New physics in ?
At present, the measurements of some observables in and decays, and of , are in disagreement with the predictions of the standard model. While most of these discrepancies can be removed with the addition of new physics (NP) in , a difference of still remains in the measurement of at small values of , the dilepton invariant mass-squared. In the context of a global fit, this is not a problem. However, it does raise the question: if the true value of is near its measured value, what is required to explain it? In this paper, we show that, if one includes NP in , one can generate values for that are within of its measured value. Using a model-independent, effective-field-theory approach, we construct many different possible NP scenarios. We also examine specific models containing leptoquarks or a gauge boson. Here, additional constraints from lepton-flavour-violating observables, - mixing and neutrino trident production must be taken into account, but we still find a number of viable NP scenarios. For the various scenarios, we examine the predictions for in other bins, as well as for the observable .
At the present time, there are a number of measurements of -decay processes that are in disagreement with the predictions of the standard model (SM). Two of these processes are governed by : there are discrepancies with the SM in several observables in BK*mumuLHCb1 (); BK*mumuLHCb2 (); BK*mumuBelle (); BK*mumuATLAS (); BK*mumuCMS () and BsphimumuLHCb1 (); BsphimumuLHCb2 () decays. There are two other observables that exhibit lepton-flavour-universality violation, involving and : RKexpt () and RK*expt (). Combining the various observables, analyses have found that the net discrepancy with the SM is at the level of 4-6 Capdevila:2017bsm (); Altmannshofer:2017yso (); DAmico:2017mtc (); Hiller:2017bzc (); Geng:2017svp (); Ciuchini:2017mik (); Celis:2017doq (); Alok:2017sui ().
All observables involve . For this reason, it is natural to consider the possibility of new physics (NP) in 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: . It is found that, if the values of the WCs obey one of two scenarios111These numbers are taken from Ref. Alok:2017sui (). Other analyses find similar results. – (i) or (ii) – the data can all be explained.
In fact, this is not entirely true. has been measured in two different ranges of , the dilepton invariant mass-squared RK*expt ():
We refer to these observables as and , respectively. At low , the mass difference between muons and electrons is non-negligible RK*theory (), so that the SM predicts flavio (). For central values of (or larger), the prediction is . The deviation from the SM is then () or (). Assuming NP is present in , one can compute the predictions of scenarios (i) and (ii) for the value of in each of the two bins. These are
In each line above, the final number is the predicted value of the observable for the best-fit value of the WCs in the given scenario. The number to the left of it (in parentheses) is the smallest predicted value of the observable within the (68% C.L.) range of the WCs. We see that the experimental value of can be accounted for [though scenario (ii) is better than scenario (i)]. On the other hand, the experimental value of cannot – both scenario predict considerably larger values than what is observed.
Now, scenarios (i) and (ii) are the simplest solutions, in that only one NP WC (or combination of WCs) is nonzero. However, one might suspect that the problems with could be improved if more than one WC were allowed to be nonzero. With this in mind, we consider scenario (iii), in which and are allowed to vary independently. The best-fit values of the WCs, as well as the prediction for , are found to be
(Note that the errors on the WCs are highly correlated.) The number in parentheses is the smallest predicted value of within the 68% C.L. region in the space of and . We see that the predicted value of is not much different from that of scenarios (i) and (ii). Evidently, NP in and/or does not lead to a sizeable effect on .
What about if other WCs are nonzero? In scenario (iv), four WCs – , , and – are allowed to be nonzero. We find the best-fit values of the WCs and the prediction for to be
Here the smallest predicted value of (the number in parentheses) is computed as follows. In scenarios (i)-(iii), we have determined that varying and does not significantly affect . Thus, for simplicity, we set these WCs equal to their best-fit values. The smallest predicted value of is then found by scanning the 68% C.L. region in - space. But even in this case, the predicted value of is still quite a bit larger than the measured value. This leads us to conclude that if there is NP only in , is predicted, which is more than above its measured value222We note that, if all four WCs (, ) are allowed to vary, one can generate a smaller value of , 0.81. This is due only to the fact that the allowed region in the space of WCs is considerably larger: when one varies two parameters, the 68% C.L. region is defined by , whereas when one varies four parameters, it is ..
