Flavour anomalies after the R K ∗ measurement

Journal of High Energy Physics, Sep 2017

The LHCb measurement of the μ/e ratio R K ∗ indicates a deficit with respect to the Standard Model prediction, supporting earlier hints of lepton universality violation observed in the R K ratio. We show that the R K and R K ∗ ratios alone constrain the chiralities of the states contributing to these anomalies, and we find deviations from the Standard Model at the 4σ level. This conclusion is further corroborated by hints from the theoretically challenging b → sμ + μ − distributions. Theoretical interpretations in terms of Z′, lepto-quarks, loop mediators, and composite dynamics are discussed. We highlight their distinctive features in terms of the chirality and flavour structures relevant to the observed anomalies.

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Flavour anomalies after the R K ∗ measurement

Received: May Flavour anomalies after the RK Guido D'Amico 1 2 5 Marco Nardecchia 1 2 5 Paolo Panci 1 2 5 Francesco Sannino 1 2 5 Alessandro Strumia 1 2 5 Riccardo Torre 1 2 3 4 Alfredo Urbano 1 2 5 Geneva 1 2 Switzerland 1 2 Odense 1 2 Denmark 1 2 Italy 1 2 0 Origins and Danish IAS, University of Southern Denmark 1 Via Dodecaneso 33, I-16146 Genova , Italy 2 Route de la Sorge , CH-1015 Lausanne , Switzerland 3 Institut de Theorie des Phenomenes Physiques, EPFL 4 INFN , Sezione di Genova 5 Theoretical Physics Department , CERN The LHCb measurement of the =e ratio RK indicates a de cit with respect to the Standard Model prediction, supporting earlier hints of lepton universality violation Beyond Standard Model; Technicolor and Composite Models - observed in the RK ratio. We show that the RK and RK ratios alone constrain the chiralities of the states contributing to these anomalies, and we nd deviations from the Standard Model at the 4 level. This conclusion is further corroborated by hints from the theoretically challenging b ! s + distributions. Theoretical interpretations in terms of Z0, lepto-quarks, loop mediators, and composite dynamics are discussed. We highlight their distinctive features in terms of the chirality and avour structures relevant to the observed anomalies. 1 Introduction 2 E ective operators and observables 3 lepton sector. Taking the ratio of branching ratios strongly reduces the Standard Model (SM) theoretical uncertainties, as suggested for the rst time in ref. [2]. The experimental result [1] is reported in two bins of di-lepton invariant mass These values have to be compared with the SM predictions [3] RK = 8< 0:660+00::101700 : 0:685+00::101639 0:024 (2m )2 < q2 < 1:1 GeV2 0:047 1:1 GeV2 < q2 < 6 GeV2 : RKSM = <8 0:906 0:028 (2m )2 < q2 < 1:1 GeV2 : 1:00 0:01 1:1 GeV2 < q2 < 6 GeV2 : { 1 { b ! s + related measurements: 1. the RK ratio [4] At face value, a couple of observables featuring a transitions. In fact, anomalous deviations were also observed in the following RK = BR (B+ ! K+ + BR (B+ ! K+e+e ) ) 2. the branching ratios of the semi-leptonic decays B ! K( ) + [5] and Bs ! 3. the angular distributions of the decay rate of B ! K + . In particular, the so-called P50 observable (de ned for the rst time in [7]) shows the most signi cant The coherence of this pattern of deviations has been pointed out already after the measurement of RK with a subset of observables in [10, 11] and in a full global analysis in [12, 13]. For the observables in points 2 and 3 the main source of uncertainty is theoretical. It resides in the proper evaluation of the form factors and in the estimate of the non-factorizable hadronic corrections. Recently, great theoretical e ort went into the understanding of these aspects, see refs. [7, 14{23] for an incomplete list of references. Given their reduced sensitivity to theoretical uncertainties in the SM, the RK and RK observables o er a neat way to establish potential violation of lepton avour universality. Future data will be able to further reduce the statistical uncertainty on these quantities. In addition, measurements of other ratios RH analogous to RK , with H = Xs; ; K0(1430); f0 will constitute relevant independent tests [2, 24]. The paper is structured as follows. In section 2 we discuss the relevant observables and how they are a ected by additional e ective operators. We perform a global t in section 3. We show that, even restricting the analysis to the theoretically clean RK , RK ratios, the overall deviation from the SM starts to be signi cant, at the 4 level, and to point towards some model building directions. Such results prompt us to investigate, in section 4, a few theoretical interpretations. We discuss models including Z0, lepto-quark exchanges, new states a ecting the observables via quantum corrections, and models of composite Higgs. 2 E ective operators and observables Upon integrating out heavy degrees of freedom the relevant processes can be described, near the Fermi scale, in terms of the e ective Lagrangian X where the sum runs over leptons ` = fe; ; g and over their chiralities X; Y = fL; Rg. New physics is more conveniently explored in the chiral basis ObX `Y = (s PX b)(` PY `): These vector operators can be promoted to SU(2)L-invariant operators, unlike scalar or tensor operators [25]. In SM computations one uses the equivalent formulation de ning dimensionless coe cients CI as He = VtbVts 4 v2 em X `;X;Y CbX `Y ObX `Y + h:c: ; cI = VtbVts 4 evm2 CI = CI (36 TeV)2 ; prediction for RK is percent. RK = jCbL+R L R j2 + jCbL+R L+R j2 : 2 jCbL+ReL R j2 + jCbL+ReL+R j 1By theoretically clean observables we mean those ones predicted in the SM with an error up to few 2In the limit of vanishing lepton masses the decay rate in eq. (2.5) takes the form [12] d (B+ ! K+ + dq2 210 5MB3 ) = G2F e2mjVtbVtsj2 3=2(MB2 ; M K2; q2) jFV j2 + jFAj2 ; where Vts = 0:040 vacuum expectation value, usually written as 1=v2 = 4GF=p2. The SM itself contributes 0:001 has a negligible imaginary part, v = 174 GeV is the Higgs as CSM bL`L = 8:64 and CSM bL`R = 0:18, accidentally implying jCbSLM`R j jCbSLM`L j. This observation suggests to use the chiral basis, related to the conventional one (see e.g. ref. [12]) by C9 = CbL L+R =2, C10 = CbR L R =2, with the approximate relation CSM 9 CbL L R =2, C90 = CbR L+R =2, C100 = C1S0M holding in the SM. To make the notation more compact, we de ne CbL R`Y CbR`L CbL`R CbR`R , and CbX ( e)Y CbX Y CbL`Y CbX eY . We now summarize the theoretically clean observables,1 presenting both the full exstandard-model (BSM) contribution, CI = CSM + CBSM. I I pressions and the ones in chiral-linear approximation. The latter is de ned by neglecting jCbSLM`R j CSM j bL`L j and expanding each coe cient CI at rst order in the beyond-theCbR`Y and CbL+R`L R CbL`L + RK revisited The experimental analysis is made by binning the observable in the squared invariant mass of the lepton system q 2 (P` + P`+ )2. Writing the explicit q2-dependence, we have contributions from the electromagnetic dipole operator, justi ed by the cut qm2in = 1 GeV2, and non-factorizable contributions from the weak e ective Hamiltonian,2 the theoretical (2.2) (2.3) (2.4) (2.5) (2.9) (2.6) and e can induce theoretical errors, bringing back the issue of hadronic uncertainties. In the chiral-linear approximation, RK becomes RK ' 1 + 2 Re CbBLS+MR( e)L ; CSM remains valid for the simpli ed expression proposed in ref. [24], expanded up to quadratic terms in new physics coe cients. The reason is that the expansion is controlled by the parameter CbBXSlMY=CbSXMlY , a number that is not always smaller than 1. This is particularly true in the presence of new physics in the electron sector in which | as we shall discuss in detail | large values of the Wilson coe cients are needed to explain the observed anomalies. For this reason, all the results presented in this paper make use of the full expressions for both RK [12] and, as we shall discuss next, RK . 2.2 Anatomy of RK Given that the K has spin 1 and mass MK = 892 MeV, the theoretical prediction for the RK ratio given in eq. (1.1) is RK = (1 (1 p)(jCbL+R L R j2 + jCbL+R L+R j2) + p jCbL R L R j2 + jCbL R L+R j 2 p)(jCbL+ReL R j2 + jCbL+ReL+R j2) + p jCbL ReL R j2 + jCbL ReL+R j 2 where p 0:86 is the \polarization fraction" [24, 27, 28], that is de ned as p = g0 + g k g0 + g + g k ? : The gi are the contributions to the decay rate (integrated over the intermediate bin) of the di erent helicities of the K . The index i distinguishes the various helicities: longitudinal where GF is the Fermi constant, (a; b; c) a2 +b2 +c2 2(ab+bc+ac), MB 5:279 GeV, MK 0:494 GeV, jVtbVtsj 40:58 10 3. Introducing the QCD form factors f+;T (q2) we have FA(q2) = C10 + C100 f+(q2) ; FV (q2) = (C9 + C90)f+(q2) + 2mb MB + MK C7 + C70 fT (q2) + hK(q2) : SM| electromagnetic dipole contributi}on non fa|ctorizable term {z {z } Notice that for simplicity we wrote the Wilson coe cient C9 omitting higher-order s-corrections [26]. Neglecting SM electromagnetic dipole contributions (encoded in the coe cients C7(0)), and non-factorizable (i = 0), parallel (i =k) and perpendicular (i =?). In the chiral-linear limit the expression for RK simpli es to RK ' RK 4p Re CbBRS(M e)L ; CSM 0:40. The formula above clearly shows that, in this approximation, a deviation of RK from RK signals that bR is involved at the e ective operator level with the dominant e ect still due to left-handed leptons. As already discussed before, eq. (2.13) is not suitable for a detailed phenomenological study, and we implement in our numerical code the full expression for RK [29]. In the left panel of gure 1, we present the di erent predictions in the (RK ; RK ) plane due to turning on the various operators assumed to be generated via new physics in the muon sector. A reduction of the same order in both RK and RK is possible in the presence of the left-handed operator CBSM (red solid line). In order to illustrate the size of the required correction, the arrows correspond to CbBLSML = (see caption for details). Conversely, as previously mentioned, a deviation of RK from RK signals the presence of CbBRSML (green dot-dashed line). Finally, notice that the reduced value of RK measured in eq. (1.4) cannot be explained by CbBRSMR and CBSM . The information bL R summarized in this plot is of particular signi cance since it shows at a glance, and before 1 an actual t to the data, the new physics patterns implied by the combined measurement of RK and RK . (s PLb)( ative values C9B;SM Before proceeding, another important comment is in order. In the left panel of gure 1, we also show in magenta the direction described by non-zero values of the coe cient C9B;SM = (CbBLSML + CbBLSMR )=2. The latter refers to the e ective operator O9 = ), and implies a vector coupling for the muon. The plot suggests that neg1 may also provide a good t of the observed data. However, it is also interesting to notice that in the non-clean observables, the hadronic e ects might mimic a short distance BSM contribution in C9B;SM. From the plot in our gure 1, it is clear that with more data a combined analysis of RK and RK might start to discriminate between C9B;SM and CbBLSML using only clean observables. However, with the present data, there is only a mild preference for CbBLSML , according to the 1-parameter ts of section 3.1 using only clean observables. It is also instructive to summarise in the right panel of gure 1 the case in which new physics directly a ects the electron sector. The result is a mirror-like image of the muon case since the coe cients CbX eY enter, both at the linear and quadratic level, with an opposite sign when compared to their analogue CbX Y . In the chiral-linear limit the only operator that can bring the values of RK and RK close to the experimental data is CbLeL > 0. As before, a deviation from RK in RK can be produced by a non-zero value of CbBRSeML . Notice that, beyond the chiral-linear limit, also CBSM bL;ReR points towards the observed experimental data but they require larger numerical values. A closer look to RK reveals additional observable consequences related to the presence of BSM corrections. RK , in a given range of q2, is de ned in analogy with eq. (2.5):  μ * *    <   ratios refer to the [1:1; 6] GeV2 q2-bin. We assumed real coe cients, and the out-going (in-going) arrows show the e ect of coe cients equal to +1 ( 1). For the sake of clarity we only show the arrows for the coe cients involving left-handed muons and electrons (except for the two magenta arrows in the left-side plot, that refer to C9B;SM = (CbBLSML + CbBLSMR )=2 = 1). The constraint from Bs ! is not included in this plot. as function of q2, the invariant mass of the `+` pair, for the SM and for two speci c values of the new-physics coe cients. The inset shows iso-contours of deviation from RK = 1 in the [0:045; 1:1] GeV2 bin as a function of new-physics coe cients, compared to their experimentally favoured values. Right: correlation between RK (horizontal axis) and [0:045; 1:1] GeV2 bin (vertical axis) of q2: a sizeable new physics e ect can be present in the low-energy bin. The numerical values of q2 are given in GeV2. measured in the [1:1; 6] GeV2 bin { 6 { 1.1 1.0 ] .10.9 1 , 5 . 