The Benefits of B --->K* l+ l- Decays at Low Recoil

Using the heavy quark effective theory framework put forward by Grinstein and Pirjol we work out predictions for B ->K* l+ l-, l = (e, mu), decays for a softly recoiling K*, i.e., for large dilepton masses sqrt{q^2} of the order of the b-quark mass m_b. We work to lowest order in Lambda/Q, where Q = (m_b, sqrt{q^2}) and include the next-to-leading order corrections from the charm quark mass m_c and the strong coupling at O(m_c^2/Q^2, alpha_s). The leading Lambda/m_b corrections are parametrically suppressed. The improved Isgur-Wise form factor relations correlate the B ->K* l+ l- transversity amplitudes, which simplifies the description of the various decay observables and provides opportunities for the extraction of the electroweak short distance couplings. We propose new angular observables which have very small hadronic uncertainties. We exploit existing data on B ->K* l+ l- distributions and show that the low recoil region provides powerful additional information to the large recoil one. We find disjoint best-fit solutions, which include the Standard Model, but also beyond-the-Standard Model ones. This ambiguity can be accessed with future precision measurements.


I. INTRODUCTION
The study of b-flavored mesons made possible our current understanding of quark flavor violation in the Standard Model (SM) [1]. It is an ongoing endeavour to map out the flavor sector at the electroweak scale and beyond, and possibly thereby gaining insights on the origin of flavor.
In this effort, flavor changing neutral current-induced exclusive B decays into dileptons are important modes because of their sensitivity to physics beyond the SM and their accessibility at current collider experiments and possible future high luminosity facilities [2].
We focus in this work on the semileptonic decaysB →K * l + l − with l = e, µ. Their branching ratios are measured at O(10 −7 − 10 −6 ) [3], consistent with the SM [4]. Beyond the rate, several observables can be obtained from the rare decays, in particular when analyzed throughB →K * (→ Kπ)l + l − [5]. The presence of multiple observables is advantageous because they are, in general, complementary in their sensitivity to the electroweak couplings, and they provide opportunities to control uncertainties. This is even more important nowadays, as flavor physics data are favoring the amount of fundamental flavor violation being at least not far away from the one in the SM, and require a certain level of precision to be observed.
Recently, data have become available onB →K * l + l − decay distributions in the dilepton invariant mass, q 2 , from the experiments BaBar [6,7], Belle [8] and CDF [9]. These experimental studies cover essentially the full kinematical dilepton mass range, with the exception of the regions around q 2 ∼ m 2 J/ψ and q 2 ∼ m 2 ψ . Here, cuts are employed to remove the overwhelming background induced byB →K * (cc) →K * l + l − from the dominant charmonium resonances (cc) = J/ψ, ψ .
Most theoretical works onB →K * l + l − decays over the past years have focussed on the region of large recoil, that is, small q 2 m 2 J/ψ . However, at low recoil (large q 2 m 2 ψ ) dedicated studies are lacking with a similar QCD-footing as the ones at large recoil, where QCD factorization (QCDF) applies [10,11]. It is the goal of this work to fill this gap and benefit from the incoming and future physics data from the low recoil region as well.
We use the heavy quark effective theory (HQET) framework by Grinstein and Pirjol [12], which is applicable to the low recoil region, where q 2 is of the order of the mass of the b-quark, m b , and the emitted vector meson is soft in the B mesons rest frame. The original application was to extract the Cabibbo Kobayashi Maskawa (CKM) matrix element V ub by relating the dilepton spectra ofB → ρlν to those inB →K * l + l − decays. The framework has also been used previously to study the implications of the sign of the forward-fackward asymmetry inB →K * l + l − decays being determined SM-like for large q 2 [13], see also [14] for relatingB →Kl + l − toB →Kνν decays. Here, we work out and analyze in detail distributions ofB →K * l + l − decays in this low recoil framework and give predictions within the SM and beyond.
The description ofB →K * l + l − decays at low recoil is based on two ingredients: the improved Isgur-Wise form factor relations [12,15], going beyond the original ones [16], and an operator product expansion (OPE) in 1/Q, where Q = (m b , q 2 ) [12]. The latter allows to include the contributions from quark loops, most notably charm loops in a model-independent way. Both ingredients are first principle effective field theory tools and allow to obtain theB →K * l + l − matrix element in a systematic expansion in the strong coupling and in power corrections suppressed by the heavy quark mass. The implementation of continuum and resonancecc effects from e + e − → hadrons data [17] suggests no large duality violation at least above the ψ , supporting the aforementioned OPE.
