Non-Perturbative Effects in $\mu \to e \gamma$

We compute the non-perturbative contribution of semileptonic tensor operators $(\bar q \sigma^{\mu \nu} q)(\bar \ell \sigma_{\mu \nu} \ell)$ to the purely leptonic process $\mu \to e \gamma$ and to the electric and magnetic dipole moments of charged leptons by matching onto chiral perturbation theory at low energies. This matching procedure has been used extensively to study semileptonic and leptonic weak decays of hadrons. In this paper, we apply it to observables that contain no strongly interacting external particles. The non-perturbative contribution to $\mu \to e $ processes is used to extract the best current bound on lepton-flavor-violating semileptonic tensor operators, $\Lambda_\text{BSM} \gtrsim 450$ TeV. We briefly discuss how the same method applies to dark-matter interactions.


Introduction
The branching ratio for the purely leptonic decay µ → eγ has the current upper bound BR(µ → eγ) < 4.2 × 10 −13 , (1.1) determined by the MEG collaboration [1,2]. In the Standard Model (SM) including flavorchanging Majorana neutrino masses, the decay rate is suppressed by (m ν /M W ) 4 ∼ 10 −48 , so the result Eq. (1.1) provides strong constraints on µ → e transitions due to new physics beyond the Standard Model (BSM). The effect of new physics at energies above the muon mass can be described by an effective field theory, and the leading operators that contribute to µ → eγ are the dipole operators 1 L = Leγ eµ e L σ µν µ R F µν + Leγ µe µ L σ µν e R F µν + h.c. If one assumes that new physics consists of particles with masses well above the electroweak scale, the Standard Model Effective Field Theory (SMEFT) provides an adequate description of BSM effects [5,6]. At the weak scale, the SMEFT is matched to the low-energy effective field theory (LEFT) [3,4]. In such a scenario, the dimension-five dipole operators in Eq. (1.2) derive from dimension-six dipole operators in the SMEFT, so their coefficients are of the same size as those of the other dimension-six operators. In this paper, we will consider the case where L eγ is formally of order 1/Λ 2 BSM , and of the same size as other dimension-six operators. In many BSM models, the dipole operators are typically induced with a coefficient L eγ ∼ em µ /Λ 2 BSM , and so the limit Eq. (1.4) is equivalent to Λ BSM > 930 TeV. The operators in the low-energy effective theory below the weak scale have been classified in Ref. [3]. We can calculate the decay rate µ → eγ in terms of LEFT operators renormalized at a scale µ ∼ 2 GeV at which perturbation theory is still valid. The heavy quarks can be integrated out of the theory, so the low-energy effective theory contains three dynamical light quark flavors, q = u, d, s.
In addition to dipole operators such as Eq. (1.2), LEFT has dimension-six q 4 , 4 , q 2 2 , and G 3 operators, where q, , and G are quark, lepton, and gluon fields. The 4 and q 2 2 operators contribute to lepton-flavor-violating processes such as µ → eγ. The 4 operators contribute to µ → eγ via a perturbative loop graph giving L eγ ∼ em /(16π 2 )L 4 , where L 4 is a typical 4 operator coefficient. In this paper, we compute the non-perturbative contribution of semileptonic q 2 2 operators to the purely leptonic process µ → eγ, and show that these are of order L eγ ∼ e(F 2 π /Λ χ )L q 2 2 , where Λ χ ∼ 4πF π is the scale of chiral symmetry breaking [7]. These contributions are significant, and the limit Eq. (1.1) on the µ → eγ branching ratio puts interesting limits on semileptonic tensor operators.
Within the LEFT, the perturbative renormalization-group equations can be used to compute the running and mixing of operators from the weak scale down to the hadronic scale around ∼ 2 GeV. Below this scale, non-perturbative QCD effects are present and must be included. At low energies, the interactions of the lightest hadrons are described by chiral perturbation theory (χPT) [20][21][22]. In order to express the low-energy constants of χPT in terms of the LEFT parameters, a non-perturbative matching has to be performed, either using lattice QCD or phenomenological input. In this paper, we study the matching of semileptonic q 2 2 operators onto χPT operators that contribute to processes with no hadrons. Table 1. Semileptonic LEFT operators involving a charged-lepton bilinear and a quark bilinear.

