Supersymmetric constraints from Bs → μ+μ− and B → K*μ+μ− observables

We study the implications of the recent LHCb limit and results on Bs → μ+μ− and B → K*μ+μ− observables in the constrained SUSY scenarios. After discussing the Standard Model predictions and carefully estimating the theoretical errors, we show the constraining power of these observables in CMSSM and NUHM. The latest limit on BR(Bs → μ+μ−), being very close to the SM prediction, constrains strongly the large tan β regime and we show that the various angular observables from B → K*μ+μ− decay can provide complementary information in particular for intermediate tan β values.


Introduction
The rare decays B s → µ + µ − and B → K * µ + µ − are sensitive probes of new particles arising in the extensions of the Standard Model (SM) and in particular Supersymmetry (SUSY). The measurements of these decays provide important constraints on the masses of new particles which are too heavy to be produced directly.
At large tan β, the SUSY contributions to the decay B s → µ + µ − is dominated by the exchange of neutral Higgs bosons, and it has been emphasised in many works [1][2][3][4][5][6][7][8][9][10] that this decay receives large enhancement, and very restrictive constraints can be obtained on the supersymmetric parameters. This decay is currently searched for by three LHC experiments: LHCb, CMS and ATLAS, and recently LHCb collaboration reported a very strong limit on the branching ratio of 4.5 ×10 −9 [11] which is only about 15% larger than the SM prediction. This improved limit further constrains the SUSY parameter space. A possible signal, although with a low significance is also reported by the CDF collaboration [12].
The decay B → K * µ + µ − on the other hand provides a variety of complementary observables as it gives access to angular distributions in addition to the differential branching fraction. Experimentally the exclusive B → K * µ + µ − decay is easier to measure compared to the theoretically cleaner inclusive mode B → X s µ + µ − . However from a theoretical point JHEP08(2012)092 of view in the exclusive mode there are large uncertainties, which come mostly from the B → K form factors. Within the QCD factorisation [13,14], simplifications can be made on the form factor description and by looking into the rich phenomenology of the various kinematic distributions, observables that have smaller dependency on the form factors can be defined [15,16]. These observables prove to be important tools to study extensions of the SM [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] and they can provide valuable information on different sectors of the theory. The full determination of the angular distributions of B → K * µ + µ − constitutes a worthwhile challenge to the present and future experiments. Several angular observables have already been measured by Belle, Babar, CDF and LHCb. The most precise measurements come from the recent LHCb analyses with 1 fb −1 of integrated luminosity [33].
In this work we study for the first time the implications of the recent measurements of B → K * µ + µ − observables on constrained SUSY scenarios and update the constraints from BR(B s → µ + µ − ). Our numerical analysis is performed with SuperIso v3.3 [34,35] and we study two constrained SUSY models: CMSSM and NUHM. To give some insight on the origin of these constraints, a mapping of the CMSSM parameter space into the Wilson coefficient subspace for the most relevant operators is also provided, which displays interesting partial correlations and hierarchies.
This paper is organised as follows: in section 2 we present a theoretical introduction to the decays B s → µ + µ − and B → K * µ + µ − , provide the SM predictions and estimate the errors. In section 3 we summarise the experimental results and section 4 contains our numerical analysis of the constraints on the SUSY models that are obtained from the recent LHCb results. Conclusions are contained in section 5.

Observables, inputs and theoretical uncertainties
The effective Hamiltonian describing the b → s + − transitions has the following generic structure 1 [39,40]: where Q 1 and Q 2 are the scalar and pseudo-scalar operators and the primed operators are chirality flipped compared to the non-primed operators. Physics contributions at scales higher than µ are summarized in the so called Wilson coefficients C i include the long distance contributions from scales lower than µ b . In the SM the primed and (pseudo-)scalar operators are either highly suppressed or absent. The most relevant operators for our work are

