Rare decays at LHCb

Flavour-changing neutral-current processes, such as $b\to s \ell^+\ell^-$, are forbidden at tree level in the Standard Model and hence might receive comparatively large corrections from new particles. This document highlights recent measurements from LHCb on $b\to s \ell^+\ell^-$ and purely-leptonic decays, including tests of lepton flavour universality and searches for lepton flavour violation.

1. Theoretical framework Rare b−hadron decays offer a rich phenomenology for indirect searches for New Physics (NP), making it possible to probe energies above the TeV scale.
b → s + − decays are flavour-changing neutral-current (FCNC) processes that can only occur at the loop level in the Standard Model (SM) [1].The presence of new particles can be spotted via precision measurements of theoretically well-known observables, such as b → sµ + µ − differential branching fractions, angular coefficients as well as ratios of b → s + − rates for different final state leptons.
Due to the non-perturbative nature of QCD interactions at the b−quark mass scale, an effective Hamiltonian can be built to factorise the high-and low-energy contributions in the limit M b M W : where G F is the Fermi constant and V CKM are the relevant CKM matrix elements.The longdistance (i.e.low-energy) contributions are described by the local operators O i , which are factorised from the short-distance contributions, encoded in the Wilson coefficients C i , at the energy scale λ.The SM local operators describing b → s + − processes mentioned in this document are the vector O A generic NP contribution at the scale Λ NP takes the form ∆H ef f = (c i /Λ 2 NP )O i , and can both alter the value of SM Wilson coefficients or introduce new operators.

Branching fractions
Differential branching fractions of various b → sµ + µ − decays have been measured at LHCb as a function of the dilepton invariant mass squared, q 2 .Some examples are shown in Fig. 1.    (10 8 ) in the high q 2 region.For the B ! Kµ + µ modes the region is defined as 15 22 GeV 2 /c 4 , while for B + !K ⇤+ µ + µ it is 15 19 GeV 2 /c 4 .Predictions are obtained using the form factors calculated in lattice QCD over the same q 2 regions.For the measurements, the first uncertainty is statistical and the second systematic.

Decay mode
Measurement Prediction   (10 8 ) in the high q 2 region.For the B ! Kµ + µ modes the region is defined as 15 22 GeV 2 /c 4 , while for B + !K ⇤+ µ + µ it is 15 19 GeV 2 /c 4 .Predictions are obtained using the form factors calculated in lattice QCD over the same q 2 regions.For the measurements, the first uncertainty is statistical and the second systematic.

Decay mode
Measurement Prediction 15.8 +3.2 2.9 ± 1.1 26.8 ± 3.6 measurements are all individually consistent with their respective predictions, they all have values below those. 9 in an S-wave configuration, are not considered in the fit and a systematic uncertainty is assigned.Integrated over the full q 2 range, signal yields, N ϕμ þ μ − , of 458 AE 12, 484 AE 13, and 1064 AE 28 are found from the simultaneous fit to the different datasets.Figure 1 distribution of the full data sample, integrated over q 2 and overlaid with the fit projections.Figures for the different data-taking periods are available as Supplemental Material [31].
The relative branching fraction measurement is affected by systematic uncertainties on the fit model and the efficiency ratio, where the latter is determined using SM simulation.A summary of the systematic uncertainties is provided in the Supplemental Material [31].The dominant systematic uncertainty on the absolute branching fraction [Eq.(1)] originates from the model used to simulate B 0 s → ϕμ þ μ − events (0.04 − 0.10 × 10 −8 GeV −2 c 4 ).The model depends on ΔΓ s , the decay width difference in the B 0 s system [32], and the specific form factors used.The effect of the model choice on the relative efficiency is assessed by varying ΔΓ s by 20%, corresponding to the difference in ΔΓ s between the default value [33] and that of Ref. [26], and by comparing the form factors in Ref. [34] with the older calculations in Ref. [35].The observed differences are taken as a systematic uncertainty.Other leading sources of systematic uncertainty arise from the limited size of the simulation sample (0.02 − 0.07 × 10 −8 GeV −2 c 4 ) and the omission of small background contributions from the fit model (0.01 − 0.04 × 10 −8 GeV −2 c 4 ).
The resulting relative and total branching fractions are given in Table I.In addition, the differential branching fraction is shown in Fig. 2, overlaid with SM predictions.These predictions are based on form factor calculations q 2 region between 1.1 and 6.0 GeV 2 =c 4 , the measured branching fraction of ð2.88 AE 0.22Þ × 10 −8 GeV −2 c 4 , lies 3.6σ below a precise SM prediction of ð5.37 AE 0.66Þ × 10 −8 GeV −2 c 4 , which uses both LCSR and LQCD calculations.A less precise SM prediction of ð4.77AE 1.01Þ × 10 −8 GeV −2 c 4 based on LCSRs alone lies 1.8σ above the measurement.To determine the total branching fraction, the branching fractions of the individual q 2 intervals are summed and corrected for the vetoed q 2 regions using ϵ q 2 veto ¼ ð65.47 AE 0.27Þ%.This efficiency is determined using SM simulation, and its uncertainty originates from the comparison of form factors from Refs.[34,35].The resulting branching fractions are Differential branching fractions of s → φµ + µ − decays measured at LHCb, compared with LCSR and lattice form factor predictions [2,3].
In both Run 1 and Run 2 data, measured rates appear to be consistently lower with respect to the SM predictions, whose uncertainty is dominated by the form factors. On the experimental side, the precision is limited by the knowledge of B → J/ψX branching fractions used for normalisation.For these reasons, the discrepancies cannot be attributed to anomalous muonic couplings: observables with reduced hadronic contributions should be investigated.

