Implication of K → πνν̄ for generic neutrino interactions in effective field theories

theories Tong Li,1, ∗ Xiao-Dong Ma,2, † and Michael A. Schmidt3, ‡ 1School of Physics, Nankai University, Tianjin 300071, China 2Department of Physics, National Taiwan University, Taipei 10617, Taiwan 3School of Physics, The University of New South Wales, Sydney, New South Wales 2052, Australia Abstract In this work we investigate the implication ofK → πνν̄ from the recent KOTO and NA62 measurements for generic neutrino interactions and the new physics scale in effective field theories. The interactions between quarks and left-handed Standard Model (SM) neutrinos are first described by the low energy effective field theory (LEFT) below the electroweak scale. We match them to the chiral perturbation theory (χPT) at the chiral symmetry breaking scale to calculate the branching fractions of Kaon semi-invisible decays and match them up to the SM effective field theory (SMEFT) to constrain new physics above the electroweak scale. In the framework of effective field theories, we prove that the Grossman-Nir bound is valid for both dim-6 and dim-7 LEFT operators, and the dim-6 vector and scalar operators dominantly contribute to Kaon semi-invisible decays based on LEFT and chiral power counting rules. They are induced by multiple dim-6 lepton-number-conserving operators and one dim-7 lepton-number-violating operator in the SMEFT, respectively. In the lepton-number-conserving s → d transition, the K → πνν̄ decays provide the most sensitive probe for the operators with ττ component and point to a corresponding new physics scale of ΛNP ∈ [47 TeV, 72 TeV] associated with a single effective coefficient. The lepton-number-violating operator can also explain the observed K → πνν̄ discrepancy with the SM prediction within a narrow range ΛNP ∈ [19.4 TeV, 21.5 TeV], which is consistent with constraints from Kaon invisible decays.

These efforts require the introduction of a new invisible degree of freedom with the mass scale being around 100 − 200 MeV.
The interpretation of the KOTO result depends on not only whether the invisible particles are viewed as neutrinos, but also the experimental uncertainties. Even if we only take into account the statistical uncertainties at 95% CL for neutrino final states, there is allowed space for heavy NP beyond the SM consistent with both K L → π 0 νν and K + → π + νν measurements and satisfying the Grossman-Nir bound. As one can see from the Fig. 1 in Ref. [4], the allowed region is rather delimited and not far away from the SM prediction. It can provide a constraint on the relevant quark-neutrino interactions and shed light on the search for generic neutrino interactions in the future. Thus, without introducing any new light particles, we focus on heavy NP contributing to the generic quark-neutrino interactions and generically confine the NP scale from the allowed region of B(K L → π 0 νν) and B(K + → π + νν) measurements. As the neutrino flavor is not measured and the fermionic nature of neutrinos is not determined, the semi-invisible Kaon decays K → πνν are sensitive probes for a range of interactions.
In this work, we will use an effective field theory approach, where NP is described by a set of non-renormalizable operators which are added to the SM Lagrangian Here O (d) i are the dimension-d (dim-d in short below) effective operators. Each Wilson coefficient C is associated with a NP scale Λ NP = (C (d) i ) 1/(4−d) . We first use the low energy effective field theory (LEFT) [17,18] to describe the interactions between quarks and left-handed SM neutrinos below the electroweak scale. Then, in order to calculate the Kaon decay rate, we match the LEFT operators to chiral perturbation theory (χPT) [19,20] at the chiral symmetry breaking scale to take into account non-perturbative QCD effects. The branching fractions of Kaon semi-invisible decays are evaluated in terms of the Wilson coefficients and neutrino bilinears as external sources.
The paper is outlined as follows. In Sec. II, we describe the LEFT basis and give the quarkneutrino operators relevant for our study. The LEFT operators are matched to χPT and we show the general expressions for the branching fractions of Kaon semi-invisible decays. We then match the results to the SMEFT in Sec. III. In Sec. IV we show the implication of K → πνν for new physics and discuss other constraints. Our conclusions and some discussions are drawn in Sec. V. Some calculation details for Kaon decays are collected in the Appendix.

