Constraints on new physics from
We study generic effects of new physics on the rare decay modes and . We discuss several cases: left-handed neutrino couplings; right handed neutrino couplings; neutrino lepton flavour violating (LFV) interactions; and interactions. The first of these cases has been studied before as it covers many new physics extensions of the standard model; the second one requires the existence of a new light (sterile) right-handed neutrino and its contribution to both branching ratios is always additive to the SM. The case of neutrino LFV couplings introduces a CP conserving contribution to which affects the rates in a similar manner as a right handed neutrino as neither one of these interferes with the standard model amplitudes. Finally, we consider new physics with interactions to go beyond the Grossman-Nir bound. We find that the rare kaon rates are only sensitive to new physics scales up to a few GeV for this scenario.
In the standard model (SM), the rare decay modes proceed dominantly via a short distance contribution from a top-quark intermediate loop. This allows a precise calculation of the rates in terms of SM parameters Hagelin:1989wt (); Littenberg:1989ix (). The effective Hamiltonian responsible for these transitions in the SM is frequently written as
It follows that the branching ratios can then be written as (we use the notation and throughout this paper),
In these equations, the hadronic matrix element of the quark current is written in terms of the well measured semileptonic rate and is part of the overall constants and . Modern calculations of the parameters in these equations result in: which includes long distance QED corrections Mescia:2007kn (), and ; the Inami-Lim function for the short distance top-quark contribution Inami:1980fz () including NLO QCD corrections Buchalla:1998ba () and the two-loop electroweak correction Brod:2010hi (), result in ; and all known effects of the charm-quark contributions Buras:2006gb (); Brod:2008ss (); Isidori:2005xm (); Falk:2000nm () in . Finally, is the usual Wolfenstein parameter.333Uncertainties for these quantities can be found in the references.
Our estimate for these branching ratios within the SM, using the latest CKMfitter input Charles:2011va (), is
These numbers are to be compared with the current experimental results for the charged Adler:2000by (); Adler:2001xv (); Anisimovsky:2004hr (); Artamonov:2009sz () (measured by BNL 787 and BNL 949) and neutral Ahn:2009gb () modes (from KEK E391a),
An interesting correlation between these two modes was pointed out by Grossman and Nir (GN), namely that which is satisfied in a nearly model independent way Grossman:1997sk (). 444It was recently noted that the GN bound applied to the experimental result for needs to treat a possible two body intermediate state separately Fuyuto:2014cya (). In this paper we revisit these modes in the context of generic new physics motivated by the new results that are expected soon for the charged mode from NA62 at CERN and for the neutral mode from KOTO in Japan.
Ii New physics with lepton flavour conserving left-handed neutrinos
In this case the effective Hamiltonian describing the effects of the new physics (NP) takes the form
where the parameters encoding the NP are collected in and the overall constants have been chosen for convenience. Notice that this form is valid for both left-handed and right-handed quark currents as only the vector current is operative for the transition. Numerically it is then possible to obtain the rates from the SM result, Eq. 2, via the substitution . This has been done in the literature for a variety of models Buras:2015yca () so we will not dwell on this case here. In Figure 1 we illustrate the results. In general and the parameterisation in Eq. 5 implies that corresponds to NP with the same phase as . The green curve corresponds to (so called MFV in Buras:2015yca ()) and its two branches correspond to constructive and destructive interference with the charm-quark contribution in Eq. 2. The tick marks on the curve mark values of . If we allow for an arbitrary phase, this type of NP can populate the entire area below the GN bound, making it nearly impossible to translate a non-SM measurement into values of and .
We illustrate two more situations: the blue line shows being minus the phase of , which corresponds to CP conserving NP which does not contribute to the neutral kaon mode. The red line shows being the same as the phase of , which corresponds to NP which doubles the SM phase. Interestingly this case nearly saturates the GN bound. For comparison, we show the purple oval representing the SM allowed region as predicted using the parameters and uncertainties in CKMfitter Charles:2011va (). For the NP, however, we have only included the SM central values in Eq. 5. Allowing the SM parameters to vary in the rates that include NP, turns the green line into an arc-shaped region as can be seen in Ref. Buras:2015yca () for example.
Finally we have included in the plot a vertical red dashed line which marks a 30% uncertainty from the SM central value. This number has been chosen as it corresponds to the statistical uncertainty that can be achieved with 10 events that agree with the SM, in the ball park of what is expected from NA62.