We work to lowest order in Λ/m b , however, the actual leading power corrections to the decay amplitudes arise only at order α s Λ/m b or with other parametric suppression factors, and amount only to a few percent.
The plan of the paper is as follows: In Section II we give the electroweak Hamiltonian responsible for b → sl + l − processes and review the observables inB →K * l + l − decays. The low recoil framework is summarized in Section III, where theB →K * l + l − transversity amplitudes and observables are computed and correlations are pointed out. SM predictions and the comparison with the data are given in Section IV. We conclude in Section V. In several appendices we give formulae and detailed input for our analysis.

II. GENERALITIES
We define the short distance couplings entering b → sl + l − decays in Section II A and introduce in Section II B the observables inB →K * l + l − decays, where the former can be tested.

A. Quark level
For the description of processes induced by b → sl + l − we use an effective ∆B = 1 electroweak Hamiltonian contributions of the order V ub V * us , hence, there is no CP violation in the SM in the decay amplitudes. We also set the strange quark mass to zero.
For the decays b → sl + l − the electromagnetic dipole (O 7 ) and semileptonic four-fermion (O 9,10 ) operators are the most relevant: where P L,R denote chiral projectors, m b is the MS mass of the b-quark and F µν (G a µν ) is the field strength tensor of the photon (gluons a = 1, ..., 8). The contributions from the gluonic dipole operator O 8 enter the semileptonic decay amplitude at higher order in the strong coupling g s , and have a significantly reduced sensitivity to New Physics as compared to those from O 7,9,10 . For the current-current and QCD-penguin operators O 1...6 we use the definitions of Ref. [18]. We call the set of operators Eq. (2.2) plus the four-quark operators O 1...6 the SM basis, and stay in this work within this basis.
The goal of this work is to extract from b-physics data the coefficients C 7,9,10 and test them against their respective SM predictions. All other Wilson coefficients are fixed to their respective SM values.
We restrict ourselves to real-valued Wilson coefficients, hence allow for no CP violation beyond the SM. We made this choice because existing CP data on the b → sl + l − transitions [3], which are consistent with our assumption, are currently quite limited, have rather large uncertainties, and the inclusion of phases doubles the number of parameters in the fit. We hope to come back to this in the future.
In the following we understand all Wilson coefficients being evaluated at the scale of the b-quark mass. In the SM at next-to-leading order their values are approximately, for µ = m b , The coefficient of O 7 is suppressed with respect to the ones of O 9,10 , a feature that holds in many extensions of the SM as well, and is also respected by the data. This hierarchy in coupling strengths is beneficial for controlling theoretical uncertainties, see Section III.
We neglect lepton flavor non-universal effects, hence, the couplings to l = e and l = µ are considered to be equal. For recent works exploiting the possibility that New Physics affects the final state electron and muon pairs differently, see, e.g., [19]. Since the decays b → sτ + τ − are experimentally difficult and have not been seen so far, we do not consider taus and can neglect the lepton masses.
B. TheB →K * l + l − observables Angular analysis offers the maximal information which is accessible from the decay viaB →K * (→ Kπ)l + l − . For an on-shellK * the differential decay width can be written as [5,20] where the lepton spins have been summed over. Here, q 2 is the dilepton invariant mass squared, that is, q µ is the sum of p µ l + and p µ l − , the four momenta of the positively and negatively charged lepton, respectively. Furthermore, θ l is defined as the angle between the negatively charged lepton and theB in the dilepton center of mass system (c.m.s.) and θ K * is the angle between the Kaon and theB in the (K − π + ) c.m.s.. We denote by p i the three momentum vector of particle i in thē B rest frame. Then, φ is given by the angle between p K − × p π + and p l − × p l + , i.e., the angle between the normals of the (K − π + ) and (l − l + ) planes.
The full kinematically accessible phase space is bounded by where m l , m B and m K * denote the mass of the lepton, B meson and the K * , respectively.