A. Generic quark-neutrino operators in LEFT basis
We consider the effective operators for neutrino bilinears coupled to SM quarks in the framework of LEFT obeying SU(3) c ×U(1) em gauge symmetry. In the basis of LEFT for neutrinos, the only dim-5 operator contributing to the neutrino magnetic moments is [27] where F µν is the electromagnetic field strength tensor and ν = P L ν denote left-handed active SM neutrinos. Its SMEFT completion has been investigated by Cirigliano et al. in Ref. [28]. There are also the dim-6 operators [17] with lepton number conservation (LNC, |∆L| = 0) and those with lepton number violation (LNV, |∆L| = 2) where u L (u R ) and d L (d R ) denote the left-(right-) handed up-type and down-type quark fields in mass basis, respectively. Note that the tensor operator ν C α σ µν ν β vanishes for identical neutrino flavors (with α = β). The flavors of the two quarks and those of the two neutrinos in the above operators can be different although we do not specify their flavor indexes here. For the notation of the Wilson coefficients, we use the same subscripts as the operators, for instance C V,xyαβ , where x, y denote the down-type quark flavors and α, β are the neutrino flavors.
In the following we will study the K → π transition and thus only consider the operators with down-type quarks s and d.

B. Matching to the leading order of χPT
The dim-6 quark-neutrino operators can be matched onto the meson-lepton interactions through the χPT formalism by treating the lepton currents together with the accompanied Wilson coefficients as proper external sources. The QCD-like Lagrangian with external sources for the first three light quarks (q = u, d, s) can be described as where the flavor space 3×3 matrices {l µ = l † µ , r µ = r † µ , s = s † , p = p † , t µν r = t µν † l } are the external sources related with the corresponding quark currents. One can extract the relevant external sources from the above dim-6 effective operators. On the other hand, based on Weinberg's powercounting scheme, the most general chiral Lagrangian can be expanded according to the momentum p and quark mass. The chiral Lagrangian with external sources at leading order reads [19,20] where U is the standard matrix for the Nambu-Goldstone bosons with the constant F 0 being referred to the pion decay constant in the chiral limit. The covariant derivative of U and χ are expressed in terms of the external sources where the constant B is related to the quark condensate and F 0 by B = − qq 0 /(3F 2 0 ). For the later numerical estimation, we take F 0 = 87 MeV [29] and B ≈ 2.8 GeV [28,30]. The Nambu-Goldstone bosons parameterized by U and the (pseudo-)scalar sources χ transform as U → LU R † By inspecting the dim-6 operators related to the s → dν ν transition (the symbol ' ' here indicating the neutrino pair can be either LNC νν or LNV νν 1 ), we find that only the LNC operators can contribute to the leading order chiral Lagrangian. The tensor operator O T dν only contributes to the next-to-leading order chiral Lagrangian at O(p 4 ). They lead to the following external sources After expanding U , i.e. U = 1 + i Φ F 0 + 1 2F 2 0 (iΦ) 2 + · · · and the insertion of the above external sources into the Lagrangian in Eq. (13), we obtain the effective Lagrangians for K 0 → π 0 ν ν and 1 Below we use K → πνν to generally denote the experimental processes. K → πν ν appears when both νν and νν final states can occur in the analytical expressions of the theoretical calculation unless a LNC or LNV process is specified in our discussion.