Matching the LEFT to chiral perturbation theory
If we consider the leptonic sector of the LEFT, at leading order in the electromagnetic coupling, the quark dipole operators, the q 4 , and G 3 operators only affect the hadronic vacuum polarization function. The most interesting non-perturbative effect on the lepton sector is given by semileptonic q 2 2 operators. We consider the subset of LEFT operators involving a charged-lepton bilinear and a quark bilinear [3], shown in Table 1.
Quark operators in the LEFT Lagrangian can be matched onto operators in the chiral Lagrangian, as long as the momentum transfer is well below Λ χ . Such a matching has been extensively used in hadronic weak decays (e.g. see Ref. [23]). The results for scalar and vector quark bilinears can be obtained from the usual χPT Lagrangian including external sources [21,22]. The extension to tensor sources has been derived in Ref. [24].
We quickly remind the reader of the construction of the chiral Lagrangian with external sources. The massless QCD Lagrangian is supplemented by quark bilinears according to where q = (u, d, s) T is the three-component column vector of the light quarks and the external sources r µ , l µ , S, and t µν are 3 × 3 matrices in flavor space. 2 By promoting the sources to spurion fields transforming under chiral rotations, one obtains a QCD Lagrangian which is formally invariant under chiral symmetry, and one then constructs the most general chiral Lagrangian based on this chiral symmetry involving the spurion source fields and the matrix In the above equations, the matrix π(x) is defined in terms of the eight Goldstone boson fields π a (x), a = 1, · · · , 8, the Gell-Mann matrices λ a , and the pion decay constant in the chiral limit, F 0 . Green functions of quark bilinears are computed with the chiral Lagrangian via functional derivatives with respect to the sources. Finally, the spurion sources are fixed to the physical values including the contribution of the semileptonic LEFT operators.
Since the semileptonic LEFT contributions to the sources include a factor 1/v 2 with v ∼ 246 GeV, we are only interested in terms linear in the LEFT sources: quadratic terms are of the same size as dimension-eight contributions in the LEFT and can be neglected. This applies to weak SM contributions to the LEFT coefficients and even more so to BSM contributions suppressed by 1/Λ 2 BSM . Therefore, we explicitly split the LEFT contribution to the spurions and only consider linear terms in the LEFT contribution, which are given e.g. bỹ (2.4) The usual spurions are fixed in the end to their physical values (in our sign convention the QED covariant derivative is given by D µ = ∂ µ + ieQA µ ): where M q = diag(m u , m d , m s ) is the light-quark mass matrix. Note that terms proportional to G F induced by the SM weak interaction are included inl µ . At order p 4 in the chiral counting, 3 the matching is easily computed from the chiral Lagrangian by shifting sources as in Eq. (2.3), and amounts to the following replacement rules for the LEFT quark bilinears. Scalar: Vector: where we have omitted O(p 4 ) terms involving αβλσ from the odd-intrinsic-parity (anomalous) sector [25][26][27][28].
and similarly for F µν L with r µ → l µ , B 0 and L i are the usual non-perturbative low-energy constants (LECs) that enter the chiral Lagrangian to order p 2 and p 4 , and Λ 1,2 are analogous parameters for tensor sources studied in Ref. [24]. The matching ofq R S † q L andq R t † µν q L is given by the Hermitian conjugates of Eqs. (2.6), (2.8), and the one ofq . When computing matrix elements to order p 4 , one has to include the contribution from one-loop diagrams with p 2 vertices and tree graphs with p 4 vertices, as usual in χPT.
The operators on the l.h.s. of Eqs. (2.6), (2.7), and (2.8) are renormalized at a scale µ at which perturbation theory is still valid. While the combination χ = −2B 0 S † and the LECs L i are independent of the perturbative scale µ [21], the LECs B 0 and Λ i are scale dependent [21,24]. They connect the perturbative scale µ with low-energy hadron dynamics.
We can now match q 2 2 operators in LEFT onto the chiral Lagrangian. Quarks and leptons do not interact if electromagnetic effects are turned off, so we can directly get the matching to lowest order in α from Eqs. (2.6)-(2.8). For example, the LEFT operators The matching of O S,RL eu , O S,RL ed is given by a similar expression with the hadronic part in Eq. (2.12) replaced by its hermitian conjugate.