0 ∗ [ where the di erential decay width d (B ! K , and takes the compact form + )=dq2 actually describes the four-body d (B ! K + dq2 ) = 3 4 (2I1s + I2c) (2I2s + I2c) : 1 4 (2.15) The angular coe cients Iia==1s;;2c in eq. (2.15) can be written in terms of the so-called transversity amplitudes describing the decay B ! K V with the B meson decaying to an on-shell K and a virtual photon or Z boson which later decays into a lepton-antilepton pair. We refer to ref. [29] for a comprehensive description of the computation. In the left panel of gure 2 we show the di erential distribution d (B ! K dilepton invariant mass q2. The solid black line represents the SM prediction, and we show in dashed (dotted) red the impact of BSM corrections due to the presence of non-zero + )=dq2 as a function of the CBSM (CbBRSML ) taken at the benchmark value of 1. bL L We now focus on the low invariant-mass range q2 = [0:045; 1:1] GeV2, shaded in blue with diagonal mesh in the left panel of g 2. In this bin, the di erential rate is dominated deviation in RK [0:045; 1:1] compared to its SM value RKSM by the SM photon contribution. It is instructive to give more quantitative comments. In the inset plot in the left panel of g 2, we show in the plane (CbBLSML ; CbBRSML ) the relative 0:9, and we superimpose the 1- and 3 con dence contours allowed by the t of experimental data (without including RK ). This comparison shows that a 10% reduction of RK in the mass-invariant bin q 2 = [0:045; 1:1] GeV2 is expected from the experimental data. The SM prediction, RKSM[0:045; 1:1] 0:9, departs from one because of QED e ects which distinguish between m and me. The observed central value RKSM[0:045; 1:1] = 0:66 can be again explained with possible e ects of new physics. The natural suspect is a new physics contribution to the dipole operator, but it can be shown that this cannot be very large because of bounds coming from the inclusive process B ! Xs , see for example ref. [30]. We can instead correlate the e ect in RKSM[0:045; 1:1] with RKSM[1:1; 6]. The results are shown in the right panel of gure 2. Here we learn that the new physics hypotheses predict values larger than the one observed in the data. However, since the experimental error is quite large, precise measurements are needed to settle this issue. In conclusion, the picture emerging from a simple inspection of the relevant formulas for RK and RK is very neat, and can be summarized as follows: New physics in the muon sector can easily explain the observed de cits in RK ,RK , and we expect a preference for negative values of the operator involving a left-handed current, CbBLSML . Sizeable deviations of RK from RK signal non-zero values for CbBRSML . New physics in the electron sector represents a valid alternative, and positive values of CbBLSeML are favoured. Sizeable deviations of RK from RK signal non-zero values for CbBRSeML . However invoking NP only the electronic channels does not allow to explain other anomalies in the muon sector such as the angular observables. There exists an interesting correlation between RK in the q2-bin [1:1; 6] GeV2 and [0:045; 1:1] GeV2. At present, all the new physics hypothesis invoked tend to predicts larger value of RK in the low bin than the one preferred by the data. In section 3 we shall corroborate this qualitative picture with quantitative ts. { 7 { 10 9 [31]. This BR can also be a ected by extra scalar operators (bPX s)( PY ), so that it is sometimes omitted from global BSM ts. (2.16) HJEP09(217) 3 Fits We divide the experimental data in two sets: `clean' and `hadronic sensitive': i ) The `clean' set includes the observables discussed in the previous section: RK , RK , to which one can add BR(Bs ! + ) given that it only provides constraints.3 The `cleanness' of these observables refers to the SM prediction, in the presence of New Physics larger theoretical uncertainties are expected. We didn't include the Q4 and Q5 observables measured recently by the Belle collaboration [33]. ii ) The `hadronic sensitive' set includes about 100 observables (summarized in the appendix A). This list includes the branching ratios of semi-leptonic B-meson decays as well as physical quantities extracted by the angular analysis of the decay products of the B-mesons. Concerning the hadronic sensitivity of the angular observables, the authors of [7] argue that the optimised variables Pi have reduced theoretical uncertainties. The rationale is to rst limit the analysis to the `clean' set of observables. In this way one can draw solid conclusions without relying on large and partially uncontrolled e ects. This approach is aligned with the spirit of this paper, and can be extremely powerful, as already shown in section 2.2. Furthermore, extracting from this reliable theoretical environment a BSM perspective could be of primary importance to set the stage for more complex analyses. In a second and third step we will estimate the e ect of the `hadronic sensitive' observables and combine all observables in a global t. 3.1 Fit to the `clean' observables only The formul summarized in the previous section allow us to t the clean observables. We wrote a dedicated Flavour Anomaly Rate Tool code (Fart). For simplicity, in our 2 ts we combine in quadrature the experimental errors on the two RK bins, using the higher error band when they are asymmetric. We checked that our results do not change appreciably if a more precise treatment is used. Let us start discussing the simplest case, in which we consider one-parameter ts to each NP operator in turn. Apart from its simplicity, this hypothesis is motivated from a experimental error is correlated with the one on BR(Bs ! + ). 