K + → π + ν ν at the leading order The above Lagrangians fit to the relation This relation is the result of the transition operators that change isospin by 1/2. By neglecting the small CP violation in K 0 −K 0 mixing, for the K L → π 0 transition, the relevant effective Lagrangian becomes where the flavor indices α, β are summed over all three neutrino generations. Note that the Wilson coefficients for the scalar operators are symmetric in the neutrino flavor indices. From the effective Lagrangian we derive the branching ratios for the decays K L → π 0 ν ν and K + → π + ν ν The details of the calculation are collected in Appendix A. The J functions parameterize the kinematics of the three-body decay and are defined as where m K L (m K + ) and Γ Exp K L (Γ Exp K + ) denote the physical mass and decay width of K L (K + ), respectively. m π 0 (m π + ) is the mass of π 0 (π + ), G F is the Fermi constant and s is the invariant squared mass of the final-state neutrino pair. From the hermiticity of the effective Lagrangian and the Cauchy-Schwarz inequality we derive the following relations for the Wilson coefficients in LEFT Note that in the second inequality we used C V,dsαβ * dν1/2 = C V,sdβα dν1/2 from the fact that the vector operator itself is hermitian. If we sum over neutrino flavors, the second relation above turns out to be α,β Based on the above inequalities, the branching ratios in Eq. (28) and Eq. (29) lead to where is defined by This result is nothing but the Grossman-Nir bound [16] expected from the isospin relation in Eq. (26) and the CP-conserving limit for neutral Kaon system. The numerical value slightly differs from the standard G-N bound value of 4.3, because we do not consider isospin breaking and electroweak correction effects beyond the mass difference in the phase space integration [31]. We obtain the G-N bound from the matching of LEFT to χPT. Thus, as expected, the Grossman-Nir bound holds for dim-6 LEFT operators in leading order χPT.
C. Dim-6 tensor operators and dim-7 operators in the chiral Lagrangian For the tensor currents in Eqs. (11), we have to go beyond the leading order of chiral La- where Λ 2 denotes the low-energy constant. In terms of the dim-6 tensor operator O T dν , the relevant tensor sources are After inserting these external sources into the O(p 4 ) Lagrangian (39), we obtain the following where We also investigate dim-7 tensor operators in LEFT. There happens to be only one such operator related with the transition K → πνν under consideration, that is which leads to the tensor sources to be By analogy we expand the O(p 4 ) Lagrangian (39) to obtain the interactions with mesons One can see that, for the next-to-leading order chiral Lagrangian with dim-6 and dim-7 tensor operators in LEFT, the isospin relation in Eq. (26) and thus the Grossman-Nir bound still hold.
Note that the Eq. (46) vanishes for massless neutrinos. This implies that the non-zero contribution from dim-7 tensor operator appears at O(p 6 ) level, and therefore is further suppressed by additional p 2 /Λ 2 χ factor. In addition, there are also two dim-7 vector-like LNV operators related to K → πνν, which we list for completeness These dim-7 operators are suppressed by p/m W compared with dim-6 operators and, like the above tensor operators, lead to sub-leading contributions. Thus, we neglect them in the following calculation and restrict us to only consider the scalar and vector dim-6 operators in LEFT.

III. MATCHING TO THE SMEFT
where the loop function X α /X t can be found in Refs. [33,34] and higher order corrections are given in Ref. [9]. We take the central values for CKM elements from CKMfitter [35], X t and X α from Ref. [36], and the rest from the PDG book [37]. Then, to the leading order in χPT, the analytical expressions in Eqs. (28) and (29) with the Wilson coefficients in Eqs. (48) predict the branching ratios of Kaon semi-invisible decays in the SM which are consistent with SM predictions quoted in the literature [8][9][10].