Of course, when one tries to simultaneously explain a number of different observables, it is not necessary that every experimental result be reproduced within . As long as the overall fit has , it is considered acceptable. This is indeed what is found in the analyses in which NP is assumed to be only in Capdevila:2017bsm (); Altmannshofer:2017yso (); DAmico:2017mtc (); Hiller:2017bzc (); Geng:2017svp (); Ciuchini:2017mik (); Celis:2017doq (); Alok:2017sui (). Still, this raises the question: suppose that the true value of is near its measured value. What is required to explain it?
This has been explored in a few papers. In Refs. Datta:2017ezo (); Altmannshofer:2017bsz (), it is argued that cannot be explained by new short-distance interactions, so that a very light mediator is required, with a mass in the 1-100 MeV range. And in Ref. Bardhan:2017xcc (), it is said that cannot be reproduced with only vector and axial vector operators, leading to the suggestion of tensor operators. In the present paper, we show that, in fact, one can generate a value for near its measured value with short-range interactions involving vector and axial vector operators.
To be specific, we show that, if there are NP contributions to , one can account for .333NP in has also been considered in some previous studies. In Refs. Capdevila:2017bsm (); Altmannshofer:2017yso (); Geng:2017svp (), it is found that the data can be explained by NP in or . A more complete analysis, similar to that performed in the present paper, is carried out in Ref. Ciuchini:2017mik (). However, there they do not focus on . Using a model-independent, effective-field-theory approach, we find that there are quite a few scenarios involving various NP WCs in and in which a value for can be generated that is larger than its measured value, but within . Indeed, if there is NP in , it is not a stretch to imagine that it also contributes to . We consider the most common types of NP models that have been proposed to explain the anomalies – those containing leptoquarks or a gauge boson – and find that, if they are allowed to contribute to , the measured value of can be accounted for (within ).
In scenario (ii) above, , so the NP couples only to the left-handed (LH) quarks and . This is a popular scenario, and many models have been constructed that have purely LH couplings. However, we find that, if the NP couplings in are also purely LH, can not be explained – couplings involving the right-handed (RH) quarks and/or leptons must be involved.
One feature of this type of NP is that it is independent of . Thus, if the WCs are affected in a way that lowers the value of compared to what is found if the NP affects only , the value of is also lowered. We generally find that, if the true value of is above its present measured value, the true value of will be found to be below its present measured value. This is a prediction of this NP explanation.
As noted above, there are a number of scenarios involving different sets of and NP WCs in which can be explained. Since NP in is independent of , each of these scenarios makes specific predictions for the values of and in other bins. Furthermore, a future precise measurement of the LFUV observable will help to distinguish the various scenarios.
The observables in and are Lepton-Flavour Dependent (LFD), while and are Lepton-Flavour-Universality-Violating (LFUV) observables. If one assumes NP only in , one uses LFUV NP to explain both LFD and LFUV observables. Recently, in Ref. Alguero:2018nvb (), Lepton-Flavour-Universal (LFU) NP was added. The LFUV observables are then explained by the LFUV NP, while the LFD observables are explained by LFUV LFU NP. Our scenarios, with NP in and , can be translated into LFUV LFU NP, and vice-versa. As we will see, the two ways of categorizing the NP are complementary to one another.
We begin in Sec. 2 with a detailed discussion of how the addition of NP in can explain . We construct a number of different scenarios using both a model-independent, effective-field-theory approach, and within specific models involving leptoquarks or a gauge boson. In Sec. 3, we examine the predictions of the various scenarios for and , and compare NP in and to LFUV LFU NP. We conclude in Sec. 4.
Ii NP in and
We repeat the fit, but allowing for NP in both and transitions. The observables used in the fit are given in Ref. Alok:2017sui (). The observables that have been measured are given in Table 1 futurebsee (). In this Table, we see that most observables have sizeable errors. The one exception is , but here the theoretical uncertainties are significant. The net effect is that NP in is rather less constrained than NP in .
Note that and have been measured in two different ranges of , [0.1-4.0] GeV and [1.0-6.0] GeV. These regions overlap, so including both measurements in the fit would be double counting. Since we are interested in the predictions for , in the fit we use the observables for in the lower range, [0.1-4.0] GeV. However, we have verified that the results are little changed if we use the observables for in the other range, [1.0-6.0] GeV.
