Charged-lepton-flavor violation in $|\Delta S|=1$ hyperon decays

Studies of lepton-flavor violation in strangeness-changing ($|\Delta S|=1$) transitions have a long tradition in the kaon sector where they provide some of the strongest limits on physics beyond the standard model. Recent hints of violation of lepton-flavor universality in $B$-meson decays have revived interest in lepton-flavor violation as the two phenomena appear simultaneously in many extensions of the standard model. At the same time, the LHCb experiment has produced new results for the hyperon process $\Sigma^+\to p\mu^+\mu^-$ and may be in a position to study other rare hyperon decay modes. With this in mind, we investigate $|\Delta S|=1$ hyperon decays into different-flavor lepton pairs $e^\pm\mu^\mp$ in a model-independent manner and contrast the coverage of parameter space that can be achieved with what is known from kaon modes. We include a comparison with selected two-body leptonic decays of charged mesons, with $K\to \pi\nu\bar\nu$ modes, with $\mu\to e$ conversion, and with lepton-flavor violating decays of other neutral mesons, all of which constrain the same parameter space in a complementary way.

The organization of this paper is as follows. In section 1 we consider the most general effective Lagrangian involving quark-lepton operators of dimension six which are invariant under the SM gauge group and can induce |∆S| = 1 transitions with LFV amongst the lightest hadrons. We then briefly discuss a possible ultraviolet completion of this effective Lagrangian in terms of leptoquarks. In section 2, we first obtain the baryonic matrix elements pertaining to our hyperon decays of interest and subsequently derive their decay rates. We also deal with their kaon counterparts as well as other processes without hyperons that are affected by the same operators as a consequence of gauge invariance, such as K → πνν and µ → e conversion in nuclei. In section 3, we present our numerical analysis and illustrate how the different processes are complementary in probing the NP of concern. We summarize and draw our conclusions in section 4. Some technical details are relegated to the appendices.

