BR(Bs to mu+ mu-) as an electroweak precision test

Using an effective-theory approach, we analyze the impact of BR(Bs to mu+ mu-) in constraining new-physics models that predict modifications of the Z-boson couplings to down-type quarks. Under motivated assumptions about the flavor structure of the effective theory, we show that the bounds presently derived from BR(Bs to mu+ mu-) on the effective Z-boson couplings are comparable (in the case of minimal flavor violation) or significantly more stringent (in the case of generic partial compositeness) with respect to those derived from observables at the Z peak.


Introduction
The rare decay B s → µ + µ − is one of the most clean low-energy probes of physics beyond the Standard Model (SM). A first experimental evidence of this rare process has recently been obtained by the LHCb collaboration [1], that reported a 3.5σ signal. The corresponding flavor-averaged time-integrated branching ratio determined by LHCb is [1] where the error is dominated by the statistical uncertainty and is expected to be improved significantly in the near future. At this level of precision there is good agreement with the SM prediction, that for the same quantity reads [2] B th SM = (3.54 ± 0.30) × 10 −9 , taking into account the effect of ∆Γ s = 0 pointed out in Ref. [3]. The effectiveness of B s → µ + µ − as a probe of physics beyond the SM is related to a double-suppression mechanism at work within the SM. One the one hand, it is a flavorchanging neutral-current (FCNC) process and, as such, it receives no tree-level contributions. On the other hand, the purely leptonic final state and the pseudoscalar nature of the initial state imply a strong helicity suppression and forbid photon-mediated amplitudes at the one-loop level. As a result of this double suppression, up to the one-loop level B s → µ + µ − receives contributions only from Yukawa and weak interactions.
This process is often advocated as a probe of models with scalar-mediated FCNCs, that are naturally predicted in models with an extended Higgs sector. However, it is also an excellent probe of the Z → bs effective coupling (see e.g. Refs. [4][5][6]). In this Letter we compare the bounds set on such coupling by B(B s → µ + µ − ) with the deviations from universality on the Z → bb coupling determined from electroweak precision observables. To this purpose, we describe the possible deviations on the Z-boson couplings to downtype quarks by means of an effective-theory approach, and we employ two motivated assumptions about the flavor structure of the theory, namely minimal flavor violation or generic partial compositeness, to relate flavor-changing and flavor-diagonal couplings.  [4,6], there exists a wide class of models where the only relevant deviations from the SM in B(B s → µ + µ − ) and Z → bb can be described in terms of modified Z-boson couplings at zero momentum transfer, defined by the following effective Lagrangian Here g is the SU (2) L gauge coupling, c W = cos θ W (s W = sin θ W ), and g ij L,R denote the effective SM couplings. In the following we employ state-of-the-art expressions to estimate the SM contributions to B(B s → µ + µ − ) and Z → bb, and use L Z eff at the tree level only to estimate the non-standard effects parameterized by δg ij L,R . For later convenience we recall the leading structure of the g ij L,R . The tree-level SM couplings are At the one-loop level the g ii L,R are gauge dependent, but they assume the following simple and gauge-independent form in the limit m t m W (or g → 0): where V ij denote the elements of the CKM matrix and v ≈ 246 GeV.
The new-physics contributions, parameterized by δg ij L,R , can be related to the couplings of a manifestly gauge-invariant Lagrangian, with the following set of dimension-six operators: Defining the flavor indices {i, j} in the mass-eigenstate basis of down-type quarks we find The set of operators in Eq. (7) is not the complete set of gauge-invariant dimensionsix operators contributing to B s → µ + µ − and Z → bb at the tree level. In principle, we can consider also four-fermion (two-quarks/two-leptons) operators, terms of the type J ν × D µ F µν , or terms of the type H † J µν × F µν , where J ν and J µν are quark bilinears, and F µν generically denotes the field-strength tensor of U (1) or SU (2) L gauge fields. However, the effects of these operators cannot be described by means of L Z eff and we lose the natural correlation between these two observables. 1 For this reason in the following we concentrate only on the set of operators in Eq. (7).
In order to relate flavor-diagonal and flavor-violating couplings we need to specify the flavor structure of the effective theory. We consider two reference frameworks: 1) the hypothesis of Minimal Flavor Violation (MFV), as defined in Ref. [7]; 2) the generic flavor structure implied by the hypothesis of Partial Compositeness (PC) [8], following the effective-theory approach described in Refs. [9,10].
