Consequences of R-Parity violating interactions for anomalies in and
We investigate the possibility of explaining the enhancement in semileptonic decays of , the anomalies induced by in and violation of lepton universality in within the framework of R-parity violating (RPV) MSSM. Exchange of down type right-handed squark coupled to quarks and leptons yield interactions which are similar to leptoquark induced interactions that have been proposed to explain the by tree level interactions and anomalies by loop induced interactions, simultaneously. However, the Yukawa couplings in such theories have severe constraints from other rare processes in and decays. Although this interaction can provide a viable solution to anomaly, we show that with the severe constraint from , it is impossible to solve the anomalies in process simultaneously.
Recent experimental data have shown deviations from standard model (SM) predictions in the ratio of with and also in induced decays. Experimental values for RD-1 (); RD-2 (); RD-3 () are larger than the SM predictionsRD-sm (). This anomalous effect is significant, at about 4 levelHFAG (). The anomalies due to induced processes show up inb2s-1 () decays. The observed branching ratios in these decays are lower than SM predictionsb2s-sm (); b2s-the-1 (). Also a deficit is shown in the ratio RK (). The SM predicts to be close to one, but experimental data giveRK () . These effects are at 2 to 3 . Needless to say that these anomalies need to be further confirmed experimentally and we also need to understand SM predictions better. The latter processes involved are rare processes and therefore are sensitive to new physics. These anomalies have attracted a lot of theoretical attentions trying to solve the problems using new physics beyond SMRD-sm (); b2s-sm (); b2s-the-1 (); RD-the-1 (); desh-menon (); leptoquark (); b2s-the-2 (); RD-b2s (); bauer-neubert (). In this work we study the possibility of using R-parity violating interaction to solve these anomalies. Previously, R-parity violation was invoked to explaindesh-menon () only . Exchange of down type right-handed squark coupled to quarks and leptons yield interactions, which are similar to leptoquark induced interactions that have been proposed to explain the and induced anomalies simultaneouslybauer-neubert (). However, the Yukawa couplings have severe constraints from other rare processes in and decays. This interaction can provide a viable solution to anomaly. But with severe constraint from , it proves to be impossible to solve the anomalies induced by process.
The most general renomalizable R-parity violating terms in the superpotentials areR-parity ()
We will assume that term is zero to ensure proton stability. Since the processes we discuss involve leptons and quarks, the term should remain. In fact the interactions induced by this term at the tree and one loop level can contribute to and induced processes. It is tempting to see if these interactions can solve the related anomalies already. Although a combination of and terms can also contribute, the resulting operators are disfavored by process.
We shall limit ourselves to exchange of right-handed down type squark, , which are expected to have the necessary ingredients to explain the anomalies in decays. This model is similar to the leptoquark exchange discussed by many authorsb2s-the-2 (), except a general leptoquark also has a right-handed couplings to singlets, which is forbidden in SUSY. These additional right-handed couplings turn out to be important for explaining the anomaly of muon, but do not play an essential role in explaining the B anomalies discussing here. The object of our paper is a careful consideration of the constraints from various and decays and analysie structure of Yukawa couplings to see if the B anomalies can be resolved simultaneously. The paper by Bauer and Neubertbauer-neubert () is closest in spirit to our paper, but we are able to bring out the tension between different experimental constraints, and find that it is impossible to solve the and anomalies simultaneously.
The and anomalies occur at tree level and loop level in the SM, respectively. To simultaneously solve these anomalies using a simple set of beyond SM interactions faces more constraintsRD-b2s (); bauer-neubert () than just solving one of them as has been done in most of the studies. We find that by exchanging right-handed down type of squark, it is possible to solve the anomaly with tree interaction provided is sizable, of order . For anomalies induced by , to obtain the right chirality for operators , one needs to go to one loop level. The allowed couplings are constrained from various experimental data, such as , and . The strongest constraint comes from making the model impossible to explain anomalies induced by .
Ii R-parity violating interactions and
Expanding the term in terms of fermions and sfermions, we have
where the “tilde” indicates the sparticles, and “c” indicates charge conjugated fields.
Working in the basis where down quarks are in their mass eigenstates, , one replaces in the above by . Here is the Kobayashi-Maskawa (KM) mixing matrix for quarks. If experimentally, the mass eigenstate of neutrino are not identified, one does not need to insert the PMNS mixing matrix for lepton sector. The neutrinos in the above equation are thus in the weak eigenstates. For leptoquark interactions discussed in eq. (6) in Ref.bauer-neubert (), the reference seems to indicate that new parameters are involved due to rotation matrix in the lepton sector. However, since neutrinos are not in the mass basis in our work, it seems that provided we are always in the weak basis, no matrix is required in the lepton sector. We will assume sfermions are in their mass eigenstate basis. For a discussion of the choice of basis see Ref.R-parity ()
Exchanging sparticles, one obtains the following four fermion operators at the tree level
In the above and are color indices.