At order α, the q 2 2 operators can no longer be treated as the product of quark and lepton bilinears: e.g. the matching of the tensor operator Eq. (2.9) includes operators such as those in Eq. (2.12) where the leptonic part is a Lorentz scalar, since both operators transform as We want to study the effect of q 2 2 operators on µ → eγ. The matrix element γ(p, )|S|0 of a scalar operator to a physical photon vanishes due to Lorentz and gauge invariance. The matrix element γ(p, )|V µ |0 of a vector operator to a physical photon has the form by gauge invariance, and so vanishes for an on-shell photon. Eq. (2.13) has the structure of a penguin amplitude, and does contribute if the virtual photon is attached to external fermion lines, as in µ → 3e. The lowest-order version of this argument is a consequence of Eq. (2.6) with U = 1 having no F µν term, and Eq. (2.7) with U = 1 being proportional to ∂ ν F µν . The absence of a one-photon matrix element for scalar and vector operators holds to all orders in the chiral expansion, but to lowest order in α, since photon exchange between the quark and lepton bilinears can change the Lorentz structure of the bilinears. Ignoring QED corrections, which can mix Lorentz structure, the only LEFT operators that contribute to µ → eγ are tensor operators, which can have a non-zero hadronic matrix element to an on-shell photon: where F µν is the photon field-strength tensor. To study the impact of Eq. (2.14), we need the value of the non-perturbative parameter Λ 1 . Naive dimensional analysis (NDA) [7,29] gives the estimate with c T of order one. Here F π = 92.3(1) MeV is the physical pion decay constant [30], and F 0 = F π to lowest order in the chiral expansion. Eq. (2.15) is the size one would expect from the graph in Fig. 1 where the loop integral is estimated using the scale Λ χ . The model estimate of Ref. [31] gives c T ≈ −3.2. If we use the lattice input from Ref. [32] for the pion tensor charge and relate Λ 1 and Λ 2 by vector-meson saturation [33], we obtain c T ≈ −1.0(2), where the error is due to uncertainties in the vector couplings F V and G V and does not include model uncertainties. Similar estimates can be obtained from [34,35]. From Eq. (2.14) we obtain the low-energy matching condition for the additional non-perturbative contribution to the dipole operator, where the LEFT coefficients L T,RR eq are evaluated at a low scale µ ∼ 2 GeV, and c T (µ) is computed from tensor operators renormalized at the same scale. Note that in (2.16), the µ dependence due to α s corrections cancels in the product of the LEC c T and the tensor operator coefficients. using Eq. (2.16) as the only contribution to the dipole operator, which translates to Λ BSM > |c T | 1/2 250 TeV ≈ 450 TeV, using |c T | = 3.2.

Constraints on electromagnetic dipole interactions
In Table 2, we collect the constraints on the dipole and tensor operator coefficients at the hadronic scale µ = 2 GeV. Apart from the above constraints from µ → eγ, the same tensor operators contribute to µ → e conversion in nuclei. Due to their Lorentz structure these operators induce spin-dependent µ → e amplitudes, and do not directly contribute to the A 2 -enhanced spin-independent rate. However, these interactions do indirectly induce a spin-independent amplitude through their contribution to the dipole operators in Eq. (2.16). 4 As a result, we can compare the limits on the tensor operators from µ → eγ and µ → e conversion, the resulting bounds from the latter are shown in the last two rows of Table 2.
