Status of rare exclusive meson decays in 2018
Abstract
This review discusses the present experimental and theoretical status of rare flavourchanging neutral current quark decays at the beginning of 2018. It includes a discussion of the experimental situation and details of the currently observed anomalies in measurements of flavour observables, including lepton flavour universality. Progress on the theory side, within and beyond the Standard Model theory is also discussed, together with potential New Physics interpretations of the present measurements.
Keywords: Flavour Physics, Rare Decays, Exclusive B Decays
Received Day Month Year
Revised Day Month Year
PACS numbers: 13.20.He 13.30.Ce 12.15.Mm 12.60.Jv
Contents
1 Introduction
The year 2018 marks a special point for the field of particle physics as it is not only the end of second period of datataking at the Large Hadron Collider(LHC) but also the startup of the Belle II experiment. In the field of flavour physics, a number of anomalies have been observed when comparing measurements to the Standard Model ( Standard Model (SM)) expectation; from which a coherent and conclusive picture emerges. Those anomalies could, if they persist, point towards certain New Physics (NP) models such as leptoquarks or bosons. In this review, we will give a comprehensive overview of the current stateoftheart of exclusive rare decays from both experimental and theoretical perspective. We will present the current experimental landscape with a focus of the observed anomalies and discuss potential new physics interpretations.
The LHCb experiment will continue to run for the next two decades and hence will be able to either confirm or rule out many of the presentday anomalies. With the imminent startup of the datataking at the Belle II experiment, these tensions will be independently crosschecked and further complementary measurements will be accessible.
As the progress in inclusive decays was limited after the end of the factories, this review focusses on exclusive measurements of rare decays of hadrons as these are excellent probes for many new physics scenarios. These rare decay processes are CKM, GIM and loopsuppressed, wherefore potential new physics effects can be large compared to the SM amplitude. Such indirect searches allow to probe NP models at much higher mass scales as are currently accessible through direct searches. In the past, exclusive decays with muons in the final state have been measured extensively by the LHC experiments, most notably by LHCb, as well as the factories. We review the status of these measurements, discuss their theoretical description in the SM and modelindependently in the presence of new physics. A special focus is set on the anomalies seen in angular distributions of and in the branching fraction measurements of all exclusive decays as well as to the tensions observed in tests of lepton flavour universality in this class of decays. Measurements of the type are not covered in this review.
On the theoretical side, the current state of the SM predictions for the discussed observables is presented. The anatomy of the amplitudes, and their dependence on various local and nonlocal hadronic matrix elements is discussed. Particular focus is set on recent developments for the nonlocal matrix elements. The modelindependent interpretation of the measurements is examined in the framework of the usual Effective Field Theory.
This review is structured as follows: Section 2 introduces the amplitudes of decays in an effective field theory approach. Section 3 discusses the experimental landscape with a focus on semileptonic decays, which is followed by a brief discussion of decays, purely leptonic and decays. The experimental part of the review closes with a brief discussion of decays.
The second part of the review, Section 4, examines the foundation for the SM predictions required in the first part of the review. Section 5 discusses then a combined interpretation of all presented measurements in the framework of global fits and possible interpretations of the observed patterns in NP frameworks.
The review closes with a discussion of the experimental outlook in Section 6.
2 Anatomy of the amplitudes
Within the SM and at the Born level there are no flavorchanging neutral currents (FCNC). As a consequence, any FCNCmediated quarkflavor transitions, such as , emerge only from virtual loop corrections, in which a is first emitted and subsequently reabsorbed. The emergence at the loop level naturally suppresses the rate at which these processes occur. In the SM, the unitarity of the CabibboKobayashiMaskawa (CKM) matrix introduces an additional suppression mechanism that renders transitions very rare . Moreover, descriptions of these rare decays in the SM are further complicated since they pose a multiscale problem: as weak decays they are mediated through the exchange of and bosons, which are much heavier than the remaining particles (with the exception of the quark). The use of an Effective Field Theory (EFT) helps our understanding of such multiscale dynamics. In the following, we will discuss the EFT used for predictions of rare decays below the electroweak breaking scale . Within the SM, this EFT captures the effects of the quark as well as the and bosons, which are nolonger dynamical degrees of freedom for scales .
The effective Lagrangian reads (see e.g. )
(1) 
where and denote the Lagrangians of the electromagnetic and strong interactions (with five quark flavours after integrating out the top quark), refers to the Fermi constant, and the last term captures the local effective operators with their effective couplings  or Wilson coefficients (WC) at the renormalisation scale . For convenience, the product of CKM matrix elements has been extracted from the definition of the WC.