3When using the Flavio [32] code, for consistency we include the observable BR(B0 ! + ), whose { 8 { 1- error, and p 2 SM theoretical viewpoint, as it captures most of the relevant features of concrete models, as we shall discuss in detail in section 4. We show the corresponding results | best- t point, 2 best | in the `clean' column in table 1. In the upper part of the table we show the cases in which we allow new physics in the muon sector. It is evident that the results of the t match the discussion of section 2.2: the left-handed coe cient CbBLSML is favoured by the measured anomalies in RK and RK , with a signi cance of about 4 . We can similarly discuss the hypothesis in which we allow for new physics in the electron sector, shown in the lower part of table 1. Three cases | CbBLSeML , CBSM bLeR and CbBRSeMR | are equally favoured by the t. However, only the operator ObLeL involving left-handed quarks and electrons can explain the observed anomalies with an order one Wilson coe cient since it dominates the new-physics corrections to both RK and RK , see eqs. (2.10), (2.13). As before, we nd a statistical preference with respect to the SM case at the level of about 4- . To simplify the comparison with the existing literature, we show in In conclusion, the piece of information that we learn from this simple t is quite sharp: by restricting the analysis to the selected subset of `clean' observables RK , RK and ), not much a ected by large theoretical uncertainties, we nd a preference for the presence of new physics in the observed experimental anomalies in B decays. In particular, the analysis selects the existence of a new neutral current that couples lefthanded b, s quarks and left-handed muons/electrons as the preferred option. E ective four-fermions operators that couple left- or right-handed b, s with righthanded electrons are also equally preferred at this level of the analysis, but they require larger numerical values of their Wilson coe cients. Needless to say, this conclusion, although already very signi cant, must be supported by the result of a more complete analysis that accounts for all the other observables related to B decays, and not included in the `clean' set used in this section. We shall return to this point in section 3.2. Before moving to the t with the `hadronic sensitive' observables, we perform several two-parameter ts using only `clean' observables. We show our results in gure 3. Allowing for new physics in muons only, the combined best- t regions are shown as yellow contours. Since there are few `clean' observables, we turn on only two new-physics coe cients in each plot, as indicated on the axes. We also show, as rotated axes, the usual C9 and C10 coe cients. We see that the key implications mentioned in section 2.2 are con rmed by this t, although here wider regions in parameter space are allowed. In the upper plot of gure 3 we show the results for new physics in the operators involving left-handed muons, CbL L and CbR R : both coe cients are xed by the `clean' data. Operators involving right-handed muons, on the other hand, do not lead to good ts. A good t is obtained by turning on only CbL L , although uncertainties do not yet allow to draw sharp conclusions. We conclude this section with a comment on the size of the theoretical uncertainties in the presence of New Physics. While there is a consensus on the small error of the Standard Model predictions, in the presence of New Physics the \clean" observables have a larger theoretical error, barring the special case where new physics violate avour universality { 9 { SM 2 best SM `clean' `HS' 2 best all 1:30 0:30 0:14 all 0:99 3:46 0:02 0:94 1:62 1:17 0:11 0:33 0:23 0:08 0:97 `clean' 2:31 1:21 4:23 6:10 0:39 0:21 4:66 6:56 1:01 1:68 0:40 1:03 0:04 0:29 0:61 0:18 `HS' 0:69 0:39 0:33 2:83 0:29 1:55 3:52 2:70 all 1:07 1:55 0:02 0:59 0:00 0:25 0:48 0:04 all 1:30 0:70 2:81 4:05 0:30 0:25 2:65 4:43 4:1 1:2 0:2 0:8 4:1 4:3 0:3 4:2 4:6 2:1 1:3 1:7 0:3 0:9 0:7 0:5 all 6:2 0:9 1:0 1:2 all 3:5 3:6 0:1 2:5 `clean' 1:27 0:64 0:05 `clean' 1:72 5:15 0:085 coe . CBSM bL L CBSM bL R CbBRSML CbBRSMR coe . CbBLSeML CbBLSeMR CbBRSeML CbBRSeMR RK , and BR(Bs ! + nd in appendix A. ), or only the `Hadronic Sensitive' observables (denoted by `HS' in the while maintaining the same chiral structure of the SM (mostly LL at large enough q2). As shown in gure 2, away from the Standard Model our errors are still of a few percent, in agreement with refs. [35, 36]. However, other groups [37] nd a much larger theoretical error in the presence of New Physics, due to a more conservative treatment of the form factor uncertainties.4 Therefore, we warn the reader that the statistical signi cance quoted in our ts may be smaller with a di erent treatment of the error. We didn't take into account another important source of error: QED radiative corrections, calculated in ref. [3]. These are of the same order or larger than the hadronic uncertainties on RK , RK in the Standard Model as predicted by Flavio. We did the 4We thank Joaquim Matias for enlightening discussions about this point. New physics in the muon sector (Vector Axial basis) coe . all all SM 2 best exercise of in ating our hadronic error by a factor of 3, nding indeed a larger error away from the Standard Model, but still of the same order of the QED corrections. 3.2 Fit to the `hadronic sensitive' observables In order to perform a global t using the `hadronic sensitive' observables we use the public code Flavio [32]. Theoretical uncertainties are dominant, and it is di cult to quantify them. We rst take theoretical uncertainties into account using the `FastFit' method in the Flavio code with the addition of all the included nuisance parameters. With this choice, the SM is disfavoured at about 5 level. Given that most `hadronic sensitive' observables involve muons (detailed measurements are much more di cult with electrons), we present a simple 2 of the 4 Wilson coe cients involving muons. This is a simple useful summary of the full analysis. In this approximation, the `hadronic sensitive' observables determine the 4 muon Wilson coe cients as5 0:07 1 0:25 0:25 0:74 CC 1 0:50 CC 0:03 The uncertainties can be rescaled by factors of O(1), if one believes that theoretical uncertainties should be larger or smaller than those adopted here. The global t of `hadronic sensitive' observables to new physics in the 4 muon coe cients is also shown as red regions in gure 3. The important message is apparent both from 5In general, within the Gaussian approximation, the mean values i, the errors i and the correlation matrix ij determine the 2 as 2 = Pi;j(Ci i)( 2 )ij1(Cj j), where ( 2)ij = i ij j . HJEP09(217) Fit to the new-physics contribution to the coe cients of the 4 muon operators (b PX s)( PY ), showing the 1; 2; 3 contours. The yellow regions with dotted contours show the best t to the `clean' observables only; due to the scarcity of data, in each plot we turn on only the two coe cients indicated on its axes. The red regions with dashed contours show the best global t to the `hadronic sensitive' observables only, according to one estimate of their theoretical