2)
JHEP08(2012)092 where P R,L = (1±γ 5 )/2 denote the chiral projectors and m b is the M S mass of the b quark. A full list of the operators as well as the analytical expressions for all the Wilson coefficients can be found in [35]. The renormalisation group equations for C 1 -C 8 are given in [41], for C Q 1 and C Q 2 in [1] and we calculated the running of C 9 which is given in appendix C.
2.1 B s → µ + µ − Stringent 95% C.L. limit on the untagged branching ratio BR untag (B s → µ + µ − ) < 4.5 × 10 −9 has recently been obtained by the LHCb collaboration [11]. Taking into account the precise measurement [42] of the fractional decay width difference between the B s heavy and light mass eigenstates, y s ≡ ∆Γ Bs /(2Γ Bs ) = 0.088 ± 0.014, this limit on the untagged branching ratio can be translated into an even stronger limit on the CP-averaged branching where we assumed an SM-like scenario. In terms of Wilson coefficients, this average branching ratio is expressed as [35,46]: In the Standard Model, only C 10 is non-vanishing and gets its largest contributions from a Z penguin top loop (75%), and from a charmed box diagram (24%). With the inputs of table 2, C 10 = −4.21, from which BR(B s → µ + µ − )| SM = (3.53 ± 0.38) × 10 −9 . The latest experimental limit thus severely restraints the room for new physics, and its proximity with the 2σ upper value calls for a discussion of the uncertainties in this SM prediction. The main uncertainty comes from the B s decay constant f Bs , which has recently been re-evaluated by independent lattice QCD groups of table 1. Their 4.3% uncertainties agree, as do their results within these uncertainties, so that we have chosen an average of these three results in what follows. This implies a 8.7% uncertainty on the branching ratio.
Notice this range covers the recently published result f Bs = 225 ± 4 MeV of McNeile et al. [51], whose lower value and striking precision dominates any weighted average including it, like the one in [52] (227 ± 4 MeV) proposed by one of the authors of [51], or the one in [53]: 227 ± 6 MeV. The smallness of the extrapolation error in this work raises a number of new questions, and we prefer to keep our naive but more conservative average. This choice has however little effect on the new physics applications we have in mind, as these depend mostly on the lower end of that range.
Another potential source of uncertainty comes from the choice of scale at which the fine structure constant is used in eq. (2.4): there is a non-negligible 4% difference between the running M S couplingsα(m b ) = 1/133 andα(m Z ) = 1/128. If the first choice may seem natural, the weak couplings involved in the top Z-penguin (or charmed box) are closer to the weak scale, and do not run below it, as discussed in ref. [54]. We thus take that last value, as well as sin 2θ W (m Z ) = 0.2312 in the expression of eq. (2.4). This may seem at odds with the conclusion of ref. [54], that choosingα(m b ) minimises the EW corrections to B → K * + − . However, the EW corrections to C 7,9 , which dominate the low q 2 ll region of this last process, are opposite to the EW corrections for C 10 , which controls B s → µ + µ − . Having made this choice for the EM-coupling, we expect EW corrections to mostly absorb the remaining scale dependence inα EM (µ), leaving a small, 2% uncertainty in the branching ratio.
The remaining theoretical uncertainties are smaller thanks to the NNLO treatment of QCD corrections: increasing the low scale µ b (or the matching scale µ W ) by a factor of 2 induces a 1.4% (or respectively 2%) effect.
Finally, parametric uncertainties from the top mass (1.3%), from the B s lifetime (1.8%) and from the CKM element V ts (5%), will reduce in the future. Adding all these (small) errors in quadrature, we thus get a Standard Model prediction assorted with an 11% uncertainty: This value is compatible with recent SM predictions for this observable, e.g. by the CKMfitter group [55], for which the uncertainties are reduced by the use of other flavour information.
Considering theK * meson to be on-shell, the differential decay distribution of theB 0 → K * (→ K − π + ) + − decay can be written in terms of three angles θ , θ K * , φ and the invariant dilepton mass squared (q 2 ) [15,22]: In the above equation θ is the angle between − andB 0 in the rest frame of the dilepton, θ K * is the angle between K − andB 0 in theK * rest frame and φ is the angle JHEP08(2012)092 m B = 5.27950 GeV [56]   between the normals of the + − plane and the K − π + in theB 0 rest frame. The angular dependence of J(q 2 , θ , θ K * , φ) can be written as J(q 2 , θ , 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 + J 3 sin 2 θ K * sin 2 θ cos 2φ + J 4 sin 2θ K * sin 2θ cos φ + J 5 sin 2θ K * sin θ cos φ + (J s 6 sin 2 θ K * + J c 6 cos 2 θ K * ) cos θ + J 7 sin 2θ K * sin θ sin φ + J 8 sin 2θ K * sin 2θ sin φ + J 9 sin 2 θ K * sin 2 θ sin 2φ . (2.7) The angular coefficients J i (see appendix A), are functions of q 2 and can be described in terms of eight transversity amplitudes The transversity amplitudes up to corrections of O(α s ) can be expressed in terms of seven independent form factors, A 0,1,2 , T 1,2,3 and V . Since these form factors are hadronic quantities they require non-perturbative calculations and hence are the main source of uncertainty in the exclusive mode. But even if the form factors were known precisely theB 0 →K * + − decay would still need corrections emerging from non-factorisable effects which are related to the current-current operators O 1 and O 2 , the QCD penguin operators O 3 -O 6 and the chromomagnetic operator O 8 . The non-factorisable corrections contribute to the decay amplitude by producing a virtual photon which decays into a lepton pair. When theK * JHEP08(2012)092 has a large energy (q 2 is small) and the decaying hadron (B 0 ) is heavy, the non-factorisable corrections can be computed in the QCD factorisation framework [13,14]. In the QCDf framework in the large recoil limit the seven independent form factors can be expanded in ratios of 1/m b and 1/E K * [60,61]. While α s corrections to the form factors in QCDf have been calculated [61], the 1/m b corrections are unknown. Neglecting 1/m b corrections, the transversity amplitudes at NLO in α s in the large recoil limit are 2 [15,22,23]: where E K * is the energy of the final vector meson in the B rest frame Further explanation on what we use for T ± ⊥, , ξ ⊥, , A 1,2 and V is given in appendix B. For the differential distribution of the CP conjugate decay we keep the definition of θ to remain the same and replace K − with K + for the definition of θ K * and consider φ to be the angle between the normals of the + − plane and the K + π − in the B 0 rest frame, we can use eq. (2.6) where J is replaced withJ. The function J(q 2 , θ , θ K * , φ) is obtained from eq. (2.7) by the replacements [62] J (c,s)