Model-independent approach
We begin from the most general effective Lagrangian that can be built out of SM fields, including gauge fields and a light Higgs, 1 and respects the gauge symmetries of the SM, as has been described before in the literature [2,55]. The operators Q k that can contribute to |∆S| = 1 transitions with LFV between down-type light fermions first occur at dimension six (dim-6). There are several such operators [2,55], and the Lagrangian containing them has the form where Λ np denotes a heavy mass scale characterizing the underlying NP interactions, C ijxy 1,...,6 are generally complex coefficients, i, j, x, y = 1, 2, 3 stand for family indices, summation over them being implicit, with l i and q i (e i and d i ) representing the left-handed doublets (right-handed singlets) of quarks and leptons, respectively, τ I=1,2,3 denoting Pauli matrices, and I being summed over. 2 Accordingly, Q ijxy † k = Q jiyx k and the Hermiticity of L np implies C ijxy * k = C jiyx k for k = 1, ..., 5. We note that there are no dim-6 SM-gauge-invariant operators comprising tensor bilinears that directly participate in down-type quark-lepton transitions, as has been observed previously [55][56][57]. 3 Moreover, the absence of tree-level flavor-changing neutral currents in the SM implies that dim-6 operators made up of a quark or lepton bilinear in combination with gauge and Higgs fields also do not contribute to L np .
For convenience, we can choose to work in the mass basis of the down-type fermions, implying that where V ckm (U pmns ) is the Cabibbo-Kobayashi-Maskawa quark (Pontecorvo-Maki-Nakagawa-Sakata neutrino) mixing matrix, P L,R = (1 ∓ γ 5 )/2, and U 1,2,3 = u, c, t, D 1,2,3 = d, s, b, ν 1,2,3 , and E 1,2,3 = e, µ, τ refer to the mass eigenstates. We can then express the part of L np containing operators Q eµ k and Q µe k which contribute to s → d transitions and do not conserve electron and muon flavors as . The Hermitian conjugates of these terms are responsible for the corresponding d → s transitions. Given that the tau lepton is too heavy to appear in the final states of light hyperon decays, we do not discuss operators with the tau field.
where ( ) = e, µ but = and V , A , S , P ,ṽ ,ã ,s , andp are dimensionless constants which can be complex. As will be seen later on, in the hyperon decay rates the couplings V , A , S , and P , which accompany the parity-even quark bilinears in eq. (5), do not interfere withṽ ,ã ,s , andp , which are associated with the parity-odd quark bilinears. These couplings are related to the coefficients defined in eq. (4) by For c k being free parameters, V , A ,ṽ , andã are therefore linearly independent, whereas only two of their (pseudo)scalar partners are, which may be taken to be S and P . We notice from eq. (7) that c 6 ± c 6 can each accompany both parity-even and -odd quark bilinears, which can of course also be understood from the explicit expressions for the relevant parts of Q 6,6 in, say, the = eµ case: Q eµ 6 ⊃ dP L s eP R µ and Q eµ 6 ⊃ dP R s eP L µ. To see what other processes can receive contributions from the operators Q eµ,µe k , in appendix A we summarize the Feynman rules that follow from them. We then see that the decays listed below can also constrain these NP couplings. 4 Changes in lepton-flavor number can take place in all of these modes, and some of them involve one or two neutrinos.
• K + → π + νν and K L → π 0 νν. In this case, the contributions of Q k have no interference with the SM ones, due to their differing lepton-flavor combinations, but change the decay rates, as the neutrinos are not detected.
• π ± → ± ν, K ± → ± ν, and D ± (s) → ± ν. Since these are helicity suppressed in the SM, the impact of NP will be most important on the electron modes, = e. Again, the NP we consider in this paper does not interfere with the SM in these processes because it produces the 'wrong' neutrino flavor. Since the neutrino flavor is not observed experimentally, these contributions increase the rates over their SM value.
• µ → e conversion in nuclei and flavor-violating π 0 , η, D 0 → + − . These serve as additional null tests of the SM, and there are ongoing searches for them.
From L lq , we can then derive LQ-mediated quark-lepton couplings at tree level which yield the operators in eq. (1), with their coefficients being given by Evidently C 1,...,5 can all be affected by the scalar and vector LQs, but C 6 only by the vector ones.
2 Hadronic matrix elements and decay rates

Hyperon decays
Our baryon decays of interest are B → B e ∓ µ ± for BB = Λn, Σ + p, Ξ 0 Λ, Ξ 0 Σ 0 , Ξ − Σ − , all involving spin-1/2 particles only, 5 and Ω − → Ξ − e ∓ µ ± , where Ω − is a spin-3/2 hyperon. To determine their amplitudes, we need the baryonic matrix elements of d γ η , γ η γ 5 , 1, γ 5 s, which can be estimated with the aid of chiral perturbation theory at leading order. Their derivation from the chiral Lagrangian is sketched in appendix B. For B → B e ∓ µ ± , we can express the results as where V B B and A B B are constants, their values for the aforesaid B B pairs collected in table 1, the us are Dirac spinors,Q = p B − p B , with p X denoting the four-momentum of X, and the other quantities are defined in appendix B. In the Ω − case, we have and Ξ − |dγ η s|Ω − = Ξ − |ds|Ω − = 0, whereq = p Ω − − p Ξ − and u η Ω is a Rarita-Schwinger spinor. 5 We do not include Σ 0 → ne ∓ µ ± because their branching fractions are expected to be comparatively much smaller due to the Σ 0 width being overwhelmingly dominated by the electromagnetic channel Σ 0 → Λγ [1].
The parameters C and F are from the lowest-order chiral Lagrangian.
In numerical work, to incorporate form-factor effects not taken into account in eq. (10), we will modify V B B and , respectively, with M V = 0.97 GeV and M A = 1.25 GeV, following the commonly used parametrization in experimental analyses of semileptonic hyperon decays [58][59][60][61] and assuming isospin symmetry. Analogously, as theq 2 range in Ω − → Ξ − e ∓ µ ± is significantly larger than theQ 2 ones, for this decay we will make the change C → C/ 1 −q 2 /M 2 A 2 . These modifications turn out to translate into increases of the decay rates ranging from a few to 20 percent.
From eq. (10), we determine the amplitude for the octet-baryon decay B → B − + to be where Hereafter we neglect the electron mass. Defining Γ X→Y − + ≡ dΓ X→Y − + /dŝ for the differential decay rate, we then arrive at where Similarly, for the Ω − decay, we find Evidently, in our approximation Ω − → Ξ − e − µ + is not sensitive to the untilded couplings V eµ and A eµ but indirectly still probes S eµ and P eµ in light of eq. (7). The differential rates of the µ − e + modes are obtainable from their e − µ + counterparts by interchanging e and µ in the subscripts of A eµ , S eµ , P eµ , andã eµ as well as applying V eµ → −V µe andṽ eµ → −ṽ µe .

