Probing leptonflavour universality with \(K\rightarrow \pi \nu \bar{\nu }\) decays
Abstract
We analyse the rare processes \(K\rightarrow \pi \nu \bar{\nu }\) in view of the recent hints of violations of leptonflavour universality (LFU) observed in B meson decays. If, as suggested by present data, the new interactions responsible for LFU violations couple mainly to the third generation of lefthanded fermions, \(K\rightarrow \pi \nu \bar{\nu }\) decays turn out to be particularly interesting: these are the only kaon decays with thirdgeneration leptons (the \(\tau \) neutrinos) in the final state. In order to relate Bphysics anomalies and K decays we adopt an effective field theory approach, assuming that the new interactions satisfy an approximate \(U(2)_q\times U(2)_\ell \) flavour symmetry. In this framework we show that O(1) deviations from the Standard Model predictions in \(K\rightarrow \pi \nu \bar{\nu }\) branching ratios, closely correlated to similar effects in \(B\rightarrow K^{(*)}\nu \bar{\nu }\), are naturally expected. The correlation of \(\mathcal {B}(K \rightarrow \pi \nu \bar{\nu })\), \(\mathcal {B}(B\rightarrow K^{(*)}\nu \bar{\nu })\), and the LFU violations in B decays would provide a very valuable tool to shed more light on this interesting phenomenon.
1 Introduction

deviations from \(\tau /(\mu ,e)\) universality in charged currents of the type \(b\rightarrow c \ell \bar{\nu }\) (observed in \(B \rightarrow D^* \ell \overline{\nu }\) and \(B \rightarrow D \ell \overline{\nu }\) decays [2, 3, 4]);

deviations from \(\mu /e\) universality in neutral currents of the type \(b\rightarrow s \ell \overline{\ell }\) (observed in \(B \rightarrow K^* \ell \overline{\ell }\) and \(B \rightarrow K \ell \overline{\ell }\) decays [1, 5]).
These deviations from the SM have triggered a series of theoretical speculations about possible new physics (NP) interpretations. In particular, attempts to provide a combined/coherent explanation for both charged and neutralcurrent anomalies have been presented in Refs. [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. One of the puzzling aspects of the present anomalies is that they have been seen only in semileptonic B decays and are quite large compared to the corresponding SM amplitudes. On the contrary, no evidences of deviations from the SM have been seen so far in the precise (permil) tests of LFU performed in semileptonic K and \(\pi \) decays, in purely leptonic \(\tau \) decays, and in electroweak precision observables. The most natural assumption to address this apparent paradox is the hypothesis that the NP responsible for the breaking of LFU is coupled mainly to the third generation of quarks and leptons, with some small (but nonnegligible) mixing with the light generations [14, 26, 28]. Within this paradigm, a motivated class of models are those based on a \(U(2)_q\times U(2)_\ell \) flavour symmetry acting on the light generations of SM fermions [29, 30], that turns out to be quite successful in addressing these anomalies while satisfying all existing bounds [26].
If NP is coupled mainly to thirdgeneration fermions, it is very difficult to detect it in K decays, which necessarily imply a transition among light quarks and, in most cases, also imply light leptons in the final states. The only exception in this respect is provided by \(K\rightarrow \pi \nu \bar{\nu }\) decays, which involve thirdgeneration leptons in the final state—the \(\tau \) neutrinos. As we will show in the following, this fact implies that \(K\rightarrow \pi \nu \bar{\nu }\) decays are a very sensitive probe of the most motivated models addressing the hints of LFU violations in B physics, as already pointed out in Refs. [18, 31] in specific models. On the one hand, \(\mathcal {B}(K\rightarrow \pi \nu \bar{\nu })\) could exhibit O(1) deviations from the SM predictions in a large area of the parameter space of such models. On the other hand, even in absence of large deviations, improved measurements (or constraints) on \(\mathcal {B}(K \rightarrow \pi \nu \bar{\nu })\) would provide a very valuable modelbuilding information.
The paper is organised as follows: in Sect. 2 we briefly review the main formulae to evaluate \(\mathcal {B}(K\rightarrow \pi \nu \bar{\nu })\) within and beyond the SM. In Sect. 3 we discuss the effective field theory (EFT) approach to LFU violations based on the \(U(2)_q\times U(2)_\ell \) flavour symmetry and, in that framework, we analyse the possible impact on \(K\rightarrow \pi \nu \bar{\nu }\) decays. In Sect. 4 we focus in particular on the expected correlations between \(K\rightarrow \pi \nu \bar{\nu }\), the \(R_{D^{(*)}}\) anomaly, and \(B\rightarrow K^{(*)}\nu \bar{\nu }\), which turn out to be closely related observables (impact and constraints from other observables are briefly mentioned at the end of the section). The results are summarised in the conclusions.
