Spin-dependent $\mu \to e$ conversion

The experimental sensitivity to $\mu \to e$ conversion on nuclei is expected to improve by four orders of magnitude in coming years. We consider the impact of $\mu \to e$ flavour-changing tensor and axial-vector four-fermion operators which couple to the spin of nucleons. Such operators, which have not previously been considered, contribute to $\mu \to e$ conversion in three ways: in nuclei with spin they mediate a spin-dependent transition; in all nuclei they contribute to the coherent ($A^2$-enhanced) spin-independent conversion via finite recoil effects and via loop mixing with dipole, scalar, and vector operators. We estimate the spin-dependent rate in Aluminium (the target of the upcoming COMET and Mu2e experiments), show that the loop effects give the greatest sensitivity to tensor and axial-vector operators involving first-generation quarks, and discuss the complementarity of the spin-dependent and independent contributions to $\mu \to e$ conversion

Introduction -New particles and interactions beyond the Standard Model of particle physics are required to explain neutrino masses and mixing angles. The search for traces of this New Physics (NP) is pursued on many fronts. One possibility is to look directly for the new particles implicated in neutrino mass generation, for instance at the LHC [1] or SHiP [2]. A complementary approach seeks new interactions among known particles, such as neutrinoless double beta decay [3] or Charged Lepton Flavour Violation (CLFV) [4].
CLFV transitions of charged leptons are induced by the observed massive neutrinos, at unobservable rates suppressed by (m ν /m W ) 4 ∼ 10 −48 . A detectable rate would point to the existence of new heavy particles, as may arise in models that generate neutrino masses, or that address other puzzles of the Standard Model such as the hierarchy problem. Observations of CLFV are therefore crucial to identifying the NP of the lepton sector, providing information complementary to direct searches.
From a theoretical perspective, at energy scales well below the masses of the new particles, CLFV can be parametrised with effective operators (see e.g. [5]), constructed out of the kinematically accessible Standard Model (SM) fields, and respecting the relevant gauge symmetries. In this effective field theory (EFT) description, information about the underlying new dynamics is encoded in the operator coefficients, calculable in any given model.
The experimental sensitivity to a wide variety of CLFV processes is systematically improving. Current bounds on branching ratios of τ flavour changing decays such as τ → µγ, τ → eγ and τ → 3ℓ [6][7][8] are O(10 −8 ), and Belle-II is expected to improve the sensitivity by an order of magnitude [9]. The bounds on the µ ↔ e flavour changing processes are currently of order ∼ 10 −12 [10,11], with the most restrictive contraint from the MEG collaboration: BR(µ → eγ) ≤ 4.2 × 10 −13 [12]. Future experimental sensitivities should improve by sev-eral orders of magnitude, in particular, the COMET [13] and Mu2e [14] experiments aim to reach a sensitivity to µ → e conversion on nuclei of ∼ 10 −16 , and the PRISM/PRIME proposal [15] could reach the unprecedented level of 10 −18 .
In searches for µ → e conversion, a µ − from the beam is captured by a nucleus in the target, and tumbles down to the 1s state. The muon will be closer to the nucleus than an electron (r ∼ αZ/m), due to its larger mass.
In the presence of a CLFV interaction with the quarks that compose the nucleus, or with its electric field, the muon can transform into an electron. This electron, emitted with an energy E e ≃ m µ , is the signature of µ → e conversion.
Initial analytic estimates of the µ → e conversion rate were obtained by Feinberg and Weinberg [16], a wider range of nuclei were studied numerically by Shankar [17], and relativistic effects relevant in heavier nuclei were included in Ref. [18]. State of the art conversion rates for a broad range of nuclei induced by CLFV operators which can contribute coherently to µ → e conversion were obtained in Ref. [19], while some missing operators were included in Ref. [20].
The calculation has some similarities with dark matter scattering on nuclei [21][22][23], where the cross-section can be classified as spin-dependent (SD) or spin-independent (SI). Previous analyses of µ → e conversion [19,20] focused on CLFV interactions involving a scalar or vector nucleon current, because, similarly to SI dark matter scattering, these sum coherently across the nucleus at the amplitude level, giving an amplification ∼ A 2 in the rate, where A is the atomic number. However, other processes are possible, such as spin-dependent conversion on the ground state nucleus, which we explore here, or incoherent µ → e conversion, where the final-state nucleus is in an excited state [17,24].
The upcoming exceptional experimental sensitivities motivate our study of new contributions to µ → e conversion induced by tensor and axial vector operators 1 , which were not considered in Refs. [19,20]. These operators couple to the spin of the nucleus and can induce "spin-dependent" µ → e conversion in nuclei with spin (such as Aluminium, the proposed target of COMET and Mu2e), not enhanced by A 2 . In addition, the tensor and axial operators will contribute to "spin-independent" conversion via finite-momentumtransfer corrections [25,26], and Renormalisation Group mixing [27,28] In an EFT framework, our analysis shows new sensitivities to previously unconstrained combinations of dimension-six operator coefficients, as we illustrate below. In the absence of CLFV, this gives new constraints on the coefficients, and when CLFV is observed, it could assist in determining its origin.
