New Physics in : Distinguishing Models through CPViolating Effects
Abstract
At present, there are several measurements of decays that exhibit discrepancies with the predictions of the SM, and suggest the presence of new physics (NP) in transitions. Many NP models have been proposed as explanations. These involve the treelevel exchange of a leptoquark (LQ) or a flavorchanging boson. In this paper we examine whether it is possible to distinguish the various models via CPviolating effects in . Using fits to the data, we find the following results. Of all possible LQ models, only three can explain the data, and these are all equivalent as far as processes are concerned. In this single LQ model, the weak phase of the coupling can be large, leading to some sizeable CP asymmetries in . There is a spectrum of models; the key parameter is , which describes the strength of the coupling to . If is small (large), the constraints from  mixing are stringent (weak), leading to a small (large) value of the NP weak phase, and corresponding small (large) CP asymmetries. We therefore find that the measurement of CPviolating asymmetries in can indeed distinguish among NP models.
I Introduction
At present, there are several measurements of decays involving that suggest the presence of physics beyond the standard model (SM). These include

: Measurements of have been made by the LHCb BK*mumuLHCb1 (); BK*mumuLHCb2 () and Belle BK*mumuBelle () Collaborations. They find results that deviate from the SM predictions. The main discrepancy is in the angular observable P'5 (). Its significance depends on the assumptions made regarding the theoretical hadronic uncertainties BK*mumuhadunc1 (); BK*mumuhadunc2 (); BK*mumuhadunc3 (). The latest fits to the data Altmannshofer:2014rta (); BK*mumulatestfit1 (); BK*mumulatestfit2 () take into account the hadronic uncertainties, and find that a significant discrepancy is still present, perhaps as large as .

: The LHCb Collaboration has measured the branching fraction and performed an angular analysis of BsphimumuLHCb1 (); BsphimumuLHCb2 (). They find a disagreement with the predictions of the SM, which are based on lattice QCD latticeQCD1 (); latticeQCD2 () and QCD sum rules QCDsumrules ().