The dependence of the decay distribution Eq. (2.4) on the angles θ l , θ K * and φ can be made explicit as J(q 2 , θ l , θ K * , φ) = J s 1 sin 2 θ K * + J c 1 cos 2 θ K * + (J s 2 sin 2 θ K * + J c 2 cos 2 θ K * ) cos 2θ l + J 3 sin 2 θ K * sin 2 θ l cos 2φ + J 4 sin 2θ K * sin 2θ l cos φ + J 5 sin 2θ K * sin θ l cos φ + J 6 sin 2 θ K * cos θ l + J 7 sin 2θ K * sin θ l sin φ + J 8 sin 2θ K * sin 2θ l sin φ + J 9 sin 2 θ K * sin 2 θ l sin 2φ, (2.6) where the angular coefficients J i . The latter can be written in terms of the transversity amplitudes A ⊥, ,0 , see Appendix A. The fourth amplitude A t does not contribute in the limit m l = 0. The transversity amplitudes at low recoil are given in the next section. The ones at large recoil can be seen, for example, in Ref. [13].
The angular coefficients J (a) i , or their normalized variants J i /(dΓ/dq 2 ) or J i /J j , are observables which can be extracted from an angular analysis. This method allows to test the SM and probe a multitude of different couplings [13,[20][21][22][23]. We focus first on rather simple observables, which can be extracted without performing a statistics intense full angular analysis. Afterwards, we point out opportunities of measuring the angular distribution.
Data onB →K * l + l − decays already exists from BaBar [6,7], Belle [8] and CDF [9] for the differential decay width dΓ/dq 2 , the forward-backward asymmetry A FB and the fraction of longitudinal polarized K * 's, F L . They are written as and are all distributions in the dilepton mass.
The experimental data on the q 2 -distributions [6][7][8][9] are currently available in q 2 -bins, i.e., the decay rate is given as a list of rates dΓ/dq 2 k , where we denote by .. k the dq 2 -integration over the k-th bin. Normalized quantities such as the forward-backward asymmetry are then delivered as J 6 k / dΓ/dq 2 k , and likewise as Note that the J 5,6,8,9 , and hence A FB are CP-odd observables, which vanish in an untagged equally mixed sample ofB and B decays in the absence of CP violation [13].
We also consider the transverse asymmetries A [21], given as , which have not been measured yet. The factor β l is given in Appendix A. Here we keep the lepton mass dependence for generality but discard it later on when discussing the low recoil region where m l is entirely neglibile.
We propose the following new transversity observables for the region of low recoil (high q 2 ) , (2.14) As will become clear in Section III, see also Appendix B, the H (i) T are designed to have very small hadronic uncertainties at low recoil. While both H T and A FB depend on J 6 and probe similar short distance physics, the former has a significantly smaller theoretical uncertainty than the latter.
Note also that the numerator J 5 of H (2) T is related to the observable S 5 which has good prospects to be measured with early LHCb data of 2 fb −1 at least in the large recoil region [24].
Different possibilities to extract the J i from single differential distributions as well have been outlined in [13].
We start in Section III A with the model-independent description of the exclusive heavy-to-light decays in the low recoil region following Grinstein and Pirjol [12,15]. After calculating and investigating theB →K * l + l − transversity amplitudes in Section III B, we work out predictions for and correlations between theB →K * l + l − observables at low recoil in Section III C. A numerical study within the SM is given in Section IV A.

A. The model-independent framework
The description ofB →K * l + l − decays at low recoil, where q 2 ∼ O(m 2 b ), is based on the improved form factor relations in this region and an OPE in 1/Q [12,15]. The latter keeps the nonperturbative contributions from 4-quark operators (sb)(qq) under control by expanding in m 2 q /Q 2 . This is most important for charm quarks, since their operators can enter with no suppression from small Wilson coefficients nor CKM matrix elements.
Following [12] we briefly sketch the derivation of the improved Isgur-Wise form factor relations to leading order in 1/m b between the vector and the tensor current. The starting point is the QCD operator identity (for m s = 0) After taking the matrix element of Eq. (3.1) using the form factors given in Appendix C one arrives at an exact relation between the form factors T 1 and V and the matrix element of the currentsi ← Dµ b. The latter can be expanded in 1/m b through matching onto the HQET currents with the heavy quark field h v : We further needs to express the HQET currents in Eq. perturbative expansion in the strong coupling, see, e.g., [12,25].