The fit can be done in two different ways. First, there is the model-independent, effective-field-theory approach. Here, the NP WCs are all taken to be independent. The fit is performed simply assuming that certain WCs in and transitions are nonzero, without addressing what the underlying NP model might be. Second, in the model-dependent approach, the fit is performed in the context of a specific model. Since the NP WCs are all functions of the model parameters, there may be relations among the WCs, i.e., they may not all be independent. Furthermore, there may be additional constraints on the model parameters due to other processes. Each approach has certain advantages, and, in the subsections below, we consider both of them.
ii.1 Model-independent Analysis
In this subsection, we examine several different cases with NP WCs, where and are respectively the number of independent NP WCs (or combinations of WCs) in and . For each case, we find the best-fit values of the NP WCs, and compute the prediction for .
ii.1.1 Cases with NP WCs
Here we consider the simplest case, in which there is one nonzero NP WC (or combination of WCs) in each of and . We are looking for scenarios that satisfy the following condition: if one varies the NP WCs within their 68% C.L.-allowed region (taking into accout the fact that the errors on the WCs are correlated), one can generate a value for that is within of its measured value.
Although many of the scenarios we examined do not satisfy this conditon, we found several that do. They are presented in the first four entries of Table 2. In each scenario, the right-hand number in the column is its predicted value for the best-fit value of the WCs. The number in parentheses to the left is the smallest predicted value of within the (68% C.L.) range of the WCs. The and columns are similar, except that the numbers in parentheses are the values of and evaluated at the point that yields the smallest value of . We also examine how much better than the SM each scenario is at explaining the data. This is done by computing the pull , evaluated using the best-fit values of the WCs.
|NP in||NP in||Pull|
|(0.76) 0.82||(0.54) 0.66||(0.76) 0.74||6.5|
|(0.75) 0.82||(0.52) 0.65||(0.77) 0.82||6.5|
|(0.78) 0.83||(0.58) 0.68||(0.77) 0.77||6.6|
|(0.78) 0.82||(0.58) 0.67||(0.77) 0.78||6.7|
|(0.80) 0.83||(0.64) 0.70||(0.70) 0.74||6.4|
|(0.81) 0.85||(0.64) 0.70||(0.68) 0.71||6.3|
|(0.85) 0.86||(0.73) 0.74||(0.73) 0.73||6.4|
In all four scenarios, the addition of NP in makes it possible to produce a value of roughly above its measured value, which is an improvement on the situation where the NP affects only . As noted in the introduction, this type of NP is independent of , so that, if one adds NP to in a way that lowers the predicted value of , it will also lower the predicted value of . Indeed, we see that the values of the NP WCs that produce a better value of also lead to a value of that is roughly below its measured value. This is then a prediction: if the true value of is near its measured value, and if this is due to NP in , the true value of will be found to be below its measured value.
Note that this behaviour does not apply to . Its measured value is RKexpt ()
which differs from the SM prediction of IsidoriRK () by . In all scenarios, the value of is accounted for, and this changes little if one uses the central values of the NP WCs or the values that lead to a lower .
The pulls for all four scenarios are sizeable and roughly equal. It must be stressed that the values of pulls are strongly dependent on how the analysis is done: what observables are included, how theoretical errors are treated, which form factors are used, etc. For this reason one must be very careful in comparing pulls found in different analyses. On the other hand, comparing the pulls of various scenarios within a single analysis may be illuminating. With this in mind, consider again scenarios (i) and (ii) [Eq. (3)], and compare them with scenarios S3 and S1, respectively, of Table 2. Below we present the pulls of (i) and (ii)444In Ref. Altmannshofer:2017fio (), using only data (i.e., data was not included), the pulls of (i) and (ii) were found to be 5.2 and 4.8, respectively. Using the same method of analysis, we added the data and found that the pulls were increased to 6.2 and 6.3, respectively., and repeat some information given previously, in order to facilitate the comparison:
We first compare scenarios (i) and S3, noting that pull[S3] pull[(i)]. What is this due to? In the two scenarios, the value of is very similar, so that the contribution to the pull of the observables is about the same in both cases. (Indeed, the dominant source of the large pull is NP in .) That is, the difference in the pulls is due to the addition of NP in in S3. Now, the observablies in Table 1 have virtually no effect on the pull; the important effect is the different predictions for . Above, we see that the prediction of scenario S3 for () is much (slightly) closer to the experimental value than that of scenario (i). (The predictions for are essentially the same.) This leads to an increase of 0.4 in the pull. The comparison of scenarios (ii) and S1 is similar.
