The Anomalies and New Physics in
Alakabha Datta firstname.lastname@example.org, Jacky Kumar email@example.com and David London firstname.lastname@example.org
: Department of Physics and Astronomy, 108 Lewis Hall,
University of Mississippi, Oxford, MS 38677-1848, USA
: Physique des Particules, Université de Montréal,
C.P. 6128, succ. centre-ville, Montréal, QC, Canada H3C 3J7
(July 3, 2019)
We investigate the implications of the latest measurement of for NP explanations of the anomalies. This measurement leads to a tension between separate fits to the and data, hinting at NP in . The previous data could be explained if the NP is in (I) or (II) . We examine the effect of adding NP in to these two options. We find several scenarios in which the addition of NP in leads to important improvements in the fits, but only in scenario (I). This suggests that scenario (II) is now somewhat disfavoured. LQ models can only generate scenarios within scenario (II), so they too are somewhat disfavoured. In addition, with the assumption of NP in , LQ models are now further disfavoured due to constraints from lepton-flavour-violating processes such as . To explain the data, it may now be necessary to consider NP models with more than one particle contributing to .
At present, there are several measurements of -decay processes involving the transition () that are in disagreement with the predictions of the standard model (SM). First, there are discrepancies with the SM in a number of observables in [1, 2, 3, 4, 5] and [6, 7]. These decays involve only . Second, the measurements of  and  also disagree with the SM predictions. These ratios involve both and . In this paper, we refer to these two sets of observables as the and observables.
Since all processes involve , it is natural to examine whether the anomalies can be explained by adding new physics (NP) to this decay. The transitions are defined via an effective Hamiltonian with vector and axial vector operators:
where the are elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and the primed operators are obtained by replacing with . The Wilson coefficients (WCs) include both the SM and NP contributions: . Following the announcement of the measurement in 2017, global fits were performed that combine the various observables [10, 11, 12, 13, 14, 15, 16, 17]. It was found that the net discrepancy with the SM is at the level of 4-6, and that the data can be explained if the nonzero WCs are (I) or (II) . In Ref. , the best-fit values of the WCs for these two scenarios were found to be (I) and (II) (other analyses found similar results). The simplest NP models involve the tree-level exchange of a leptoquark (LQ) or a boson. Scenario (II) can arise in LQ or models, but scenario (I) is only possible with a .
The measurement of was made in 2014 by the LHCb Collaboration using the Run 1 data . For , where is the dilepton invariant mass-squared, the result was
Second, the Run 2 data was analyzed:
Combining the Run 1 and Run 2 results, the LHCb measurement of is
This is closer to the SM prediction, though the discrepancy is still due to the smaller errors.
The LHCb measurement of was 
Recently, Belle announced its measurement of :
The errors are quite a bit larger than in the LHCb measurement.
In this paper we examine the effect of these new measurements – especially that of [Eq. (5)] – on the NP explanations of the anomalies.
The first step is to simply combine all the observables, and update the global fit performed in Ref. . (We refer to this paper for a description and the measured values of all the (CP-conserving) observables.) The results are shown in Table 1. We present the best-fit values of the WCs, as well as the predictions for , , and . ( is the observable in with the largest discrepancy with the SM.) For comparison, the measured central values are444The experimental results for , and are given in Eqs. (5) and (6), repeated here for convenience, while that of is taken from Ref. .
The pull is defined to be , i.e., it quantifies how much better the SM + NP fit is than the fit with the SM alone. Thus, a pull of 5.6 indicates that (i) the discrepancy between the experimental data and the predictions of the SM is at least , and (ii) the addition of NP improves the agreement with the measurements by . Based on the results of Table 1, we conclude that scenario (I) provides a slightly better explanation of the data than scenario (II). This can be understood qualitatively by looking at the predictions. The predictions of scenario (II) for and are closer to the experimental measurements than thos of senario (I), but its prediction for is far worse. The net effect is that scenario (I) has a larger pull than scenario (II). Of course, this does not exclude the possibility of finding an even larger improvement over the SM in another NP scenario.