Purely-leptonic decays
Purely-leptonic final states allow for more precise theoretical predictions.For example, the SM branching fraction of B 0 s → µ + µ − decays depends only on the Wilson coefficient C 10 and on a single hadronic constant, known at ≈ 0.5% [4], leading to the precise prediction B(B 0 s → µ + µ − ) = (3.66 ± 0.14) × 10 −9 [5].The most-precise LHCb measurement, shown in Fig. 2 (left), agrees with this prediction within one standard deviation [6,7].A recent precise measurement from the CMS experiment confirms SM compatibility [8], and is reported in Fig. 2 (right) together with the latest LHC results.
with boundaries 0.0, 0.25, 0.4, 0.5, 0.6, 0.7, and 1.0; candidates having BDT < 0.25 are not included in the fit to the dimuon mass distribution.The mass distribution of the The BDT distributions of B 0 ðsÞ → μ þ μ − decays are calibrated using simulated samples which have been reweighted to improve the agreement with the data.The The mass distributions of the B 0 s → μ þ μ − and B 0 → μ þ μ − signals are described by two-sided Crystal Ball functions [42] with core Gaussian parameters calibrated from the mass distributions of s → μ þ μ − γ decays is described with a threshold function modeled on simulated events that were generated using the theoretical predictions of Refs.[14,15], convoluted with the experimental resolution.
The signal branching fractions are determined using the relation where N B 0 ðsÞ →μ þ μ − is the signal yield determined in the mass fit, N norm is the number of selected normalization decays dates (black dots) with BDT > 0.5.The result of the fit is overlaid and the different components are detailed:  Other fully-leptonic decays are analysed at LHCb, leading to the world's best upper limits summarised in Tab. 1.
Table 1.Upper limits on the branching fraction of various fully-leptonic decays at LHCb.A light scalar with m(a) = 1 GeV is assumed.

Channel
B upper limit (95% CL) Reference

Angular observables
The differential decay rate of a B meson to a vector meson (e.g. a K * 0 ) and two leptons can be described by the q 2 and the three angles − → Ω = (φ, θ K , θ L ), defined in Fig. 3 (left): i.e. a superposition of angular moments f i with coefficients I i .
The leading hadronic uncertainty cancels in the combination of angular coefficients where S 5 is a CP-averaged coefficient and F L represents the longitudinal polarisation fraction of the K * 0 [12].
The most-precise LHCb measurement of P 5 is reported in Fig. 3 (right) from the angular analysis of B 0 → K * 0 µ + µ − decays performed on Run 1 and a portion of Run 2 data [13].The results show local deviations in the fourth and fifth q 2 bins of 2.5 σ and 2.9 σ with respect to the SM prediction.The data can be explained with a ∼ 3 σ shift on the real part of the Wilson coefficient C 9 , although this deviation depends on the choice of the SM nuisance parameters such as form factors and long-distance contributions from the charmonium modes.
A similar trend in P 5 is observed in the LHCb analysis of B + → K * + µ + µ − decays [14], whereas a recent measurement of the observable F L on B 0 s → φµ + µ − decays agrees with the SM prediction [15].

