Lepton-flavor-violating semileptonic $\tau$ decay and $K\to\pi\nu\bar\nu$

We consider lepton-flavor violation in strangeness changing ($|\Delta S|=1$) semileptonic $\tau$-lepton decays arising from new physics encoded in a standard model effective Lagrangian. Its invariance under the standard model gauge group entails the relevance of other processes which can serve as complementary probes of the new physics operators. We show in particular that for some of them the bounds implied by current data on the rare kaon decays involving a neutrino pair, $K\to \pi\nu\bar\nu$, are stronger than the existing limits from direct searches for lepton-flavor-violating semileptonic $\tau$ decays. We discuss additional processes affected by the same operators and find that certain leptonic charged-meson decays also provide stricter constraints on a few more of them. Upcoming results of ongoing experiments such as Belle II and NA62 will further test the new physics parameter space.

Interactions manifesting lepton-flavor violation (LFV) do not occur in the standard model (SM) with zero neutrino mass but are relatively common in new physics (NP) scenarios. There is a renewed interest in studying LFV for both theoretical and experimental reasons. On the theoretical side, the so-called 'B-physics anomalies' constitute suggestive evidence for leptonflavor universality violation [1]. Model building to account for them often gives rise to LFV as well. On the experimental side, there are a number of ongoing and forthcoming efforts that will improve upon the existing limits on LFV. Amongst them are LHCb [1], BESIII [2], Belle II [3], and COMET [4].
Following our earlier work [5], here we adopt a model-independent approach that starts from the most general effective Lagrangian involving dimension-six operators which respect the SM gauge symmetry and can generate |∆S| = 1 tau-flavor-violating interactions. The resulting operators also contribute to other processes, in particular to ones where the lepton flavor is carried by a neutrino. We explore the impact of these operators at tree level on various lowenergy processes and map the constraints that can be extracted from the data available at the moment. We find that the so-called golden rare kaon decays, K → πνν, impose bounds on a number of the operators that are stricter by up to two orders of magnitude than the limits from direct searches for LFV in semileptonic τ decay. Likewise, certain leptonic charged-meson decays also provide stronger restrictions on a few more of the operators. The KOTO and NA62 experiments [24] can further tighten the constraints from K → πνν in the near future, and Belle II after achieving an integrated luminosity of 50 ab −1 may improve upon the current limits on τ couplings exhibiting LFV by as much as an order of magnitude [3].

Effective Lagrangian
The most general effective Lagrangian constructed from SM fields, including an elementary Higgs, and invariant under the SM gauge group exists in the literature [25,26]. The part of this Lagrangian containing the operators Q k pertinent to our discussion can be written as where Λ np stands for a heavy mass scale associated with the NP interactions, the coefficients C ijxy 1,...,6 are in general complex, and the family indices i, j, x, y = 1, 2, 3 are implicitly summed over. Explicitly, 2 Amplitudes and rates 2.1 |∆S| = 1 semileptonic τ decays We treat first τ decay into a charged lepton plus a pseudoscalar meson P or a vector meson V , on which there are direct search data. For τ − → − P and τ − → − V the amplitudes have the general forms, respectively, which lead to the decay rates where K(x, y, z) = (x − y − z) 2 − 4yz andk(x, y, z) = (x − y) 2 + (x + y)z − 2z 2 .
For τ − → − K S the hadronic matrix elements which do not vanish are where f K is the kaon decay constant and B 0 = m 2 K 0 /(m d + m s ). Applying them to the operators in Eq. (4) for the amplitude in Eq. (5), with the approximation whereṽ Similarly, for τ − → − K * 0 , − K * 0 the nonzero mesonic matrix elements are where ε K * and f K * are, respectively, the polarization vector and decay constant of K * , and so from Eqs. (4) and (6) follow with Clearly, τ ± → ± K * 0 , ± K * 0 can access only the coupling combinations V τ,τ and A τ,τ , while τ ± → ± K S cannot probe them.
We turn now to the three-body modes τ − → − π ± K ∓ , on which empirical information also exists. The relevant hadronic matrix elements in this case are given by where f + and f 0 denote form factors which are functions ofq 2 , Accordingly, the amplitude for where wherê The differential rate of τ − → − π + K − is also given by Eq.
. We observe that, unlike τ → K * 0 , K * 0 , these three-body modes are sensitive to all the operators with parity-even quark parts.