While this is an interesting result, we note that the global fit does not contain all the important NP implications of the experimental data. Let us instead separate the data into and observables, and perform separate fits on these two data sets. The results are shown in Table 2. We see that there is now a tension between the NP WCs required to explain the and data: in scenario (I), the two best-fit values differ by , while in scenario (II) the difference is , where is defined by adding the errors of the two solutions in quadrature. Once again, this can be understood qualitatively by looking in Table 1 at the predictions of the two scenarios for . Previously, the experimental value of was 0.745 [Eq. (2)], and both scenarios were in reasonable agreement with this. However, now the value has increased to 0.846 [Eq. (5)], which is larger than the predictions. This all suggests that NP in alone is now not quite sufficient to explain the anomalies – NP in may be required. With this in mind, we consider a variety of scenarios in which some NP WCs are taken to be nonzero, in order to see if this tension can be removed, and the fit improved.
In a recent paper , a similar observation was made about the different NP implications of the and data. However, rather than focusing on NP in and , there this analysis is done in terms of lepton-flavour-universal (LFU) and lepton-flavour-universality-violating (LFUV) NP . It is worthwhile comparing the two ways of classifying the NP. First, technically LFU NP includes NP contributions to . However, such effects are not discussed in Ref. , nor in the present paper, and so we ignore them. Thus, LFU NP simply implies that there are contributions to the same operators in and , and LFUV NP refers to the non-LFU contributions to and . But this means that, if one invokes LFU NP, as is done in Ref. , there are additional contributions to both and . In this case, the fits to both and observables will be affected. In our case, since we are interested in , we consider NP only in . Still, we will occasionally use the LFU and LFUV labels.
We begin by examining the addition of NP in to scenario (I). We consider three different scenarios, shown in Table 3. Scenario S0 uses LFU NP (the same operators in both and ). However, for this scenario the best-fit value of the WC is consistent with zero, as in Table 1. This is reflected in the fact that the pull is also unchanged from Table 1. Thus, S0 is no better than the original scenario (I), and we discard it. On the other hand, in scenarios S1 and S2, both LFUV, nonzero values of the WCs are preferred. Furthermore, these scenarios are clear improvements. The predictions for are close to the experimental value, and the pull has increased. These scenarios demonstrate that, by adding NP to , one can improve the agreement with the data.
For scenarios S1 and S2, in Fig. 1 we show the allowed and regions of the and new observables individually, as well as the combined fit, all as functions of the WCs. In both cases, we see that the combined global fit prefers nonzero values of the WC. We also see how the new measurement of has moved the parameter space of the combined fit.
We now add NP in to scenario (II). The three different scenarios considered are shown in Table 4. There are similarities and differences among these scenarios. First, we note that S3 and S4 are LFU scenarios, while S5 is LFUV. Second, while S3, S4 and S5 all prefer a nonzero value of the WC(s), the statistical significance varies. The WCs are nonzero at 1.4, 1.5 and 1.8 for S3, S4 and S5, respectively. This is correlated with the predicted value of , which is smallest (largest) for S3 (S5). Finally, in all cases, the pulls are little changed compared to Table 1. This is because, while the agreement with experiment has improved for , which was the main goal of adding NP in , it has decreased for .
From this, we conclude that scenario (II) is now somewhat disfavoured, since it cannot be improved with the addition of NP in , while scenario (I) can. Even so, these solutions are in no way ruled out, and so should not be discarded. In Fig. 2 we show the allowed regions of the S3, S4 and S5 scenarios in the parameter space of the WCs.
We note that it was argued in Ref.  that scenario (II) can be improved with the addition of LFU NP. This is not in disagreement with us. As noted earlier, the addition of LFU NP implies new contributions to both and . As we found above, the NP in will not improve scenario (II). However, the addition of new contributions to may have an important effect. After all, this is equivalent to taking independent of (which means that, technically, it is no longer scenario (II)). With this increased freedom, it would not be a surprise if the prediction for were improved.
We now turn to a model-dependent analysis. As noted earlier, the simplest NP models that contribute to involve the tree-level exchange of a LQ or a boson. With the previous data, both of these NP models were viable. Does this still hold with the present data? We begin by looking at LQs.
There are three types of LQ that can contribute to at tree level. They are an -triplet scalar (), an -singlet vector (), and an -triplet vector () . Their couplings are
Here, in the fermion currents and in the subscripts of the couplings, and represent left-handed quark and lepton doublets, respectively, while , and represent right-handed up-type quark, down-type quark and charged lepton singlets, respectively. 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 left-handed (or ) and a left-handed (or ). These couplings are relevant for or (and possibly ).