: The ratio has been measured by the LHCb Collaboration in the dilepton invariant masssquared range 1 GeV GeV RKexpt (), with the result
(1) This differs from the SM prediction of IsidoriRK () by , and thus is a hint of lepton flavor nonuniversality.
While any suggestions of new physics (NP) are interesting, what is particularly intriguing about the above set of measurements is that they can all be explained if there is NP in ^{1}^{1}1Early modelindependent analyses of NP in can be found in Refs. bsmumuCPC () (CPconserving observables) and bsmumuCPV () (CPviolating observables).. To be specific, transitions are defined via the effective Hamiltonian
(2) 
where the are elements of the CabibboKobayashiMaskawa (CKM) matrix. The primed operators are obtained by replacing with , and the Wilson coefficients (WCs) include both SM and NP contributions. Global analyses of the anomalies have been performed DescotesGenon:2013wba (); Altmannshofer:2014rta (); BK*mumulatestfit1 (); BK*mumulatestfit2 (). It was found that there is a significant disagreement with the SM, possibly as large as , and it can be explained if there is NP in . Ref. BK*mumulatestfit1 () gave four possible explanations: (I) , (II) , (III) , (IV) .
Numerous models have been proposed that generate the correct NP contribution to at tree level^{2}^{2}2The anomalies can also be explained using a scenario in which the NP enters in the transition, but constraints from radiative decays and  mixing must be taken into account, see Ref. AlexLenz ().. Most of them use solution (II) above, though a few use solution (I). These models can be separated into two categories: those containing leptoquarks (LQs) CCO (); AGC (); HS1 (); GNR (); VH (); SM (); FK (); BFK (); BKSZ (), and those with a boson CCO (); Crivellin:2015lwa (); Isidori (); dark (); Chiang (); Virto (); GGH (); BG (); BFG (); Perimeter (); CDH (); SSV (); CHMNPR (); CMJS (); BDW (); FNZ (); AQSS (); CFL (); Hou (); CHV (); CFV (); CFGI (); IGG (); BdecaysDM (); Bhatia:2017tgo (). But this raises an obvious question: assuming that there is indeed NP in , which model is the correct one? In other words, short of producing an actual LQ or experimentally, is there any way of distinguishing the models?
A first step was taken in Ref. RKRDmodels (), where it was shown that the CPconserving, leptonflavorviolating decays and are useful processes for differentiating between LQ and models. In the present paper, we compare the predictions of the various models for CPviolating asymmetries in and .
CPviolating effects require the interference of two amplitudes with a relative weak (CPodd) phase. (For certain CPviolating effects, a relative strong (CPeven) phase is also required.) In the SM, is dominated by a single amplitude, proportional to [see Eq. (2)]. In order to generate CPviolating asymmetries, it is necessary that the NP contribution to have a sizeable weak phase. As we will see, this does not hold in all NP models, so that CPviolating asymmetries in and can be a powerful tool for distinguishing the models. (The usefulness of CP asymmetries in for identifying NP was also discussed in Ref. BHP ().)
We perform both modelindependent and modeldependent analyses. In the modelindependent case, we assume that the NP contributes to a particular set of WCs (and we consider several different sets). But if a particular model is used, one can work out which WCs are affected. In either case, a fit to the data is performed to establish (i) whether a good fit is obtained, and (ii) what are the bestfit values and allowed ranges of the real and imaginary pieces of the WCs. In the case of a good fit, the predictions for CPviolating asymmetries in and are computed.
The data used in the fits include all CPconserving observables involving transitions. The processes are , , , , , , and . For the first process, a complete angular analysis of was performed in Refs. BHP (); BK*mumuCPV (). It was shown that this decay is completely described in terms of twelve angular functions. By averaging over the angular distributions of and decays, one obtains CPconserving observables. There are nine of these. Most of the observables are measured in different bins, so that there are a total of 106 CPconserving observables in the fit.
For the modelindependent fits, only the data is used. However, for the modeldependent analyses, additional data may be taken into account. That is, in a specific model, there may be contributions to other processes such as ,  mixing, etc. The choice of additional data is made on a modelbymodel basis. Because the modelindependent and modeldependent fits can involve different experimental (and theoretical) constraints, they may yield significantly different results.
CPviolating asymmetries are obtained by comparing and decays. In the case of , there is only the direct partial rate asymmetry. For , one compares the and angular distributions. This leads to seven CP asymmetries. There are therefore a total of eight CPviolating effects that can potentially be used to distinguish among the NP models.
For the LQs, we will show that there are three models that can explain the data. The LQs of these models contribute differently to , so that, in principle, they can be distinguished by the measurements of . However, the constraints from these measurements are far weaker than those from , so that all three LQ models are equivalent, as far as the data are concerned. We find that some CP asymmetries in can be large in this single LQ model.
In models, there are and couplings, leading to a treelevel contribution to . In order to explain the anomalies, the product of couplings must lie within a certain (nonzero) range. If is small, must be large, and viceversa. The also contributes at tree level to  mixing, proportional to . Measurements of the mixing constrain the magnitude and phase of . If is large, the constraint on its phase is significant, so that this model cannot generate sizeable CP asymmetries. On the other hand, if is small, the constraints from  mixing are not stringent, and large CPviolating effects are possible.
The upshot is that it may be possible to differentiate and LQ models, as well as different models, through measurements of CPviolating asymmetries in .
We begin in Sec. 2 with a description of our method for fitting the data and for making predictions about CP asymmetries. The data used in the fits are given in the Appendix. We perform a modelindependent analysis in Sec. 3. In Sec. 4, we perform modeldependent fits in order to determine the general features of the LQ and models that can explain the anomalies. We present the predictions of the various models for the CP asymmetries in Sec. 5. We conclude in Sec. 6.
Ii Method
The method works as follows. We suppose that the NP contributes to a particular set of WCs. This can be done in a “modelindependent” way, in the sense that no particular underlying NP model is assumed, or it can be done in the context of a specific NP model. In either case, all observables are written as functions of the WCs, which contain both SM and NP contributions. Given values of the WCs, we use flavio flavio () to calculate the observables. By comparing the computed values of the observables with the data, the can be found. The program MINUIT James:1975dr (); James:2004xla (); James:1994vla () is used to find the values of the WCs that minimize the . It is then possible to determine whether or not the chosen set of WCs provides a good fit to the data. This is repeated for different sets of WCs.
We are interested in NP that leads to CPviolating effects in . As noted in the introduction, this requires that the NP contribution to have a weak phase. With this in mind, we allow the NP WCs to be complex (other fits generally take the NP contributions to the WCs to be real), and determine the bestfit values of both the real and imaginary parts of the WCs.
In the case where a particular NP model is assumed, the main theoretical parameters are the couplings of the NP particles to the SM fermions. At low energies, these generate fourfermion operators. The first step is therefore to determine which operators are generated in the NP model. This in turn establishes which observables are affected by the NP. The fit yields preferred values of the WCs, and these can be converted into preferred values for the real and imaginary parts of the couplings.
We note that caution is needed as regards the results of the modelindependent fits. In such fits it is assumed that the NP contributes to a particular set of WCs. One might think that the results will apply to all NP models that contribute to the same WCs. However, this is not true. The point is that a particular model may have additional theoretical or experimental constraints. When these are taken into account, the result of the fit might be quite different. That is, the “modelindependent” fits do not necessarily apply to all models. Indeed, in the following sections we will see several examples of this.
Finally, for those sets of WCs that provide good fits to the data, we compute the predictions for the CPviolating asymmetries in and .
ii.1 Fit
The is a function of the WCs , and is constructed as follows:
(3) 
Here are the theoretical predictions for the various observables used as constraints. These predictions depend upon the WCs. are the the corresponding experimental measurements.
We include all available theoretical and experimental correlations in our fit. The total covariance matrix is obtained by adding the individual theoretical and experimental covariance matrices, respectively and . The theoretical covariance matrix is obtained by randomly generating all input parameters and then calculating the observables for these sets of inputs flavio ().The uncertainty is then defined by the standard deviation of the resulting spread in the observable values. In this way the correlations are generated among the various observables that share some common parameters flavio (). Note that we have assumed to be independent of the WCs. This implies that we take the SM covariance matrix to construct the function. As far as experimental correlations are concerned, these are only available (bin by bin) among the angular observables in BK*mumuLHCb2 (), and among the angular observables in BsphimumuLHCb2 ().
For minimization, we use the MINUIT library James:1975dr (); James:2004xla (); James:1994vla (). The errors on the individual parameters are defined as the change in the values of the parameters that modifies the value of the function such that . However, to obtain the and CL 2parameter regions, we use equal to 2.3 and 6.0, respectively pdg ().
The fit includes all CPconserving observables. These are