Here we used the expressions of Refs. [11,36], which are collected in App. A, together with the current and projected experimental constraints [37][38][39][40] Br(Au) ≤ 7 × 10 −13 , and employed the same hadronic and nuclear input as Ref. [11]. The constraints in Table 2 are derived by turning on a single operator at a time at the scale µ = 2 GeV. In this scenario, the current constraints from µ → eγ are stronger than the constraints from µ → e conversion, while the projected bounds are comparable. Note, however, that in a realistic BSM scenario, there is no reason to believe that only a single operator is present at the hadronic scale and in general the constraints have to be applied to the linear combination L eγ + δL eγ . In a second step, we consider the constraints on the operator coefficients at the scale µ = M W by taking into account the renormalization-group equations (RGEs) discussed in App. B. Detailed analyses of these RGEs have been presented in Refs. [10,11,13,19,[41][42][43]. We evaluate the bounds from µ → eγ and µ → e conversion by turning on single operators at a time at the scale µ = M W . These constraints almost directly translate to limits on operators in the SMEFT, using the tree-level matching [4], In the case of the tensor operators, the running and mixing from µ = M W down to µ = 2 GeV leads to a combination of two effects at the hadronic scale: on the one hand, the tensor operators contribute to the effective dipole operators through non-perturbative effects as discussed in Sect. 2, leading to a contribution proportional to c T . On the other hand, the RGE mixing induces perturbative contributions to the coefficients of the dipole operators which are independent of c T . Therefore, the bounds on µ → eγ constrain the linear combinations of terms proportional of c T and independent of c T as shown in the upper half of Table 3. Assuming c T = O(1), the non-perturbative contribution dominates for the couplings to up and down quarks, while it is expected to be of the same order of magnitude as the perturbative term for the strange-quark couplings.
In the case of µ → e conversion, the RGEs have more important consequences: the tensor operator coefficients also mix into the coefficients of semileptonic scalar operators L S,RR ed,eu [3,11]. As discussed in Ref. [43], these mixing effects from µ = M W down to µ = 2 GeV turn out to be large and give a significant contribution to the spin-independent amplitude. 5 As in the case of µ → eγ, the bounds on µ → e conversion constrain a combination of the couplings at 5 In principle there are additional contributions to the spin-independent amplitude that also appear at O(e 2 /(4π) 2 ). Firstly, a spin-independent (N N )(μe) interaction can be induced in the chiral Lagrangian by the combination of the tensor operator with two electromagnetic currents. In addition, there are chiral loops involving photon exchange around the leading-order spin-dependent (N N )(μe) interaction that is generated by L T,RR ed,eu . As these contributions involve unknown LECs we neglect them here and use the RGE-induced contribution as an estimate of the full amplitude.  Table 3.
Note that the current constraints from µ → e conversion are of the same order of magnitude as the limits from µ → eγ, if we use the model estimate c T ≈ −3. In this case, the µ → e conversion limits are somewhat stronger for the couplings involving the up and down quarks, while the strange-quark couplings are more stringently constrained by the µ → eγ limit. It should be mentioned that these comparisons are currently rather uncertain due to the fact that only NDA and model estimates exist for c T . In addition, the two observables depend on different linear combinations of the tensor interactions. In particular, the combination of couplings that enters µ → eγ only depends on the quark charges, while the spin-independent µ → e conversion rate also involves the matrix elements N |qq|N . This implies that the two observables are complementary probes, as one would generally expect BSM physics to generate multiple operators in the LEFT.
The CP-odd parts of the dipole moments can be constrained by EDM measurements, which, in the case of the muon [72] and tau [73] do not lead to significant limits. The electron EDM is more relevant and is currently most stringently constrained by measurements on the paramagnetic ThO molecule [74]. This system receives contributions from the electron EDM as well as from scalar interactions between electrons and nucleons [75][76][77] ω ThO = (120.6 ± 4.9) d e 10 −27 e cm + (181.6 ± 7.3) × 10 7 Z A C showing that the non-perturbative contributions are of similar size as those induced through the RGEs. Using the experimental upper bound, ω ThO ≤ 10.5 mrad/s, we obtain Λ BSM 10 3 TeV for the couplings to up and down quarks, while for the interaction with strange quarks we have Λ BSM 3 × 10 3 TeV.
The EDM of Mercury receives direct contributions from tensor interactions as it is a diamagnetic system. Following the analysis of [80] for the atomic [81] and hadronic [82] input and using the current experimental constraint [83,84] leads to more stringent constraints on the up and down quark couplings. In particular, for the coupling to the up (down) quark, the mercury EDM leads to a constraint on Λ BSM that is stronger by a factor 4 (10) than those mentioned above. In contrast, the coupling to the strange quark is more stringently constrained by the ThO measurement. This implies that although the tensor interactions contribute to the mercury EDM directly, the contributions to paramagnetic molecular systems can be significant, in particular in the case of strange-quark couplings.