Ratios of branching fractions
Lepton Flavour Universality (LFU), i.e. the fact that lepton electroweak couplings are equal, is a property of the SM which has been verified in several channels [16].However, the discrepancies highlighted in the previous sections prompted the verification of LFU in b−hadron decays.Penguin decays of b−hadrons can serve as LFU tests via measurements of the ratios of branching fractions: with H being a strange hadron.These observables are free from QCD contributions and are predicted in the SM with percent precision due to high-order QED effects [17]. ).
I System fully described by q 2 and three angles ⌦ = (cos ✓l, cos ✓K , )
Since LFU in J/ψ → + − decays is verified with per mille precision [16], the LHCb observables are built as double ratios with respect to the J/ψ resonant modes, in order to profit from the cancellation of detection differences between electrons and muons.In the case of B + → K + + − decays, the ratio is defined as: where resonant and rare modes are distinguished by their q 2 values.Fig. 4 shows the mass spectra of the electron and muon modes from the latest R K measurement at LHCb [18], yielding the most precise LFU test in this sector.Several other

NATURE PHYSICS
has the B + → K + μ + μ − yield and R K as fit parameters and the resonant decay mode yields incorporated as Gaussian-constraint terms.The resonant yields are determined from separate fits to the mass, m J/Ψ (K + ℓ + ℓ − ), formed by kinematically constraining the dilepton system to the known J/ψ mass 2 and thereby improving the mass resolution.
Simulated events are used to derive the two ratios of efficiencies needed to form R K using equation (2).Control channels are used to calibrate the simulation to correct for the imperfect modelling of the B + production kinematics and various aspects of the detector response.The overall effect of these corrections on the measured value of R K is a relative shift of (+3 ± 1)%.When compared with the 20% shift that these corrections induce in the measurement of r J/ψ , this demonstrates the robustness of the double-ratio method in suppressing systematic biases that affect the resonant and non-resonant decay modes similarly.
The systematic uncertainty (Methods) from the choice of signal and background mass-shape models in the fits is estimated by fitting pseudo-experiments with alternative models that still describe the data well.The effect on R K is at the 1% level.A comparable uncertainty arises from the limited size of the calibration samples, with negligible contributions from the calibration of the B + production kinematics and modelling of the selection and particle-identification efficiencies.Systematic uncertainties that affect the ratios of efficiencies influence the measured value of R K and are taken into account using constraints on the efficiency values.Correlations between different categories of selected events and data-taking periods are taken into account in these constraints.The combined statistical and systematic uncertainty is then determined by scanning the profile likelihood, and the statistical contribution to the uncertainty is isolated by repeating the scan with the efficiencies fixed to their fitted values.
The determination of the r J/ψ ratio requires control of the relative selection efficiencies for the resonant electron and muon modes and does not therefore benefit from the cancellation of systematic effects in the double ratio used to measure R K .Given the scale of the corrections required, comparison of r J/ψ with unity is a stringent cross-check of the experimental procedure.In addition, if the simulation is correctly calibrated, the measured r J/ψ value will not depend on any variable.The r J/ψ ratio is therefore also computed as a function of different kinematic variables.Even though the non-resonant and resonant samples are mutually exclusive as a function of q 2 , there is significant overlap between them in the quantities on which the efficiency depends, such as the laboratory-frame momenta of the final-state particles or the opening angle between the two leptons.This is because a given set of values for the final-state particles' momenta and angles in the B + rest frame will result in a distribution of such values when transformed to the laboratory frame.
The value of r J/ψ is measured to be 0.981 ± 0.020.This uncertainty includes both statistical and systematic effects, where the latter dominate.The consistency of this ratio with unity demonstrates control of the efficiencies well in excess of that needed for the determination of R K .In the measurement of the r J/ψ ratio, the systematic uncertainty is dominated by the imperfect modelling of the B + production kinematics and the modelling of selection requirements, which have a negligible impact on the R K measurement.No significant trend is observed in the differential determination of r J/ψ as a function of any considered variable.An example distribution, with r J/ψ determined as a function of B + momentum component transverse to the beam direction, p T , is shown in Fig. 3. Assuming that the observed r J/ψ variation in such distributions reflects genuine mis-modelling of the efficiencies, rather than statistical fluctuations, and taking into account the spectrum of the relevant variables in the non-resonant Data 9 fb -1 Total fit

