# Footprints of leptoquarks: from \( R_{K^{(*)}} \) to \( K \rightarrow \pi \nu \bar{\nu }\)

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## Abstract

Rare \(K \rightarrow \pi \nu \bar{\nu }\) decays, being dominated by short distance contributions within the standard model (SM), open a window for new physics (NP) searches at low energies. The \( K \rightarrow \pi \nu \bar{\nu }\) branching ratios are expected to be measured with \( \sim 10\% \) accuracies by NA62/CERN and KOTO/JPARC. The theoretical uncertainties of branching ratios within the SM are well under control. In the *B* sector, it is tentative to explain the *B*-meson anomalies \( R_{D^{(*)}} \) and/or \( R_{K^{(*)}} \) by effects of physics beyond the SM. Although NP seems to be present in the third fermion generation it might also manifest in the flavor changing neutral current transition \( s \rightarrow d \). Together with the anticipated good experimental sensitivities and accurate theoretical predictions for \( K\rightarrow \pi \nu \bar{\nu }\), this motivates studies of correlated effects of NP in rare \( K\rightarrow \pi \nu \bar{\nu }\) and \( B \rightarrow K^{(*)} \mu ^+ \mu ^- \) decays. Here we consider the loop induced effects in \( K \rightarrow \pi \nu \bar{\nu }\) in two leptoquark models designed to address lepton-flavor universality violation in the \( R_{K^{(*)}} \) anomalies.

## 1 Introduction

There has been an increasing interest in extensions of the Standard Model (SM) incorporating Lepton Flavor Universality Violation (LFUV), motivated by the measurements of the LFU sensitive observables \( R_{K^{(*)}}, R_{D^{(*)}} \) in *B*-meson decays. Individually, to mention first neutral current processes, the tension in the measurements of \( R_K \) and \( R_{K^*} \) (testing lepton universality among muons and electrons), the latter over two distinct di-lepton invariant mass bins, are 2.1–2.6\(\sigma \) away from the predictions made in the SM [1, 2]. The tension is even more significant for charged current processes (testing lepton universality among taus and the light charged leptons), with individual tensions in the range \( \sim (2-3.4) \sigma \), see Refs. [3, 4, 5, 6, 7, 8]. Very recently, the measurement of the ratio \( R_{J/\psi } \), with a tension with the SM of \( \sim 2\sigma \) [9], adds a further piece of evidence for LFUV in \(b \rightarrow c\) transitions, while tensions are also found in branching ratios and angular observables of the \( b \rightarrow s \mu \mu \) transition [10, 11, 12, 13, 14, 15, 16, 17]. Though it is certainly valid to be cautious on interpreting these tensions as true manifestations of New Physics (NP), their good theoretical control (hadronic uncertainties largely cancel out in the ratios) and coherency in terms of a LFUV picture justifies different attempts to extend the SM.^{1}

New effects in flavor changing neutral current (FCNC) \( b \rightarrow s \) and/or charged current \( b \rightarrow c \) semi-leptonic decays are likely to be accompanied by new effects in other quark-flavor transitions. The correlation depends strongly on the specific NP model, and a good candidate for explaining \( R_{K^{(*)}} \) and/or \( R_{D^{(*)}} \) must satisfy other experimental constraints, such as low-energy flavor data, high-precision electroweak observables, as well as high-energy collider data. More concretely, the authors of [18] have adopted an Effective Field Theory approach assuming that NP couples dominantly to the third generation. They focused on the transitions involving \(\tau \) leptons and \(\tau \) neutrinos and found that branching ratios for \(K \rightarrow \pi \nu \bar{\nu }\) could exhibit large deviations from the SM predictions, if the ratios \(R_{D^{(*)}}\) get modified by \(\sim 20\%\) relative to the SM prediction. Correlations of different flavor sectors, including \(K \rightarrow \pi \nu \bar{\nu }\), have also been systematically investigated in Ref. [19], for the case where a particular class of dimension six operators is added on top of the SM. Also, the correlation between \(\epsilon '/\epsilon \) and rare kaon decays in the context of leptoquark models has been explored recently in Ref. [20]. The authors of Ref. [21] explored the flavor structure of custodial Randall–Sundrum (RS) models in the light of observed deviations in the *B* decays and found that the \(K \rightarrow \pi \nu \bar{\nu }\) decays might distinguish among different scenarios of lepton compositeness.