At the tree level, besides the SM contributions to , there are also R-parity violating contributions, they are given by the term proportional to in the above equation. Including the SM contributions one obtainsdesh-menon ()
where are elements in .
Identifying different charged leptons in the final states, we find the ratio of branching ratios compared with SM predictions to be given by
One can define a similar quantity for and , and have
Experimentally, deviations from SM predictions is small, that is , therefore we require , to be close to zero, which can be achieved by setting , so that no linear terms in contribute to . No large deviation has been observed in . However in induced anomalies involves couplings, we will bare in mind that effect may have some impact for . One may even contemplate that a somewhat enhanced must be there if one tries to solve the anomalies simultaneously. Although such a large deviation has not been established, theoretical calculations for the absolute values for the SM predictions and the experimental measurements may have some errors, so a certain level of deviation can be tolerated. We will take a conservative attitude to only allow up to 10% deviation from SM value, in . We find that even such modest requirement put stringent constraint and making the attempt of simultaneously solve the two types of anomalies difficult.
Defining , we have
Changing to , one can obtain the R-parity vilating contributions to . With the same approximation as above, we have
The linear terms in and are proportional to
respectively. Note that there is a large enhancement factor for the first term in the expression for compared with . This may cause potential problem for a small deviation from 1 in to a large deviation in . One can avoid such a large enhancement by setting to be much smaller than other terms. In our later discussions we will set to be zero. The is also constrained to be small from decay to be discussed in the following. But may play some important role in decay. We will keep it in our discussions.
The SM predictions and experimental measurements for areHFAG ()
The R-parity violating contributions to both and occur in a similar way, we use the averaged of and to represent the anomaly. In the SM, . To obtain a within the region, is typically of order . This large coupling makes it worrisome for this scenario from unitarity consideration. In more general terms, the unitarity limits concern the upper bound constraints on the coupling constants imposed by the condition of a scale evolution between the electroweak and the unification scales, free of divergences or Landau poles for the entire set of coupling constants. If so, the R-parity couplings are constrained to be about one at TeV scaleR-parity (). A value of 3 is not consistent. The requirement of no Landau pole up to unifications scale may be not necessary if some new physics appear. One cannot for sure rule out the possibility of reaching unitarity bound of at a lower energy. However when attempt to also solve induced anomalies, the model become much more constrained.
Iii Constraints from other tree level processes
Several other rare processes may receive tree level R-parity violating contributions. The constraints from these processes should be taken into account. We now study a few of the relevant ones: , , and .
The possible terms generating these decays are
If is non-zero for restricted to only one value, the two terms on the second line in the above equation will not induce the decays in question. For simplicity, we will work with this assumption
decay in the SM is extremely small. In our case, there are tree contributions which are therefore constrained severly. We have
The decay width is given by
where MeVfD () is the decay constant.
Using experimental upper boundPDG () at 90% C.L. for , we have . With set to zero, is give by . We have . is only very loosely constrained from . If just or is non-zero, they are constrained as
These constraints on and , make their effects on small. Later we will show that even a small may play some important role in having a better coherent explanation of and anomalies.
For , the ratio of is given byddh ()
Combining the SM predictionkpiuu-sm () for the branching ratio and experimental informationPDG () , at level, are constraint to be less than a few times of . Since we will set , this process is not affected at tree level.
The expressions for and of and can be obtained from Eq.(14) by replacing to and , respectively. The corresponding are
For , since we have set , it is again not affected by R-pairty violating interactions in this model.
The process will be affected. We have the following non-zero
Iv Loop contributions for induced anomalies
The anomalous effects in induced processes are only 2 to 3 effects and need to be confirmed further. They may be due to our poor understanding of hadronic matrix elements involved, and may also be caused by new physics beyond SM. We now discuss how R-parity violating interaction may help to solve the problems.
New physics contributes to can be parametrized as . Some of the most studied operators are
The SM predictions are . A global analysis shows that to solve the anomalies in decays induced by , there are few scenarios where the anomalies can be solved with high confidence level and all cases need to be around b2s-sm (). For example with and , with a 4.5 pull; the cases with , the best fit values are: and others equal to zero with a 4.8 pull; And the case with , the best fit values are: and others equal to zero with a 4.2 pull. Here the number of “pulls” indicates by how many sigmas the best fit point is preferred over the SM point for a given scenario. The higher the pull, the better fit between theory and experimental data is reached. In our case, the R-parity violating contribution to be discussed belongs to the last case. For this case, the allowed range isb2s-sm (), . With negative value for , the new physics contribution reduces and therefore helps to explain why branching ratios and are smaller than those predicted by SM.
There is a potential contribution to at tree level due to a term proportional to . However, since we assume that there is only one non vanishing value for , is not induced by this contribution.
One needs to include one loop contributions. At one loop level, exchanging in the loop, contributions with can be generated with
where is the up type quark mass. The first term is induced by exchanging a boson and a sparticle , and the second term is by exchanging two sparticles in the loops. The term of interest corresponds to , , and for the process . One can relabel them with different numbers for other process.
The first term is dominated by , its contribution to is about