: The CPaveraged differential angular distribution for can be derived using Refs. P'5 (); BHP (); BK*mumuCPV (); it is given by BK*mumuLHCb2 ()
(4) Here represents the invariant mass squared of the dimuon system, and represents the solid angle constructed from , and . There are therefore nine observables in the decay: the differential branching ratio, , , , , , , and , all measured in various bins. The experimental measurements are given in Tables 6 and 7 in the Appendix.
In the introduction it was mentioned that the main discrepancy with the SM is in the angular observable . This is defined as P'5 ()
(5) 
The differential branching ratio of . The experimental measurements Aaij:2014pli () are given in Table 8 in the Appendix.

The differential branching ratio of . The experimental measurements Aaij:2014pli () are given in Table 9 in the Appendix. When integrated over , this provides the numerator in . Thus, the measurement of [Eq. (1)] is implicitly included here^{3}^{3}3Previous studies (Ref. RKRDmodels () and references therein) have indicated that the anomaly can be accommodated sidebyside with several other anomalies in if new physics only affects transitions involving muons. Following this lead, in this paper we therefore study models that modify the transition while leaving the decays unchanged..

The differential branching ratio of . The experimental measurements Aaij:2014pli () are given in Table 10 in the Appendix.

: The experimental measurements of the differential branching ratio and the angular observables BsphimumuLHCb2 () are given respectively in Tables 11 and 12 in the Appendix.

The differential branching ratio of . The experimental measurements Lees:2013nxa () are given in Table 13 in the Appendix.
In computing the theoretical predictions for the above observables, we note the following:

For and , we use the form factors from the combined fit to lattice and lightcone sum rules (LCSR) calculations QCDsumrules (). These calculations are applicable to the full kinematic region. In LCSR calculations the full error correlation matrix is used, which is useful to avoid an overestimate of the uncertainties.

In , we use the form factors from lattice QCD calculations Bailey:2015dka (), in which the main sources of uncertainty are from the chiralcontinuum extrapolation and the extrapolation to low . In order to cover the entire kinematicallyallowed range of , we use the modelindependent expansion given in Ref. Bailey:2015dka ().

The decay has special characteristics, namely (i) there can be (timedependent) indirect CPviolating effects, and (ii) the  width difference, , is nonnegligible. These must be taken into account in deriving the angular distribution, see Ref. DescotesGenon:2015hea (). In flavio flavio (), the width difference is taken into account, but all observables correspond to timeintegrated ones (so no indirect CP violation).