uncertainties; in this t, we turn on all 4 muon operators at the same time and, in each plot, we marginalise over the coe cients not shown in the plot. The green regions show the global t, again turning on all 4 muon operators at the same time. In gure 5 we turn on the extra 4 electron operators too. (vertical axis): left-handed in the left panel, and right-handed in the right panel. Regions and contours have the same meaning as in gure 3: `clean' data can be tted by an anomaly in muons or electrons; `hadronic sensitive' data favour an anomaly in muons. the gure and from eq. (3.1): `hadronic sensitive' observables favour a deviation from the SM in the same direction as the `clean' observables, i.e. a negative contribution CBSM 1 to the Wilson coe cient involving left-handed quarks and muons. `Clean' observables and `hadronic sensitive' observables | whatever their uncertainty is | look consistent and bL L favour independently the same pattern of deviations from the SM. 3.3 Global t We are now ready to combine `clean' and `hadronic sensitive' observables in a global t, using both the Flavio and Fart codes. The result is shown as green regions in gure 3, assuming that new physics a ects muons only. The global t favours a deviation in the SM in CbBLSML , and provides bounds on the other new-physics coe cients. Using the Gaussian approximation for the likelihood of the muon coe cients, the global t is summarized as 0:26 1 0:17 An anomaly in muons is strongly preferred to an anomaly in electrons, if we adopt the default estimate of the theoretical uncertainties by FLAVIO. This is for example shown in gure 4, where we allow for a single operator involving muons and a single operator involving electrons. In view of this preference, and given the scarcity of data in the electron sector, we avoid presenting a global t of new physics in electrons only. We instead perform a global combined t for the muon and electron coe cients (which should be interpreted with caution, given that `hadronic sensitive' observables are dominated by theoretical uncertainties). We nd the result shown in gure 5, which con rms that | while electrons can be a ected by new physics | `hadronic sensitive' data favour an anomaly in muons. The latter result has been obtained by a global Bayesian t to the observables listed in tables 6, 7 in addition to the clean observables. We used the Flavio code to calculate the likelihood, and we sampled the posterior using the Emcee code [38], assuming for the 8 Wilson coe cients (at the scale 160 GeV) a at prior between 10 and 10. In this global t, we choose to marginalize over 25 nuisance parameters only, to keep computational times within reasonable limits. The nuisances (form factors related to B decays) are selected in the following way. For each observable, we de ne theoretical uncertainties due to changing each nuisance within its uncertainty, keeping the others xed at their central values. Then, we choose to marginalize only over the parameters which give a theoretical uncertainty larger than the experimental error on the observable. 4 Theoretical interpretations We now discuss di erent theoretical interpretations that can accommodate the avour anomalies. We start with the observation that an e ective (s can be mediated at tree level by two kinds of particle: a Z0 or a leptoquark. Higher-order induced mechanisms are also possible. These models tend to generate related operators cbLbL (s PLb)2 + c L ( PL )( PL ) ; (4.1) and therefore one needs to consider the associated experimental constraints. The rst operator a ects Bs mass mixing for which the relative measurements, together with CKM ts, imply cbBLSbML = ( 0:09 0:08)=(110 TeV)2 , i.e. the bound jcbBLSbML j < 1=(210 TeV)2 [11, 39]. The second operator is constrained by CCFR data on the neutrino trident cross section, yielding the weaker bound jcBLSM j < 1=(490 GeV)2 at 95% C.L. [40]. Furthermore, new physics that a ects muons can contribute to the anomalous magnetic moment of the muon. Experiments found hints of a possible deviation from the Standard Model with a = (24 Models featuring extra Z0 to explain the anomalies are very popular, see the partial list of references [42{61]. Typically these models contain a Z0 with mass MZ0 savagely coupled to [gbs(s PLb) + h.c.] + g L ( PL ) : (4.2) The model can reproduce the avour anomalies with cbL L = in gure 6a. At the same time the Z0 contributes to the Bs mass mixing with cbLbL = gbsg L =MZ20 as illustrated gb2s=2MZ20 . The bound from MBs can be satis ed by requiring a large enough g L in order to reproduce the b ! s`+` anomalies. Left-handed leptons are uni ed in a SU(2)L doublet L = ( L; `L), such that also the neutrino operator c L = However the latter does not yield a strong constraint on g L . g2L =MZ20 is generated. .0 0 .0 0 .0 3 .0 0 .5 1 − SBMbeLL 3 CbBLSeML 4 . 2. 1 at tree level: a Z0 or a lepto-quark, scalar or vector. Another possibility is for the Z0 to couple to the 3-rd generation left-handed quarks with coupling gt and to lighter left-handed quarks with coupling gq. The coupling gbs arises as gbs = (gt gq)(UQd )ts after performing a avour rotation UQd among left-handed down quarks to their mass-eigenstate basis. The matrix element (UQd )ts is presumably not much larger than Vts and possibly equal to it, if the CKM matrix V = UQu U Qyd is dominated by the rotation among left-handed down quarks, rather than by the rotation UQu among Unless gq = 0, the parameter space of the Z0 model gets severely constrained by combining perturbative bounds on g L . In addition the LHC bounds on pp ! Z0 ! can be relaxed by introducing extra features, such as a Z0 branching ratio into invisible A characteristic feature of Z0 models is that they can mediate e ective operators involving di erent chiralities. In fact, gauge-anomaly cancellations also induce multiple chiralities: for example a Z0 coupled to L L is anomaly free [44], where the Le contribution is avoided because LEP put strong constraints on 4-electron operators. The chiralities inanomalies can be determined trough more precise measurements volved in the b ! s`+` of `clean' observables such as RK and RK . 