Motivated by these efforts, we are interested here in FCNC \( s \rightarrow d \nu \bar{\nu }\) (and \(s \rightarrow d \ell ^+ \ell ^-\)) transitions. In particular, the branching ratio for \( K^\pm \rightarrow \pi ^\pm \nu \bar{\nu }\) has been measured with \( \sim 100\% \) uncertainty, while for the \( K_L \rightarrow \pi ^0 \nu \bar{\nu }\) decay rate only an upper bound two orders of magnitude above the SM expectation value is known at present. This situation will improve in the coming years, since the experiments NA62 at CERN and KOTO at JPARC will achieve experimental accuracies of around ten-percent. On the theoretical front, the uncertainties are well under control due to the absence of long-distance effects, that conversely plague the analogous \( s \rightarrow d \ell ^+ \ell ^- \), or radiative \( s \rightarrow d \), transitions (and analogous annihilation topologies).

In this paper, we focus our attention on leptoquark (LQ) models. More specifically, we consider two distinct LQ attempts to explain LFUV data, whose phenomenological aspects have been discussed recently in the literature: one model where the new contributions to \( b \rightarrow s \mu ^+ \mu ^- \) appear first at the loop-level, and a second LQ model where a contribution to \( b \rightarrow s \mu ^+ \mu ^- \) is present already at tree-level.

The paper is organized as follows. In Sect. 2 we introduce the LQ models under consideration and comment on their roles in *B*-meson decays. Then, in Sect. 3, we discuss their contributions to the decays \( K^\pm \rightarrow \pi ^\pm \nu \bar{\nu }\) and \( K_L \rightarrow \pi ^0 \nu \bar{\nu }\). In Sect. 4 we conclude with our final comments. In Appendix A we provide some numerical values and in Appendix B we discuss technical issues related to the loop-functions, together with a brief discussion of the process \( K^\pm \rightarrow \pi ^\pm \mu ^+ \mu ^- \) in the light of the two LQ models.

## 2 Framework

The *B*-meson puzzles have been interpreted by means of an effective Lagrangian approach [22, 23] and the structure of the four-fermion Lagrangian explaining \( R_{D^{(*)}} \) and/or \( R_{K^{(*)}} \) anomalies has been well studied. However, it was found in a full-fledged model, matched onto the effective Lagrangian, that it is very difficult to accommodate within \(1\sigma \) the large value of \(R_{D^{(*)}}\) [24]. On the other hand it seems that \( R_{K^{(*)}} \) can be explained by a number of leptoquarks which contribute to \(b\rightarrow s \mu ^+ \mu ^-\) transition either at the tree-level [22, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] or at the loop-level [40, 41, 42].

### 2.1 New physics in \(R_{K^{(*)}}\)

*B*-meson decays. While to full generality NP could be present in semi-leptonic operators involving left- and/or right-handed currents, semi-leptonic operators involving electrons, or still electromagnetic dipole operators, in the NP cases discussed here only the left-handed currents operator \( (\bar{s} \gamma _\mu P_L b) \times (\bar{\mu } \gamma ^\mu P_L \mu ) \) will be present. We then define the effective Lagrangian at the scale 4.8 GeV

^{2}

^{3}To explore the correlations of \( K \rightarrow \pi \nu \bar{\nu }\) rates with \( R_K \), we consider the linearized expression [45]