Taking then the matrix element of Eq. (3.2) yields After working out the corresponding formulae involving the axial currents, the improved Isgur-Wise relations to leading order in 1/m b including radiative corrections are obtained as Here, subleading terms of the order m K * /m B , Λ/m B are dropped and a naively anticommuting γ 5 matrix is used. The latter allows to relate the HQET Wilson coefficients of currents without a γ 5 matrix to those containing one by replacings withs(−γ 5 ) in the matching equations. We also suppress the renormalization scale dependence of the penguin form factors T i and of the coefficient The relations Eq. (3.6) are consistent with the ones derived in [12] at lowest order in 1/m b after changing to the Isgur-Wise form factor basis [16].
The inclusion of the 4-quark and gluon dipole operators leads to the effective couplings, C eff 7,9 [12]. They read and we recall that we use the 4-quark operators O 1...6 as defined in [18]. The functions A, B, C and F 8 , F can be seen in [26] and [10], respectively. 1 The lowest order charm loop function is given as which is simply the perturbative quark loop function for massless quarks. The m 2 c /Q 2 corrections are given by the last line of Eq. (3.9). Loops with b quarks stemming from penguin operators are taken into account by the function We stress that the effective coefficients Eqs. (3.9)-(3.10) are different from the ones used in the low q 2 region given in [10].
The product m b κ C eff 7 is independent of the renormalization scale [12]. As we will see in the next section, this is important because contributions from C eff 7 enter theB →K * l + l − amplitudes in exactly this combination. The µ-dependence of C eff 9 is very small and induced at the order α 2 s C 1,2 and α s C 3,..6 .
The heavy quark matrix elements K * |si ← Dµ (γ 5 )h v |B are the only new hadronic input required at order Λ/m b for both the form factor relations and the matrix elements related to the electromagnetic current, C eff 7,9 [12]. However, we refrain from including these explicit Λ/m b corrections.
Firstly, the requisite additional matrix elements are currently only known from constituent quark model calculations [15,28] bringing in sizable uncertainties. More importantly, the leading power corrections to the form factor relations are parametrically suppressed, see Section III B. Note that the ones to the OPE arise only at O(α s Λ/m b , m 4 c /Q 4 ). Hence, the power corrections have a reduced impact on the decay observables. Quantitative estimates are given in Section IV A.
Note that explicit spectator effects are power suppressed and absent to the order we are working.
They only appear indirectly in the form factors, lifetime and meson masses. Hence, the formulae can be used for charged and neutralB →K * l + l − decays, andB s → φl + l − decays after the necessary replacements.

B. The transversity amplitudes
Application of the form factor relations in Eq. (3.6) and using the effective coefficients Eqs. (3.9)-(3.10) yields the low recoil transversity amplitudes to leading order in 1/m b as where the form factors enter as 16) and the normalization factor reads Here, we switched to the dimensionless variablesŝ = q 2 /m 2 . We also suppressed for brevity the dependence on the momentum transfer in the form factors and the effective coefficients. We further neglected subleading terms of order m K * /m B in the C eff 7 -term only. Interestingly, within our framework (SM basis, lowest order in Λ/m b ) the transversity amplitudes Eqs. (3.13)-(3.15) depend in exactly the same way on the short distance coefficients. Consequently, only two independent combinations of Wilson coefficients can be probed, related to |A L i | 2 ± |A R i | 2 , since A L and A R do not interfere for massless leptons, see Appendix A. The independent combinations can be defined as ρ 1 and ρ 2 are largely µ-scale independent. The dominant dependence on the dilepton mass in ρ 1,2 stems from the 1/ŝ-factor accompanying C eff 7 . The short distance parameter ρ 1 equals up to Λ/m b corrections the parameter N eff introduced in Ref. [12].
The relation between all three transversity amplitudes makes the low recoil region overconstrained and very predictive. We work out the corresponding implications in Section III C. Note that in the large recoil region two amplitudes are related as A X = −A X ⊥ by helicity conservation up to corrections in 1/E K * in the SM basis [29]. We simulate the effect of the 1/m b corrections by dimensional analysis when estimating theoretical uncertainties in Section IV A.

C. Observables and predictions
We begin with low recoil predictions of some basic distributions. At leading order they can be written in terms of the transversity amplitudes A ⊥, ,0 given in Eqs. (3.13)-(3.15) as: and A The new high q 2 transversity observables read as All observables factorize into short distance coefficients ρ 1,2 and form factor ones f 0,⊥, .
