Enhanced electromagnetic correction to the rare $B$-meson decay $B_{s,d} \to \mu^+ \mu^-$

We investigate electromagnetic corrections to the rare $B$-meson leptonic decay $B_{s,d} \to \mu^+\mu^-$ from scales below the bottom-quark mass $m_b$. Contrary to QCD effects, which are entirely contained in the $B$-meson decay constant, we find that virtual photon exchange can probe the $B$-meson structure, resulting in a"non-local annihilation"effect. We find that this effect gives rise to a dynamical enhancement by a power of $m_b/\Lambda_{\rm QCD}$ and by large logarithms. The impact of this novel effect on the branching ratio of $B_{s,d}\to\mu^+\mu^-$ is about $1\%$, of the order of the previously estimated non-parametric theoretical uncertainty, and four times the size of previous estimates of next-to-leading order QED effects due to residual scale dependence. We update the Standard Model prediction to $\overline{{\cal B}}(B_s \to \mu^+\mu^-)_{\rm SM} = (3.57 \pm 0.17) \cdot 10^{-9}$.

Rare leptonic decays B q → ℓ + ℓ − of neutral B mesons (q = d, s and ℓ = e, µ, τ ) provide important probes of flavour-changing neutral currents, since the decay rate in the Standard Model (SM) is predicted to be helicityand loop-suppressed. Both suppressions can be lifted, for example, in models with extended Higgs sectors, in which case the leptonic decays constrain the scalar masses far above current direct search limits.
Only the muonic decay B s → µ + µ − has been observed to date [1,2]. The most recent measurement of the LHCb experiment for the untagged time-integrated branching ratio finds B(B s → µ + µ − ) LHCb = (3.0 +0. 7 −0.6 ) · 10 −9 [3], compatible with the SM prediction [4] B(B s → µ + µ − ) SM = (3.65 ± 0.23) · 10 −9 . (1) With higher experimental statistics and improvement in the knowledge of SM parameters, the accuracy of both results is expected to increase in the future, eventually providing one of the most important precision tests in flavour physics. The neutral B-meson leptonic decays are indeed well suited for precision physics, because long-distance stronginteraction (QCD) effects, which cannot be computed with perturbative methods, are under exceptionally good control. This follows from the purely leptonic final state and the fact that the decay is caused by the effective local interaction The strong interaction effects are therefore confined to the matrix element which defines the B-meson decay constant. f Bq can be computed non-perturbatively with few percent accuracy within the framework of lattice QCD [5].
In this Letter, we report on an investigation of electromagnetic (QED) quantum corrections to the leptonic decay which even at the one-loop order reveals a surprisingly complex pattern. As a consequence, the suppression of the correction due to the small electromagnetic coupling is partially compensated by a power-like enhancement in the ratio of the B-meson mass m B ≈ 5 GeV and the strong interaction scale Λ QCD ≈ 200 MeV. While logarithmic enhancements due to collinear and soft radiation are well-known in QED and also appear in the process under consideration, the power-like enhancement arises due to a dynamical mechanism that to our knowledge has not been observed before. A virtual photon exchanged between the final-state leptons and the light spectator antiquarkq in theB q meson effectively acts as a weak probe of the QCD structure of the B meson. The scattering "smears out" the spectator-b-quark annihilation over the distance 1/ m B Λ QCD inside the B meson, as opposed to the local annihilation through the axial-vector current in Eq. (3). This provides powerenhancement and also shows that at first order in electromagnetic interactions, the strong interaction effects can no longer be parameterized by f Bq alone. Our calculation below shows that the effect is of the same order as the non-parametric theoretical uncertainty previously assumed to obtain Eq. (1).
Before discussing the main result, we briefly review the computations and theoretical uncertainties entering Eq. (1), referring to Ref. [4] for further details. The general framework employs the effective weak interaction Lagrangian, which generalizes the Fermi theory to the full SM, includes all short-distance quantum effects systematically by matching, and sums large logarithms between the scale m W of the W -boson mass and µ b ∼ m b of the order of the bottom-quark mass, m b . The SM prediction (1) includes next-to-leading order (NLO) electroweak (EW) [6] and next-to-next-to-leading order QCD [7] corrections and the resummation of large logarithms ln(µ W /µ b ) due to QCD and QED radiative correc-tions by means of the renormalization-group (RG) evolution [8,9] down to µ b at the same accuracy. Relevant to this work is the observation that unlike QCD effects, which are contained in f Bq to any order, QED corrections below the bottom mass scale µ b have not been fully considered even at NLO.