Secondly, the dim-6 SMEFT operators in the Warsaw basis [22] and the dim-7 SMEFT operators in the basis given in Refs. [23,24] can induce the operators in the LEFT by integrating out the SM particles at tree-level. The LNC operators O V dν1/2 are obtained through matching with the dim-6 SMEFT operators in addition to the SM contribution in Eq. (48). To linear order in the SMEFT Wilson coefficients, the matching results are where D is the unitary matrix transforming left-handed down-type quarks between flavor d L and mass eigenstates d L , d L = Dd L . We choose D to be approximately the identity matrix and neglect its effect in the following, i.e. the weak interaction eigenstates are the same as the mass eigenstates and the mixing originates from the up-type quarks. The convention for the Wilson coefficients is taken from Ref. [22], with the corresponding SMEFT operators being O (1) The σ I are the Pauli matrices, and with the matching result for the Wilson coefficients at the electroweak scale We use subscripts 1, 2, and x to represent the SM quark generation. The indices α or β denote the SM lepton flavor. As the operator violates quark flavor, the contribution to neutrino masses is suppressed and does not pose a stringent constraint. Note that the O S dν2 operator cannot be induced at tree-level from SMEFT. In the following we derive constraints on the SMEFT operators from K → πνν and compare to the existing measurements of other related processes.
A brief comment on renormalization group corrections in LEFT is in order. As neutrinos neither couple to gluons nor photons, we only have to consider QCD corrections. Due to the QCD Ward identity, there are no QCD corrections to the vector operators at one-loop order and the running of the scalar operator can be simply obtained by noting that m ff P L,R f is invariant under QCD renormalization group corrections. Hence, the running of the scalar operator can be directly related to the QCD correction to the quark masses, C S (µ) = C S (m W )m q (m W )/m q (µ).

IV. IMPLICATION OF K → πνν FOR NEW PHYSICS AND OTHER CONSTRAINTS
In this section, based on the above LEFT coefficients in the leading-order chiral Lagrangian and the matching to SMEFT, we evaluate the constraints on new physics above the electroweak scale from the K → πνν measurements and other rare decays. According to the decay branching ratios in Eqs. (28) and (29), to the leading order of the chiral Lagrangian, both vector and scalar LEFT operators contribute to the decays K → πν ν. They correspond to dim-6 LNC operators and one dim-7 LNV operator in the SMEFT, respectively. We will separately discuss the constraints on them below.

A. Constraint on the LNC operators
From the branching ratios in Eqs. (28)(29), and the matching results in Eqs. (52-55), we split the contributions to the amplitude into the SM part given in Eq. (48) and the NP part as follows where the NP part is the linear combination of the Wilson coefficients of dim-6 LNC operators in the SMEFT in Eqs. (52-55) Taking the splitting in Eq. (61) and the experimental results in Eqs. (1, 2), we find ∈ [0.4 × 10 −9 , 6.2 × 10 −9 ],(63) The following generic relations can be immediately obtained α,β More importantly, as the component with (α, β) = (τ, τ ) does not participate in any tau lepton rare decays or leptonic charged Kaon decays at tree-level, K → πνν provides a unique opportunity to probe the SMEFT Wilson coefficients entering C V,sdτ τ dν,dim−6 . If the NP contribution to Kaon semiinvisible decays originates only from the operator with (α, β) = (τ, τ ), we have The first term describes the SM contribution from decays to electron and muon neutrinos and the second term describes the decay to tau neutrinos and receives contributions from both the SM and NP. The LFV contributions are neglected as stated above.
We further require the above results fall within the KOTO and NA62 sensitivity in Eqs. ( If we denote the Wilson coefficient as The SM has no interference with LFV contribution in this case. After neglecting the above LFC contribution from NP and assuming the coefficient with only one set of lepton flavors is switched on at a time, a lower limit on the NP scale associated with the LFV Wilson coefficients can be obtained as where the NA62 result for K + → π + νν at 95 (90)% CL is taken. A stronger bound can be set if we assume lepton flavor universality (LFU) In Refs. [38][39][40], there are similar analyses for LFV coefficients using the limit on B(K + → π + νν) from PDG. Their limits can be translated into a bound on NP scale in our convention as 56.8 TeV in Ref. [38] and 50 TeV in Refs. [39,40]. We can see that the new NA62 result pushes the NP scale higher. The bound obtained above is the most stringent one for the coefficients with τ flavor, compared to the bound from τ lepton LFV rare decays [40]. For the coefficients with (α, β) = (e, µ) or (µ, e), the most stringent bound with Λ NP ≥ 259 TeV is from the charged lepton decay modes of Kaon, i.e. K L → µ − e + , µ + e − . See also the derivation in Appendix B.