In the MFV framework there is a strict correlation between flavor-diagonal (but nonuniversal) and flavor-violating couplings of the operators listed in Eq. (7). Restricting to the contributions relevant to this correlation, the effective couplings can be decomposed as follows: where a nL,R are unknown O(1) couplings and Y u,d are the SM Yukawa couplings. The last equalities in Eqs. (9), (10) hold after rotating the Yukawa matrices in the mass-eigenstate basis of down-type quarks, where Y u = V † λ u and Y d = λ d , with λ u,d diagonal matrices [7]. As a result, we can parameterize all the δg ij L,R in terms of two flavor-blind parameters, δg L,R , defined by The normalization has been chosen such that in order to identify δg L,R with the usual definition of the modified Z → bb couplings [12].
As can be seen, in the left-handed sector the flavor structure is identical to the one of the leading one-loop contribution within the SM, reported in Eq. (5). In the right-handed sector the structure is different but the effects are expected to be very small due to the strong suppression of down-type masses. Indeed the overall normalization implies In the PC framework the correlation between flavor-diagonal and flavor-violating couplings is determined up to unknown O(1) parameters, related to the hypothesis of flavor anarchy in the composite sector. In this case, following the notation of Ref. [10], we expect where the q,d i parameterize the mixing of the SM fermions with the composite sector, and {m ρ , g ρ } are the reference mass and coupling characterizing the composite sector. On the r.h.s. of Eqs. (13), (14) we have eliminated the q,d i in favor of quark masses and CKM angles by means of the relations [9,10] M h = 125 GeV [14] ∆α Table 1: Input parameters relevant for the Z → bb constraints. Quantities without an explicit reference are taken from Ref. [18]. We do not show the errors for quantities whose uncertainty has a negligible impact on our numerical analysis.
As can be seen, up to O(1) factors the flavor structure of the left-handed couplings is the same as in the MFV framework. On the other hand, the structure is significantly different in the right-handed sector, where larger effects are now possible in the flavor-violating case. Ignoring O(1) factors, we parameterize the structure of the two couplings in the PC framework as follows: where again the normalization has been chosen in order to satisfy Eq. (12). (For recent studies of the same correlation within specific PC setups, see Ref. [11].) With such choice, the overall normalization implies

Analysis and discussion
The previous considerations can be summarized by stating that, within the two reference frameworks of MFV or PC, possible departures from the SM predictions in the Zbb couplings and in B(B s → µ + µ − ) can be parameterized in terms of the two couplings δg L,R defined in Eq. (11) or Eq. (16). Concerning Z-peak observables, the δg L,R shifts are constrained by R b , A b and A 0b FB . The state-of-the-art SM calculations for these quantities, to which it is straightforward to add the generic shifts in Eq. (12), can be implemented following Ref. [12] (taking also into account the recent SM estimate of R b in Ref. [13]). These quantities can then be fitted to the averages of experimental results collected in Table 1, where we also report the main inputs necessary for their evaluation beyond the lowest order.
The resulting allowed regions at 68% CL and 95% CL in the δg R -δg L plane are shown in Fig. 1. As can be noticed, for both δg L and δg R the fit prefers positive non-zero values, and the SM point (δg R = δg L = 0) is outside the 95% CL region. The upper limits for the two parameters are in good agreement with the results recently reported in Ref. [19]. Let us now compare these limits with those obtained from the B(B s → µ + µ − ) measurement within the frameworks of MFV or PC. The δg 32 L,R couplings shift linearly the Z-penguin contribution to the B(B s → µ + µ − ) amplitude. These shifts can easily be translated into shifts on the short-distance function appearing in the SM formula for the branching ratio (see e.g. Ref. [2]). To good accuracy, the effect can simply be described by where Y SM ≈ 0.957. 2 Using the 95% CL range on the flavor-averaged branching ratio reported by LHCb [1] 1.1 × 10 −9 < B exp < 6.4 × 10 −9 , and the central value of the SM prediction in Eq. (2) (at this level of accuracy the theoretical error is negligible), one obtains the following bounds on δg L and δg R : These bounds have been obtained considering the effects of the two couplings separately (i.e. barring the possibility of cancellations between δg L and δg R , on which we will comment at the end of this section) and ignoring the fine-tuned configuration where the non-standard amplitude is about twice, and opposite in sign, compared to the SM one (a possibility that is highly disfavored by the Z → bb constraints [6]). These bounds are also depicted in Fig. 1 as horizontal or vertical bands delimited by solid lines. From the figure it is evident that, even with its large error, the recent evidence for B(B s → µ + µ − ) provides a constraint on |δg L | -under either of the MFV or PC hypotheses -more stringent than the one obtained from the Z → bb observables. Furthermore, the constraint on the |δg R | coupling within PC is stronger than the one obtained from the Z → bb by more than two orders of magnitude. This circumstance is well represented by the right panel of Fig. 1, where the thickness of the B(B s → µ + µ − )-allowed band (vertical blue 'line') is not resolved at the scale of the electroweak-fit ellipse. This implies that, within anarchic PC models, the B(B s → µ + µ − ) bound forbids any significant contribution to Z → bb observables able to decrease the existing tension between data and theoretical predictions.