In the calculation of the branching ratio of the inclusive decay , the dominant perturbative contributions are calculated up to NNLO precision following Refs. Asatryan:2002iy (); Ghinculov:2003qd (); Huber:2005ig (); Huber:2007vv ().
The above observables are used in all fits. However, a particular model may receive further constraints from its contributions to other observables, such as ,  mixing, etc. These additional constraints will be discussed when we describe the modeldependent fits.
ii.2 Predictions
Eq. (4) applies to decays. Here the seven angular observables , , , , , and are obtained by averaging the angular distributions of and decays. However, one can also consider the difference between and decays. This leads to seven angular asymmetries: , , , , , and BHP (); BK*mumuCPV (). For , there is only the partial rate asymmetry .
In general, there are two categories of CP asymmetries. Suppose the two interfering amplitudes are and , where the are the magnitudes, the the weak phases and the the strong phases. Direct CP asymmetries involving rates are proportional to . On the other hand, CP asymmetries involving Todd triple products of the form are proportional to . Both types of CP asymmetry are nonzero only if the interfering amplitudes have different weak phases, but the direct CP asymmetry requires in addition a nonzero strongphase difference. In the SM, the weak phase () and strong phases are all rather small, and the NP strong phase is negligible DatLon (). From this, we deduce that (i) large CP asymmetries are possible only if the NP weak phase is sizeable, and (ii) triple product CP asymmetries are most promising for seeing NP since they do not require large strong phases.
In order to compute the predictions for the CP asymmetries, we proceed as follows. As noted above, we start by assuming that the NP contributes to a particular set of WCs. We then perform fits to determine whether this set of WCs is consistent with all experimental data. In the case of a modelindependent fit, the data involve only observables; a modeldependent fit may involve additional observables. We determine the values of the real and imaginary parts of the WCs that minimize the . In the case of a good fit, we then use these WCs to predict the values of the CPviolating asymmetries  in and in .
In Ref. BHP (), it was noted that , , and are direct CP asymmetries, while , and are triple product CP asymmetries. Furthermore, is very sensitive to the phase of . We therefore expect that, if NP reveals itself through CPviolating effects in , it will most likely be in , with being particularly promising.
Iii ModelIndependent Results
In Refs. Altmannshofer:2014rta (); BK*mumulatestfit1 (), global analyses of the anomalies were performed. It was found that there is a significant disagreement with the SM, possibly as large as , and that it can be explained if there is NP in . Ref. BK*mumulatestfit1 () offered four possible explanations, each having roughly equal goodnessoffits:
(6)  
In this section we apply our method to these four scenarios. There are several reasons for doing this. First, we want to confirm independently that, if the NP contributes to these sets of WCs, a good fit to the data is obtained. Note also that the above solutions were found assuming the WCs to be real. Since we allow for complex WCs, there may potentially be differences. Second, the main idea of the paper is that CPviolating observables can be used to distinguish the various NP models. We can test this hypothesis with scenarios IIV. Finally, it will be useful to compare the modelindependent and modeldependent fits.
iii.1 Fits
The four scenarios are modelindependent, so that the fit includes only the observables. The results are shown in Table 1. In scenarios II and III, there are two bestfit solutions, labeled (A) and (B). In both cases, the two solutions have similar bestfit values for Re(WC), but opposite signs for the bestfit values of Im(WC). In all cases, we obtain good fits to the data. The pulls are all , indicating significant improvement over the SM. Indeed, our results agree entirely with those of Ref. BK*mumulatestfit1 ().
Scenario  [Re(WC), Im(WC)]  pull 
(I)  [, ]  4.2 
(II)  (A) [, ]  4.2 
(B) [, ]  4.0  
(III)  (A) [, ]  4.4 
(B) [, ]  4.4  
(IV)  [, ]  4.1 
iii.2 CP asymmetries: predictions
For each of the four scenarios, the allowed values of Re(WC) and Im(WC) are shown in Fig. 1. In all cases, Im(WC) is consistent with 0, but large nonzero values are still allowed. Should this happen, significant CPviolating asymmetries in can be generated. To illustrate this, for each of the four scenarios, we compute the predicted values of the CP asymmetries , and in . The results are shown in Fig. 2. From these plots, one sees that, in principle, one can distinguish all scenarios. If a large asymmetry is observed, this indicates scenario II, and one can differentiate solutions (A) and (B). A large asymmetry at low indicates scenario IV, while a large asymmetry at high indicates scenario III (here solutions (A) and (B) can be differentiated). Finally, if no or asymmetries are observed, but a sizeable asymmetry is seen at low , this would be due to scenario I.
This then confirms the hypothesis that CPviolating observables can potentially be used to distinguish the various NP models proposed to explain the anomalies. This said, one must be careful not to read too much into the modelindependent results. If NP is present in decays, it is due to a specific model. And this model may have other constraints, either theoretical or experimental, that may significantly change the predictions. That is, since the modelindependent fits have the fewest constraints, the CPviolating effects shown in Fig. 2 are the largest possible. In a particular model, there may be additional constraints, which will reduce the predicted sizes of the CP asymmetries. For this reason, while a modelindependent analysis is useful to get a general idea of what is possible, real predictions require a modeldependent analysis. We turn to this in the following sections.
Iv Modeldependent Fits
Many models have been proposed to explain the anomalies, of both the LQ CCO (); AGC (); HS1 (); GNR (); VH (); SM (); FK (); BFK (); BKSZ () and CCO (); Crivellin:2015lwa (); Isidori (); dark (); Chiang (); Virto (); GGH (); BG (); BFG (); Perimeter (); CDH (); SSV (); CHMNPR (); CMJS (); BDW (); FNZ (); AQSS (); CFL (); Hou (); CHV (); CFV (); CFGI (); IGG (); BdecaysDM (); Bhatia:2017tgo () variety. Rather than considering each model individually, in this section we perform general analyses of the two types of models. The aim is to answer two questions. First, what are the properties of models required in order to provide good fits to the data? Second, which of these goodfit models can also generate sizeable CPviolating asymmetries in ? We separately examine LQ and models.
iv.1 Leptoquarks
The list of all possible LQ models that couple to SM particles through dimension operators can be found in Ref. AGC (). There are five spin0 and five spin1 LQs, denoted and respectively, with couplings
(7)  
In the fermion currents and in the subscripts of the couplings, and represent lefthanded quark and lepton doublets, respectively, while , and represent righthanded uptype quark, downtype quark and charged lepton singlets, respectively. The LQs transform as follows under :
(8) 
Note that here the hypercharge is defined as .
In Eq. (7), the LQs can couple to fermions of any generation. To specify which particular fermions are involved, we add superscripts to the couplings. For example, is the coupling of the LQ to a lefthanded (or ) and a lefthanded . Similarly, is the coupling of the LQ to a righthanded and a lefthanded . These couplings are relevant for (and possibly ). Note that the and LQs do not contribute to .
A number of these LQs, and their effects on and other decays, have been analyzed separately. For example, in Ref. Sakakietal (), it was pointed out that four LQs can contribute to . They are: a scalar isosinglet with , a scalar isotriplet with , a vector isosinglet with , and a vector isotriplet with . These are respectively , , and . In Ref. Sakakietal (), they are called , , and , respectively, and we adopt this nomenclature below.
The LQ has been studied in the context of in Refs. HS1 (); GNR (); VH (); SM (). has been examined in Refs. CCO (); RKRDmodels (). In Ref. FK (), the LQ was proposed as an explanation of the anomalies. Finally, in Refs. BFK (); BKSZ () it was claimed that the treelevel exchange of a LQ can account for the results.
There are therefore quite a few LQ models that contribute to , several of which have been proposed as explanations of the decay anomalies. We would like to have a definitive answer to the following question: which of the LQs in Eq. (7) can actually explain the anomalies? Rather than rely on previous work, we perform an independent analysis ourselves.
iv.1.1 LQ fits
The difference between modelindependent and modeldependent fits is that, within a particular model, there may be contributions to new observables and/or new operators, and this must be taken into account in the fit. In the case of LQ models, the LQs contribute to a variety of operators. In addition to [Eq. (2)], there may be contributions to
(9)  
contributes to , while and are additional contributions to . Based on the couplings in Eq. (7), it is straightforward to work out which Wilson coefficients are affected by each LQ. These are shown in Table 2 AGC (). Although the scalar LQs do not contribute to , some vector LQs do. For these we have and .
LQ  