Kaon decays
For K L,S → e ∓ µ ± the pertinent hadronic matrix elements are with f K being the kaon decay constant, while for where f +,0 represent form factors which are functions of q 2 Kπ . Additional required matrix elements 2 under the assumption of isospin symmetry, which also implies π − d γ η s K − = π + ūγ η s K 0 . This allows us to adopt (10) and λ 0 = 0.0142(23) from K L → π + µ − ν measurements [1] as well as f + (0) = 0.9681(23) from lattice computations [62]. It is simple to check that the baryonic and mesonic matrix elements detailed above satisfy the free quark relations After the absolute square of the amplitude is summed over the final spins, there is no interference between the S and P terms. Hereafter we neglect the electron mass. This leads to the decay rates The expressions for S K L,S eµ and P K L,S eµ have been relegated to appendix C.

Other modes
As mentioned earlier, there are other modes that can be influenced by Q eµ,µe k in eq. (4). The relevant observables are affected as follows.
• The most important modification to the leptonic decay M + → + ν of a pseudoscalar meson M + ∼ ud (u = u, c and d = d, s) from NP with LFV induced by (pseudo)scalar operators which are not helicity suppressed. In our case, they are of the form This yields the biggest impact if = e, in which case the SM rate is helicity suppressed the most. With 0|dγ 5 u|M + = if M m 2 M /(m u + m d ) and the M decay constant f M , the modification to the rate is then analogously to eq. (22), the lepton masses having been ignored. Note that there is no interference with the SM contribution as the neutrino is of the wrong flavor [14]. From the Feynman rules in appendix A, we infer • µ → e conversion in nuclei and π 0 , D 0 → e ∓ µ ± . These arise from some of the operators responsible for K L → e ∓ µ ± discussed above but, according to table 4, are not affected by the scalar operators Q 6,6 . From the general formulas in ref. [65], we find the rate of µ − → e − conversion in nucleus N to be where V (p,n) N are dimensionless integrals representing the overlap of e and µ wave-functions for N and incorporating appropriate proton (p) and neutron (n) densities, and ω N capt is the rate of µ capture in N . For the meson decays, we obtain , 3 Numerical results