4.2 Models with lepto-quarks The anomalous e ects in b ! s`+` transitions might be due to the exchange of a LeptoQuark (LQ), namely a boson that couples to a lepton and a quark. Concerning lepton avour, in general a LQ can couple to both muons and electrons. However, simultaneous sizeable couplings of a LQ to electrons and muons generates lepton avour violation which is severely constrained by the time-honoured radiative decay typically assumes that LQs couple to either electrons or muons. (Here sizeable means an e ect which has an impact on the anomalous observables). The coupling to muons allows ! e . For this reason one to t the anomalies in b ! s + distributions, as well as the RK and RK =e ratios. The gauge quantum numbers of scalar LQs select a speci c chirality of the SM fermions involved in the new Yukawa couplings, and thereby generate a unique characteristic operator in the e ective Lagrangian in the chiral basis of eq. (2.1), as illustrated in gure 6b. The correspondence is given by Coe cient Lepto-Quark Yukawa couplings CbL`L CbL`R CbR`L CbR`R S3 R2 ~ R2 ~ S1 (3; 3; 1=3) y QL S3 + y0 QQ S3y + h.c. (3; 2; 7=6) y U L R2 + y0 QE R2y + h.c. where ` can be either an electron or a muon. In parentheses we report the SU(3) SU(2)L U(1)Y gauge quantum numbers, and we follow the notations and conventions from ref. [63] for LQ names. Q; L (U; D; E) denote the left-handed (right-handed) SM quarks and leptons. Given that each LQ mediates e ective operators with a given chirality, we can draw conclusions from our one parameter ts of the b ! s`+` anomalies of table 1. Assuming new physics in the muon sector, the measurement of RK selects a unique scalar leptoquark: S3 (3; 3; 1=3), which is a triplet under SU(2)L. It is remarkable that this is obtained with just the information coming from `clean' observables while the inclusion of the remaining observables (with our speci ed treatment of the errors) reinforces this hypothesis. The explanation of the anomalies in terms of S3 has been rstly proposed after the measurement of RK in ref. [10] switching on only those couplings needed to reproduce the e ect. In ref. [64] the LQ has been identi ed as a pseudo-Goldstone boson associated to the breaking of a global symmetry of a new strongly coupled sector [65]. In refs. [64, 65] it has also been suggested that a rationale for the size of the various avour couplings could be dictated by the mechanism of partial compositeness [66]. Another motivated pattern of couplings has been suggested in ref. [67] using avour symmetry. Also ref. [68] makes use of S3 as mediator of the b ! s + transition. A potential issue with S3 is the danger of extra renormalizable couplings with diquarks (denoted collectively by y0 in the Lagrangians above) which may induce proton decay. Baryon number conservation has to be invoked to avoid this issue. Motivated by this, in refs. [69, 70], the LQ R~2 (which respects the global symmetry U(1)B accidentally at the renormalizable level) has been considered leading to the prediction RK > 1, which is now disfavoured by the LHCb data. The other two options S~1 and R2 were already disfavoured after the measurement of RK [10, 71]. The situation is di erent if LQs couple to electrons, rather than to muons, such that only the anomalies in the `clean' observables can be reproduced. `Clean' observables can be reproduced by all chiralities, with the only exclusion of CbR`L , which is mediated by the R~2 LQ. From the t, we notice that the S~1 and R2 LQs can only t the anomalies by giving a large contribution to the Wilson coe cients, comparable to the SM contributions: this happens because these LQs couple to right handed electrons, with little interference with the SM. One the other hand, S3 couples to left-handed leptons, such that the sizeable interference with the SM allows to reproduce the observed anomalies with a smaller new physics component. We brie y comment on the possible interpretation of a LQ as a supersymmetric particle in the MSSM. The only sparticle with the same gauge quantum numbers as a LQ is the left-handed squark Q~ R~2. However, even if it has R-parity violating interactions, this LQ gives the wrong correlation between RK and RK , disfavouring the supersymmetric interpretation of the anomalies. We move now to the discussion of the exchange of vector LQs at tree level, illustrated in gure 6c. There are 3 cases: U3 (3; 3; 2=3), V2 (3; 2; 5=6) and U1 (3; 1; 2=3). Their relevant interactions are: LU3 = y Q LV2 = y D LU1 = y Q L U3 + h.c. L V2 + y0 Q L U1 + y2 D E V2 + y00 Q E U 1 + h.c. U V2y + h.c. (4.4a) (4.4b) (4.4c) Spin Quantum Clean observables Clean observables Number new physics in e new physics in observables X X X S3 R2 ~ R2 ~ S1 U3 V2 U1   0 0 0 0 1 1 1 (3; 3; 1=3) (3; 2; 7=6) dressed by further new composite dynamic contributions. . In Fundamental Composite Higgs models these diagrams will be The vector LQ V2 and U1 can contribute to the anomalous observables trough multiple chiral structures. In general, if both y and y0 are sizeable, dangerous scalar operators may be generated. If one of the two couplings dominates, we can again restrict to our one parameter t, with the following correspondence: CbL`L can be generated by U3; CbL`R or CbR`L can be generated by V2; CbL`L or CbR`R can be generated by U1. Similar phenomenological considerations to explain the B-meson anomalies as in the case of the scalar LQ apply, we summarise the relevant options in table 3. Models featuring vector LQs models in order to explain the avour anomalies appeared recently in the literature [72{75], typically as new composite states. The presence of these states signals that the theory in isolation is non-renormalizable, meaning that loop e ects of the vectors are UV divergent, for a recent re-discussion see ref. [76]. Naive dimensional analysis shows that one-loop contributions to physics observables such MBs might be problematic. A careful study of this topic is a model dependent issue and it requires extra information on the UV embedding of the LQ in a complete theory. 