B. Constraint on the LNV operator
In this section we assume the NP contribution from dim-6 LNC operators is negligible and therefore only keep the SM contribution in the LNC case. Under this assumption, we focus on the LNV NP contribution. As discussed above, the scalar LEFT operators from one dim-7 LNV operator in the SMEFT play an important role in the Kaon semi-invisible decays.
The Kaon invisible decays can entail constraint on the above Wilson coefficients. The effective Lagrangian for K L → νν at the leading order is and that for K S → νν decay is One can see that the processes are only induced by |∆L| = 2 dim-6 operators in LEFT since they flip the helicity of one of neutrinos to allow the pseudoscalar Kaon to decay invisibly. The |∆L| = 0 dim-6 operators O V dν1/2 are severely suppressed by the neutrino mass because they are subject to helicity-suppression. As seen in the above subsection, only the O S dν1 operator is induced by one dim-7 SMEFT operator at tree-level. By including the one-loop QCD running result for C S dν1 from electroweak scale Λ EW ≈ m W to the chiral symmetry breaking scale Λ χ ≈ 2 GeV, we obtain We further assume the Wilson coefficients in Eqs. (59-60) at Λ EW scale as where Λ NP denotes the NP scale above the electroweak scale. Combining Eqs. (76-77), the matching results in Eqs. (59-60), and the naive assumption in Eq. (79), we find that there is no tree-level contribution to K S decay. For K L invisible decay, we obtain the branching ratio of invisible decay where the factor 2 accounts for the anti-neutrino case, the factor 3 for the 3 generations of neutrinos, the factor 1/2 for the identical neutrinos in final states, and 1/16π for the phase space, respectively. The experimental bounds for the Kaon invisible decay were estimated in Ref. [41] as The above K L bound translates into a rather weak lower limit on the new physics scale Given the matching results in Eqs. (59) and (60) together with the assumption in Eq. (79), the branching ratios of K → πν ν in Eqs. (28) and (29) can be simplified as There is obviously no interference between the SM contribution and the LNV contribution. If we require those results to satisfy the region allowed by the KOTO and NA62 upper bounds, the NP contribution resides along the pink line in Fig. 2  The operator in Eq. (58) can also contribute to neutrinoless double beta (0νββ) decay. The NP scale from this process is constrained to be larger than O(100 TeV) [28,42]. Constraints from K → πνν are complementary and provide a similar sensitivity, because they constrain the quark flavor violating Wilson coefficients with an sand a d-quark and any arbitrary generations of the lepton fields in the operator after relaxing the assumption in Eq. (79).

V. DISCUSSIONS AND CONCLUSIONS
In the above analysis, we focus on the contact interactions from effective operators composed of s, d quarks and two neutrinos for s → dν ν transition, that is the so-called short-distance (SD) contribution. In addition, there exist the long-distance (LD) contributions to K → πνν from One the other hand, we assume the NP contribution from dim-6 LNC operators is negligible and therefore only keep the SM prediction in the LNC case. As a result, the scalar LEFT operators from one dim-7 LNV operator in the SMEFT dominates the K → πν ν decay. We find that the K L invisible decay K L → νν places a weak bound on the new physics scale for the LNV operator. As there is no interference with the SM contribution, the constraint on the NP scale from K → πνν is rather precise and resides in a narrow range Λ NP ∈ [19.4 TeV,21.5 TeV].