Hyperon and kaon constraints
Integrating Γ B→B e − µ + over m 2 µ ≤ŝ ≤ (m B − m B ) 2 , we arrive at the branching fractions Compared to these results, the corresponding numerical factors in B Ξ 0,− → Σ 0,− e ± µ ∓ turn out to be roughly at least two orders of magnitude lower, partly due to smaller phase space, and hence are not shown. On the other hand, the Ω − decay having comparatively greater phase space, its numbers are bigger by an order of magnitude or more, All the results in eqs. For the two-body kaon decays, we calculate the branching fractions to be having employed the central value of f K = 155.6(4) MeV [1]. For K → πe ∓ µ ± , integrating their differential rates in appendix C over m 2 µ ≤ŝ ≤ (m K − m π ) 2 , we obtain B K + → π + µ − e + = 8.7 |V eµ | 2 + |A eµ | 2 + 10 |S eµ | 2 + |P eµ | 2 We see that K → e ± µ ∓ K → πe ± µ ∓ are not sensitive to V and A ṽ andã but can still probe S and P s andp in light of eq. (7).
Currently there is not much empirical information on the lepton-flavor-violating decays of strange hadrons. The only data available are the limits [1] B K L → e ± µ ∓ < 4.7 × 10 −12 , B K L → π 0 e ± µ ∓ < 7.6 × 10 −11 , and B K L → π 0 π 0 e ± µ ∓ < 1.7 × 10 −10 , all at 90% confidence level. We will not employ the bound from K L → π 0 π 0 e ± µ ∓ as it has smaller phase space than the other modes and probes the same couplings as K L → e ± µ ∓ . The numbers in eq. (41) |V µe | 2 + |A µe | 2 + 10 |S µe | 2 + |P µe | 2 + 3.6 Re A * µe P µe + V * µe S µe To illustrate how the different bounds may constrain the couplings, we consider a few examples in which the couplings are real and only two of the independent ones are nonzero at a time. In figure 1, we display for Λ np = 1 TeV the allowed regions of V eµ and V µe (top-left plot),ṽ eµ and v µe (top-right plot), A eµ and P eµ (bottom-left plot), andã eµ andp eµ (bottom-right plot), subject to the kaon bounds in eq. (41). In the bottom-left (-right) plot, the vertical axis implies that s eµ S eµ , which equals −P eµ −p eµ according to eq. (7), is also nonvanishing and consequently affects K L → e ± µ ∓ K L → π 0 e ± µ ∓ , leading to the extra constraint depicted by the orange (light cyan) area on the left (right). , all taken to be real, for Λ np = 1 TeV, allowed by the experimental limits on the branching-fractions of K L → π 0 e ± µ ∓ , K + → π + e − µ + , K + → π + µ − e + , and K L → e ± µ ∓ indicated by K L → πeµ, K + eµ , K + µe , and K L → eµ, respectively). In the left (right) plot at the bottom, the bound from K L → e ± µ ∓ K L → πe ± µ ∓ is included because they are affected bys eµ = −P eµ S eµ = −p eµ , from eq. (7). In each of the four cases, all the other couplings are set to zero.
For comparison, given that there are still no direct-search restrictions on hyperon LFV, we entertain the possibility of future experimental limits of 10 −10 on all of the branching fractions in eqs. (31)- (34), inspired by the aforementioned LHCb finding on Σ + → pµ + µ − [51]. Under this assumption, we acquire the areas in figure 2, which reveals that these constraints are still much weaker than the kaon ones if fine cancelations do not occur among the couplings. If future hyperon measurements could achieve branching-fraction limits of 10 −12 instead, the allowed regions would be reduced by a factor of 10, from which one can infer that for limits better than 10 −12 the hyperon bounds would start to become comparable to their kaon counterparts. and Ω − , respectively. The bottom plots take into account eq. (7). In each case the other couplings are set to zero. 6 The numbers we use to illustrate possible future LHCb bounds are based on the following. Their single event sensitivity (ses) for Σ + → pµ + µ − with 3 fb −1 is (2.2 ± 1.2) × 10 −9 [51]. With expected collection of 50 fb −1 in the Phase-I upgrade and assuming that the ses to modes with one muon and one electron is within a factor of five or so, limits of order 10 −10 would be possible. A further collection of 300 fb −1 in the Phase-II, combined with expected improvements in trigger efficiency [53], leads us to speculate on possible 10 −12 future limits.