4.3 The RK anomaly can be reproduced by one loop diagrams involving new scalars S and new fermions F with Yukawa couplings to SM fermions that allow for the Feynman diagram on the left in gure 7 [39, 77] | see also ref. [78]. In this particular example, one generates an operator involving left-handed SM quarks and leptons, denoted respectively by Q and L. The needed extra Yukawa coupling to the muon must be large, yL 1:5. This also explains why the MSSM does not allow for an explanation of the RK ; RK anomalies: a possibile box diagram containing Winos and sleptons predicts yL g2 1:5, where g2 is the SU(2)L gauge coupling. In section 4.4 we will consider renomalizable models of composite dynamics featuring extra elementary scalars, where we will show that the extra particles S and F can be identi ed with the constituents of the Higgs boson, and that their Yukawa couplings are the source of the SM Yukawa couplings, giving rise to a avour structure similar to the SM structure. Then, the one loop Feynman diagrams of gure 7 are dressed by the underlying composite dynamic. Fundamental composite Higgs Y = 1=2 and Y = 0. couplings Models in which the Higgs is a composite state are prime candidates as potential source of new physics in the avour sector [79{81]. Fundamental theories with a Higgs as a composite state that are also able to generate SM fermion masses appeared in ref. [82]. These theories feature both techni-scalars S and techni-fermions F .6 In models of fundamental composite Higgs: i) it is possible to replace the standard model Higgs and Yukawa sectors with a composite Higgs made of techni-particles; ii) the SM fermion masses are generated via a partial compositeness mechanism [66] in which the relevant composite techni-baryons emerge as bound states of a techni-fermion and a techni-scalar. The composite theory does not address the SM naturalness issue and it is fundamental in the sense that it can be extrapolated till the Planck scale [82]. Having a fundamental theory of composite Higgs, we use it to investigate the avour anomalies. The gauge group and the eld content of a simple model are summarised in table 4. Here the new strong group is chosen to be SU(NTC) with NTC = 3 and we list the gauge quantum numbers of the new vectorial fermions and scalars that can provide a composite Higgs with Yukawa couplings to all SM fermions L; E; Q; U; D. Three generations of techni-scalars are introduced in order to reproduce all SM fermion masses and mixings, while having a renormalizable theory with no Landau poles below the Planck scale. The hypercharge Y of the FL fermion is free. We assume the minimal choices Y = 1=2, The matrices of SM Yukawa couplings y`, yu, yd are obtained from the TC-Yukawa LY = yL LFLSEc + yE EFN SEc + (yD DF Nc + yU U F Ecc )SDc + yQ QFLSDc + h.c. (4.5) c becomes strong, gTC 4 =pNTC, at the scale TC as y` yLyET =gTC, yd yQyDT=gTC, yu yQyUT =gTC, where the new gauge coupling gTC gTCfTC, forming composite particles 6Composite theories including TC scalars attempting to give masses to the SM quarks appeared earlier in the literature [83{88] for (walking) TC theories that didn't feature a light Higgs. name spin generations SU(3)c SU(2)L FL c FN SEc SDc FEc 1=2 1=2 1=2 0 0 1 1 1 3 3 1 1 1 1 3 2 1 1 1 1 Y Y Y = Y 1=2 F Nc ; FL; FEc with conjugated gauge quantum numbers such that the fermion content is vectorial are implicit. Names are appropriate assuming the value Y = 1=2 for the hypercharge Y of FL; however generic values are allowed. HJEP09(217) with mass of order TC and condensates hF F ci fT2C TC. In view of the resulting breaking of the TC-chiral symmetry, the Higgs doublet H (identi ed with pseudo Goldstone bosons of the theory) and other composite scalars remain lighter. Lattice simulations [89{ 91] of the most minimal fundamental composite theories [92{94], without techni-scalars, have demonstrated the actual occurrence of chiral symmetry breaking with the relevant breaking pattern, and furthermore provided the spectrum of the spin one vector and axial techni-resonances with masses mV = 3:2(5) TeV= sin and mA = 3:6(9) TeV= sin where is the electroweak embedding angle to be determined by the dynamics, that must be smaller than about 0:2. The TC-Yukawa couplings accidentally conserve lepton and baryon numbers (like in the SM) and TC-baryon number; depending on the value of Y the lightest TC-baryon can be a neutral DM candidate. We require TC-scalar masses and TC-quartics to respect avour symmetries so that the BSM corrections to avour observables abide the experimental bounds. At one loop7 in the TC-Yukawas one obtains the following operators involving 4 SM fermions f;f0 L;E;Q;U;D (yfy yf )ij (yfy0 yf0 )i0j0 (fi fj00 )(fi00 X 2 2 gTC TC fj ) + (yLyyE)ij (yQyDT)i0j0 (Li 2 2 gTC TC Qi0 )(Ej Dj0 ): (4.6) All SM fermions and their chiralities are involved. These operators are phenomenologically viable if the fundamental TC-Yukawa couplings have the minimal values needed to reproduce the SM Yukawa couplings: yE yL pgTCy`, and similarly for quarks. However, when the TC-Yukawas (say, yL) are enhanced the impact on new physics is also enhanced. The observed SM Yukawa couplings are reproduced when the corresponding TC-Yukawas (say, yE) are reduced. Consequently, in this scenario new physics manifests prevalently in leptons of one given chirality. Because data prefer new physics to emerge prevalently in left-handed muons it is natural to consider here an enhanced muon coupling yL and a correspondingly reduced right-handed yE. 7The loop analysis, in the composite scenario, is merely a schematic way to keep track of the relevant factors stemming from the TC dynamics when writing SM four-fermion interactions. cbL L cbLbL a gZ L NTC One-loop result (yLyLy) (yQyQy)bs (4 )24M 2 F (x; y) NTC (4(yQ)2y8QyM)b22s F (x; x) FL FL m2 (yLyLy) NTC (4 )2M 2 FL (2Y 1)F7(y) + 2Y F7(1=y) y MZ2 (yLyLy) NTCg2 2(4 )2(1 2s2W)M 2 F9(Y; y) FL Non-perturbative estimate (yLyLy) (yQyQy)bs g2 fermion and TC-scalar estimate (second column) along with their NDA analysis counterpart (third column). The NDA result for a modi es in the presence of a TC-fermion condensate to m v(yLyET) =gTC 2TC. = Λ µ ->ν µ+µ δ µ Δ µ Δ † ) ( Higgs models. The model generates an e ective operator that can simultaneously account for both RK and RK , so only RK is plotted. We summarise in table 5 the coe cients of the relevant avour-violating e ective operators, both within a naive one-loop approximation (adopting the results from refs. [11, 39]) and Naive Dimensional Analysis (NDA) in the composite theory. We de ned x = SD FL M 2c =M 2 , y = M 2c =M 2 SE FL and the loop functions F (x; y) = F7(y) = (1 y 3 1 x)(1 y) 6y2 + 6y ln y + 3y + 2 12(1 y)4 x2 ln x x)2(x ; + y) (1 y2 ln y y)2(y x) (4.7a) (4.7b) G9(y) = 2y3 + 6 ln y + 9y2 36(y 1)4 ; 7 ; F9(Y; y) = s2W(2Y 1)F9(y) (1 s2W(2Y + 1))G9(y) (4.7c) (4.7d) (4.7e) that equal F (1; 1) = 1=3, F7(1) = F~7(1) = 1=24, F9(1) = 1=24, G9(1) = 1=8, for degenerate masses. The latter entry in table 5 is the correction to the Z coupling to left-handed muons g L , written in terms of the weak mixing angle sW = sin W. The LEP bound at the Z pole is j gZ L j 0:8% g2 at 2 [95]. We can neglect TC-penguin diagrams [11]. We can always work in a basis where yL = diag(yLe ; yL ; yL ) is diagonal, such