4 { V'tb\1t:[C1Qi + C2Q2 J.=.1:;Q;J + V'tuv:s [C1 (Qi - Q'[) + C2 (Q2 - Q!t)] }

Transcription

1 B ---+ K*µµ: theory interpretation or A hitchiker's guide to B ---+ K*µµ optimised observabletf S. Descotes-Genon, L. Hofer2, J. Matias2 and J. Virto3 Laboratoire de Physique Thtiorique, CNRS/Univ. Paris-Sud (UMR 8627} 94 Orsay Cedex, France 2 Universitat Autbnoma de Barcelona, 893 Bellaterra, Barcelona 3 Theoretische Physik, Naturwissenschaftlich-Technische Fakultat, Univ. Siegen, 768, Germany We discuss the role and advantages of the different B -+ K* µµ optimised observables, defined to have little sensitivity to hadronic (form factor) input. We focus on the sensitivity of each observable to short-distance Wilson coefficients. Motivation and interest of B ---+ K*µµ Flavour-Changing Neutral Currents (FCNC ) have been prominent tools in high-energy physics to search for new degrees of freedom, due to their quantum sensitivity to energies much higher than the external particles involved. n the current context where the LHC has discovered a scalar boson completing the Standard Model ( SM) picture but no additional particles that would go beyond this framework, FCNC can be instrumental in order to determine in which direction to look for New Physics ( NP). One particularly interesting instance of FCNC is provided by b ---+ scc and b ---+ S"f transitions, which can be probed through various decay channels, currently studied in detail at the LHC, in particular at the LHCb and CMS experiments. ndeed, recent experimental results have shown interesting deviations from the Standard Model. n 23, the LHCb collaboration announced the measurement of angular observables describing the decay B ---+ K* µµ in both regions of low and large-k* recoils. Two observables, P2 and P, were in significant disagreement with the SM expectations in the large-k* recoil region 2. A few months later, an improved measurement of the branching ratio for B ---+ Kµµ at large recoil turned out to be on the low side compared to theoretical expectations 3, as well as the branching ratios of B ---+ K* µµ and B <f>µµ at low recoil 3,4, Another measurement has also raised a lot of attention recently, namely RK Br(B ---+ Kµµ)/Br(B ---+ Kee), measured by LHCb, and showing a significantly lower result than its SM prediction equal to (up to a very good accuracy) 6. n the Moriond 2 conference, a new analysis of B ---+ K*µµ was presented with an extended data set (3 fb- ), confirming the pattern of deviations observed with a restricted set of data 7. The presence of very different scales for the external states ( at most O (mb )) and the internal degrees of freedom (O(Mw) or above) allows for model-independent analyses relying on the effective Hamiltonian approach. The latter is obtained by focusing on b ---+ s transitions and integrating out all heavy degrees of freedom, leading to ( in the case of the SM) : -l8m- 4 { V'tb\t:[CQi + C2Q2 J..:;Q;J + V'tuv:s [C (Qi - Q'[) + C2 (Q2 - Q!t)] } () "Based on talks given by J. Matias at Moriond 2 Electroweak session and by S. Descotes-Genon at Moriond 2 QCD session.