*B*-anomalies. In our study the ratios \(R^{(*)}_{\nu \nu }\) are defined as

*K*and \(K^*\) channels [46, 47] (both at \(90\%\) CL)

### 2.2 Scalar leptoquarks

Leptoquarks are particularly interesting since they allow interactions between quarks and leptons (see e.g. [48] for a review), thus suggesting a unification of these two classes of fermions. Although we could consider scalar or vector leptoquarks, the phenomenological study of vector LQs would require either the introduction of a UV cut-off scale for calculations beyond the tree-level, or the consideration of full fledged models [22, 49, 50, 51]. In view of this feature, we prefer not to discuss the phenomenological aspects of vector LQ models regarding \( s \rightarrow d \nu \bar{\nu }\) transitions, but rather note that Refs. [27, 52] discuss some tree-level effects.

According to the classification in [48], scalar leptoquarks which are doublets of \(SU(2)_L\) cannot destabilize the proton via the diquark coupling, since \(3 B+L =0\), *B* and *L* being baryon and lepton numbers,^{4} and therefore can be relatively light. Moreover, it was pointed out in [24, 55] that the weak triplet leptoquark, although having \(3B +L = 2\), when embedded in a *SU*(5) GUT model does not destabilize the proton. The weak doublet leptoquarks may have weak hypercharges 7 / 6 or 1 / 6. In what follows, different leptoquarks are denoted by their transformation under \((SU(3), SU(2)_L, U(1)_Y)\) and we adopt here the notation introduced in Ref. [48].

#### 2.2.1 \(R_2 (\mathbf {3}, \mathbf {2}, 7/6) \) scalar LQ

New physics exclusively in semi-leptonic processes naturally evokes LQ frameworks since flavor changing processes involving four-lepton and four-quark transitions are loop suppressed. Despite that, it might seem sensible to invoke a LQ model that contributes to \(b \rightarrow s \ell \ell \) process at one-loop level (with contributions to \( b \rightarrow c \ell \nu \) at tree-level) and thus mimics the SM amplitudes hierarchy, and may therefore be adequate to address \( R_{K^{(*)}} \) and \( R_{D^{(*)}} \) anomalies simultaneously. An approach along these lines has been followed in [40] but large couplings needed to explain \(R_{K^{(*)}}\) at loop-level and \(R_{D^{(*)}}\) at tree-level are difficult to reconcile with all available flavor constraints [56]. Here we follow the approach of [41] where the anomalies in \( b \rightarrow s \ell \ell \) decays are accounted for at one-loop level.

*K*decays, and then the PMNS matrix reduces to the identity matrix, \( \mathbf {1}_3 \). In order to explain \( R_{K^{(*)}} \) anomalies it was suggested in Ref. [41] that the Yukawa matrices \( g_R \) and \( g_L \) have the following textures

With the ansatze for \(g_R\) and \(g_L\) in Eq. (8) and the aforementioned constraints, tree-level contributions to \( b \rightarrow c \ell \nu \), and contributions to \( b \rightarrow s \ell \ell \) involving right-handed currents, are absent. Note, however, that a different coupling texture was used in Refs. [57, 58] to explain the \( R_{D^{(*)}} \) anomalies. See also [59] for a different texture of LQ Yukawa couplings.

- (a)
in order to explain the \( R_{K^{(*)}}\) anomalies we are compelled to adjust the NP Wilson coefficient \( \delta C_{9 \mu } = - \delta C_{10 \mu } \),

- (b)
the constraint coming from the difference between the experimental and theoretical results for the muon anomalous magnetic moment \(a_\mu ^\mathrm {exp}-a_\mu ^\mathrm {SM} =\Delta a_\mu = (2.88 \pm 0.63 \pm 0.49)\times 10^{-9} \) [60, 61] (see, e.g., [62] for the breakdown of uncertainties),

- (c)
the precise experimental value of \( \mathrm {Br}(Z \rightarrow \mu \mu ) \), measured at LEP (cf. Ref. [63]).