We note the following: • The only two independent combinations of Wilson coefficients, ρ 1 and ρ 2 , enter the decay rate dΓ/dq 2 and the forward-backward asymmetry A FB , respectively.
• The observables F L , A T does not depend on Wilson coefficients either. Its simple prediction Eq. (3.24) holds beyond the SM and provides a null test of the framework.
• The set of observables Eqs. (3.20)-(3.24) and (B10) with two short distance and three form factor coefficients is heavily overconstrained. Measurements can directly yield either products ρ i f j f k or ratios ρ 2 /ρ 1 and f j /f k , but not the f i or the ρ i alone.

IV. EXPLOITING DATA
We give numerical SM predictions forB →K * l + l − decay observables in Section IV A, with emphasis on the low recoil region. In Section IV B we confront the distributions with existing data and work out constraints for the Wilson coefficients. Next, we combine low with large recoil regions and point out complementarities.

A. SM predictions
The low recoil predictions are obtained using the formulae given in Section III. The framework applies to the region where theK * is soft in the heavy mesons rest frame, i.e., has energy E K * = m K * + Λ. In terms of dilepton masses, this corresponds to large values, up to the kinematical endpoint. We use, unless otherwise stated, transitions [33]. Note that there is lattice and experimental information available on B → ρ form factors at low recoil [34,35], however, to use this for B → K * would require knowledge of the size of SU (3) flavor breaking. More details on the form factors and a comparison with existing lattice results for T 1,2 [36,37] are given in Appendix C. We use the parameters given in Table I.  of order Λ/m b , and the neglected kinematical factors of m K * /m B in the term ∼ κ C eff 7 are accounted for by three real scale factors for A ⊥, ,0 with ±20 % (IWR). Note however, that the latter are additionally suppressed in the SM by 2 C eff 7 /C eff 9 . The uncertainties due to the CKM parameters V tb V * ts correspond to their 1σ ranges (CKM), which cancel in the normalized quantities and thus appear in the branching ratio only. The uncertainties due to the µ-dependence and the t-and b-quark masses (at 1 σ) concern the short distance couplings ρ 1,2 only, and are subsumed under the label (SD). The variation with the scale µ ∈ [µ b /2, 2µ b ] (with central value µ b = 4.2 GeV) is small, as expected.
In Fig. 1 we show ρ 1 and the ratio ρ 2 /ρ 1 with error bands from different sources. The t-pole mass and b-MS mass dependence (at 3σ) are comparable in size and amount to about 5 % each.
For the SM predictions at large recoil [10,11] we follow closely [13], with the updates of the numerical input given in Table I. In this kinematical region, spectator effects arise and for concreteness, we give predictions for neutralB decays.
We estimate the uncertainties due to the two large energy form factors ξ ⊥, by varying them separately -for an improved treatment of this source of uncertainty using directly the LCSRs the reader is refered to [22]. Furthermore, we estimate uncertainties due to subleading QCDF corrections of order Λ/m b by varying a real scale factor for each of the transversity amplitudes A L,R ⊥, ,0 within ±10 % separately and adding the resulting uncertainties subsequently in quadrature. The latter constitute the numerically leading uncertainties in the observables A where form factor uncertainties cancel at leading order in QCDF [21].
The differential branching ratio dB/dq 2 , the forward-backward asymmetry A FB and the longitudinal polarization F L in the SM in both the low and large recoil regions are shown in Fig. 2. The vertical grey bands are the regions vetoed by the experiments to remove backgrounds from intermediate charmonia, J/ψ and ψ decaying to muon pairs for 8.68 GeV 2 < q 2 < 10.09 GeV 2 and 12.86 GeV 2 < q 2 < 14.18 GeV 2 [8,9]. Within QCDF, the region of validity is approximately within (1 − 7) GeV 2 . We mark the large recoil range (below the J/ψ) outside this range by dashed lines.
In Fig. 3 we show the SM predictions for B, A FB and F L next to the available data. Note that the physical region of F L is between 0 and 1. The data are consistent with the SM, although they allow for large deviations from the SM as well given the sizeable uncertainties. In particular, the data for B at low q 2 and A FB at high q 2 show a trend to be slighly below the SM. The shape of A FB at low q 2 is currently not settled and allows for either sign of the dipole coefficient C 7 while having the others kept at their SM values. In the future the LHCb collaboration expects to surpass the precision of the existing B-factory A FB measurements after an integrated luminosity of 0.3 fb −1 [39], and may shed light on this matter.