Iii A light right handed neutrino
In models which contain a light right handed neutrino the effective Hamiltonian can be written as
where the first term is the SM, the new physics is parameterised by and its coupling to quarks can be through either a left or right handed current. In writing Eq. 6 we have assumed that there is only one new neutrino and that its mass is negligible. The rates for the rare kaon decay modes follow immediately,
where the accounts for the fact that we have only one right handed light neutrino (a factor of 3 from summing over the left-handed neutrinos is hiding in and ). In the result, Eq. 7, we see that this type of NP can only increase the rates, as it does not interfere with the SM, and this is illustrated in Figure 2. As in the previous case, we have chosen a parameterisation in Eq. 6 in which and corresponds to the NP having the same phase as . The green line in the figure corresponds to and the tick marks show that a maximum value of is allowed by the current BNL 90% c.l. limit on the charged rate, and that this number can be reduced to with about ten events. The pink region covers the parameter space with an arbitrary phase, and we show two more lines near the boundary of this region. The red line is obtained for ; whereas the blue line occurs for , for which there is no new contribution to the neutral mode.
Within the specific model detailed in the Appendix, the effect of the additional neutrino contributes both via a flavour changing tree-level exchange and a one-loop penguin and can be written as,
The overall strength of the coupling is parameterised by , where the upper limit arises from requiring the interaction to remain perturbative He:2002ha (). This, combined with the CMS limit on a that decays to tau-pairs TeV Khachatryan:2016qkc (), implies that the factor in the first bracket of Eq. 8 can be of order one. The tree-level contribution (second term in the second bracket) is constrained to be small by -mixing and -mixing, He:2006bk (). The Inami-Lim factor appearing in the penguin, , is less constrained and can be of order 10 He:2004it (). All in all, in our model the magnitude of can be order one but its phase is limited by the size of the tree contribution. This provides an example of NP in which a measurement of the two rates can be mapped to parameters in the model.
The existence of an additional light neutrino can, in general, have other observable consequences. As we show in Ref. He:2012zp (), the invisible width constrains the mixing between the and bosons in our model. This mixing, however, does not alter the leading contributions to shown in Eq. 8. In essence the width does not constrain this additional light neutrino because it is sterile as far as the SM interactions are concerned. A new light right-handed neutrino also contributes to the effective number of neutrino species which is constrained by cosmological considerations. In Ref. He:2017bft () we show that this constraint can also be evaded if the new neutrino mixes dominantly with the tau-neutrino and not with the muon or electron neutrinos.
Iv Neutrino lepton flavour violating interactions
Another possibility consists of interactions that violate lepton flavour conservation in the neutrino sector. These are particularly interesting because they can yield CP conserving contributions to the decay. In this case it is convenient to write
to normalise the strength of the NP to that of the SM but without inserting the SM phase into the new couplings. This then results in
where again a factor of compensates for the factor of 3 hiding in and . These lepton flavor violating contributions (proportional to , ) produce a very similar pattern of corrections as the case of the right handed neutrino Eq. 6. This LFV contribution to the neutral mode is maximised when
and we illustrate this scenario in Figure 3. The green line corresponds to the case and the dots mark values of . The allowed region when only is allowed to be non-zero and satisfying with arbitrary phases is shown in pink. The blue line, where the neutral kaon rate is unaffected, occurs for .
Neither the LFV nor the right-handed neutrino scenarios interferes with the SM amplitude, so they both result in additive corrections to the rates. We can illustrate the correspondence between the two cases by considering the red line of Figure 2 for which the phase of plus the phase of equals and therefore . This line matches the green line of Figure 3 for , and is equivalent to .
Figure 3 indicates that this model can have important effects for . In terms of the leptoquark couplings shown in the Appendix, is of order
implying that for leptoquark couplings of electroweak strength, these rare kaon modes are sensitive to leptoquark masses up to about 80 TeV.
V Beyond the Grossman-Nir bound
The hadronic transition between a kaon and a pion can be mediated in general by an operator that changes isospin by or by . The ratio of matrix elements follows from the Clebsch-Gordan coefficients
and the GN bound follows from the first of these equations, appropriate for the isospin structure of dimension six effective Hamiltonians of the cases discussed so far. Long distance contributions in the SM can violate this isospin relation but they are known to be small Lu:1994ww (). Long distance contributions within the SM can also produce CP conserving contributions to due to different CP properties of the relevant operators but these effects are also known to be small Buchalla:1998ux ().
When the transition is mediated by a vector current, as in the short distance SM of Eq.1, the decay is CP violating due to the CP transformation properties of the current: . In the same manner is CP conserving when mediated by a scalar density as Kiyo:1998aw ().555The operator discussed in this reference, , can be generated by leptoquark exchange at tree level in models which also have right handed neutrinos. Its effects satisfy the GN bound and, as it does not interfere with the SM, produces changes to the rates similar to the ones already discussed for LFV interactions.