NATURE PHYSICS
has the B + → K + μ + μ − yield and R K as fit parameters and the resonant decay mode yields incorporated as Gaussian-constraint terms.
The resonant yields are determined from separate fits to the mass, m J/Ψ (K + ℓ + ℓ − ), formed by kinematically constraining the dilepton system to the known J/ψ mass 2 and thereby improving the mass resolution.Simulated events are used to derive the two ratios of efficiencies needed to form R K using equation (2).Control channels are used to calibrate the simulation to correct for the imperfect modelling of the B + production kinematics and various aspects of the detector response.The overall effect of these corrections on the measured value of R K is a relative shift of (+3 ± 1)%.When compared with the 20% shift that these corrections induce in the measurement of r J/ψ , this demonstrates the robustness of the double-ratio method in suppressing systematic biases that affect the resonant and non-resonant decay modes similarly.
The systematic uncertainty (Methods) from the choice of signal and background mass-shape models in the fits is estimated by fitting pseudo-experiments with alternative models that still describe the data well.The effect on R K is at the 1% level.A comparable uncertainty arises from the limited size of the calibration samples, with negligible contributions from the calibration of the B + production kinematics and modelling of the selection and particle-identification efficiencies.Systematic uncertainties that affect the ratios of efficiencies influence the measured value of R K and are taken into account using constraints on the efficiency values.Correlations between different categories of selected events and data-taking periods are taken into account in these constraints.The combined statistical and sys-The determination of the r J/ψ ratio requires control of the relative selection efficiencies for the resonant electron and muon modes and does not therefore benefit from the cancellation of systematic effects in the double ratio used to measure R K .Given the scale of the corrections required, comparison of r J/ψ with unity is a stringent cross-check of the experimental procedure.In addition, if the simulation is correctly calibrated, the measured r J/ψ value will not depend on any variable.The r J/ψ ratio is therefore also computed as a function of different kinematic variables.Even though the non-resonant and resonant samples are mutually exclusive as a function of q 2 , there is significant overlap between them in the quantities on which the efficiency depends, such as the laboratory-frame momenta of the final-state particles or the opening angle between the two leptons.This is because a given set of values for the final-state particles' momenta and angles in the B + rest frame will result in a distribution of such values when transformed to the laboratory frame.
The value of r J/ψ is measured to be 0.981 ± 0.020.This uncertainty includes both statistical and systematic effects, where the latter dominate.The consistency of this ratio with unity demonstrates control of the efficiencies well in excess of that needed for the determination of R K .In the measurement of the r J/ψ ratio, the systematic uncertainty is dominated by the imperfect modelling of the B + production kinematics and the modelling of selection requirements, which have a negligible impact on the R K measurement.No significant trend is observed in the differential determination of r J/ψ as a function of any considered variable.An example distribution, with r J/ψ determined as a function of B + momentum component transverse to the beam direction, p , is shown in Fig. 3. Assuming that the observed   )

Conclusions
In recent years, several deviations with respect to the SM predictions have been observed in b → s + − decays at LHCb.Although individual significances do not exceed 2 − 3 standard deviations, a coherent pattern seem to emerge: further experimental investigation is required to clarify these anomalies.

Figure 2 :
Figure 2: Di↵erential branching fraction results for the B + !K + µ + µ , B 0 !K 0 µ + µ and B + !K ⇤+ µ + µ decays.The uncertainties shown on the data points are the quadratic sum of the statistical and systematic uncertainties.The shaded regions illustrate the theoretical predictions and their uncertainties from light cone sum rule and lattice QCD calculations.