The largest uncertainties in the SM prediction are of parametric origin: 4% from the B s meson decay constant f Bs , 4.3% from the quark-mixing element V cb [25], and 1.6% from the top-quark mass. These uncertainties will reduce as lattice QCD calculations and measurements of SM parameters improve. Non-parametric uncertainties are due to the omission of higher-order corrections α 3 s , α 2 em , α s α em in the QCD and QED couplings α s and α em , respectively, and also m 2 b /m 2 W from higherdimension operators in the weak effective Lagrangian. Altogether, the non-parametric uncertainties have been estimated to be about 1.5% [4]. Among these, the renormalization scale dependence of B(B q → ℓ + ℓ − ) due to higher-order QED corrections accounts for only 0.3%. In view of such extraordinary precision, it is necessary to exclude the existence of unaccounted theoretical effects at the level of 1%.
Although NLO electromagnetic effects above the bquark mass scale µ b are completely included in Eq. (1), this is not the case for photons with energy or virtuality below this scale. Since the decay involves electrically charged particles in the final state, only a suitably defined decay rate Γ(B q → ℓ + ℓ − ) + Γ(B q → ℓ + ℓ − + n γ) cut including photon radiation and virtual photon corrections is infrared finite and well-defined. Energetic photons are usually vetoed in the experiment and accordingly neglected on the theory side. Soft-photon emission from the final-state leptons is accounted for by experiments [1][2][3]. Initial-state soft radiation has been estimated to be very small based on heavy-hadron chiral perturbation theory [10]. The quoted measured branching fraction is corrected for soft emission and actually refers to the nonradiative branching ratio [11], as does Eq. (1). For the purpose of the SM prediction [4] it was assumed that other NLO QED corrections below µ b can not exceed the natural size of α em /π ∼ 0.3%. However, as we discuss now, the true size of so far neglected QED effects is substantially larger and in fact of the same order as the non-parametric theoretical uncertainty of 1.5%.
The primary challenge of NLO QED computations below µ b consists in the reliable computation of non-local matrix elements. For example, a virtual photon connecting the spectator quark with one of the final-state leptons involves the QCD matrix element where j QED = Q qq γ µ q is the electromagnetic quark current and L ∆B=1 denotes the (QCD part of the) weak effective Lagrangian for ∆B = 1 transitions. This matrix element bears close resemblance to the hadronic tensor that contains the strong-interaction physics of B + → ℓ + ν ℓ γ decay, which is known to be highly nontrivial (for example, Ref. [12]) despite its apparently purely non-hadronic final state.
In the following we focus on the muonic final state µ + µ − . We have analyzed the complete NLO electromagnetic corrections below the bottom mass scale µ b , counting the muon mass m µ and spectator quark mass m q as m µ ∼ m q ∼ Λ QCD ≪ m b to organize the result in an expansion in Λ QCD /m b . We then find that the electromagnetic correction to the decay amplitude is enhanced by one power of m B /Λ QCD compared to the pure-QCD amplitude. In the following we discuss only this formally dominant power-enhanced contribution, leaving the analysis of the complete QED correction to a separate publication. Note that the standard collinear and soft electromagnetic logarithms belong to these further, non powerenhanced terms, and are therefore not discussed here.
We then find that the leading-orderB q → ℓ + ℓ − decay amplitude plus the electromagnetic correction can be represented as where the overall factor contains CKM quark-mixing elements, the Fermi constant G F , and Q ℓ = −1, Q q = −1/3 denote the lep-ton and quark electric charge, respectively. We use the short-handsl =ū(p ℓ − ), ℓ = v(p ℓ + ) for the external lepton spinors. Omitted terms are power-suppressed. The two terms in the electromagnetic correction in the above equation arise from the four-fermion operator Q 9 = αem 4π (qγ µ P L b)(lγ µ ℓ) and the electric dipole operator Q 7 in the effective weak interaction Lagrangian with the effective operators Q i as defined in Ref. [13]. The effective short-distance coefficients [14,15] account for the quark-loop induced contributions. The relevant Feynman diagrams are shown in Fig. 1. An important observation on Eq. (5) is that the nonperturbative strong-interaction physics is no longer contained in the B-meson decay constant f Bq alone. Rather, the exchange of an energetic photon between the lepton pair and the spectator antiquarkq probes correlations between the constituents in the B meson separated at large but light-like distances. The corresponding stronginteraction physics is parameterized by the inverse moment of the B-meson light-cone distribution amplitude (LCDA) λ B , introduced in Ref. [16], and the first two inverse-logarithmic moments, which we define as in Ref. [12] with fixed µ 0 = 1 GeV. These parameters have frequently appeared in other exclusive Bmeson decays. In the numerical analysis below we shall adopt [12] λ B (1 GeV) = (275 ± 75) MeV, σ 1 (1 GeV) = 1.5 ± 1, and σ 2 (1 GeV) = 3 ± 2. The non-locality of qb annihilation due to the photon interaction removes a suppression factor of the local annihilation process. The enhancement of the electromagnetic correction by a factor m B /Λ QCD in Eq. (5) arises from There is a further single-logarithmic enhancement of order ln m b Λ QCD /m 2 µ ∼ 5 for the C eff 9 term, and even a double-logarithmic enhancement of the C eff 7 term. We obtained Eq. (5) in two different ways. First, from a standard computation of QED corrections to the four-point amplitude with two external lepton lines, one heavy-quark and one light-quark line, and second, from a method-of-region computation [17] in the framework of soft-collinear effective theory (SCET) [18,19]. The second method is instructive as it reveals the origin of the enhancement from the hard-collinear virtuality O(m b Λ QCD ) of the spectator-quark propagator. A further single-logarithmic enhancement arises from the contribution of both hard-collinear and collinear (virtuality Λ 2 QCD ∼ m 2 ℓ ) photon and lepton virtuality. The double logarithm in the C eff 7 term is caused by an endpointsingularity as u → 0 in the hard-collinear and collinear convolution integral for the box diagrams, whereby the hard photon from the electromagnetic dipole operator becomes hard-collinear. The singularity is cancelled by a soft contribution, where the leptons in the final state interact with each other through the exchange of a soft lepton. The relevance of soft-fermion exchange is interesting by itself since it is beyond the standard analysis of logarithmically enhanced terms in QED. We shall therefore return to a full analysis within SCET in a detailed separate paper.