Note that LHC constraints of flavored processes at high energies are becoming sensitive to flavor couplings employed in low-energy flavor phenomenology, see, e.g., [66]. Their analysis sets constraints on effective four-fermion operators contributing to \( p p \rightarrow \mu ^+ \mu ^- \) at the tail of the di-lepton invariant mass spectrum. Therefore, the results apply directly for a NP spectrum beyond \( \mathrm {few} \) TeV, but they may still remain indicative at the region 1 TeV, in which case a rough estimate of \(g_L^{c \mu }\) from their study gives \( | g_L^{c \mu } | \lesssim 0.8 \). This is not very different from the values used in our analysis, but for a more precise knowledge of this coupling, dedicated analyses of LHC data would clarify the situation.

#### 2.2.2 \(S_3({\bar{\mathbf {3}}}, \mathbf {3}, 1/3)\) scalar LQ

We now discuss a different mechanism for explaining the anomalies in \( b \rightarrow s \ell \ell \) data. In Refs. [24, 55], a model with two LQs has been considered, namely \( S_3 = ({\bar{\mathbf {3}}}, \mathbf {3}, 1/3) \) and \( \tilde{R}_2 = (\mathbf {3}, \mathbf {2}, 1/6) \). The authors of that model noticed that these two leptoquark states are important for the one-loop neutrino mass mechanism within the framework of grand unification theory (GUT) [24, 55]. In Ref. [24] an attempt was done to explain all *B*-meson anomalies in this GUT setup, accounting for all the existing low-energy constraints. As already noticed by the authors of [34, 47] the current bounds on \(R_{\nu \bar{\nu }}^{(*)}\) are rather restrictive for the leptoquark models and since \(S_3\) and \(\tilde{R}_2\) contribute to \(B \rightarrow K^{(*)} \nu \bar{\nu }\) at tree-level they cannot fully accommodate the \(R_{D^{(*)}}\) anomaly. The role of \(\tilde{R}_2\) in LFU anomalies in this setting has been shown to be minor and thus we omit the \(\tilde{R}_2\) state from further discussion and focus on scenarios with light \(S_3\) as studied in [24].

*y*assumes the following texture [24],

## 3 Leptoquarks in rare *K* decays

### 3.1 \(R_2 (\mathbf {3}, \mathbf {2}, 7/6) \) scalar LQ in \(K \rightarrow \pi \nu \bar{\nu }\)

*W*boson and \( R_2 \) have been integrated out at the energy scale \( \mu _{tW} = \mathcal {O} (m_t, M_W, M_{R_2}) \), one is left with the operator structures \( (\bar{s} \gamma ^\mu P_L q') \times (\bar{q} \gamma _\mu P_L d) \), \( (\bar{q}' \gamma ^\mu P_L q) \times (\bar{\nu }_\ell \gamma _\mu P_L \nu _\ell ) \), \( q, q' = u, c \), and \( (\bar{s} \gamma ^\mu P_L d) \times (\bar{\nu }_\ell \gamma _\mu P_L \nu _\ell ) \), similarly to the SM case after the top, the

*W*and the

*Z*bosons have been integrated out from the full theory. The following factor summarizes short-distance QCD corrections to the charm box contribution (after neglecting the bottom quark threshold), taking leading order (LO) anomalous dimensions from [67]