The matrix elements of the effective operators need to be evaluated at a low scale , which minimises logarithms in the perturbative expansion of the matrix elements. Consequently, one requires the WC evaluated at . Problems arising from large logarithms are resummed through Renormalisation Group (RG) improved running. Contemporary analyses use resummation of QCDinduced large logarithms up to NexttoNexttoLeadingLogarithm (NNLL). This requires knowledge of the relevant anomalous dimensions at the fourloop level, and the matching conditions at the threeloop level . The electroweak effects to NLL, which are numerically subleading, are known only for a subset of the WC .
For a consistent treatment at the leading order in the electromagnetic coupling , all operators of the following set  usually called the SM basis  are required (see e.g. ). The SM basis consists of the currentcurrent operators ()
(2) 
the QCDpenguin operators
(3)  
the electromagnetic and chromomagnetic dipole operators
(4) 
and the semileptonic operators
(5) 
In the above, we abbreviate , , , and sums over run over all active quark flavours , , , , and . Through using , universal QED corrections at NLO are taken care of .
Searches for NP effects at energies smaller than have so far not discovered either new interactions or new particles. Assuming that no such lowmass fields exist, the EFT framework, which is necessary for accurate theory predictions, can also be used to systematically describe NP effects. To this end, the effective Lagrangian Eq. (2) has to be modified in the following way :

[]

the set of effective field operators is enlarged to include all operators allowed by field content and Lorentz symmetry up to a given mass dimension (typically mass dimension six),

their WC are assumed to be independent and uncorrelated parameters.
When limiting this enlarged set to only semileptonic operators, a basis of all operators up to and including mass dimension six can be formed by also including the chiralityflipped operators,
(6) 
the (pseudo)scalar operators
(7)  
and the tensor operators
(8) 
A complete and nonredundant set of dimensionsix operators, including their oneloop anomalous dimensions
in both QCD and QED is compiled in .
This bottom up approach probes modelindependently for deviations from
the SM: the WCs are sensitive to NP effects of new particles
that are too heavy to be produced directly. Rare semileptonic decays therefore
provide information that is complementary to the “direct” searches for new
interactions and particles that are carried out at the Large Hadron Collider.
The above basis is further reduced when one assumes a manifest invariance of NP effects
under the SM gauge group within a SMlike (i.e. linear ) Higgs model.
For the operators, this was explicitly
demonstrated in .^{b}^{b}b
Note, however, that a nonlinear representation
fully restores the basis of dimension six operators to the set introduced above .
The effective Lagrangian in Eq. (2) can be matched onto the SM Effective Field Theory (SMEFT)
(see e.g. ),
whose operators are manifestly invariant under the SM gauge group. The required matching
formulas are compiled in to leading nontrivial loop level.
Matching, basis transformations and RenormalizationGroupEquation (RGE) evolution can be conveniently
achieved through a variety of computational tools, including but not limited to:
the “DsixTools” Mathematica package ;
the “Wilson Coefficient Exchange Format” and its reference Python implementation ;
and the “Wilson” Python package .
Schematically, the matrix elements for all exclusive processes can now be expressed as
(9) 
where refers to the invariant mass squared of the dilepton pair. The refer to hadronic matrix elements of local currents as induced by the operators , while denotes matrix elements of timeordered products involving fourquark operators and the chromomagnetic operator together with the electromagnetic current. In the presence of NP effects in the semileptonic and radiative operators, the hadronic matrix elements remain unchanged. However, the coefficients multiplying the latter are then modified. For the complete anatomy of the amplitudes of decays in the presence of NP operators of mass dimension six we refer to . A similar study for is presented in .
3 Experimental measurements
In this section, we draw a picture of the current experimental status of semileptonic (including a brief discussion of ) decays, purely leptonic and radiative transitions. Hereby, we will discuss measurements of the various hadronic final states, e.g. kaon and states, giving access to a broad range of observables such as branching fractions, and isospin asymmetries as well as angular observables. Within this section, the LHCb measurements refer to the full Run 1 dataset corresponding to 3 collected during the years 2011 and 2012, the BaBar results were obtained on their full dataset of 424 and the Belle publications exploit their complete dataset of 711 if not otherwise stated. Presented results integrated over the whole range have been obtained by vetoing the charmonium resonances and interpolating over this vetoed region. It should be noted that the exact range to veto the charmonium resonances depends on the respective analysis.
3.1 Semileptonic decays
Particular interest was raised throughout the past years by various measurements of semileptonic decays, in which several tensions between the SM predictions and experimental measurements have been observed. These deviations are mostly in the range of two to three standard deviations and show a consistent pattern, which can be explained by lowering the WC with respect to its SM value as detailed further in Section 5.1.