Other modes
The required SU(2) L -gauge-invariance of Q τ k and Q τ k in Eq. (4) implies that some of these operators involve left-handed quark and/or lepton doublets and therefore can influence additional processes. The related couplings are summarized in appendix A and can generate transitions with one or two neutrinos. Here we discuss the extra modes which may offer complementary restrictions on the couplings.
The operators in Eq. (4) with a pair of left-handed lepton doublets provide the (ds)(ν ν τ ) interaction listed in Table 2, as well as itsν τ ν counterpart. Since they have neutrino-flavor combinations that are different from those in the SM amplitudes, the former have no interference with latter and cause the K → πνν rates to exceed their SM values, as the neutrinos are not detected. The resulting modifications ∆B K + and ∆B K L to the SM branching fractions can be inferred from Eqs. (9)-(10) in Ref. [27] to be where [28]κ + = 5.17 × 10 −11 and κ L = 2.23 × 10 −10 .
These are induced by the (ūu) ¯ τ couplings in Table 2 from the operators with a pair of lefthanded quark doublets. The pertinent mesonic matrix elements are π 0 uγ η γ is the pion (ρ meson) decay constant and ε ρ is the ρ polarization vector. The τ → π, ρ rates have, respectively, the forms in Eq. (7) with These are suppressed by the CKM factor |V ud V us | 0.22 compared to their counterparts in Eqs. (9) and (12).
Like the preceding case, Eq. (4) includes the (cc) ¯ τ,τ interactions listed in Table 2 which bring about the charmonium decay J/ψ → ∓ τ ± . Thus, the amplitude for J/ψ → − τ + also has a suppression factor of |V cd V cs | 0.22. Its rate in the m = 0 limit is where the J/ψ decay constant f J/ψ is defined by 0|cγ κ c|J/ψ = ε κ J/ψ f J/ψ m J/ψ .
The couplings in the last four rows of Table 2 and/or their partners with and τ interchanged can affect the SM-dominated leptonic decay P + → + ν of a pseudoscalar meson P + ∼ ud, where u = u, c and d = d, s. The biggest impact comes from the (pseudo)scalar operators, which are not subject to helicity suppression, with = e, in which case the SM amplitude is the most helicity-suppressed. With only O eτ,τ e 6( ) being present, we derive the modification ∆Γ P + →e + ν to the SM rate of P + → e + ν for P = π, K, D, D s to be [5] with the P decay constant f P being defined by 0|dγ 5 u|P + = if P m 2 P /(m u + m d ) and the lepton masses ignored. Note that there is no interference with the SM contribution as the neutrino is of the wrong flavor [6].

|∆S| = 1 semileptonic τ decays
We treat the two-body modes with the amplitude terms in Eqs. (9) and (12) and the rates in Eq. (7). The required decay constants are f K = 155.6(4) MeV [29] and f K * = 206(6) MeV, the latter having been extracted from the data on τ − → νK * − under the assumptions of isospin symmetry and no NP in this channel. 1 For hadron and quark masses, we use the central values supplied by Ref. [29]. Thus, we arrive at the branching fractions To evaluate the three-body case with Eq. (19), we need the πK form-factors f + and f 0 as functions ofŝ. Assuming isospin symmetry, we adopt the π − K S invariant-mass spectrum which has been extracted from the study of τ − → νπ − K S by the Belle Collaboration [31]. 2 Thus, after integrating the differential rate over (m π + + m K + ) 2 ≤ŝ ≤ (m τ − m ) 2 , we obtain for = e, µ As already mentioned, B(τ − → − π + K − ) is equal to B(τ − → − π − K + ) except the subscript τ of the couplings is replaced with τ and a minus sign is added to S τ .
From Eqs. (27)- (28), we notice that these modes do not all probe the same set of couplings and hence are complementary in their sensitivity to the NP contributions. The existing experimental limits [at 90% confidence level (CL)] for these decays are [29]

Constraints on c τ,τ k
We entertain the possibility that only one of the coefficients c τ,τ k is nonzero at a time. In this case, after comparing the calculated branching fractions and their experimental data described in the preceding two subsections, we collect in Table 1 the best upper-bound, and the process supplying the corresponding constraint, on each coefficient. Evidently, K + → π + νν and π + , K + → e + ν produce the strongest restrictions to date on a number of these couplings. If NA62 reaches its goal of testing the SM prediction with 10% precision [24], the bound from K + → π + νν in Table 1 will be improved by roughly a factor of 4. The bounds derived from lepton-flavor-violating τ decays may be lowered as much as 10 times by Belle II, which aims at reducing their branching-fraction limits by 2 orders of magnitude with its expected full dataset [3].  if only one of them is nonzero at a time and the processes which provide the constraints. Note that the lepton-flavor index (f 1 or f 2 ) can be carried by the neutrino.

Conclusions
We have outlined the existing constraints on LFV in |∆S| = 1 semileptonic τ decays. To do this, we first parametrized the NP responsible for LFV with all the dimension-six operators in the effective Lagrangian that respects the gauge symmetry of the SM and is appropriate for an elementary Higgs. We subsequently computed all the |∆S| = 1 semileptonic τ decays with an electron or muon plus one or two mesons in the final state that have been searched for. We finally extracted the constraints on the parameters of the effective Lagrangian using the current 90%-CL upper limits on their respective rates.
Noticing that the gauge symmetry of the SM relates these τ decay modes to other processes, we then studied those other modes. We found that the golden rare kaon decay K + → π + νν places the most stringent constraint available on several of the NP couplings and that this can be further improved by the expected NA62 results in the near future. Moreover, the measured π + → e + ν and K + → e + ν rates imply the strictest limits to date on a couple other of the NP couplings. Our numerical findings are summarized in Table 1. Table 2: Feynman rules from L np in Eq. (4). In the second column, each entry is to be furnished with an overall factor Λ −2 np and with the spinors of the fermions in the first column, V U i D j = (V ckm ) ij from Eq. (3), and we have defined L η = γ η P L , R η = γ η P R ,L = P L , andR = P R . Since the neutrinos are nearly massless and not detected in decays, we display their weak eigenstates ν E i = j (U pmns ) ij ν j in the first column.