2 up to contributions suppressed by additional powers of mb /Mw. The Wilson coefficients C; describe the short-distance physics (function of mt, mw... in the SM) whereas the local opera tors Q; correspond to long-distance physics involving only light/soft degrees of freedom. n this framework, b -+ s transitions are mainly described by Q e/(47r) 2 suµv (l + ')' ) Fµv b, related to the emission of a real or soft photon, Q9 e2 /(47r) 2 s'yµ(l - ')' )b l')'µf involved in b -+ sµµ via the emission of a Z boson or a hard photon, and Q e2 /(47r) 2 s'yµ(l - 'Y )b l')'µ')' involved in b -+ sµµ via the emission of a Z boson. The value of the Wilson coefficients can be obtained by matching the SM at a high-energy scale µo O(mt) and evolving down at µref O(mb ) (usually 4.8 GeV, with typical values cfm -.29, cm 4., CfoM -4.3). The presence of NP can alter this picture by modifying the value of the Wilson coefficients C,9,, but also by allowing new long-distance operators Q;, which would be very suppressed or absent in the SM. This yields the chirally-flipped operators Q7',9', l' (obtained for instance by a heavy spin- boson coupling to right-handed fermions), scalar and pseudoscalar operators Qs,S',P,P' (induced, e.g., by the exchange of charged scalar or pseudoscalar Higgs-like bosons) or tensor operators Qr,T' (allowed in principle, but difficult to generate in viable models). These NP contributions are expressed as C; CfM + 8C; at µref. An accurate extraction of short-distance physics requires a good understanding of long-distance physics, and in particular QCD effects. n some kinematic configurations (either low or large recoil of the K* meson), one can use effective theories to separate soft and hard physics, in order to build observables with a limited sensitivity to hadronic uncertainties, or so-called optimised observables 8,9,l, l l, l 2. Additional studies have been performed in order to assess long-distance effects that could contribute, in particular the charm resonances and loop contributions, the form factors, and the power corrections in the effective-field theory approach l3,l4, _ These elements have been combined in global fits of b -+ sl!e Wilson coefficients, with different sets of observables and statistical approaches 6, l7, 8. They point towards a large negative contribution to Cg, amounting to 2% of its SM value, leaving open the possibility of large contributions to other Wilson coefficients. Each optimised observable does play a different role in such fits, pulling the Wilson coeffi cients in different directions in order to increase the agreement of predictions with LHCb data compared to the SM result. We are going to review their role in this proceeding. 2 B -+ K*µµ observables: Optimised basis Due to its particularly rich kinematics 9, the B -+ K* µµ decay provides 2 angular coefficients, corresponding to interferences between 8 transversity amplitudes, generally labelled in relation to the polarisation of the K*-meson and/or the chirality of the dilepton pair (A,'f, and At, As ). However, if lepton masses or scalars are not considered, not all angular coefficients carry inde pendent information: the corresponding redundancies can be worked out thanks to the analysis of symmetries of the transversity amplitudes that leave the angular coefficients invariant zo,2. n this scenario one can show that only 8 independent observables can be built, out of which 6 op timised observables can be chosen, namely P, P2, P3, P, Pf,, P. The remaining two observables can be chosen, for instance, to be the (differential) branching ratio and longitudinal polarisation. For a complete phenomenological description, the previous set of P-wave observables should be complemented with a set of S-wave independent observables 22,23,24 associated to B -+ Kµ+µ where K is a broad scalar resonance. We will now focus on the optimised observables of the P-wave sector, which exhibit the largest sensitivity to short-distance physics. 2. P or A The definition in terms of amplitudes is 8 (2)

3 where in this definition and all the following ones, it should be understood that each term has associated the corresponding CP conjugated term, with the notation A;l 2 Af l 2 + Afl 2 + Af 2 + Afl 2 Lepton, scalar and tensor terms are neglected. This observable is particularly suited to detect the presence of right-handed currents. The left-handed structure of the SM implies that the s quark produced in the decay of the b quark will be in a helicity state of -/2 (neglecting the s-quark mass). The combination of the s quark with the spectator quark generates a K*-meson with helicity - or, but not +l. The suppression of H+l (Al_ + A )/V2 "" implies Al_ "" -Au and consequently PfM "". Deviations from this prediction would signal contributions from a new right-handed structure. n Table we present how (P ) [.,.98] > (P ) [6,8] and (P)[, 9] are affected by shifting one Wilson coefficient at a time. Only significant changes are indicated, and shifts improving the agreement with data are indicated in boldface. As expected, changing Wilson coefficients for SM operators (not carrying a right handed structure) does not induce any sizeable shift. The first bin of P exhibits the largest sensitivity to Cf. Contrary to most observables, P is also rather sensitive to New Physics at low recoil. 2.2 p The definition is 2 (3) Together with Pf,, this observable establishes bounds on P or enters in consistency relations. n particular, the bound (4) works very efficiently in two bins: [6,8] and low-recoil. n the first case, the preference of data for P 2': in the [6,8] bin requires P :<:'., in agreement with 2 data (notice that in 23 data P was positive in the bin [4.3,8.68]). n the second case, taking the central values of the low-recoil bin as an illustration, one finds that -.4 :<:'. P :<:' again in the right ballpark as compared to the measurement P "" -.. Strictly speaking, this is a bound on the unbinned observables, but they can be adapted for binned observables in the case where the observables are slowly varying with, providing important cross-checks among the LHCb measurements P2 The definition is 2 () This observable is the optimised and clean version of the forward-backward asymmetry, and it was originally called A'f 2P2. t highlights the correlation among AFB and FL, with a low sensitivity to choices of form factors compared to these observables. ndeed, the prediction for AFB and FL depends strongly on the parametrisation of form factors used, and the ratio of errors between two commonly used parametrizations, for some bins of AFB or FL can be as large 2 as a factor 3 or 4. On the contrary, in the case of P2, besides a shift in central values due to the different central value predictions of the form factors (induced by leading-order power corrections included in our predictions), the ratio of errors is near one showing its robustness and low dependence of this observable on the details of the parametrization. The observable P2 contains three relevant elements of information: the position of its zero q'fi, the position qf of its maximum, and the value of P2 at qf. At leading order, assuming no

