MSSM Higgs Decays to Bottom Quark Pairs revisited

We present an update of neutral Higgs boson decays into bottom quark pairs in the minimal supersymmetric extension of the Standard Model. In particular the resummation of potentially large higher-order corrections due to the soft SUSY breaking parameters A_b and mu is extended. The remaining theoretical uncertainties due to unknown higher-order SUSY-QCD corrections are analyzed quantitatively.

Although we investigate only the SUSY-QCD corrections, it should be noted that the electroweak corrections can be important, too, and yield an additional contribution to the uncertainties. The full one-loop electroweak corrections were computed in Ref. [9], and later refined in [10] including the two-loop contributions to the Higgs boson propagator matrix. Section 2 summarizes the present theoretical status of Higgs decays into bottom quark pairs and sets the basis for the resummation, which is described in Section 3. In Section 4 we analyze the remaining theoretical uncertainties originating from the SUSY-QCD corrections in detail for representative MSSM scenarios. In Section 5 we conclude.

Higgs decays into bottom quark pairs 2.1 QCD corrections
The partial decay widths of the neutral Higgs bosons Φ = h, H, A into bottom quark pairs, including QCD corrections, can be cast into the form where regular quark mass effects are neglected. The large logarithmic part of the QCD corrections has been absorbed in the running MS bottom quark mass m b (M Φ ) at the scale of the corresponding Higgs mass M Φ . The QCD corrections ∆ QCD and the top quark induced contributions ∆ Φ t read as [11] ∆ QCD = 1 + 5.67 where N F = 5 active flavours are taken into account. In the intermediate and large Higgs mass regimes the QCD corrections reduce the bb decay widths by about 50% due to the large logarithmic contributions.

SUSY-QCD corrections
In the MSSM the full SUSY-QCD corrections to the fermionic decay modes have been computed at NLO [9,12]. In the low-energy limit M φ , M Z , m b ≪ mb i , mg the results can be cast into the simple form Γ QCD (φ → bb) denotes the QCD-corrected decay width of Eq. (6). It should be noted that NLO-terms involving the trilinear mixing parameter A b are absent in Eq. (8).

