure 2: Two-loop diagram of sbottom-self-energy insertions contributing to the SUSY-QCD corrections to the bottom-quark self-energy involving bottom quarks b, bottom squarksb and gluinosg.
In this work we adopt the renormalization program of Ref. [21], i.e. the counter term for the electroweak contributions is modified for the trilinear coupling A t that is defined in the MS scheme leading to a vanishing counter term for A t at the (leading) order we are calculating.

Strange Yukawa Couplings
The translation of the results for the bottom-Yukawa couplings to the Higgs boson couplings to strange quarks requires a careful investigation of the corresponding quark-mass contributions. Since in the calculation of the bottom-Yukawa coupling the bottom quark is treated strictly massless and the external momentum dependence is omitted, there is no difference for the individual two-loop diagrams, if the bottom parameters are replaced by their corresponding strange parameters. Care must be taken for the proper summation over all quark/squark flavours for the diagrams with gluino-self-energy insertions since the strange-squark mass coincides with the left-and right-handed squark masses of the first two generations and the sbottom and stop masses of the third generation are independent. Another difference to the bottom-quark case is the absence of sizeable SUSY-electroweak contributions to the strange-Yukawa coupling, since we are neglecting the charm Yukawa coupling λ c . The final result can be cast into the form where δ s denotes the NNLO SUSY-QCD corrections to the strange Yukawa couplings and ∆ s,1/2 are defined in Eq. (6). The expression above for ∆ s is then inserted into the effective Lagrangian of Eq. (5)

Results
The results of this work have been implemented in the program HDECAY [22], which calculates the MSSM Higgs masses and couplings according to the RG-improved results of Ref. [23] and all partial decay widths and branching ratios including the relevant higher-order corrections [11].
For large values of tgβ the decays of the neutral Higgs bosons are dominated by the decays into bb and τ + τ − . Their branching ratios have been studied with the one-loop expressions of the correction ∆ b of Eq. (6) in Ref. [12].

Higgs Decays into Bottom and Strange Quarks
The partial decay widths of the neutral Higgs bosons Φ = h, H, A into bottom-quark pairs, including QCD and SUSY-QCD corrections, are given by [12] where m b (M Φ ) denotes the MS bottom-quark mass at the scale of the corresponding Higgs mass M Φ and quark mass effects beyond O(m 2 b ) are neglected. The QCD corrections δ QCD and the top quark induced contributions δ Φ t have been calculated [24] and can be found in Ref. [11] in compact form. The QCD corrections δ QCD are taken into account up to N 4 LO and the corrections δ Φ t at the NNLO level in HDECAY. The dominant part of the SUSY-QCD corrections [25] has been absorbed in the resummed bottom-Yukawa couplingsg φ b of Eq. (12). The remainder δ rem SQCD is small in phenomenologically relevant scenarios for large values of tgβ [12]. In our analysis we include the two-loop corrected mixed top-Yukawa-coupling-induced SUSY-QCD/electroweak corrections ∆ elw b,2 (1 + δ 2 ) in the couplingsg φ b , too. The strange Yukawa coupling plays a phenomenological role for charged Higgs decays into charm and strange quarks H + → cs. Neglecting regular quark mass effects 3 this partial decay width can be expressed as [26] Γ with the same QCD-correction-factor δ QCD as in Eq. (21). The small remainder of the genuine SUSY-QCD corrections after absorbing the dominant part in the effective strange-Yukawa couplingg A s is neglected.

Numerical Results
We perform our numerical analysis of the MSSM Higgs boson decays into bottom and strange quarks for two MSSM benchmark scenarios [27] as representative cases:  [21] as implemented in HDECAY.
In the following we will present the impact of the new results on the bottom-and strange-Yukawa couplings as well as related observables. In Fig. 3 the scale dependence of the ∆ b and ∆ s terms is shown with and without the A b , A s contributions for the m mod+ h and in Fig. 4 for the τ -phobic scenario. The ∆ b and ∆ s corrections amount to about 10% in the m mod+ h scenario and about 40-60% in the τ -phobic scenario for tgβ = 30. The scale dependence is reduced significantly from one-to two-loop order to the few-per-cent level at NNLO, while the additional contributions of the A b , A s terms are small as can be inferred from the differences between the red and blue curves. However, in general the sizes and signs of the total ∆ b and ∆ s contributions depend on the MSSM scenario, in particular on the sign and size of µ and the value of tgβ. The central scale choices equal to the average of the corresponding SUSY masses, i.e. µ 0 = (mq 1 + mq 2 + mg)/3 for ∆ QCD b , ∆ s and µ 0 = (mt 1 + mt 2 + µ)/3 for ∆ elw b at NNLO, turns out to be reasonably close to the maximum of the scale dependence at NNLO and thus suitable as the natural central scale choice.
As particular examples we analyze the partial decay widths of the heavy neutral MSSM Higgs bosons into bb pairs and of the charged Higgs boson into cs in Fig. 5 for the τ -phobic scenario, respectively. The two-loop corrections to ∆ b (∆ s ) reduce the partial decay widths to bb (cs) pairs for the central scale choices by O(10%). The bands at NLO (dashed blue curves) and NNLO (full red curves) are generated by varying the renormalization scales of ∆ b and ∆ s between 1/2 and 2 times the corresponding central scales µ 0 . A significant reduction of the dashed one-loop bands of O(10%) to the full two-loop bands at the per-cent level can be inferred from these results. All NNLO results are positioned at the lower ends of the NLO error bands. The small gap for the H + → cs decay between the NLO and NNLO bands is due to the cross over of the NLO and NNLO scale dependences of ∆ s in Fig. 4 Figure 5: Partial decay widths of the heavy scalar H and the pseudoscalar A Higgs bosons to bb (upper two plots) and charged Higgs decays to cs (lower plot) in the τ -phobic scenario. The dashed blue bands indicate the scale dependence at one-loop order and the full red bands at two-loop order by varying the renormalization scales of ∆ b and ∆ s between 1/2 and 2 times the central scale given by the corresponding average of the SUSY-particle masses.

Conclusions
We have calculated the NNLO corrections to the effective bottom-and strange-quark-Yukawa couplings within the MSSM, extending previous analyses to non-leading terms that are mediated by the soft SUSY-breaking trilinear couplings A b , A s for large values of tgβ. The leading parts of the SUSY-QCD corrections originate from factorizable contributions induced by virtual squark and gluino exchange, that can be absorbed in effective Yukawa couplings in a universal way. We have calculated the two-loop SUSY-QCD corrections to these effective bottom-and strange-Yukawa couplings beyond the leading µ tgβ approximation.
In summary, the significant scale dependence of O (10%) of the NLO predictions for processes involving the bottom-and strange-quark-Yukawa couplings of MSSM Higgs bosons necessitate the inclusion of NNLO corrections. For the NNLO-corrected Yukawa couplings, we find a decrease of the scale dependence to the per-cent level. These results known for the leading µtgβ-terms of the bottom-Yukawa couplings have been established for the non-leading A b terms and the strange-Yukawa couplings in this work, too. The improved NNLO predictions for the bottom-and strange-Yukawa couplings provide a quantitative basis for experimental analyses at the LHC and future e + e − colliders as the ILC.