3.1.1 decays
The differential branching fractions of the decays and versus the invariant mass squared of the dimuon pair, , were determined by the LHCb collaboration with the averaged isospin asymmetry defined as
(10)  
(11) 
with the partial widths and branching fractions of the corresponding decay channels and the ratio of to lifetimes . The analysis finds values individually compatible with the SM prediction. However, the entity of measurements lies systematically below the predictions as shown in Fig. 1. In the same analysis, the averaged isospin asymmetry has been measured and is depicted in Fig. 1. In a similar analysis, the asymmetry
(12) 
was determined under the assumption of no direct violation in the control mode to be , where the uncertainties are statistical and systematic, respectively , and this measurement is consistent with the SM prediction .
The double differential decay rate of decays is given by
(13) 
where the angle between the () and the oppositely charged kaon () of the () decay is denoted by , is the socalled flat term, and is the forwardbackward asymmetry of the dimuon system.
As the flavour of the neutral meson cannot be determined for the selfconjugate final state , the double differential decay rate is extracted as a function of the absolute value of as
(14) 
with the constraint of enforcing the expression to be positivedefinite for all values of . The results of and are compatible with the SM predictions and are shown in Fig. 2.
Interference effects between the short and longdistance contributions were studied in decays at low recoil . The LHCb collaboration reported the first observation of the decays and the subsequent decay with , which is determined under the assumption of lepton flavour universality and hence the second uncertainty stems from the known branching fraction. No significant signal was observed for the resonance, and an upper limit at C.L. is set. The mean and the width of the resonance were measured to be and , respectively, where the uncertainties comprise both statistical and systematic sources. The interference between the observed resonant and the nonresonant decay in the large region amounts to leading to renewed interest in previous estimates of quarkhadron duality violation for these processes as discussed in Section 4.2.
By assuming a model of hadronic resonances, the LHCb collaboration fits the differential branching fraction of to data. The branching fraction of the shortdistance component is determined by setting the and resonance amplitudes to zero, and is found to be , where the uncertainties are statistical (including the form factor uncertainties) and systematic. In addition, the phase difference between the shortdistance and the narrowresonance amplitudes in decays was determined by performing a fit to the mass of the dimuon pair; of which one possible solution is illustrated in Fig. 3. As the values of the phases are compatible with , the interference with the shortdistance contributions far from the pole masses is small.
3.1.2 decays
In addition to the aforementioned studies of and states, channels with a pair of mesons in the final
state have received particular attention over the past years as those decays show a rich phenomenology. Amongst these decays, events with a invariant mass
close to the vector resonance (892) are the subject of numerous studies . Here and in the following, we discuss
treating the (892) as a quasistable particle, and decays should be interpreted as the decay chain
if not otherwise
stated.
The asymmetry defined in Eq. (3.1.1) was measured by the
LHCb collaboration to be , where the uncertainties are statistical
and systematic, respectively , and the measurement
is found to be consistent with the SM expectation of a small asymmetry in this decay.
The differential branching fraction of a charged meson into a final state, , has been measured by the LHCb collaboration; the results are shown in Fig. 4 in several bins of . A study of the neutral decay mode was performed, which resulted in the measurement of the isospin asymmetry as defined in Eq. (3.1.1) illustrated in Fig. 4. As the differential branching fractions of the decay had been previously reported in , the values were not updated in until later in . The latter results are as well depicted in Fig. 5, and the analysis yields the most precise measurement to date of the averaged branching ratio
(15) 
with (in the bin ) where the uncertainties are statistical, systematic and from the uncertainty on the branching fraction of the normalisation channel . In the region and for the invariant mass range , the Swave component is measured to be , where the uncertainties are of statistical and systematic origin. Assuming the absence of highorder waves e.g. D and Fwaves, the pure Pwave component of the differential branching fraction has been determined for the first time. However, in previous analyses, the Swave fraction was not taken into account as is the case in a measurement of the differential branching fraction published by the CMS collaboration , whose results are shown in Fig. 5.
The fourdimensional differential decay rate of () decays is expressed as a function of the invariant mass of the lepton pair , and the angles , and , where refers to the angle between the and the () flight directions in the dilepton rest frame, and to the angle between the () and the () flight directions in the rest frame and corresponds to the angle between the planes defined by the dimuon and the kaon and pion in the rest frame . The fully differential decay rate is then given by
(16)  
(17) 
where the () terms depend on products of the spin amplitudes and the functions are the corresponding angular distribution functions^{c}^{c}cThe differential decay rate taking the full basis including scalar and tensor operators has been derived in . assuming an onshell meson . As the theoretical calculations define with respect to the negatively charged muon in both and decays, the angular distributions between theory and experiment differ. A translation scheme between the two conventions is given in . More common than expressing the differential decay rates as functions of and is using the averaged and asymmetric observables and as introduced in .