Construction of the effective Lagrangian
The result of Eq. (8) can be derived from the effective low-energy Lagrangian [8] 3 after expansion up to NLO. The symbol φ 0 1 (φ 0 2 ) denotes the neutral components of the Higgs doublet coupling to down-(up-)type quarks. The parameter tgβ = v 2 /v 1 is defined as the ratio of the two vacuum expectation values, and v 2 = v 2 1 + v 2 2 = 1/ √ 2G F is related to the Fermi constant G F . The would-be Goldstone field G 0 is absorbed by the Z boson and generates its longitudinal component. The SUSY-QCD corrections turn out to be significant for large values of tgβ and moderate or large µ values. In order to improve the perturbative result all terms of O [(α s µ tgβ) n ] have been resummed [8]. The correctly resummed effective Lagrangian is given by Eq. (9). The correction ∆m b is non-decoupling in the sense that scaling all SUSY parameters mb 1,2 , mg, µ in Eq. (10) leaves ∆m b invariant. However, its contribution develops decoupling properties [13], as we will discuss later on.
Apart from the correction ∆m b there is a second class of potentially large (nondecoupling) contributions at higher orders which may spoil the perturbative reliability of the results: The trilinear mixing parameter A b can be of similar size as µtgβ as e.g. in no-mixing scenarios of the sbottom particles. In the low-energy limit of Eq. (8) such terms are absent. However, they arise at higher orders. In the following we develop an approach to include A b terms in the resummation of Eq. (9). For this purpose we start from the unrenormalized effective Lagrangian in the low-energy limit at leading order Including higher-order corrections in the low-energy limit, the pole mass m b of the bottom quark is given by where the self-energy Σ b (m b ) can be decomposed as The leading parts in A b and µ are finite at NLO, Inserting these two expressions in Eq. (12) leads to the well-known result that the radiative corrections to the bottom mass are proportional to A b − µtgβ, i.e. the off-diagonal components of the sbottom mass matrix of Eq.
(2). The structure of the self-energy beyond NLO can be derived from general arguments based on the asymptotic behaviour of the corresponding Feynman-diagrams in the lowenergy limit. The terms involving A b or µ are generated by mass-insertions in the virtual sbottom propagators. At NLO the diagrams of Fig (for M SU SY ∼ mb 1 ∼ mb 2 ∼ mg) coinciding with the explicit results of Eq. (14). At NNLO the leading contributions involving A b and µ are generated by e.g. the diagrams of Fig. 2. The diagrams (a) and (b) behave asymptotically as while diagram (c) develops the low-energy behaviour Thus, the diagrams of Fig. 2 contribute to the same order as the pure QCD corrections to the NLO results and do not generate leading terms of . This power-counting argument can be applied to all other two-loop diagrams involving µ and A b , too. Any further mass-insertion is suppressed by another power of m b /M SU SY , and is therefore non-leading.
These arguments can be extended to any perturbative order. Due to the KLN theorem [14,15] Therefore, the low-energy behaviour of the mass-inserted diagram is modified by an additional power of m b (A b − µtgβ)/M 2 SU SY . Consequently, the diagrams of Fig. 2 constitute the leading contributions in A b and µtgβ at NNLO. These arguments prove that the results of Eq. (14) include all leading powers of α s A b and α s µtgβ. This is confirmed by the explicit two-loop results of Ref. [16].
In order to obtain the effective low-energy Lagrangian from the expression Eq. (12) for the bottom mass, we have to perform the replacements v 1 → √ 2φ 0 1 and v 2 → √ 2φ 0 * 2 in the corresponding bottom mass operator. These replacements lead to the exact interactions with non-propagating Higgs fields, i.e. in the low-energy limit of small Higgs momentum [17]. The final expression of the effective Lagrangian can be cast into the form which differs from previous results by the new factor (1+∆ 1 ) in front of φ 0 1 . This expression has to be matched with the renormalized low-energy Lagrangian yielding the relations 5 ] are resummed by means of the simple replacement in the effective Lagrangian of Eq. (9). This proof confirms and extends the resummation presented in Ref. [8] and explains the absence of any A b terms in Eq. (8) in terms of a clear physical interpretation: the leading A b terms are absorbed in the definition of the effective Yukawa coupling λ b in the low-energy effective Lagrangian. In a Feynman diagrammatic approach this corresponds to a cancellation of the A b terms in the bottommass counterterms and the genuine irreducible three-point diagrams. This cancellation is exact at zero-momentum transfer, but a mild dependence on A b appears when keeping all external momenta on-shell due to the momentum dependence of the one-particleirreducible (1PI) three-point functions.