As far as the bounds on the effective scale of new physics are concerned, in both frameworks the constraints derived from the |δg L | bound in Eq. (21) are largely dominant. They can be summarized as follows: the equality of the numerical coefficient in the two cases being an accident due to the approximate relation m t |V tb | ≈ v/ √ 2. It is also worth mentioning the m ρ bound implied by |δg R | in PC, that becomes relevant in the limit q 3 1, in which the bound from |δg L | gets weaker.
While the bounds in Eq. (21) are per se interesting, the present experimental error on B(B s → µ + µ − ) does not do full justice to the sensitivity of this observable to possible modified Z-boson couplings. Therefore, we also considered the case of a B(B s → µ + µ − ) measurement with central value as in Eq. (2) and error of ±0.3 × 10 −9 , that can be considered a realistic estimate of the experimental sensitivity on this observable around 2018. This statement takes into account the LHCb projections from Ref. [20], and the fact that CMS will likely produce a measurement with similar accuracy. We also assume a still subleading theoretical error, as expected by the the steady progress in the lattice determination of the B s decay constant [21]. With these assumptions on the projected total error on B(B s → µ + µ − ), the 95% CL bounds on δg L,R become and the bounds in Eqs. (22) and (23) improve by a factor of about two. The comparison between Eq. (24) and Eq. (18) illustrates the potential of uncovering even tiny new-physics deviations in the Z-boson couplings to down-type quarks via B(B s → µ + µ − ). Note that, in the pessimistic case where no deviations from the SM prediction are observed in B(B s → µ + µ − ), even the bound on δg R within MFV will become more stringent compared to the one obtained from the Z → bb observables. Besides improvements in the B(B s → µ + µ − ) measurement, a further avenue towards reducing the error on the Z → bs effective coupling is, in principle, that of combining the constraints from other b → s decays, most notably B → K * µ + µ − and B → Kµ + µ − (recent attempts in this direction can be found in Ref. [22]; see also Ref. [23] for other related studies). However, the extraction of information about the Z → bs effective coupling from these decays is not as pristine as in the B(B s → µ + µ − ) case. In fact, on the one side, and at variance with B s → µ + µ − , these processes receive, already within the SM, substantial contributions from amplitudes other than the Z-penguin. In addition, the definition of observables related to these processes comes with inevitable theoretical assumptions, related to the dependence on additional hadronic form factors.
Finally, as anticipated, the bounds in Eq. (21) and Eq. (24) do not take into account the possibility of cancellations in the case where both δg L and δg R are switched on simultaneously. In practice, admitting such possibility does not lead to any significant changes in the plots of Fig. 1. As expected from the hierarchical nature of the bounds in Eqs. (21) or (24), the allowed region in the case of simultaneously non-zero δg L and δg R is dominated by the region allowed by the strongest constraint, namely δg L in the case of MFV and δg R in the case of PC.

Conclusions
The long-standing discrepancy between experimental data and SM predictions for the Z → bb observables (A 0b FB and, to a lesser extent, also R b ) has often been advocated as a possible hint of physics beyond the SM. If this is the case, under reasonable assumptions about the flavor structure of the new-physics model, sizable non-standard contributions should also be expected in B s → µ + µ − .
A first attempt to relate flavor-changing and flavor-diagonal constraints on the Z-boson couplings, under the assumption that they provide the dominant new-physics contribution to both B(B s → µ + µ − ) and Z → bb, was made in Ref. [6]. At that time, the information from Z → bb observables was used to derive possible upper bounds on B(B s → µ + µ − ) and other FCNC processes. The situation is now reversed: the experimental precision reached on B(B s → µ + µ − ) is such that this observable sets the dominant constraints on possible modified Z-boson couplings.
In MFV models, where sizable deviations are expected only in the left-handed couplings of the Z boson, the bound presently derived from B(B s → µ + µ − ) is only slightly more stringent with respect to the one derived from Z → bb. However, the situation is likely to improve soon with the foreseen experimental progress on B(B s → µ + µ − ), see Fig. 1 left. In generic models with partial compositeness, B(B s → µ + µ − ) sets a constraint on possible modifications of the right-handed coupling considerably more stringent than Z → bb, see Fig. 1 right. This constraint forbids any significant contribution to Z → bb observables able to decrease the existing tension between data and theoretical predictions.
More generally, our results illustrate how a measurement of B(B s → µ + µ − ) with the expected accuracy of order 10% is able to unveil even tiny new-physics deviations in the Z-boson couplings to down-type quarks.