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whereJ is equal to J with all the weak phases conjugated. The change of sign for J 5,6,8,9 can be understood by considering that the CP conjugate decay leads to the transformations θ → π − θ and φ → −φ in eq. (2.7).

Observables
Integrating eq. (2.6) over all angles, the dilepton mass distribution can be written in terms of the angular coefficients [22,23]: (2.14) For the normalised forward-backward asymmetry A F B we use [22,23]: The longitudinal polarisation fraction F L can be constructed as the ratios of the transversity amplitudes and therefore contains less theoretical uncertainty from the form factors. For the longitudinal polarisation, F L we have [15,22]: An interesting observable is the zero-crossing of the forward-backward asymmetry (q 2 0 ). Neglecting the lepton masses and not considering the chirality flipped and scalar operators for the zero of A F B we obtain To calculate the zero of A F B in our numerical analysis we have directly used the relation for A F B (eq. (2.15)). Another observable which is rather independent of hadronic input parameters is the isospin asymmetry which arises from non-factorizable effects. The non-factorizable effects depend on the charge of the spectator quark, and hence depending on whether the decaying B meson is charged or neutral, there will be a difference in the contribution of these effects to the decay width which can cause an isospin asymmetry. The (CP averaged) isospin asymmetry is defined as [20] dA In the SM, dA I /dq 2 is at the percent level [14,20], the smallness of the isospin asymmetry makes it sensitive to isospin-violating NP effects.
Other observables of interest are the transverse amplitudes which have a small dependence on the form factors and a large sensitivity to right-handed currents via C 7 . They are defined as [15,16,22,63] A (2) as well as which is sensitive to the complex phases, but very small (O(10 −3 )) in the SM.
All the observables can also be expressed in terms of the CP averaged angular coefficients S i [22]. In particular, S 3 , which has recently been measured by LHCb, can be related to F L and A in the limit of vanishing lepton masses. For the high q 2 region we follow [26], where the heavy quark effective theory (HQET) of [64] is used. In that framework improved Isgur-Wise form factor relations are used to extrapolate the lacking high q 2 form factors from the calculations of the low q 2 form factors and an operator product expansion in powers of 1/Q 2 , (Q = (m b , q 2 )), is used for the estimation of the long-distance effects from quark loops. For the low recoil region (high q 2 ), the following observables are proposed by [26] H (1) (2.26) These transversity observables have small hadronic uncertainties at high q 2 .