To construct a operator one needs at least four quarks, and they have to take a current-current form which then leads to a CP conserving 666An four-quark operator of the form current-scalar-density would lead to a CP violating , but this has non-vanishing matrix elements only if it is .. Operators with these properties can occur beyond the SM as we parameterise in the appendix, where we show that the effect on the modes can be written as (when added to the SM)
These relations are illustrated in Figure 4 where the range covered by the rates of Eq. 14 is shown in pink along with the BNL result in green and the GN exclusion in grey. The SM central values are shown as the large red dot (the one sigma SM region is small on the scale of this plot) and the dashed vertical lines correspond to from the central SM value of . The green curve for and the blue curve for are chosen to illustrate values that can produce while keeping near its SM value.
When interactions are present, the GN bound is no longer valid. In addition, with the pattern of NP appearing in Eq. 14 and illustrated in Figure 4, it is possible to keep the charged rate close to the SM while making the neutral rate as large as desired. Interestingly, the GN bound is violated via two different effects: the factor of 2 present in the second line of Eq. 14 due to the nature of the operator, and the fact that the new contribution to the neutral rate is CP conserving. As such, a operator of the current-current form also violates the GN bound as can be checked by removing the factor of 2 present in the second line of Eq. 14. In both cases the new operator produces a CP conserving contribution to the neutral kaon decay and interferes with the SM. These properties result in a new contribution that can cancel the SM for the charged mode but not for the neutral mode.
Considering the dimension eight operator of the appendix, Eq. 28, the NP coupling reads,
Figure 4 shows that the rare kaon rates are sensitive to . With this then implies they are sensitive to a NP scale of order GeV. Given that this scale is only a few times larger than , our result is the same for the different types of possibilities discussed in the appendix, and it shows that even though this scenario is possible in principle, its effects are extremely small in most models. The conclusion is that a violation of the GN bound is completely implausible without a new few GeV new particle that carries isospin.
Appendix A A model with a right handed neutrino
The model has been described in detail elsewhere He:2002ha (); He:2003qv (), here we summarise its salient features. The gauge group is , but the three generations of fermions are chosen to transform differently to single out the third generation. In the weak interaction basis, the first two generations of quarks , , transform as , and , and the leptons , transform as and . The third generation, on the other hand, transforms as , , and . In this way acts only on the third generation.
To separate the symmetry breaking scales of and , there are two Higgs multiplets and with respective vevs and . An additional bi-doublet scalar with vevs is needed to provide mass to the fermions. Since both and are required to be non-zero for fermion mass generation, the and gauge bosons of and will mix with each other. In terms of the mass eigenstates and , the mixing can be parameterized as
In the mass eigenstate basis the quark-gauge-boson interactions are given by,
where , , is the Kobayashi-Maskawa mixing matrix and with the unitary matrices which rotate the right handed quarks and from the weak to the mass eigenstate basis.
The model has three left-handed neutrinos and one right-handed neutrino . Additional scalars and with vevs are needed to generate neutrino masses. In order for this model to contribute to the rare kaon decay modes discussed here, we need the right-handed neutrino to be light and thus requires to be small. The mass eigenstates are related by a unitary transformation to the weak eigenstates as
In our model , and and are , , and matrices, respectively.
Rotating the charged leptons from their weak eigenstates to their mass eigenstates , with , the lepton interaction with and becomes
In terms of neutrino mass eigenstates,
The new operators in this model that contribute to the rare kaon decay occur at tree level with new FCNC couplings at one-loop with new ZÕ penguin He:2004it (). They are
Both contributions couple to the right-handed neutrino so they do not interfere with the SM. In the quark current, only the vector term contributes to a transition so both LH and RH contribute in the same manner to Eq. 6.
Appendix B Models with leptoquarks
The interest of leptoquarks in kaon decays has been recently revived in connection to the B-anomalies Kumar:2016omp (); Fajfer:2018bfj (), here we will conduct a model independent analysis as in earlier papers Davies:1990sc (); Davidson:1993qk (). The scalar and vector leptoquark couplings to SM fermions which include a left-handed neutrino are,
where the leptoquark fields and their transformation properties under the SM group are given by
Exchange of these leptoquarks at tree-level generates effective operators of the form that will induce the rare kaon decays. We find with the aid of the identities
an effective four-fermion interaction of the form
For decays they combine to give
Of these leptoquarks all but contribute to processes , , and . This usually means that their effects in the kaon sector are severely constrained by which places their mass in the hundreds of TeV for couplings of electroweak strength and above 1000 TeV for Pati-Salam leptoquarks Valencia:1994cj (). On the other hand does not contribute to processes and its effects in the kaon sector are mostly constrained by lepton universality in and decays, and as we show here, by . Leptoquark models produce both LFC and LFV interactions in general so their contribution to the rare kaon rates are generally of the form
The parameters appearing here are versions of the in Eq. 26 but with a different normalisation, . In the main text we only consider the effect of the LFV couplings as the LFC ones fall under the same type of NP as Eq. 5.
Appendix C transitions
To change the GN relation we construct a operator to mediate the transition. This requires four quark fields, and an example of a dimension eight operator consistent with the symmetries of the SM that accomplishes this is