Comment on dark-matter interactions
A very similar situation to the semileptonic tensor operators arises in the context of darkmatter interactions. For dark matter consisting of weakly interacting massive particles that couple to SM particles only through a heavy mediator, an effective field theory can be used to describe the dark matter interaction with SM particles [85][86][87][88]. Under the assumption that the dark matter particles carry a conserved quantum number, the dark matter fields always appear in bilinears and give rise to operators that are similar to those in the LEFT. For example, at dimension five the dipole operators arise, where χ denotes the dark matter fermion. At dimension six, the EFT contains fourfermion operators (χΓχ)(qΓq), e.g. the tensor operators For direct-detection experiments, the interaction of dark matter with nucleons has to be studied. The constraints of chiral symmetry have been analyzed in [86,89] for the case of vector, axial-vector, scalar, and pseudo-scalar interactions. The tensor operators (3.9) lead to spindependent interactions that are not enhanced by A 2 . The leading contribution to the nucleon tensor form factors [90,91] is given by the tensor charges and requires lattice QCD input [82]. Through perturbative RGE mixing, the tensor operators induce the electromagnetic dipole operators (3.8) [92], which leads to a spin-independent contribution to the cross section [93,94]. In addition to these perturbative RGE effects, again a non-perturbative contribution is generated in analogy to the semileptonic dipole operators discussed in Sect. 2. Given the similarities to the case of µ → eγ, the linear combination of the contributions due to the RGEs and those due to c T is the same as those in the first row of Table 3 (where L eγ eµ and L T,RR ed,eu eµqq are to be identified with C (5) and C (6) q , respectively). As a result, for c T = O(1), the contributions of the couplings to up and down quarks will be dominated by the non-perturbative piece, while it is expected to be of similar size as the RGE term for the strange-quark couplings.
Apart from the contribution proportional to c T , another non-perturbative contribution is given by the coupling of the tensor current to pion exchange between two nucleons or to pion loops at the one-nucleon level. 8 The coupling of the tensor current to the pion is parametrized by the LEC Λ 2 in (2.8), which can be estimated using the lattice result [32] for the pion tensor charge, Λ 2 = 6.0(3) MeV.

Conclusion
In this paper, we have discussed the non-perturbative contributions of semileptonic LEFT operators to the purely leptonic process µ → eγ. Although scalar and vector operators do not induce this process (without additional QED corrections), q 2 2 tensor interactions do generate this decay through a non-perturbative contribution to the leptonic dipole operators, L eγ ∼ e(F 2 π /Λ χ )L q 2 2 . To quantify this contribution we have matched the semileptonic tensor operators to χPT, and estimated the required low-energy constant using NDA and the model estimate of Ref. [31]. These estimates show that the non-perturbative contribution is expected to dominate over the perturbative RGE effects for the couplings to up and down quarks, while it is of the same order of magnitude for the coupling to the strange quark. The constraints on the scale of BSM physics that can be set using the experimental limit on the µ → eγ branching ratio are of the order of Λ BSM 450 TeV, as discussed in Sect. 3. These limits are of similar size as those obtained from µ → e conversion in nuclei for the couplings to up and down quarks, while they currently provide the most stringent constraints on the couplings to strange quarks.

A µ → e conversion in nuclei
Here we discuss in more detail the contributions of the operators in Table 1 to µ → e conversion in nuclei. The rate for this process can be written as [11,Eq. (3.5)] The couplings C SL q can be obtained from C SR q by replacing the the labels eµqq → µeqq and taking the complex conjugate. It is conventional to define a branching ratio through Br(A) = Γ(µA → eA)/Γ capt (A), where Γ capt is the muon capture rate. We follow Ref. [11] for the hadronic [78,79] and nuclear input [95].
In order to obtain the non-perturbative tensor operator contribution to µ → e conversion, one has to substitute L eγ → L eγ + δL eγ with δL eγ given in (2.16).

B Anomalous dimensions
The RGEs for the dipole, scalar and tensor operators relevant for µ → eγ from Ref. [3,4] are summarized below. We treat L eγ ∼ em µ /Λ 2 BSM so that terms quadratic in L eγ have been dropped. These equations agree with earlier results in Ref. [11]. There are some differences with the results in Refs. [10,41,42].