Standard Model values and theoretical uncertainties
For our numerical analysis we have used the NLO relations for the transversity amplitudes in the (1 < q 2 < 6 GeV 2 ) region where we are sufficiently below the charm resonance threshold (q 2 = m 2 c ) and far enough from the kinematic minimum where the decay amplitude is dominated by the photon pole. The input parameters can be found in .
(2.28)   [1,6] −0.06 ±0.04 ±0.02 ±0.01 -- [1,6] 0.71 ±0.13 ±0.01 ±0.01 --  We estimate the theoretical uncertainties for the SM values in the low q 2 region using two methods: one follows the approach of [65], and the second is based on a Monte Carlo method. Both methods give similar results. We consider five different sources of uncertainty. The errors from the form factors (FF) have been calculated by adding in quadrature the uncertainties due to V , A 1 and A 2 (11%, 13% and 14%, respectively [58]).  table 4. For the Standard Model predictions the primed coefficients (C i,Q 1 ,Q 2 ) as well as the scalar coefficients (C Q 1,2 ) have been put to zero.
For the high q 2 region we have used the relative errors of table 2 in [26].

Experimental results
At present, the best upper limit for BR(B s → µ + µ − ) measured in a single experiment comes from LHCb [11]: at 95% C.L. This upper limit is followed by the result from CMS [66]:

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Observable SM prediction Experiment 10 7 GeV 2 × dBR/dq 2 (B → K * µ + µ − ) [1,6] 0.47 ± 0.27 0.42 ± 0.04 ± 0.04 [1,6] 0.71 ± 0.13 0.66 +0.06+0.04 −0.06−0.03  The CDF collaboration obtains a 95% C.L. upper limit [12]: together with a one sigma interval coming from an observed excess over the expected background. For our numerical evaluations, we consider the LHCb limit, and accounting for 11% theoretical uncertainty (as explained in section 2.1), we impose the following limit at 95% C.L. 4 (3.5) For B → K * µ + µ − related observables we also use the latest LHCb results which correspond to an integrated luminosity of 1 fb −1 [33]. The results are summarised in table 5 where the experimental uncertainties are statistical and systematic. For comparison, the SM predictions with the corresponding theoretical errors (from the five sources of errors mentioned in table 4, added in quadrature) are also provided.
In addition to the observables in table 5, three other observables have also been measured using 1 fb −1 of LHCb data namely S 3 and A Im [33] and also very recently the isospin asymmetry A I [67]. The reported results are:

7)
However, with the current experimental accuracy, these observables are not sensitive enough to probe SUSY parameters.