^{5}There is a large dependence of the LO value of \( \bar{\eta }^{(R_2)}_{cc} (\mu _{tW}) \) on the value of \( \mu _{tW} \): varying \( \mu _{tW} \) over the interval \( [M_W / 2 , 1~\mathrm{TeV}] \), results in \( \bar{\eta }^{(R_2)}_{cc} (\mu _{tW}) \) in the range [0.4, 0.9] , where the smaller value corresponds to \( \bar{\eta }^{(R_2)}_{cc} (1~\mathrm{TeV}) \). The situation is simpler in the cases of the contributions proportional to \( \lambda _t = V_{td} V^*_{ts} \), resulting from the box with two top quarks, and \( V^*_{ts} V_{cd} \) or \( V^*_{cs} V_{td} \), resulting from the boxes with top and charm quarks, since there only the operator \( (\bar{s} \gamma ^\mu P_L d) \times (\bar{\nu }_\ell \gamma _\mu P_L \nu _\ell ) \) is required. Note that for \( K_L \rightarrow \pi ^0 \nu \bar{\nu } \), the top contribution is the only one relevant; for different reasons, the top contribution is largely dominant for the transition \( b \rightarrow s \ell ^+ \ell ^- \).

A few further precisions are required. To calculate the interference of the LQ contributions with the SM, we employ the values \( X^e_c = (11.31 \pm 0.74) \times 10^{-4} \) and \( X^\tau _c = (7.77 \pm 0.62) \times 10^{-4} \) [76] (with uncertainties estimated from a very conservative uncertainty for the mass of the charm-quark), and \( \delta P_{c, u} \approx 0.04 \pm 0.02 \). Varying these values over their quoted uncertainty ranges, and the ones provided in Appendix A, we obtain modulations of the relative LQ contributions below \( 1\% \), that we neglect in our calculations. Furthermore, to make the LQ effects transparent in the plots that will be shown, we do not include the SM uncertainties in Eq. (15), to which the NP contribution is also partly sensitive, which are \( \sim 10\% \) for the charged channel and \( \sim 5\% \) for the neutral mode.

In Fig. 2 we show the correlation between \( R_K [1.1, 6.0] \) and \( \mathrm {Br}( K^+ \rightarrow \pi ^+ \nu \bar{\nu } ) \), which is due to the common couplings \( g_L^{c \mu } \) and \( g_L^{t \mu } \). With respect to the SM values, the highest variations are \( 8\% \) and \( 5\% \) for \( K^+ \rightarrow \pi ^+ \nu \bar{\nu } \) and \( K_L \rightarrow \pi ^0 \nu \bar{\nu } \), respectively. Compared to the analogous contribution accommodating the \(b \rightarrow s \ell ^+ \ell ^- \) anomalies, the effects in \(K \rightarrow \pi \nu \bar{\nu }\) are smaller due to different CKM factors. In particular, the contributions to the charged mode proportional to the large LQ coupling to the charm do not overcome the overall suppression of the LQ mass. Compared to the analogous contribution accommodating the \(b \rightarrow s \ell ^+ \ell ^- \) anomalies, the effects in \(K \rightarrow \pi \nu \bar{\nu }\) are slightly smaller for the charged channel, but of similar relative size.

### 3.2 \(S_3({\bar{\mathbf {3}}}, \mathbf {3}, 1/3) \) scalar LQ in \(K \rightarrow \pi \nu \bar{\nu }\)

#### 3.2.1 Tree-level effects

*B*anomalies explanation require finite \( y_{s \mu }\) and \(y_{s\tau }\). This would then make the lepton flavor violating decay \( \tau \rightarrow \mu K\) an important constraint. The experimental bound for \( \tau \rightarrow \mu K_S^0\) [62, 90] implies

In Scenario II, non-zero \(y_{d \tau }\) could potentially induce \(s \rightarrow d \nu \bar{\nu }\) via \(y_{s\tau } y_{d\tau }\). In this case, \(y_{d\tau }\) does not interfere with the SM contribution to \(\pi \rightarrow \mu \bar{\nu }\) and therefore the bound on \(y_{d\tau } y_{s\tau }\) is of order 1 in the \(1\sigma \) parameter space of Scenario II. Much better sensitivity is expected in \(s \rightarrow d \nu \bar{\nu }\) processes where this term does interfere with the SM amplitude.