4 Table : mpact for a given observable of the shift of one of the Wilson other Wilson coefficients keeping their SM value). The first row correspon coefficients by an amount lic, (the shift lic, and the second row to a negative shift by the same amount. ds to the variation due to a positive of predictions with the 2 LHCb data are written in boldface. Double The changes improving the agreement "-" means variations below.3, only those in (P2 ) 2.,4] are provided explicitly. (P) [o.,o. 9sJ - l <Ci l (P)[6,SJ +l oci l - l oci l (P )[, 9] (P,! )[6,sJ l <C7. loc. loc9 locg l loc l oc. loc9 l <Co l l <C. -.3 l<l'cg l loc +.9 loc l 'C. -.7 loc l 'Co l loc'. loc9' l l l <C'. locg' -.6 loc7,. -.Dl -.3 loc'. l <C' (P )[4,6J - l <C; j l <C. -. locg l l <C (P) [6,8J loc. -.4 l <Cg l loc9 +.6 l <Cg, -. l <C9' +.7 loco' loc' -.2 loc' loc' loc l <C loc' -.4 locg' -. locgl loc'. -. loc. +.4 loc' +.4 locg l -.2 loco' loc9' loc loc9, loc' (P2 ) [2.,4] +l oci l (P) [,9] l<c'. +.2 loc. loc. (P) [.,.98] +l oci l l<co l -.3 (P4) [,9] (P2 )[6,s] + loci l l<c9 +.6 loc9, +.8 l oco' l oc' loc'. -. locg' loc -.7 7' locg' -.8 loc' loc +.3 loc'. loc9, loc'

5 ' <(' 2.o r LS Lo LS i o. s i i + o.o r + - o.s i - LO rt: LO, :: [ -.6 '.4- LO :. -. os /(', +. q -+ t +t -.S c.2 t+ 2.,. li LO'..2 :. -. ' i _ r '+ ' --r--: _ '. 8 k;' t ' t -. - Lo! t : l.6,.4.2 Figure - Set o f optimised B -t K*µ µ observables: crosses represent the latest LHCb results and boxes our SM predictions computed using KMPW form factors and including a long-distance charm contribution.

6 contribution from right-handed currents, i.e., q the first two quantities are given by: (6) where for the position of the maximum we have neglected a term of O(m(Cff) 2 ) following 22. These expressions illustrate that a NP contribution to C9 and C7 would shift both the zero and the maximum of P2, but with a different magnitude. The position of the maximum can be also shifted via a Cm contribution. t was found in m.22 that a NP contribution to the SM Wilson coefficients C,9, can only shift the position of the maximum but not the value of the observable that is fixed at P:f'ax /2. On the contrary, the presence of NP contributions into the chirally flipped operators would reduce the maximum below /2, though not by a large amount. Unfortunately, a fluctuation of the (FL) [2.,4 bin has induced a large experimental error in the corresponding bin of P2, and the discussion remains inconclusive for the moment. n Table we show the sensitivity to shifts of Wilson coefficients in the [6,8] bin. The sensitivity of this observable to NP at low recoil is small. At large recoil, one should remark that the same shifts of the Wilson coefficients moving (P2 ) [6,8 towards data also improves the agreement of (P2 ) [2.,4 with data (assuming that data is above the SM) except for Cm (whose impact is small in any case). On the other hand, chirally flipped coefficients (positive or negative) shift down the value of the observable in this bin (though by a relatively small amount). ')h, r mauy, r- uuern iunner consrni;ency cnecks oasea on i;ne reiai; n 2 T"'\" r f',,,,, ' - (7) The first check stems from the reality of the square root in the previous equation. f we take a bin where P2 -E (with E > ), one must have (8) Considering the [6,8] bin and taking the central values (P2 ) [6, E and (P4) 6,8.2, one obtains (P) [6,s] :S: -.4, in fair agreement with the measurement (P)6,8 -.. More generally, eq. (8) implies a specific order 2 for (P2 ) 6,8 and (P)[6,8 which is nicely fulfilled by the current data. A similar reasoning holds in the [4,6] bin. A second check is related to the zero in eq. (7). At the position q of the zero of P2 (or AFB ) the following relation should be fulfilled 26 : (9) with rl(q) [P[ + P (P42 - P2 )]q6 f we consider the [4,6] bin, (P2 ) [4,G] is close to zero. As an illustration, let us assume that q is near the center of the bin, i.e., GeV2 (this will be measured with precision using the amplitude method analysis 27 ). Considering the central values as a raw estimate to test this relation (P)[4,6] -.3, (P4)[4,6] +.9 and (P ) [4,6 +.8, the left-hand side of eq. (9) yields.9 while the right-hand side is.84. Even though there is a good agreement, let us remind again that this relation is valid for the unbinned observables and so the binning induces an error, besides the necessary inclusion of the error bars. 2.4 Pf, The definition is 2 Re(AAi_* - AA.f*) () JJ Ao J 2 ( J A- J 2 + JA J 2 ) n the current data from LHCb, this observable exhibits the largest deviations with respect to the SM prediction in some bins, the so-called "anomaly" 2 illustrated in Fig.. nterestingly, p h