In Fig. 4 we show A T is strongly suppressed, in fact, vanishes up to 1/E K * corrections by helicity conservation [29] for low dilepton masses, but is order one for large ones. The size of T at low q 2 can be used as an indicator for the correctness of our assumptions: in the presence of chirality-flipped operators beyond those in Eq. (2.2), the aforementioned suppression of A (2) T would be lifted. Note that A T is proportional to 1/ λ and diverges at the endpointλ → 0. On the other hand, A (4) T ∝ λ is finite in this limit and vanishes at maximum q 2 . The q 2 -behaviour of both the new, transverse observables H (2,3) T can be obtained from Fig. 1, where ρ 2 /ρ 1 is shown in the SM.

B. Constraining new physics
To confront the available data with the SM we perform a parameter scan over −15 ≤ C 9,10 ≤ 15 for 60 × 60 points and check the goodness-of-fit for each of the observables listed in Table II in every point (C 9 , C 10 ). We implement every observable analytically with the single exception theB → X s γ branching ratio, for which we use the numerical SM results given in [40]. Contributions to the latter from physics beyond the SM are implemented at leading order. The integrated observables The constraints on C 9 and C 10 fromB →K * l + l − at large recoil andB → X s l + l − for C 7 = C SM 7 (a) and C 7 = −C SM 7 (b) using Belle [8,42], BaBar [43] and CDF [9] data at 68% CL (red areas) and 95% CL (red and blue areas). The (green) square marks the SM value of (C 9 , C 10 ).
(q 2 max ). In particular we calculate with the theoretical prediction of the i-th observable X i,T ≡ X i,T ( C j ) and its upper (lower) uncertainty ∆ i,T ( C j ) as described in Section IV A. The experimental result from experiment E for the i-th observable is denoted by X i,E and its error σ i,E is obtained by adding linearly the statistical and systematic errors and subsequent symmetrization. From here we calculate the likelihood L as The constraints on C 9 and C 10 fromB →K * l + l − low recoil data [8,9] only for C 7 = C SM 7 (a) and (b) at 68% CL (red areas) and 95% CL (red and blue areas). The (green) square marks the SM value of (C 9 , C 10 ).
narrow range of values around |C SM 7 |, however without determining the sign of C 7 . For this reason, we present in the following our scans for C 7 = ±C SM 7 . In Fig. 5 we show the constraints in the C 9 − C 10 plane fromB →K * l + l − decays at large recoil and B → X s l + l − data, without use of the low recoil information. On the other hand, taking into account theB →K * l + l − data at low recoil only, we arrive at the constraints given in Fig. 6. We see that the latter low recoil constraints are presently much more powerful than the others. An important ingredient for this are the A FB measurements at low recoil constraining A FB ∝ Re{C 9 C * 10 } to be SM-like, the benefits of which have already been pointed out in [13]. The individual constraints, overlaid on top of each other, are given at 68% CL in Fig. 7. The data are consistent with each other.
The global constraints, obtained after summing over the χ 2 -values of all aforementioned data, are shown in Fig. 8. Two disjoint solutions are favored, around (C SM 9 , C SM 10 ) or in the vicinity of (−C SM 9 , −C SM 10 ). There appears to be space for order one deviations from either solution, regardless of the sign of C 7 . Note that the flipped-sign solution around (−C SM 9 , −C SM 10 ) for C 7 = C SM 7 is disfavored, see Fig. 7. Varying C 7 between -0.5 and +0.5 and imposing theB → X s γ constraint (b) using Belle [8,42], BaBar [43] and CDF [9] data. The (grey) square marks the SM value of (C 9 , C 10 ). See the color key at the top for the different constraints.
leads to barely noticable larger contours in the C 9 − C 10 plane than the ones in Fig. 8 a (for C 7 < 0) and Fig. 8 b (for C 7 > 0), and are not shown.