Other constraints
The branching fractions of other modes that can restrict the NP encoded in eq. (4) are [1] B(π + → e + ν) = (1.230 ± 0.004) × 10 −4 , B(K + → e + ν) = (1.582 ± 0.007) × 10 −5 , where the limits are at 90% CL. All of these modes supply much weaker constraints than the ones already obtained from the kaon sector in the previous subsection. An illustrative list is shown in table 2 where we compare constraints, at 90% CL, on the coefficients of (pseudo)scalar operators Q 6,6 from only two-body decays, including K L → e ± µ ∓ , assuming that c 6,6 are equal, real, and the only nonvanishing coefficients. In the cases where the decays are observed, the limits are obtained from the quoted experimental errors. For the first four modes in this table, the coupling bounds are computed using eqs. (27) and (28) with the values of decay constants and particle masses from ref. [1] and CKM matrix elements from ref. [62]. 7 In this particular scenario, with the maximal c eµ 6 from the K L limit in the last row of the table, the corresponding hyperon branching fractions turn out to be less than 2 × 10 −17 .
A comparison with µ → e conversion in nuclei and π 0 , D 0 → e ± µ ∓ is also instructive. Since Q 6,6 do not affect them, in this instance we suppose that c eµ 1,5 = c µe 1,5 and that these coefficients are real and the only ones being nonzero. Based on eq. (29), the existing experimental limits on µ → e conversion in various nuclei [1], and the corresponding overlap integral and ω N capt values [65], we expect N = Au to provide the most consequential constraints. To evaluate B(µ − N → e − N ) for this nucleus, we adopt V (p) Au = 0.0974, V (n) Au = 0.146, and ω Au capt = 13.07 × 10 6 /s from ref. [65], and the result is displayed in table 3. Therein we also collect the bounds from π 0 , D 0 → e ± µ ∓ , 7 Online updates available at http://ckmfitter.in2p3.fr.

Process
Upper limit at 90% CL on branching fraction Upper bound on c eµ Table 3: Bounds from processes without hyperons on c eµ 1,5 = c µe 1,5 if these coefficients are real and the only ones nonvanishing.
their rates being given in eq. (30) with c 2 = 0. Evidently, the current data on µ → e conversion and K L → e ± µ ∓ can yield similarly strong constraints on c 1,5 . In this specific case, with the bound from the K L decay quoted in the last row of the table, we find that the hyperon branching fractions do not exceed 4 × 10 −15 .

Concluding remarks
We have studied charged-lepton-flavor violation in strangeness-changing, |∆S| = 1, transitions, paying special attention to the decays of hyperons. We start from the most general effective Lagrangian involving dimension-six operators which are invariant under the SM gauge group and can induce |∆S| = 1 processes with LFV. We illustrate how the operators would appear from the exchange of leptoquarks. We then explore the contributions of these operators to the hyperon decays as well as their kaon counterparts. This allows us to contrast the coverage of parameter space that may be achieved in the hyperon sector with what is known from kaon modes. In addition, we consider other processes that are affected by the same LFV operators when written in an SU(2) L -gauge-invariant form. Our main results from these comparisons can be summarized as follows.
• The current experimental exclusion limit on K L → µ ± e ∓ places the strongest constraint on LFV operators with a pseudoscalar |∆S| = 1 quark bilinear. Hyperon decays can only be competitive in this case if an exclusion at the 10 −16 level is reached for the Ω − mode. In the left panel of figure 3 we illustrate this scenario (the vertical axis) and for the comparison use B Ω − → Ξ − µ ± e ∓ < 10 −12 . Other hyperon decay modes are even less competitive, as can be inferred from figure 2. Indirectly, by implication of eq. (7), the same can be said of LFV operators with a scalar |∆S| = 1 quark bilinear.
• Nevertheless, the left panel of figure 3 also reveals that in some instances there are combinations of the (pseudo)scalar coefficients which can evade the K L → µ ± e ∓ restriction (the horizontal axis in this example) but which can be constrained by the hyperon modes as well as by K → πµ ± e ∓ . The situation, which is less extreme than that in the preceding scenario, is depicted in the left panel showing that an Ω − limit at the 10 −12 level is already starting to be competitive to the currently strictest limit from K + → π + e − µ + .
• For axial-vector |∆S| = 1 quark bilinears, the situation is also not as extreme and can be seen in the right panel of figure 3. In this case the constraint from B Ω − → Ξ − µ ± e ∓ < 10 −12 is only ∼ 17 times weaker than the K L → µ ± e ∓ one and the hyperons already become competitive at the 10 −14 level.
• For vector |∆S| = 1 quark bilinears, K L → µ ± e ∓ no longer offers a constraint. Presently the best restrictions on them are from K L → π 0 e ∓ µ ± and K + → π + e − µ + , as exhibited in the top-left plot of figure 1. Although the Ω − mode is insensitive to the vector quark bilinears, the spin-1/2 hyperons, especially Σ + and Ξ 0 , can probe them, but branching-fraction limits of order 10 −13 are required to be competitive to the kaon ones, as may be inferred from comparing the top-left plots in figures 1 and 2.
• The most important constraints from other rare decays correspond to K + → π + νν and to µ − → e − conversion in gold. Concerning the former, the impact of the couplings is realized via eq. (25), and so we impose |W | < 2 based on the findings of ref. [63]. Figures 4 (right panel) and 3 (right panel) place the limits from these two processes in context.   Figure 4: Comparative constraints on selected LFV couplings in eq. (4), for Λ np = 1 TeV, from current 90%-CL upper bounds on NP effects in K L → µ ± e ∓ , K + → π + νν, and µ − → e − conversion in gold and a possible future bound of B Ω − → Ξ − µ ± e ∓ < 10 −12 , under the general assumption that the couplings are real and c eµ k = c µe k . The specific choices for the nonzero ones are described in the text. part by the NSFC (Grant Nos. 11575111 and 11735010) of PRC. G.V. thanks the Physics Department at National Taiwan University for their hospitality and partial support while this work was completed. We thank Jeremy Dalseno for helpful communications.