JHEP08(2012)092 4 Constraints on SUSY
To illustrate the impact of the recent limit on BR(B s → µ + µ − ) and the measurements of the angular distributions and differential branching fraction of the B → K * µ + µ − decay by LHCb, we consider constrained MSSM scenarios. The observables are calculated as described in section 2 using SuperIso v3.3 [34,35].
We focus on two specific scenarios, both assuming SUSY breaking mediated by gravity and invoking unification boundary conditions at a very high scale m GUT where the universal mass parameters are specified. The first model is the CMSSM, characterised by the set of parameters {m 0 , m 1/2 , A 0 , tan β, sgn(µ)}. Here m 0 is the universal mass of the scalars, m 1/2 the universal gaugino mass, A 0 the universal trilinear coupling, and tan β the ratio of the vacuum expectation values of the Higgs doublets. The second model we consider involves non-universal Higgs masses (NUHM). This model generalises the CMSSM, allowing for the GUT scale mass parameters of the Higgs doublets to have values different from m 0 . The two additional parameters can be traded for two other parameters at a lower scale, conveniently the µ parameter and the mass M A of the CP -odd Higgs boson implying that the charged Higgs boson mass can be treated essentially as a free parameter, contrary to the CMSSM. In both models, the flavour-changing effects we wish to study are effectively suppressed by the degeneracy of SUSY-breaking squark mass terms (∼ m 2 0 I) and trilinear couplings (∼ A 0 I) at the high scale, which suppresses the gluino (and neutralino) loop contributions. Only charged Higgs and chargino loops then remain, the latter dominating for large tan β > 30. In this limit, the scalar penguin amplitudes get the strongest tan β enhancement. We thus expect the SUSY deviations from the SM to be maximised at large tan β for observables involving scalar penguin diagrams with a chargino-up-squark loop, for which the lightest stop mass should be a decisive parameter. Enlightening discussions of the large tan β limit and of the associated subtleties can be found in refs. [36][37][38].
In  Next we consider the constraints from B → K * µ + µ − observables. In order to study the maximal effects we consider tan β=50 but show also the results for tan β=30, and investigate the SUSY spread in function of the lightest stop mass. We start with the averaged differential branching ratio as defined in table 4. The results in CMSSM are displayed in figure 3 for the low q 2 region and in figure 4 for the high q 2 region, where the solid red lines correspond to the LHCb central value, while the dashed and dotted lines represent the 1 and 2σ bounds respectively, including both theoretical and experimental errors (added in quadrature). At low q 2 , this branching ratio excludes Mt 1 below ∼ 250 GeV for tan β=50 and ∼ 150 GeV for tan β=30. In the high q 2 region the branching ratio is doing slightly better, as the Mt 1 below ∼ 300 GeV and ∼ 200 GeV are excluded for tan β=50 and tan β=30 respectively. As this light stop region is already excluded by the direct SUSY searches for the same scenario, BR(B → K * µ + µ − ) does not provide additional information. The main reason of the limited constraining power of the branching ratio is the large theoretical uncertainties (mainly due to form factors) from which this observable is suffering. The results are shown for two values of A 0 (=0 and -1000 GeV) for comparison. As can be seen from the figures, the constraints with smaller A 0 are slightly stronger. Contrary to the branching ratio, angular distributions, in which the theoretical uncertainties are reduced, can in principle provide more robust constraints on the SUSY parameter space. In particular, we consider in the following the forward-backward asymmetry A F B , the zero-crossing q 2 0 of A F B , F L , as well as S 3 , A Im for which the LHCb results with 1 fb −1 of data are available. The two latter observables do not provide any constraint with the current results and accuracy. The SUSY spread in function of the stop mass of A F B , q 2 0 and F L is given in figures 5-7. As can be seen, A F B provides the most stringent constraints among these observables, and excludes Mt 1 800 GeV and 600 GeV at tan β=50 and tan β=30 respectively. q 2 0 on the other hand excludes Mt 1 550 GeV (for tan β=50) and 400 GeV (for tan β=30) while F L excludes Mt 1 200 GeV (for tan β=50) and 150 GeV (for tan β=30). The impressive constraining power of A F B is mainly due to the fact that the measured central value is smaller than the SM prediction and in addition the reported experimental errors are more than twice smaller than the previous results.

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Same observables at high q 2 have less impact on the SUSY parameters and therefore their results are not reproduced here.
To illustrate the sensitivity of the observables to other SUSY parameters, in figure 8 we present the variation of A F B (which has the largest impact as we have shown) with respect to the pseudo scalar Higgs mass and the gluino mass.
Another observable of interest for which LHCb has announced the measurement very recently [67] is the isospin asymmetry A I , defined in eq. (2.18). Figure 9 illustrates the JHEP08(2012)092      SUSY spread of A I in function of Mt 1 . Since the experimental measurement (eq. (3.8)) has an error larger than the SUSY spread, the current result does not provide any information on the SUSY parameters.
In figure 10, we show the SUSY spread in function of the lightest stop mass, for A T , A T , A   A comparison between these observables in the plane (m 1/2 , m 0 ) is given in figure 11. As expected, A F B is the most constraining observable also in this plane. All the observables show more sensitivity at larger tan β and more negative A 0 . It should be noted that other indirect observables like BR(B → X s γ) briefly discussed below, or the branching ratio A 0 /H 0 → τ + τ − direct searches [71,72] can independently bring comparable constraints [70].
To trace the origin of these constraints, it is interesting to present them in terms of Wilson coefficients. The variation of the observables in SUSY is driven on the one hand by the additional contributions to the Wilson Coefficients, in particular C 7 , C 8 , C 9 and C 10 , and on the other hand by new contributions through the scalar and pseudo scalar coefficients C Q 1 and C Q 2 . The variation of the Wilson coefficients in a full CMSSM scan is presented in figure 12, ignoring existing constraints on SUSY parameters or Wilson coefficients. As can be seen, C 7 and C 8 can have both signs in SUSY and their correlation is visible in the figure: the same chargino-stop (or charged Higgs-top) loop gives the leading contribution, regardless of whether a photon (C 7 ) or gluon (C 8 ) is attached to it. As for C 9 , it varies only by very little while C 10 can have a larger spread. This feature can be understood once we notice that box diagrams are suppressed with respect to scalar-or Z-penguin diagrams, giving δC 9 /δC 10 ∼ (g V /g A ) µ ∼ 1 − 4 sin 2 θ W . On the other hand, scalar coefficients can receive very large contributions in SUSY, as already known. An interesting information can be obtained from the zero-crossing of A F B since it depends on JHEP08(2012)092 the relative sign of C 7 and C 9 as can be seen in eq. (2.17), and excludes the positive sign of C 7 as can be seen in figure 13.
Finally, in figure 14 we show the correlations of BR(B → X s γ) with BR(B → K * µ + µ − ) and A F B (B → K * µ + µ − ). In the SM, BR(B → X s γ) is dominated by contributions from C 7 . The SM prediction for this branching ratio is (3.08 ± 0.22) × 10 −4 [35,73,74] while the latest combined experimental value from HFAG is (3.55 ± 0.24 ± 0.09) × 10 −4 [75]. As expected, there are strong correlations between BR(B → X s γ) and B → K * µ + µ − branching ratio and forward backward asymmetry, especially for small and intermediate values of tan β, where the scalar and pseudo scalar contributions have a limited effect.