2 The \(K\rightarrow \pi \nu \bar{\nu }\) decays
3 The EFT approach to LFU violations based on \(U(2)_q\times U(2)_\ell \)
As already anticipated, the Bphysics anomalies observed so far point toward NP coupled mainly to the third generation of SM fermions with some small (but nonnegligible) mixing with the light generations. In addition, all effects observed so far are well compatible with NP only involving lefthanded currents. Lefthanded fourfermion operators are also the most natural candidates to build a connection between anomalies in charged and neutralcurrent semileptonic processes. These observations have led to identify the EFT approach based on the \(U(2)_q\times U(2)_\ell \) flavour symmetry as a convenient framework (both successful and sufficiently general) to analyse Bphysics anomalies and discuss possible correlations with other lowenergy observables [14, 25, 26].
 1.
the field content below the NP scale \(\Lambda > (G_F)^{1/2}\) is the SM one;
 2.
the Lagrangian is invariant under the flavour symmetry \(U(2)_q\times U(2)_\ell \), apart from the breaking induced by the spurions \(V_q\) and \(V_\ell \);
 3.
NP is directly coupled only to lefthanded quark and lepton singlets in flavour space (i.e. only operators containing only \(q_{3L}\) or \(\ell _{3L}\) fields are affected by treelevel matching conditions at the NP scale \(\Lambda \)).
4 Physical observables
4.1 The \(R_{D^{(*)}}\) anomaly
4.2 LFUviolating contributions to \(K\rightarrow \pi \nu \bar{\nu }\)
4.3 Correlations between \(B\rightarrow K^{(*)}\nu \bar{\nu }\) and \(K\rightarrow \pi \nu \bar{\nu }\)
4.4 Constraints and connections to other observables
\(b\rightarrow s \tau ^+\tau ^\). FCNC decays of B mesons with a \(\tau ^+\tau ^\) pair in the final state arise at leading order in the breaking of \(U(2)_q\times U(2)_\ell \). This implies that these processes can be directly related both to the \(R_{D^{(*)}}\) anomalies, and to the two neutrino modes discussed above. The current experimental limits on \(\mathcal {B}(B\rightarrow K\tau ^+\tau ^)\) are four orders of magnitude larger than the corresponding SM prediction (which lies in the \(10^{7}\) range). The Belle II experiment is expected to improve these limits by at least one order of magnitude, reaching the \(10^{4}\) level [47]. While the value predicted in the SM would still be out of reach, this sensitivity could be interesting in the NP framework introduced above. The relevant NP Wilson coefficient are \(C_{9,\tau }^\mathrm{NP} = C_{10,\tau }^\mathrm{NP} = C_{9,\mu }^\mathrm{NP}/\epsilon _\ell ^2\), and the branching ratio depends quadratically on \(R_0 (1+c_{13})\) in the limit where the NP contribution is large. Setting \(R_{D^{(*)}}\) to the central value in (4.7), and imposing the constraints on \(\theta _q\) from \(\mathcal {B}\rightarrow K^{(*)}\nu \bar{\nu }\) and \(K\rightarrow \pi \nu \bar{\nu }\), one gets an enhancement of a factor \(10^{2}\div 10^{3}\) in \(\mathcal {B}(B\rightarrow K\tau ^+\tau ^)\) if \(\left \frac{1c_{13}}{1+c_{13}}\right \lesssim 20\%\) (which is a rather natural choice of parameters). Finally, it is interesting to note that the observation of \(b\rightarrow s\tau ^+\tau ^\) transitions, together with \(s\rightarrow d\nu \bar{\nu }\) and \(b\rightarrow s\nu \bar{\nu }\), would allow to fix the three dimensionless parameters \(c_{13}\), \(\theta _q\), and \(\phi _q\) entering the Lagrangian (3.7), thus completely determining the leading free parameters of the EFT.
Loop effects. The running from the scale \(\Lambda \) to the electroweak scale, starting from the NP semileptonic Lagrangian (3.7), does generate nonvanishing contributions to fourquark and fourlepton operators. The contributions to \(K\bar{K}\) and \(B_s\bar{B}_s\) mixing, as well as to flavourchanging \(Zq\bar{q}\) interactions, are suppressed at least by the \(\tau \) mass, and turn out to be several orders of magnitude below present experimental constraints.
It is on the other hand well known that running effects due to quark loops, leading to purely leptonic operators [42, 43], are potentially more problematic because of precise constraints from leptonic \(\tau \) decays. In concrete models, additional UV contributions to the same effective operators will arise from the matching at the scale \(\Lambda \). These contributions can help satisfying the \(\tau \) decay constraints, but can also constitute a problem for meson mixing. On general grounds, satisfying all the constraints in a concrete UV completion that incorporates both the \(b\rightarrow c\tau \bar{\nu }\) and the \(b\rightarrow s\ell ^+\ell ^\) anomalies is not straightforward. However, as shown in [26], this result can be achieved with a moderate tuning of parameters. Given the modeldependence of the radiative constraints, we do not take them into account in the present analysis whose main focus is on semileptonic decays.