(18)  
(19) 
One can construct a complete set of observables with a reduced formfactor dependence in the large energy limit. They emerge from combinations of and , where is the most notable and hence relevant for further discussions in this review.
In the angular analysis of decays by the CMS collaboration on a dataset of recorded in 2012, both and were determined , by using a folded differential decay rate  an approach originally developed by the LHCb collaboration . After the folding, a multidimensional fit is performed, with the parameters of interest , and along with signal and background yields, whereas the values of the longitudinal polarisation of the meson, , and and the interference between S and Pwave, , have been fixed to values determined in a previous analysis performed on the same dataset . The results of and are shown in Fig. 6 and are consistent with the SM predictions and previous measurements.
The aforementioned folding technique has also been employed by the Belle collaboration to extract and from the full dataset for both and decays, as well as the combination of both leptonic final states . The results on and are shown in Fig. 7. Tensions between measurement and SM prediction are observed for for the muon final state in the region of , whereas the electron mode differs by in the very same region, leading to a combined tension of . In addition to the observables, the socalled observables are determined for the first time; any deviation of these observables from zero would be a clear sign for new physics . However in the Belle analysis, no deviation from zero is observed as can be seen in Fig. 7.
A more complete angular analysis was performed by the ATLAS collaboration on its 20.3 dataset from 2012 by exploiting four different folding schemes allowing to extract a set of four observables for each scheme . However neither nor can be measured with the chosen approach. The values for and , which are common to each folding scheme, have been compared between the four fits and have found to be consistent. The publication comprises results on , , and , from which and are shown in Fig. 8. The obtained results are consistent with the different SM predictions within less than three standard deviations.
In contrast to previous analyses, the LHCb collaboration has performed a full angular analysis, and has determined the set of angular observables by employing three different methods: by a maximum likelihood fit, by exploiting the angular principal moments , and in addition the zerocrossing points of and were determined from a fit to the decay amplitudes . The full angular analysis gives access to the correlation matrices, which provide crucial input for global fits to theoretical models. The results of the maximum likelihood and the method of angular principal moments are found to be compatible. As the maximum likelihood yields the most precise results, we restrict the discussion to this approach. However, we note in passing that the angular principal moments allow for smaller bin widths in and therefore provide more information of the shape of the angular observables compared to the maximum likelihood method. In this analysis performed by the LHCb collaboration, the fraction of the Swave component was taken into account. Neglecting theoretical correlations, the observables appear to be compatible with the SM predictions. The theory correlations can be displayed best in the space of the observables with the most notable deviation of the SM in as can be seen from Fig. 9: in the regions and , deviations of the measured values from the SM prediction of and can be observed, respectively. This confirms the tension seen in a previous LHCb analysis in the region , which had a local significance of .
A fit to the complete set of averaged observables, namely , and , as determined from the maximum likelihood fit is performed by the LHCb collaboration using the EOS software package for observables in the ranges below and . The fit result illustrated in Fig. 10 indicates a tension of between the measurements and the SM prediction of alone.
When the fully differential decay rate (see Eq. (3.1.2)) is integrated over and , the resulting expression depends on and , which is what the BaBar collaboration exploited to measure those parameters on their full dataset and to extract . The studied decay channels include not only and decays but also their charged counterparts. The results are summarised in Fig. d and most results are compatible amongst each other and with the SM prediction. In the low region, a tension is observed in between and decays as well as the SM prediction. In the same region, there appears to be a small tension for .
Although most previous analyses focus on the muonic final states, several analyses measure e.g.. the branching fraction or angular observables for decays. On the 2011 LHCb dataset corresponding to 1, the branching fraction of the electron mode has been determined in the dielectron mass range to be , where the uncertainties are statistical, systematic and from the normalisation mode. This result is found to agree with SM predictions. On the 3 dataset, an angular analysis has been performed in the effective range yielding
(20)  
(21)  
(22)  
(23) 
where the uncertainties are statistical and systematic, respectively, and these results are in agreement with the SM expectations.
3.1.3 decays beyond the ground state
Apart from kaon and states, one can consider higher resonances above the mass range. The LHCb collaboration reported the first observation of the decay channels and , whose branching fractions are
(24)  
(25) 
where the uncertainties are statistical, systematic and originating from the normalisation mode . For the channel, the differential branching fractions illustrated in Fig. 12 could be measured.
In the region above the mass around , the following resonances decaying to a final state contribute to the spectrum: the Swave , the Pwaves and and the Dwave . The LHCb collaboration has, for the first time, studied the differential branching fraction of the decay and performed a full angular analysis including S, P, and Dwave contributions in the region . The former results are shown in Fig.