Conclusions
The rare decays B s → µ + µ − and B → K * µ + µ − are of great importance in exploring the footprints of physics beyond the SM. We have presented in this paper an update of the constraints by the recent LHCb limit on BR(B s → µ + µ − ) in two constrained MSSM scenarios, CMSSM and NUHM.
We also demonstrated that B → K * µ + µ − provides a large number of complementary observables from the full angular distribution. In particular while B s → µ + µ − lose very JHEP08(2012)092  rapidly its sensitivity by lowering down tan β, the constraints by B → K * µ + µ − observables weaken only mildly and thus act as additional tools to explore SUSY parameters. The forward backward asymmetry is in particular a very powerful observable which provides competitive information to the direct SUSY searches. To display the origin of these constraints in the CMSSM, we have shown the possible values and correlations between the JHEP08(2012)092 Figure 14. The correlation between BR(B → X s γ) and averaged BR(B → K * µ + µ − ) in the left, and between BR(B → X s γ) and A F B (B → K * µ + µ − ) in the right.
Wilson coefficients of the various operators relevant for these processes. Among those, the Wilson coefficient C 9 is a parameter for which these angular observables provide a unique handle. However, we have shown that the sensitivity of C 9 to CMSSM effects is suppressed by the muon vector coupling, because they proceed mainly through a Z penguin diagram. The constraining power of other observables, like the isospin asymmetry whose measurement is under way, have been shown to display their interest in the framework of Minimal Flavour Violation models for which the CMSSM provides an interesting prototype.
With reduced theoretical and experimental errors for these observables, in particular for the forward backward asymmetry and its zero-crossing which have been recently measured for the first time by the LHCb experiment, one would have access to very powerful observables in constraining supersymmetric models. The information obtained in this way can also serve as consistency checks with the results of the direct searches.

B Form factors
The T ⊥, (which include non-factorisable corrections) are described in terms of the soft form factors and can be found in [13,14] where, in order to get T ± ⊥, , C eff 7 appearing in T ⊥, should be replaced with (C eff 7 ± C 7 ) [23]. To obtain the soft form factors we have used the factorisation scheme used in [14]: The full form factors V and A 1,2 have been taken from light-cone sum rule (LCSR) calculations [58]: where the fit parameters r 1,2 , m 2 R and m 2 f it are given in table 6.

C Renormalisation group equations for C 9
To obtain the evolved Wilson coefficients from the matching scale (µ 0 ) to µ b , the renormalisation group equation (RGE) for C has to be solved:  Table 6. Fit parameters describing the q 2 dependence of the form factors V and A 1,2 in the LCSR approach [58].
whereγ is the Anomalous Dimension Matrix (ADM). A general solution for the RGE is whereÛ is the evolution matrix T g is the g(= √ 4πα s )-ordering operator such that the coupling constants increase from right to left.γ and β can be expanded in ĝ The Wilson coefficients and the evolution matrix expanded in α s arê Following the conventions used in [41], the evolution matrix is Using the above relations for the different orders of C 9 (µ b ) we havẽ l (µ 0 ) +    To obtain the evolution matrix we have followed [76] and [77]. Takingγ (0) ,γ (1) and γ (2) from [41] and [78], we have produced the necessary "magic numbers" (m kli ) for the evaluation of U 9l . Since O 9 does not mix with O 7,8 the magic numbers m 97i and m 98i are all zero. The a i 's and the rest of the m kli can be found in tables 7-13.