A Feynman Rules
The various four-fermion couplings with (2quark)(2lepton) flavor structures due to Q eµ k in eq. (4) are listed in table 4. Those with the lepton flavors interchanged can be immediately obtained from the corresponding entries in the table by applying the change c eµ k → c µe k . The Hermitian conjugates of these couplings are additional ones with the quarks interchanged. Table 4: Feynman rules arising from Q eµ k in eq. (4). In the second column, each entry is to be multiplied by i/Λ 2 np and completed with the Dirac spinors of the fermions in the first column, we have defined L η = γ η P L , R η = γ η P R ,L = P L , andR = P R , and the element V U i D j corresponds to (V ckm ) ij in eq. (3). The neutrinos being nearly massless and unobserved in decays, we display their weak eigenstates ν e = (U pmns ) 1j ν jL and ν µ = (U pmns ) 2j ν jL in the first column.

B Correspondences between quark and hadron transitions
From the chiral Lagrangian which is at lowest order in the derivative and s-quark-mass (m s ) expansions and describes the strong interactions among the lightest octet baryons and mesons and decuplet baryons [66][67][68], one can extract correspondences between quark densities or currents and hadronic transitions [69]. From the results of ref. [69] pertaining to the |∆S| = 1 processes under discussion, one can infer [70] where m N,Σ,Ξ are isospin-averaged masses of the nucleons, Σ ±,0 , and Ξ 0,− , respectively,m is the average mass of the u and d quarks, B 0 = m 2 K /(m + m s ), with m K being the average mass of K 0 and K − , the free parameters D, F , and C occur in the leading-order chiral Lagrangian and can be fixed from baryon decay data, f = f K / √ 2, and the ellipses represent terms irrelevant to our analysis.
At the same order in the chiral expansion, the baryonic matrix elements ofd γ η , 1 γ 5 s also receive contributions from kaon-pole diagrams involving 0|d γ η , 1 γ 5 s|K 0 from eqs. (49) and (50) and vertices from the lowest-order strong chiral Lagrangian L s . In the latter, the pertinent terms are given by [70] From this and the preceding paragraphs, we arrive at the matrix elements in eqs. (10), (12), (19), and (20) in the limit that f +,0 = 1.