Validity of the low-energy approximation
The expression in Eq. (19)  ] to all orders in perturbation theory. However, there are other kinds of non-decoupling terms in the 1PI self-energies, as can be inferred already from the NNLO expressions of Eqs. (16,17). The question about the numerical size of these non-leading terms arises, and whether the 5 It should be noted that the bottom wave-function renormalization constants do not contain any leading non-decoupling contribution in A b and µ. Moreover, it should be emphasized that the combination A b − µtgβ only appears in the definition of the bottom mass, while A b and µtgβ contribute in a different way to the bottom Yukawa coupling and Higgs decay processes. 6 In order to avoid an artificial singularity in Γ(h → bb) for vanishing α the remainder proportional to (C φ − C LE φ ) is multiplied by the unresummed Yukawa coupling g φ b .
NNLO resummation is necessary in practical applications. Eqs. (15)(16)(17) imply that the irreducible NNLO corrections ∆ {1,2} | < ∼ O(1%), so that the NLO corrections are small, and the size of the NNLO corrections is of the same order as the deviation of the full results from the zero-momentum approximation. This argument can be extended to higher orders in perturbation theory. At the n-loop level the non-decoupling 1PI diagrams originate from a single vacuum insertion (analogous to the diagrams of Fig. 2 . Hence, they are negligible, because either they are much smaller than the n-loop reducible contribution, or the numerical value of the leading corrections is small already at NLO. The trilinear mixing parameter A b cannot be much larger than M SU SY , since otherwise the color and charge symmetries would be broken [18]. Thus, the contribution ∆ 1 of Eq. (14) reaches maximal values of O(10%), while the term ∆m b can be larger by an order of magnitude.
In Fig. 3 we compare the relative NLO corrections including the resummation of ∆m b with the novel NNLO contributions ∆ 1 of Eq. (10) as a function of the pseudoscalar Higgs mass M A for all three neutral Higgs states in the following MSSM scenario with large A b : The relative corrections are normalized to the QCD-corrected decay widths Γ QCD (φ → bb) of Eq. (6) in both cases. While the ∆m b effects are of O(10%) and thus of moderate size, the novel ∆ 1 contributions turn out to be of O(1%) apart from the small heavy scalar Higgs mass range, where they can reach a similar magnitude as the ∆m b terms. This particular scenario, however, has to be considered as an extreme case. In general the ∆ 1 terms are small, confirming the previous qualitative discussion.

Numerical Results
The numerical analysis of the neutral Higgs boson decays into bottom quark pairs is performed for the 'small α ef f ' MSSM scenario [19] as a representative case: We use the RG-improved two-loop expressions of Ref. [20]. The bottom quark pole mass has been chosen to be M b = 4.62 GeV, which corresponds to a MS mass m b (m b ) = 4.28 GeV. The strong coupling constant has been normalized to α s (M Z ) = 0.119. The resummation effects discussed in the previous section have been derived in the low-energy limit M 2 φ , M 2 Z , m 2 b ≪ M 2 SU SY . The question arises, how reliable this approximation works in phenomenological applications. In particular, the magnitude of SY ) terms matters for sizeable masses of the lowenergy particles. This can be tested explicitly by comparing the approximate results of Eq. (8) with the full one-loop result. A typical example is depicted in Fig. 4 for the 'small α ef f ' scenario, where the relative difference between the full and approximate one-loop contributions [see Eq. (8)] is presented for all neutral Higgs particles as a function of the pseudoscalar Higgs mass M A . It is clearly visible that the approximation turns out to be sufficient for the heavy neutral Higgs particles H, A, but fails for the light scalar Higgs boson h in the decoupling limit [21]. However, in the decoupling limit the size of the approximate SUSY-QCD corrections strongly decreases, since tgα → −1/tgβ and thus so that the SUSY-QCD corrections become negligible. Due to this behaviour the lowenergy approximation is sufficient for most phenomenological applications. This also explains the failure of the approximation in this case: the large non-decoupling contributions from ∆m b cancel to a large extent in the lightest Higgs boson couplings, leaving a small remainder of the same order as the non-leading contributions. On the other hand, this cancellation does not occur for the heavy Higgs bosons, and the effective Lagrangian approach yields a good approximation. There are two basic sources of systematic uncertainties originating from the SUSY-QCD contributions: (i) The MSSM masses and couplings involved in the NLO SUSY-QCD corrections will only be known with a sizeable uncertainty at the LHC, while future e + e − linear colliders in the 500 GeV to 1 TeV range will enable precision measurements of the SUSY masses and couplings. These errors in the input parameters generate systematic uncertainties for the prediction of the partial decay widths. (ii) Due to missing higher order results the scale dependence of the strong coupling constant α s will not be compensated. The scale variation yields an estimate of the purely theoretical SUSY-QCD uncertainty, which will be analyzed quantitatively in this section. 7 The central scale µ 0 of the strong coupling constant appearing in the SUSY-QCD corrections will be chosen as the average mass of the involved SUSY particles, i.e.
In order to estimate the residual scale dependence the scale of α s will be varied between µ 0 /3 and 3µ 0 . The usual QCD corrections have been included up to the three-loop order so that the residual purely QCD-induced scale dependence ranges below the per-mille level and can thus safely be neglected.
The results for the partial decay widths are shown in Fig. 5a for the light scalar Higgs boson, in Fig. 5b for the heavy scalar Higgs boson and in Fig. 5c for the pseudoscalar Higgs boson. These results include the QCD corrections up to NNNLO of Eq. (6) and the full NLO SUSY-QCD corrections of Eq. (8) with the resummation of the leading ∆m b and ∆ 1 terms according to Eqs. (9,21). It can clearly be inferred from these figures that the remaining uncertainties due to the scale choice are typically of the order of 10%. However, they are significantly enhanced in regions where the SUSY-QCD corrections become large, as in the 'small α ef f '-scenario, which develops a strongly suppressed partial decay width Γ(h → bb) for pseudoscalar masses M A ∼ 150 GeV 8 . This, however, corresponds only to a tiny region in the light scalar Higgs mass M h close to its upper limit for large M A within the 'small α ef f '-scenario. The theoretical uncertainties turn out to be large at M A ∼ 150 GeV.
The uncertainties in the partial decay widths Γ(φ → bb) translate into systematic errors in the corresponding branching ratios. They are depicted in Figs. 6a-c for the three neutral Higgs bosons. These results have been obtained with the program HDECAY [22] after including the results obtained in this analysis. Since the partial decay into bb pairs is dominant in nearly the entire Higgs mass ranges, its uncertainty due to the scale choice above reduces to a level of O(1%). However, the scale dependence of Γ(φ → bb) develops significant systematic errors in the non-leading branching ratios into τ + τ − , gluon and tt pairs. These can reach a level of O(10%) and are larger than the expected experimental accuracy at future e + e − linear colliders, which clearly calls for a NNLO calculation of the SUSY-QCD part. These theoretical errors have to be added to the uncertainties due to inaccuracies of the input parameters as presented in [23] and the theoretical errors of the Higgs masses and couplings [3] 9 . They constitute a significant source of uncertainty. An analogous analysis is required for the theoretical uncertainties due to the SUSYelectroweak corrections beyond NLO. However, this is beyond the scope of our paper.