0  0  0  0  
0  0  0  
0  0  
0  0  0  
0  0  
0  0  0  0  
0  0  
0  0  0  
0  0  
0  0  0  0  
0  0  
0  0  
0  0  0  
0 
There are several observations one can make from this Table. First, not all of the LQs contribute to : contributes only to . Second, has two couplings, and . If both are allowed simultaneously, scalar operators are generated, and these can also contribute to . This must be taken into account in the modeldependent fits. The situation is similar for . Finally, the and LQs both have ; they are differentiated only by their contributions to .
At this stage, we can perform modeldependent fits to determine which of the LQ models can explain the data. First of all, the SM alone does not provide a good fit. We find, for 106 degrees of freedom, that
(10) 
We therefore confirm that the anomalies suggest the presence of NP.
For the scalar LQs, the results of the fits using only the data are shown in Table 3 (we address the data below). For the LQ, there are two bestfit solutions, labeled (A) and (B). (The two solutions have the same bestfit values for Re(coupling), but opposite signs for the bestfit values of Im(coupling).) From this Table, we see that only the LQ provides an acceptable fit to the data. Despite the claims of Refs. BFK (); BKSZ (), the LQ does not explain the anomalies.
LQ  Coupling  [Re(coupling), Im(coupling)]  pull 

(A) [, 