For such a small coupling, effects induced in the up-type sector by \( y_{d \mu } \ne 0 \) (cf. Eq. (9)) are negligible and the analysis in [24] is not modified. Moreover, new contributions to atomic parity violation, and corrections to meson-mixing in the system of kaons [91] are also negligible. Further note that since we do not allow for couplings to electrons, \( e - \mu \) conversion receives no contribution from \( S_3 \) exchanges up to one-loop corrections.

#### 3.2.2 Radiative Yukawas and boxes

*W*boson changes flavor of the down-type quark external leg, (V) a vertex correction to the coupling of \( S^{1/3}_3 \) where a

*W*is exchanged, and (VT) a second vertex correction to \( S^{1/3}_3 \) where a coupling to the

*W*gauge boson,

*W*and a \( S^{2/3}_3 \) LQ are exchanged. Note that the crossed box, (Box’), is only possible for charged leptons in the final state, i.e., in the process \(s\rightarrow d \ell \ell \).

*g*and powers of \( y_{ij} \) (\( i=s,b \) and \( j=\mu ,\tau \)). It is interesting to note that (SE), (V) and (VT) with an internal top have a relative factor compared to the SM top contribution of \( y_{s \tau } y_{b \tau }/ (g^2 \lambda ^2)\). Together with the loop-functions, possibly resulting in logarithmic enhancements, this sets up the hierarchy of NP contributions.

^{6}

Flavor and gauge coupling factors present in the calculation of the LQ contributions to \( K \rightarrow \pi \nu \bar{\nu }\) compared to the SM, with internal quark-flavors given by the columns and loop-topologies by the lines. The external flavors of the neutrinos are dominantly \( \nu _\tau \) (in Scenario II discussed in the text). The unitarity of the CKM matrix has been employed to include the contributions of the up-quark into those of the charm- and top-quarks. We assume \( y_{s \tau } \gg \lambda ^2 y_{b \tau } \) and \( y_{b \tau } \gg \lambda ^2 y_{s \tau } \). The gauge coupling constant of the gauge group \( SU(2)_L \) is denoted by *g*. (For the flavor factors appearing in the calculation of the LQ contributions to \( K \rightarrow \pi \mu ^+ \mu ^- \), one replaces the tauonic flavor for the muonic one.)

| | |
| | |
---|---|---|---|---|---|

(SM) | \( g^4 \lambda \) | \( g^4 \lambda ^5 \) | |||

(SE), (V), (VT) | \( g^2 \lambda y_{s \tau }^2 \) | \( g^2 \lambda ^3 y_{s \tau } y_{b \tau } \) | |||

(Box) | \( g^2 \lambda y_{s \tau }^2 \) | \( g^2 \lambda ^3 y_{s \tau } y_{b \tau } \) | \( g^2 \lambda ^5 y_{b \tau }^2 \) |

^{7}

^{8}The above numerical values are also enhanced by a constructive interference among the two vertex topologies (in the unitary gauge). Despite many enhancements, resulting in the large numerical pre-factors seen above, the net effect is of order 10% for both modes, due to the strong constraints the different LQ couplings are subjected to, cf. Sect. 2.2.2. This shows the importance of the phenomenological analysis of [24] on studying the effects of \( S_3 \) in rare kaon decays. This is illustrated in Fig. 4 for \( K_L \rightarrow \pi ^0 \nu \bar{\nu }\), where we show the correlation with \( R^*_{\nu \nu } \).