5 Conclusions
Recent Bphysics data hints toward violations of Lepton Flavour Universality in charged and neutralcurrent semileptonic processes. The most natural explanation of these phenomena, if both will be confirmed as evidences of physics beyond the SM, is the hypothesis of a new interaction in the TeV range that couples mainly to thirdgeneration fermions. If a CKMlike relation connects NP effects in B and K physics, it is natural to expect sizeable deviations from the SM in \(K\rightarrow \pi \nu \bar{\nu }\) decays, which are the only \(s\rightarrow d\) transitions that involve thirdgeneration leptons in the final state.
To quantify possible NP effects in \(K\rightarrow \pi \nu \bar{\nu }\) decays in sufficiently general terms, being motivated by present Bphysics anomalies, we have considered an EFT based on the hypothesis of a \(U(2)_q\times U(2)_\ell \) flavour symmetry acting on the light generations of lefthanded fermions, broken in the quark sector by the small CKMlike spurion \(V_q\) connecting third and light generations (and similarly broken by a small spurion \(V_\ell \) in the lepton sector). We further assumed that NP is coupled only to the lefthanded thirdgeneration flavour singlets (\(q_{3L}\) and \(\ell _{3L}\)). Because of the freedom in the choice of the flavour basis, the spurions \(V_{q,\ell }\) can enter the definition of the flavour singlets with an arbitrary mixing parameter of order one. The latter control the communication of NP effects from processes with thirdgeneration fermions only, to processes with light generations. This setup is not the most general one compatible with the \(U(2)_q\times U(2)_\ell \) flavour symmetry, but it covers a wide class of the most motivated explicit models so far proposed to address Bphysics anomalies.
In this framework, we focussed attention on semileptonic transitions involving only \(\tau \) leptons and \(\tau \) neutrinos. These processes are completely determined by four real parameters: the overall scale of the new interactions \(\Lambda \), the two modeldependent real (mixing) parameters \(\theta _q\) and \(\phi _q\) defining the (quark) flavour basis, and the relative strength of the electroweaktriplet and singlet NP interactions \(c_{13}\). The measurement of the LFU ratios \(R_{D^{(*)}}\) can be used to fix the NP scale \(\Lambda \) in terms of \(\theta _q\) and \(\phi _q\). This allows one in turn to study the neutrino FCNC transitions \(K\rightarrow \pi \nu \bar{\nu }\) and \(B\rightarrow K^{(*)}\nu \bar{\nu }\), as well as \(B\rightarrow K^{(*)}\tau ^+\tau ^\), as functions of the three remaining parameters naturally expected to be of O(1).
We have shown that, for natural values of the free parameters, sizeable and closely correlated deviations from the SM of both neutrino modes are expected. The electroweaktriplet operator alone necessarily causes a suppression of \(\mathcal {B}(K^+\rightarrow \pi ^+\nu \bar{\nu })\), due to the interference of NP with the SM amplitude which is always destructive. This suppression could be as large as \(30\%\), relative the SM value. If, on the other hand, also an electroweak singlet interaction is present, arbitrary modifications of \(\mathcal {B}(K^+\rightarrow \pi ^+\nu \bar{\nu })\) are possible. The strongest constraint on the allowed size of these deviations comes from the present bounds on \(\mathcal {B}(B\rightarrow K^{(*)}\nu \bar{\nu })\), which, however, do not exclude O(1) enhancements in \(\mathcal {B}(K^+\rightarrow \pi ^+\nu \bar{\nu })\), as illustrated in Fig. 2.
Order of magnitude enhancements of \(b\rightarrow s\tau ^+\tau ^\) compared to the SM are possible in this class of NP models. However, these transitions are very challenging from the experimental point of view. In principle, the combined measurement of \(R_{D^{(*)}}\), \(\mathcal {B}(K^+\rightarrow \pi ^+\nu \bar{\nu })\), \(\mathcal {B}(B\rightarrow K^{(*)}\nu \bar{\nu })\), and \(\mathcal {B}(B\rightarrow K\tau ^+\tau ^)\) would allow one to completely determine the leading parameters of the EFT. The correlation with other observables is less straightforward: violations of \(\mu /e\) universality in \(b\rightarrow s\ell \bar{\ell }\) transitions are a natural prediction of this framework; however, their size and the correlation with NP effects in the neutrino modes are controlled by additional free parameters.
Summarising, \(K\rightarrow \pi \nu \bar{\nu }\) decays could be significantly affected by the nonstandard LFUviolating interactions hinted by present Bphysics data. The forthcoming measurement of \(\mathcal {B}(K^+\rightarrow \pi ^+\nu \bar{\nu })\) by the NA62 experiment at CERN will provide an important insight on this class of NP models. The general expectation is a sizeable deviation from the SM, which, however, could result also into a significant suppression. Should a deviation from the SM prediction be observed in this channel, its correlation with NP effects in \(\mathcal {B}(B\rightarrow K^{(*)}\nu \bar{\nu })\) and, possibly, \(\mathcal {B}(B\rightarrow K\tau ^+\tau ^)\), would allow one to reveal the flavour structure of this new interaction.
Footnotes
Notes
Acknowledgements
We thank Admir Greljo and David Marzocca for many useful discussions. This research was supported in part by the Swiss National Science Foundation (SNF) under contract 200021159720.
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