Conclusions
In this paper we have reanalyzed the neutral scalar Higgs decays into bb pairs in the MSSM with particular emphasis on the SUSY-QCD corrections and their theoretical uncertainties. We have extended the resummation of large non-decoupling SUSY-QCD corrections of O(α s µtgβ/M SU SY ) by the inclusion of non-decoupling terms of O(α s A b /M SU SY ) which have not been taken into account in previous analyses. We have shown that these terms are absent at NLO in the effective Lagrangian but arise at NNLO and beyond. This can easily be traced back to the renormalization of the bottom Yukawa coupling in the lowenergy limit, where the heavy SUSY particles are integrated out. We have obtained the important result that these novel contributions hardly affect the theoretical predictions for the partial decay widths into bb pairs so that they do not endanger the reliability of the perturbative result in contrast to the leading terms of O(α s µtgβ/M SU SY ).
We investigated the remaining theoretical uncertainties generated by the SUSY-QCD corrections quantitatively. While the theoretical errors of the partial decay widths Γ(φ → bb) turn out to be of O(10%), this effect cancels to a large extent in the branching ratios BR(φ → bb) due to its dominance. It appears, however, as a sizeable increase in the systematic uncertainties of the non-leading branching ratios into τ + τ − , gluon and tt pairs, which appear to be larger than the anticipated experimental accuracies at future linear e + e − colliders. This clearly calls for a NNLO calculation of the SUSY-QCD part, which is beyond the scope of this work.