We find that at 2σ the allowed values of C 10 are within 0.5 ≤ |C 10 | ≤ 8. This gives branching ratios forB s → µ + µ − decays enhanced or lowered with respect to the SM one, within the interval . This is consistent with the current upper limit on this mode, B(B s → µ + µ − ) < 3.6 × 10 −8 (95% CL) [3]. Similarly, the values of the transversity observables H As the experimental precision improves over time, especially with the LHCb data at the horizon, there will be opportunities to resolve the 4-fold ambiguity of the current solutions presented in Fig. 8. Firstly, knowing whether A FB has a zero for low q 2 as in the SM or not, fixes the sign of Re {C 7 C * 10 }, thereby eliminating two of the four possible solutions. Alternatively, the sign of the interference term Re{C * 7 C 9 } in B(B → X s l + l − ) can be extracted from precision measurements. In the SM, this term decreases the branching ratio. These two effects are correlated within our framework, i.e., the existence of an A FB zero crossing implies a destructive interference term in the branching ratio and vice versa. (b) using Belle [8,42], BaBar [43] and CDF [9] data at 68% CL (red area) and 95% CL (red and blue areas). The (green) square marks the SM value of (C 9 , C 10 ).  At this point, there would still be two possible solutions left. Assuming, for instance, a confirmation of the A FB zero, these solutions are C 7,9,10 having SM-like signs, or C 7,9,10 having opposite signs with respect to their SM values. This last ambiguity can be resolved with precision measurements at the level where one becomes sensitive to the (known) difference between the Wilson coefficients C i and the effective ones C eff i . Then, the additional contribution breaks the symmetry in the observables under sign reflection. Since the contribution of C 7 to the decay amplitudes is small at large q 2 , promising observables to resolve the final sign issue are those at low dilepton masses.

V. CONCLUSIONS
Discrepancies between b physics predictions and measurements can be caused by new physics beyond the SM or by an insufficiently accounted for background from strong interaction bound state effects. Due to the decays simple transversality structure at low recoil, these QCD and electroweak effects can be disentangled inB →K * l + l − angular studies.
In fact, to leading order in the power corrections with subleading terms being further suppressed, all contributing transversality amplitudes exhibit the same dependence on the short distance electroweak physics, which moreover factorizes from the hadronic matrix elements. This in turn allows to define new observables, H  [33], which could be compared to (future) lattice results.
Exploiting data we find that the constraints from the low recoil region add significant new information, while being consistent with the large recoil and inclusive decays data, and the SM. Large deviations from the SM are, however, allowed as well due to the current experimental uncertainties.
Our findings are summarized in Figs. 3 and 8. Improved measurements of the forward-backward asymmetry or precision data on the inclusiveB → X s l + l − branching ratio can resolve the present ambiguities in the best-fit solution.
Since the decayB s → φµ + µ − has been seen [9], it becomes relevant in the near future as well.
The low recoil framework and our analysis applies to B s decays with the obvious replacements of masses and hadronic input.
To conclude, we obtained from the existing data onB →K * l + l − decays at low recoil new and most powerful constraints. The proposed angular studies offer great opportunities, both in terms of consistency checks and precision, to explore further the borders of the SM. distance couplings ρ i using the observables
LCSR provide the form factors at large recoil, q 2 14 GeV 2 [32]. There, the outcome of the LCSR calculation is fitted to a physical q 2 dependence, of pole or dipole structure. It is conceivable that the form factor parametrization obtained in this way are valid at low recoil as well.
For completeness, we give here the parametrization of the form factors V, A 1,2 from [32], which we  use at both low and large recoil.
where the fit parameters r 1,2 , m 2 R and m 2 fit are given in Table III. The resulting form factors are shown in Fig. 9. For the uncertainty we use 15 % as follows from the LCSR calculation.
In Fig. 10 we compare the LCSR fit against the lattice results, which exist for T 1,2 [36]. The agreement is reasonable, given the substantial uncertainties. There is consistency as well with the preliminary unquenched findings of Ref. [37], which are not shown.
How well do the LCSR form factors from [32] satisfy the low recoil form factor relations Eq. (3.6)?
In Fig. 11 we show the ratios which in the symmetry limit should all equal κ, which is also shown. Note, that in the large energy limit E K * Λ the form factors obey to lowest order in the strong coupling very similar relations R 1,2 = 1 + O(m K * /m B ) and T 3 /A 2 = 1 + O(m K * /m B ) [29,44]. We learn that the improved Isgur-Wise relations work reasonably well for the extrapolated LCSR form factors with the exception of the one for T 3 . The agreement improves here somewhat if the factor q 2 /m 2 B is replaced by one, its leading term in the heavy quark expansion.
For the low q 2 form factors we employ a factorization scheme within QCDF where the ξ ⊥, are related to the V, A 1,2 as [11]