### 3.3 Comparison of models

Regarding correlations with the *B* sector, we note that in the \( R_2 \) model we have the same couplings for the processes \( b \rightarrow s \mu ^+ \mu ^- \) and \( s \rightarrow d \nu \bar{\nu }\). This is due to the requirement that Yukawa couplings of \(R_2\) to tau neutrinos are suppressed in order to enhance couplings to muons and thus the neutrinos have muonic flavors. In the \( S_3 \) model, the correlation among \( b \rightarrow s \mu ^+ \mu ^- \) and \( s \rightarrow d \nu \bar{\nu }\) depends on the way corrections to rare kaon decays are generated: when they first arrive at the tree-level as discussed above, a correlation shows up due to the coupling of \( S_3 \) to the strange-quark present in both processes; in the case rare kaon decays are generated at one-loop level, the main contribution is due to the couplings to taus and the correlation with \( b \rightarrow s \ell ^+ \ell ^- \) is not significant, but instead a correlation with \( b \rightarrow s \nu \bar{\nu }\) is relevant. To summarize the effects of the two LQ models in rare kaon decays, we show in Fig. 5 the correlation of the two kaon channels.

We now shift to a short discussion of the model in [34], which contains the triplet \( S_3 \) together with a weak-singlet \( S_1 = (\bar{\mathbf {3}}, \mathbf {1}, 1/3) \). By construction, in [34] there is no contribution at tree-level to \( s \rightarrow d \nu \bar{\nu }\). This is not valid anymore after one-loop corrections are taken into account. For \( S_3 \), we have exactly the same contribution as in our present case, i.e., the same one-loop generated coupling of \( S_3 \) to a down-quark and a neutrino. For \( S_1 \), the discussion is very similar: the relevant topologies are the vertex diagram called “V” on Fig. 3 and the self-energy diagram while the diagram with *W* boson coupling to \(S_1\) is absent. Here as well, we have checked that our calculation is gauge invariant (see Appendix B for more details). With the numerical values found in [34] and both LQ masses degenerate at 1 TeV, \( K^\pm \rightarrow \pi ^\pm \nu \bar{\nu }\) gets suppressed by \( \approx 24\% \), while \( K_L \rightarrow \pi ^0 \nu \bar{\nu }\) gets suppressed by \( \approx 34\% \) with respect to the SM prediction. These are larger effects with respect to the case \( S_3 \) treated above due to the large values of \( y_{b \tau }, y_{s \tau } \) found in [34].

### 3.4 Comment on the process \( K \rightarrow \pi \mu ^+ \mu ^- \)

In Ref. [92] the authors noticed that four-fermion operators that can explain the *B*-physics anomalies leave their imprint also on the analogous \( s \rightarrow d \ell ^+ \ell ^- \) and \( s\bar{d} \rightarrow \ell ^+ \ell ^- \) transitions. In the SM, the process \( K^+ \rightarrow \pi ^+ \ell ^+ \ell ^- \) is dominated by long-distance effects. Despite that, small differences in the electron with respect to the muon channel could provide additional evidence for LFUV NP. However, the NP effects discussed here provide modulations much below actual experimental errors. See Appendix B.3 for further details.

## 4 Conclusion

In case the Lepton Flavor Universality violating observables in *B*-physics – \( R_{D^{(*)}} \), \( R_{K^{(*)}} \) – are confirmed as true New Physics messengers, it becomes of fundamental importance to look for their signatures in other low-energy processes, in order to unveil their flavor structure. In this respect, the rare kaon decays \( K \rightarrow \pi \nu \bar{\nu }\), in both charged and neutral channels, are very clean theoretical probes of New Physics. Moreover, in the coming years the measurement of their branching ratios is envisaged with \( \sim 10\% \) precision.

We have considered two scenarios of scalar leptoquarks which explain the \(R_{K^{(*)}}\) anomaly. In the first scenario, the weak-doublet leptoquark \(R_2\) contributes to \(B \rightarrow K^{(*)} \mu ^+ \mu ^-\) decay amplitude at loop level. In the second scenario the weak triplet leptoquark \(S_3\) contributes to \(B \rightarrow K^{(*)} \mu ^+ \mu ^-\) at tree level.