We now proceed to the numerical evaluation of the power-enhanced QED correction. Let us denote m B times the curly bracket in Eq. (5) by ∆ QED . Since the scalarlℓ term in the amplitude A does not interfere with the pseudoscalar tree-level amplitude, the QED correction can be included in the expression for the tree-level B s → ℓ + ℓ − branching fraction [26], by the substitution We calculate the Wilson coefficients C i (µ b ) entering ∆ QED at the scale µ b = 5 GeV at next-to-next-to-leading logarithmic accuracy in the renormalization-group evolution from the electroweak scale, evaluate the convolution integrals in Eq. (5) with m b = 4.8 GeV, and express them in terms of λ B (1 GeV), σ 1 (1 GeV), σ 2 (1 GeV) specified above. We then find where the large range is entirely due to the independent variation of the poorly known parameters of the B-meson LCDA. In this result the total effect is reduced by a factor of three by a cancellation between the C eff 9 (q 2 ) and C eff 7 term. With C 10 = −4.198, this results in a (0.3 − 1.1)% reduction of the muonic B s → ℓ + ℓ − branching fraction. We update the SM prediction to which supersedes the one from Eq. (1). To obtain this result we proceeded as in Ref. [4] and used the same numerical input except for updated values of the strong coupling α s (m Z ) = 0.1181(11) and 1/Γ s H = 1.609(10) ps [20], f Bs = 228.4(3.7) MeV (N f = 2 + 1) [5], |V * tb V ts /V cb | = 0.982(1) [21] and the inclusive determination of |V cb | = 0.04200(64) [22]. The parametric (±0.167) and non-parametric non-QED (±0.043) uncertainty and the uncertainty from the QED correction ( +0.022 −0.030 ) have been added in quadrature. Quite surprisingly, the QED uncertainty (which itself is almost exclusively parametric, from the B-meson LCDA) is now almost as large as the non-parametric non-QED uncertainty.
The generation of a scalarlℓ amplitude in Eq. (5) leads to further interesting effects. The time-dependent rate asymmetry for B s decay into a muon pair µ + λ µ − λ in the λ = L, R helicity configuration is given by where all quantities are defined in Ref. [23]. For example, the mass-eigenstate rate asymmetry A λ ∆Γ equals exactly +1, if only a pseudo-scalar amplitude exists, and is therefore assumed to be very sensitive to new flavour-changing interactions, with essentially no uncertainty from SM background. We now see that the SM itself generates a small "contamination" of the observable, given by where r ≡ αem 4π Q ℓ Qq C10 and η L/R = ±1. Present measurements [3] set only very weak constraints on the deviations of A λ ∆Γ from unity, and C λ , S λ have not yet been measured, but the uncertainty in the B-meson LCDA is in principle a limiting factor for the precision with which New Physics can be constrained from these observables.
The power-enhanced QED correction reported here may appear also relevant to the leptonic charged Bmeson decay B + → ℓ + ν ℓ , but cancels due to the V-A nature of the charged current. While we discussed only the case ℓ = µ above, the other leptonic final states ℓ = e, τ are also of interest. However, whereas the muon mass is numerically of the order of the strong interaction scale, the much larger mass of the tau lepton, and the much smaller electron mass imply that the results are not exactly the same. We therefore conclude that the systematic study of hitherto neglected electromagnetic corrections to exclusive B decays reveals an unexpectedly complex structure. Its further phenomenological and theoretical implications are currently under investigation.