We also note that, in all scenarios, the pull of the fits evaluated at the (68% C.L.) point that yields the smallest value of is only smaller than the central-value pull. That is, if NP is added to the WCs, it costs very little in terms of the pull to improve the agreement with the measured value of .
In scenario S5 of Table 2, when the NP is integrated out, the four-fermion operators and are generated. That is, the NP couples to the LH quarks and , but to the RH . In scenario S6, one has the four-fermion operators and , so that the NP couples to the LH quarks and , but to the RH quarks and . We have not included either of these among the satisfactory scenarios, since the smallest value of possible at 68% C.L. is 0.80 or 0.81, which are a bit larger than above the measured value of . However, it must be conceded that this cutoff is somewhat arbitrary, so that these scenarios, and others like them, should be considered borderline.
Finally, in scenario S7 of Table 2, the NP four-fermion operators are and , i.e., the NP couples only to LH particles. This is a popular choice for model builders. However, here the smallest predicted value for is still almost above its measured value, so this cannot be considered a viable scenario.
ii.1.2 Cases with more than NP WCs
We now consider more general scenarios, in which there are () nonzero NP WCs (or combinations of WCs) in (), with , and . As discussed in the introduction, we know that varying the NP WCs has little effect on . We therefore fix these WCs to their central values and vary the NP WCs within their 68% C.L.-allowed region to obtain the smallest predicted value of . We find that there are now many solutions that predict a value for that is within roughly of its measured value. In Table 3 we present four of these. Scenarios S8 and S9 have and , while scenarios S10 and S11 have .
|NP in||NP in||Pull|
|(0.79) 0.83||(0.61) 0.69||(0.69) 0.75||6.5|
|(0.75) 0.82||(0.53) 0.65||(0.79) 0.76||6.4|
|(0.78) 0.84||(0.59) 0.71||(0.63) 0.75||6.8|
|(0.77) 0.83||(0.55) 0.66||(0.77) 0.76||6.8|
We see that, despite having a larger number of nonzero independent NP WCs, at 68% C.L. these scenarios predict similar values for as the scenarios in Table 2. Furthermore, the NP WCs that produce these values for also predict values for that are below its measured value. Finally, as was the case for scenarios with NP WCs, all scenarios here explain , even for values of the NP WCs that lead to a lower .
As was the case with the scenarios of Table 2, here the pulls are again sizeable. And again, it is interesting to compare similar scenarios without and with NP in . Consider scenarios (iii) [Eq. (4)] and S10:
The values of the NP WCs are very similar in the two scenarios, so that the difference in pulls is due principally to the addition of NP in in S10. Looking at , we see that the predictions of scenario S10 for , and are all slightly closer to the experimental values than the predictions of (iii). This leads to an increase of 0.2 in the pull.
ii.2 Model-dependent Analysis
There are two types of NP models in which there is a tree-level contribution to : those containing leptoquarks (LQs), and those with a boson. In this subsection, we examine these models with the idea of explaining by adding a contribution to . To be specific, we want to answer the question: can the scenarios in Tables 2 and 3 be reproduced within LQ or models? In the following, we examine these two types of NP models.
There are ten LQ models that couple to SM particles through dimension operators AGC (). There include five spin-0 and five spin-1 LQs, denoted and respectively. Their couplings are
where, 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 subscripts of the LQs indicate the hypercharge, defined as .
In the above, 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 ). Similarly, is the coupling of the LQ to a right-handed and a left-handed . These couplings are relevant for or (and possibly ). Note that the , and LQs do not contribute to . In Ref. Sakakietal (), , and are called , and , respectively, and we adopt this nomenclature below.
In a model-dependent analysis, one must take into account the fact that, within a particular model, there may be contributions to additional observables. In the case of LQ models, in addition to () [Eq. (1)], there may be contributions to the lepton-flavour-conserving operators