In the scenario with \(R_2\) leptoquark, new contributions to \(s\rightarrow d\nu \bar{\nu }\) transitions are also radiatively suppressed and imply mild corrections of the order \(10\%\) relative to the Standard Model predictions for the branching ratios of \(K \rightarrow \pi \nu \bar{\nu }\). In scenario with \(S_3\) leptoquark, tree-level contributions to \( s \rightarrow d \nu \bar{\nu }\) transitions are in principle possible and may largely enhance the \(K \rightarrow \pi \nu \bar{\nu }\) rates. In case the tree-level \(S_3\) contributions are set to zero, the \( 1/\lambda ^2 \) Cabibbo-enhanced loop contributions could provide large enough shifts in \( K \rightarrow \pi \nu \bar{\nu }\) to be observed in future experiments. Furthermore, when we allow for tree-level Yukawas of \(S_3\) to down quarks, sizable enhancement of the branching ratio for \( K^\pm \rightarrow \pi ^\pm \nu \bar{\nu }\) cannot be excluded. More generally, the combined measurement of \( K^\pm \rightarrow \pi ^\pm \nu \bar{\nu }\) and \( K_L \rightarrow \pi ^0 \nu \bar{\nu }\) may provide evidence in favor of one or the other model. This discussion may be generalized to other leptoquark states, as it has been briefly pointed out for the singlet+triplet model of Ref. [34]. In addition to the leptoquark effects on the branching ratios for both decays \( K^\pm \rightarrow \pi ^\pm \nu \bar{\nu }\) and \( K_L \rightarrow \pi ^0 \nu \bar{\nu }\) , we suggest to make correlation between the branching ratios of these decays and the ratios \(R_K\) and \(R_{K^*}\), as well as with \(R_{\nu \nu }\).

The setup presented in this work could help to distinguish among leptoquark scenarios, designed to resolve the *B*-meson anomalies. The future results of both NA62 and KOTO experiments will shed more light on understanding the nature of the observed lepton flavor universality violation.

## Footnotes

- 1.
In the SM, the source of LFUV comes from Yukawa couplings of the SM Higgs to the charged leptons.

- 2.
Note that Ref. [41] used slightly different value of \(\delta C_{9\mu ,10\mu }\).

- 3.
At the energy scale 4.8 GeV, \( C^\mathrm{SM}_{9} \approx 4 \) and \( C^\mathrm{SM}_{10} \approx -4 \) [44].

- 4.
- 5.
Although the LO anomalous dimension matrix has been considered, we use the expression of the strong coupling constant up to the Next-to-Leading Order (NLO).

- 6.
We note that the same \( 1/\lambda ^2\) enhancements with respect to the SM are not possible for the processes \( B \rightarrow K^{(*)} \ell ^+ \ell ^- \) for \( S_3 \), nor for \( {R_2} \): in both LQ scenarios, the NP contributions to \(b\rightarrow s \nu \bar{\nu }\) follow the \( V_{tb} V^*_{ts} \) structure of the SM.

- 7.
Note that complex phases in the CKM matrix are

**CP**-odd while those from the loop-functions are**CP**-even. - 8.When considering the branching ratios in Eqs. (24)–(25), note that \( S_3 \) contributes to the process \( K \rightarrow \pi \mu \nu _\mu \), which is used in the extraction of the parameters \( \kappa _{+,L} \) discussed at the beginning of Sect. 3. Its short-distance contribution can be absorbed into a redefinition of the CKM matrix element \( V_{us} \) (at the tree-level):resulting in a shift which is, in any case, negligible.$$\begin{aligned} V_{us} \rightarrow V_{us} - \frac{M_W^2}{g^2 \, M_{S_3}^2} (V_{us} y_{s\mu } + V_{ub} y_{b\mu }) \, y_{s\mu } , \end{aligned}$$(26)

## Notes

### Acknowledgements

We acknowledge support of the Slovenian Research Agency through research core funding No. P1-0035. We thank Olcyr de Lima Sumensari for useful communication, and Ilja Doršner, Darius Alexander Faroughy and José Ocariz for discussions.

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