In LQ models, there may be contributions to lepton-flavour-conserving operators in addition to () [Eq. (1)]. They are
contributes to , while and are additional contributions to . There may also be contributions to the lepton-flavour-violating (LFV) operators
where , with . , and contribute to and . Using the couplings in Eq. (9), one can compute which WCs are affected by each LQ. These are shown in Table 5 for , and it is straightforward to change one or both to an . Finally, there may also be a 1-loop contribution to the LFV decay :
All LFV operators can arise if there is a single LQ that couples to both and . Since the constraints from LFV processes are extremely stringent, we therefore anticipate that it may be difficult for a single LQ to both explain the anomalies via couplings to and and satisfy the LFV constraints.
The analysis of the LQ models has the following ingredients:
and : All LQs have , . In principle, the LQ could also produce . However, if these primed WCs are sizeable, so too are the scalar WCs and (see Table 5). Now, the scalar operators [Eq. (10)] contribute significantly to . The present measurement of [27, 28], in agreement with the SM, and the upper bound (90% C.L.)  constrain to be small.
LFV processes: The contributions of LQs to LFV processes were examined in detail in Ref. . It was found that the most important LFV process is , with (90% C.L.) . Even though the LQ contributes only at the 1-loop level, the very small upper limit on the branching ratio places stringent constraints on the model. The relevant WCs are 
Here , is given in the caption of Table 5, and , 2 and for the , and LQ models, respectively. In computing the constraints on the LQ models from , we conservatively take , as it leads to the weakest constraints.
Given that LQs can only contribute to , , the only one of the scenarios in Tables 3 and 4 that can be generated by LQs is S3. In Fig. 3, we again plot the - and -allowed regions for the and observables, as well as the regions from the combined fit, as functions of the NP WCs and . We then overlay the region for the , and LQs allowed by the stringent constraint from . Focusing on the -allowed region of the combined fit, we see that the LQ is excluded in all but a small corner of parameter space. The and LQs are still allowed, but in less than half of the allowed region. In particular, these LQ models are viable only if is small. We therefore conclude that, with the new data, explanations of the anomalies involving a single LQ are now doubly disfavoured: (i) within scenario (II), LQ solutions are excluded in over half of the parameter space, and (ii) scenario (II) is somewhat disfavoured compared to scenario (I).
We now turn to models. As was the case for LQs, other processes may be affected by exchange, and these produce constraints on the couplings. In particular, the coupling is constrained by - mixing and the coupling is constrained by the production of pairs in neutrino-nucleus scattering, (neutrino trident production). These constraints are discussed in detail in Ref. . There it is found that, when these constraints are taken into account, the expected sizes of the NP WCs are .
In the most general case, the couplings of the to the various pairs of fermions are independent. For and transitions, the couplings that interest us are , , , , and , which are the coefficients of , , , , and , respectively. Defining and , we can then write
where is given in the caption of Table 5.
With these expressions, it is straightforward to see that scenarios S1, S2 and S5 of Tables 3 and 4 cannot be produced with a . On the other hand, scenarios S3 and S4 can. Both scenarios require and , while scenario S3 (S4) requires (). In addition, the WCs roughly satisfy , which is required by the constraints from - mixing and neutrino trident production. This shows that scenarios S3 and S4 can be generated in a model with a gauge boson. Still, S3 and S4 are part of scenario (II), which we have argued is somewhat disfavoured.
To summarize, of the NP models containing a single new particle that contributes to at tree level, LQ models are now doubly disfavoured, while models with a are somewhat disfavoured. This leads one to consider the possibility of more than one NP contribution. Indeed, realistic NP models often contain a variety of new particles. To investigate the possibilities, it is useful to approach this question from the SM Effective Field Theory (SMEFT) [32, 33] point of view.
Any NP model must respect the gauge symmetries of the SM. When this NP is integrated out, one produces operators involving only the SM particles, but these must also be invariant under the SM symmetries. There are, of course, many possible operators, but we are interested only in those that contribute to the WCs ( or ) at low energy. Restricting ourselves to dimension-six NP operators that contribute to at tree level, there are two categories. First, there are four-fermion operators:
Second, there are operators involving the Higgs field:
The WCs can be written in terms of the coefficients of these operators . The NP four-fermion operators generate