2 2 . 9 ±Figure 2 :
Figure 2: Di↵erential branching fraction results for the B + !K + µ + µ , B 0 !K 0 µ + µ and B + !K ⇤+ µ + µ decays.The uncertainties shown on the data points are the quadratic sum of the statistical and systematic uncertainties.The shaded regions illustrate the theoretical predictions and their uncertainties from light cone sum rule and lattice QCD calculations.
samples, respectively.A mass resolution of about 22 MeV=c 2 is determined by interpolating the measured resolutions of charmonium and bottomonium resonances decaying into two muons.The radiative tails are obtained from simulation [43].Small differences in the resolution and tail parameters of the mass shape for the different BDT regions are taken into account.The mass distribution of the B 0 the corresponding branching fraction[44], and ϵ sig (ϵ norm ) is the total efficiency for the signal (normalization) channel.For each signal mode, the two single event sensitivities, α norm B 0 ðsÞ →μ þ μ − , are then averaged in a combined α B 0 ðsÞ →μ þ μ − taking the correlations into account.The fraction f dðsÞ indicates the probability for a b quark to fragment into a B 0 ðsÞ meson.The value of f s =f d has been measured by LHCb to be ffiffi ffi p FIG. 1. Mass distribution of the selected B 0ðsÞ → μ þ μ − candidates (black dots) with BDT > 0.5.The result of the fit is overlaid and the different components are detailed:B 0 s → μ þ μ − (red solid line), B 0 → μ þ μ − (green solid line), B 0 s → μ þ μ − γ (violet solid line), combinatorial background (blue dashed line), B 0 ðsÞ → h þ h 0− (magenta dashed line), B 0 → π − μ þ ν μ , B 0 s → K − μ þ ν μ , B þ c → J=ψμ þ ν μ ,and Λ 0 b → pμ − νμ (orange dashed line), and B 0ðþÞ → π 0ðþÞ μ þ μ − (cyan dashed line).The solid bands around the signal shapes represent the variation of the branching fractions by their total uncertainty.

Figure 2 .
Figure 2. Left: Dimuon invariant mass spectrum for the LHCb analysis of B 0 (s) → µ + µ − decays.Right: measurements of the B 0 s → µ + µ − branching fraction from LHC experiments compared to the SM prediction.

FIG. 2 .
FIG.2.Results for the CP-averaged angular observables F L , A FB , S 5 , and P 0 5 in bins of q 2 .The data are compared to SM predictions based on the prescription of Refs.[43,44], with the exception of the P 0 5 distribution, which is compared to SM predictions based on Refs.[73,74].

1 BFig. 2 |
Fig.2| Candidate invariant mass distributions.Distribution of the invariant mass m(J/ψ)(K + ℓ + ℓ − ) for candidates with electron (left) and muon (right) pairs in the final state for the non-resonant B + → K + ℓ + ℓ − signal channels (top) and resonant B + → J/ψ( → ℓ + ℓ − )K + decays (bottom).The fit projection is superimposed, with dotted lines describing the signal contribution and solid areas representing each of the background components described in the text and listed in the legend.Part.reco.refers to partially reconstructed B hadron decays.In the resonant-mode distributions, some fit components are too small to be visible.Uncertainties on the data points are statistical only and represent one standard deviation, calculated assuming Poisson-distributed entries.The y axis in each panel shows the number of candidates in an interval of the indicated width.

Fig. 2 |
Fig. 2 | Candidate invariant mass distributions.Distribution of the invariant mass m(J/ψ)(K + ℓ + ℓ − ) for candidates with electron (left) and muon (right) pairs in the final state for the non-resonantB + → K + ℓ + ℓ − signal channels (top) and resonant B + → J/ψ( → ℓ + ℓ − )K + decays (bottom).The fit projection is superimposed, with dotted lines describing the signal contribution and solid areas representing each of the background components described in the text and listed in the legend.Part.reco.refers to partially reconstructed B hadron decays.In the resonant-mode distributions, some fit components are too small to be visible.Uncertainties on the data points are statistical only and represent one standard deviation, calculated assuming Poisson-distributed entries.The y axis in each panel shows the number of candidates in an interval of the indicated width.

Table 2 .
LFU measurements at LHCb.The first uncertainty is statistical and the second systematic.