that (yLyLy) = yL2 . Figure 8 shows that, in order to reproduce the b ! s`+` anomalies and the muon g 2 anomaly, a relatively large Yukawa coupling yL 1:5 is needed, like in models with perturbative extra fermions and scalars. In the composite model such values of TC-Yukawa coupling have natural sizes. This is corroborated by a RGE analysis for yL that features an extra contribution involving the gTC gauge coupling: (4 ) L 2 3 NT2C 2NTC 1 gT2CyL ; In the presence of the rst term only, setting NTC = 1, the Yukawa coupling grows with energy. Perturbativity up to a scale max implies jyL j < 2 =pln( max= TeV), with jyL j MPl. In the presence of the second term a larger yL gTC is compatible with the requirement that all couplings can be extrapolated up to the Planck scale. This is similar to how the strong coupling g3 allows for yt 1 in the SM. In the fundamental composite Higgs model, the large couplings yt and yL contribute to the prediction for the Higgs mass parameter in terms of Lepton- avour violation is absent as long as the yE matrix is diagonal in the same basis where yL is diagonal. Then y` = yL` yE` =gTC for ` = fe; ; g. In general, there can be a avour-violating mixing matrix in the lepton sector. In particular, the mixing angle e generates ! e , but only when e ects at higher order in the Yukawa couplings are included [82]. Focusing on e ects enhanced by the large coupling yL one has BR( ! e ) 4 emv6yE2e yL6 e2 gT6Cm2 4 TC yE2e yL6 e2 2 TeV TC 4 that in the electron sector too one has a large yLe and a small yEe ye The experimental bound BR( ! e ) < 0:6 10 12 [96] is satis ed even for e 1 provided Finally, we mention an e ect that can enhance the new-physics correction to some avour-violating operators. While the fermion condensates induced by the strong dynamics are known, the scalar condensates are not known (although perhaps they are computable, for example by dedicated lattice simulations). Possible scalar condensates could break the accidental avour symmetry among scalars, leading to extra lighter composite pseudoGoldstone bosons. The state made of SEc SDc behaves as a lepto-quark: if light it would mediate at tree level some e ective operators, analogously to the S~1 lepto-quark considered in section 4.2. (4.8) (4.9) We found that the new measurement of RK together with RK favours new physics in left-handed leptons. Furthermore, adding to the t kinematical b ! s + (a ected by theoretical uncertainties), one nds that they favour similar deviations from distributions the SM in left-handed muons. However, even if the experimental uncertainties on RK , RK will be reduced, a precise determination of the new-physics parameters will be prevented by the fact that these are no longer theoretically clean observables, if new physics really a ects muons di erently from electrons. We next discussed possible theoretical interpretations of the anomaly. One can build models compatible with all other data: One extra Z0 vector can give extra new-physics operators that involve all chiralities of SM leptons. The simplest possibility motivated by anomaly cancellation is a vectorial coupling to leptons. However, unless the Z0 is savagely coupled to bs quarks, a Z0 coupled to ss and bb is disfavoured by pp ! Z0 ! contraints. searches at LHC and other One lepto-quark tends to give e ects in muons or electron only (in order to avoid large avour violations), and only in one chirality. One can add extra fermions and scalars such that they mediate, at one loop level, the desired new physics. Their Yukawa coupling to muons must be larger than unity. While the e ective 4-fermion operators that can account for the b ! s`+` anomalies need to be suppressed by a scale 30 TeV, the actual new physics can be at a lower scale, with obvious consequences for direct observability at the LHC and for Higgs mass naturalness. Acknowledgments We thank Marina Marinkovic, Andrea Tesi and Elena Vigiani for useful discussions. We thank Javier Virto for further useful comments. The work of A.S. is supported by the ERC grant NEO-NAT, the one of F.S. is partially supported by the Danish National Research Foundation Grant DNRF:90, and the one of G. D'A. thanks Andrea Wulzer and the Lattice QCD group of CERN, and in particular Agostino Patella, for sharing their computing resources. R.T. is supported by the Swiss National Science Foundation under the grant CRSII2-160814 (Sinergia). R.T. thanks INFN Sezione di Genova for computing resources, Alessandro Brunengo for computing support, and David Straub for clari cations about the Flavio program. A List of observables used in the global t In table 6 and 7 we summarize the observables used in addition to the `clean' observables. All bins are treated in the experimental analyses as independent, even if overlapping. It is Observable hFLi(B0 ! K hS3i(B0 ! K hS4i(B0 ! K hS5i(B0 ! K hS7i(B0 ! K hS8i(B0 ! K hS9i(B0 ! K hAF B i(B0 ! K hP1i(B0 ! K hP50 i(B0 ! K hFLi(B0 ! K hS3i(B0 ! K hS4i(B0 ! K hS5i(B0 ! K hS7i(B0 ! K hS8i(B0 ! K hP1i(B0 ! K hP40 i(B0 ! K hP50 i(B0 ! K hP60 i(B0 ! K hP80 i(B0 ! K ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Angular observables LHCb B ! K clear that a correlation should exists between measurements in overlapping bins, however this is not estimated by the experimental collaborations. For this reason we include in our t the measurements in all relevant bins, even if overlapping, without including any correlation beyond the ones given in the experimental papers. Notice that, for instance in the case of the LHCb analysis [9], the result in the bin [1:1; 6] GeV2 has a smaller error than the measurements in the bins [1:1; 2:5]; [2:5; 4]; [4; 6] GeV2, even when the information from these three bins is combined. In fact, we veri ed that the bin [1:1; 6] GeV2 has a stronger impact on our ts than the three smaller bins. This shows that even if the measurements are potentially largely correlated, the largest bin dominates the t, so that the e ect of the unknown correlation becomes negligible. Observable ddq2 BR(B ! K ddq2 BR(B0 ! K ddq2 BR(B ! Kee) ddq2 BR(B ! K ddq2 BR(B0 ! K ddq2 BR(Bs ! ddq2 BR(B ! Xs ddq2 BR(B ! Xsee) ddq2 BR(B ! Xsll) ddq2 BR(B ! Xsll) ) ) ) ) ) ) Branching ratios LHCb B ! 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Guido D’Amico, Marco Nardecchia, Paolo Panci, Francesco Sannino, Alessandro Strumia, Riccardo Torre, Alfredo Urbano. Flavour anomalies after the R K ∗ measurement, Journal of High Energy Physics, 2017, 10, DOI: 10.1007/JHEP09(2017)010