Appendix A: The amplitudes and partial widths of K → πν ν In this appendix we present details of the calculation of K → πν ν. For the process of , the amplitudes for K L decay are and those for K + decay are where s = (p 1 + p 2 ) 2 and t = (p 2 + p π ) 2 . Here α ≤ β means that for |∆L| = 2 we take αβ = ee, eµ, eτ, µµ, µτ, τ τ flavor configurations. One should note that the flavor indices α, β are implicitly summed over in the Lagrangian, while in the above amplitudes they denote the specific neutrino flavors in final states. The overall factor 2 in the second line for K L decay is because the contributions from M ∆L=2 1 (K L ) and M ∆L=−2 1 (K L ) are the same for any pair of (α, β). The −1/2δ αβ accounts for the double counting of final state phase space for identical particles.
The partial decay width can be expressed as where the integration interval of t is The partial widths of Kaon semi-invisible decays are obtained by performing the t integral at 90% CL. The lepton-flavor-conserving modes of K L decays have been measured [37] B(K L → e + e − ) = 9 +6 −4 × 10 −12 , B(K L → µ + µ − ) = (6.84 ± 0.11) × 10 −9 . (B2) Their SM predictions are [45][46][47] B(K L → e + e − ) SM ≈ 9 × 10 −12 , We match the SMEFT Wilson coefficients in Eq. (56), relevant for Kaon physics, to the four vector operators in LEFT The matching condition is given by 2 The lepton-flavor-conserving decay widths are where X = S, L and F K is the physical Kaon decay constant. I.T. stands for the interference term between the SM part and the NP part. The NP contribution is the linear combination of Wilson coefficients in Eqs. (B6-B9) and takes the form as Considering the physical lifetime of K S and K L and the experimental constraints in Eqs. (B1, B2), the stronger limit on NP scale is set by the K L decays. After neglecting the interference term in and µµ coefficients are constrained to be Finally, we quote the result for the LFV mode The upper limit on the lepton-flavor-violating decay of B(K L → e ± µ ∓ ) < 0.47 × 10 −11 at 90% CL [37] leads to a constraint on the NP scale of with ( , ) = (e, µ) or (µ, e).
Appendix C: Long-distance contributions from dim-6 operators In this Appendix, we estimate the long distance (LD) contributions to K → πνν from the heavy NP parameterized by the dim-6 LNC operators in SMEFT. The LD contributions are mediated by light charged leptons, neutrinos or light meson propagators in the χPT picture. In Fig. 3 we categorize the possible topologies for the LD contribution from the dim-6 two-quark-two-lepton operators O(qqLL) and four-quark operators O(qqqq) in SMEFT. The dashed and solid lines represent the possible meson and lepton fields, respectively.
The LD contributions mediated by neutrinos are suppressed compared to the SD contribution.
In the Feynman diagrams for this kind of LD contribution, the vertex connecting the Kaon state involves the same Wilson coefficients as the SD case and the other vertex leads to one additional suppression factor G F . Hence, we find that they are suppressed by F 2 0 G F ∼ 10 −7 . The LD contributions mediated by charged leptons can be induced by charged-current vector and/or scalar operators. The contribution from scalar operators is strongly constrained by charged The second term is suppressed by a factor of O(10 −5 ) and consequently the LD scalar contribution is sub-dominant.
In the SM, the LD contribution induced by vector operators is suppressed by O(10 −4 ) relative to the SD contribution [48]. As the NP contribution to charged current operators is at most of the same order as the SM contribution, the LD contribution from vector operators is negligible.
Similarly, the meson-mediated tree-level contributions (LD3 and LD4) are suppressed by O(10 −4 ) with respect to the SD contribution [48,49] in the SM, while the one-loop contribution LD2 is of the same order as the LD1 contribution in the SM [50,51]. To our knowledge, there is no general LEFT analysis of LD contributions to K → πνν. As four quark operators with ∆S = 1 directly contribute to hadronic Kaon decays of which many have been measured at sub-percent level precision, we expect that similar conclusions hold for NP contributions mediated by meson exchange. In summary, currently it is safe to neglect long-distance contributions to K → πνν.