7 this observable can receive large NP contributions without spoiling the good agreement of P4 data with SM predictions: in Table the large impact of a variation of Cg in P corresponds to a negligible effect on P4 in the [6,8] bin. Two mechanisms may enforce a larger impact of NP in P with respect to P,i. The first mechanism consists in weakening the suppression of the right-handed amplitudes with respect to the left-handed amplitudes, in order to profit from the relative minus sign between the two terms in the numerator of P (compared to the plus sign in P4). The SM suppression of the right-handed amplitudes is due to the numerical coincidence CM -CJ3JA, which is altered if only one of the two coefficients, say Cg, receives a NP contribution. The second mechanism consists in introducing a NP contribution reducing the size of Ai in the numerator but keeping the other transversity amplitudes untouched, leading to a significant (minor) change in P (P4). n Table we show the sensitivity to shifts of Wilson coefficients for the [4,6], [6,8] and low-recoil bins. One notices the large sensitivity of (Ps ) [6,8] to an additional cfjp as compared to (P4)[6,8], in agreement with the data. Moreover, all Wilson coefficients have a large impact in this bin as compared to other bins and observables. Similar results are found for (P) [4,6]. The first large-recoil bin exhibits an interesting sensitivity to c7, even though lepton mass effects 24,28 affect this first bin (as well as other observables in the same bin). At low recoil, (P) [, gl is more sensitive to NP than other observables in this region, but not as much as at large recoil. 2. P3, P6 and Pf; The last optimised observables are defined as 2 and lm(aaf* - AAf*) -v'2 m(a Ai* + A Af*), p6 -yrn2 ---c Pf; J Ao l 2 (A.L l 2 + A l 2 ) JAo l 2 (A.L l 2 + A l 2 ) () (2) They are mainly sensitive to phases (strong or weak, SM or beyond). A more direct test of new weak phases is the measurement of the PPP observables 2. Present data is compatible with the SM with large error bars including local fluctuations (up to 2 u for some of the P6 measurements) that are expected to disappear with more data. This set of observables also are required to fulfill bounds such as : Pf,,2 - ::; P ::; P62 (following the same reasoning as the bounds in 26 ). - 3 Conclusion Optimised observables for B --+ K* µµ play a prominent role in the search for NP in b --+ s tran sitions. Several analyses have been performed, including some presented at this conference. Our own analysis following our earlier work 2 is under way 28, including experimental and theoretical correlations (they were not included in our results presented at the Moriond 2 sessions). This study must be performed carefully in order to gauge the impact of correlations for the analysis at different levels (soft form factors, power corrections... ). We will consider the above optimised observables, as well as the branching ratios of B --+ Kµµ, Bs --+ µµ, B --+ Xs'Y, B --+ X8µµ, together with observables related to B --+ K* (SK'-y and A). The list of observables to be included is not closed yet: for instance, electronic modes should also be considered 2g. We will take advantage of new determinations of form factors 3 and improved studies of charm effects. This should yield a more complete picture of the Wilson coefficients describing radiative b --+ s transitions, and hopefully, this will allow us to disentangle Standard Model and New Physics contributions in these decays.

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