Estimating the µ → e conversion rate -Our starting point is the effective Lagrangian [4] where Y ∈ {L, R} and O ∈ {V, A, S, T } and the operators are explicitly given by (P L,R = 1/2(I ∓ γ 5 )) While our primary focus is on the tensor (O qq T,Y ) and axial (O qq A,Y ) operators, we include the vector, scalar, dipole and gluon operators because the first three are induced by loops, and the last arises by integrating out heavy quarks.
At zero momentum transfer, the quark bilinears can be matched onto nucleon bilinears where the vector charges are G p,u V = G n,d V = 2 and G p,d V = G n,u V = 1, and for the axial charges we use 1 We leave out the light-quark pseudoscalar operators and gluon operators such as GG that can be induced by heavy-quark pseudoscalar operators at the heavy quark thresholds. The effect of this class of operators in a nucleus is suppressed both by spin and momentum transfer. 2 The analogous mixing of SD to SI dark matter interactions was discussed in [29,30].
the results inferred in Ref. [22] by using the HERMES measurements [31], namely G p,u (18). For the tensor charges we use the lattice QCD results [32] in the MS scheme at µ = 2 GeV, namely (3), and G p,s T = G n,s T = .008 (9), Finally, for the scalar charges induced by light quarks we use a precise dispersive determination [33], (3), and an average of lattice results [34] for the strange charge: G p,s S = G n,s S = mN ms 0.043 (11). In all cases, we take central values of the MS quark masses at µ = 2 GeV, namely m u = 2.2 MeV, m d = 4.7 MeV, and m s = 96 MeV [35].
Taking the above matching into account, the nucleonlevel effective Lagrangian has the same structure of (1) with the replacementsqΓ O q →N Γ O N and with effective couplings given by 3 However, we remove the tensor operators, because their effects can be reabsorbed into shifts to the axial-vector and scalar operator coefficients. In fact, to leading order in a non-relativistic expansion N σ ij N = ǫ ijk N γ k γ 5 N , so that the spin-dependent nucleon effective Lagrangian for µ → e conversion reads where N ∈ {n, p}, X, Y ∈ {L, R}, X = Y and Furthermore, at finite recoil the tensor operator induces a contribution to the SI amplitude, since u N (p)σ 0i u N (p − q) contains a term proportional to q i /m N [25,26], which contracts, in the amplitude, with the spin of the helicity-eigenstate electron. The net effect is tantamount to replacing the coefficient of the scalar operator with We write the conversion rate Γ = Γ SI + Γ SD , where Γ SI is the A 2 -enhanced rate occuring in any nucleus, and Γ SD is only relevant in nuclei with spin. The usual SI branching ratio reads [4,19] where B 0 = G 2 F m 5 µ (αZ) 3 /(π 2 Γ cap ), Γ cap is the rate for the muon to transform to a neutrino by capture on the nucleus (0.7054 × 10 6 /sec in Aluminium [36]), and the form factors F p,n (| k|) = d 3 xe −i k· x ρ p,n (x) can be found in Eq. (30) of Ref. [19].
In the evaluation of Γ SD from (5) we treat the muon as non-relativistic and the electron as a plane wave. Both are good approximations for low-Z nuclei; for definiteness we focus on Aluminium (Z = 13, A = 27, J = 5/2) the proposed target for the COMET and Mu2e experiments. After approximating the muon wavefunction in the nucleus to its value at the origin and taking it outside the integral over the nucleus [16], the nuclear part of the spin-dependent µ → e amplitude corresponds to that of "standard" spin-dependent WIMP nucleus scattering. At momentum transfer q, this is The µ → e amplitude is then obtained by multiplying by the appropriate lepton current and coefficients 4 . By analogy with WIMP scattering [22,23,37], we obtain: The spin expectation values S Al N are defined as S Al N = J Al , J z = J Al |S z N |J Al , J z = J Al , where S z N is the z component of the the total nucleon spin, and the expectation value is over the nuclear ground state. They can be implemented in our QFT notation (with relativistic state normalisation for Al) by setting Eqn. (9) The axial structure factor S A (| q|) [23,37] reads where a L,± = C pp A,L ± C nn A,L . The S Al N and S ij (q) have been calculated in the shell model in Refs. [37,38]. At | q| ≡ q = 0 the conversion rate is controlled by the spin expectation values; we use S Al n = 0.030 and S Al p = 0.34 [38]. At finite momentum transfer q = m µ , the structure factors provide a non-trivial correction. Using dominance of the proton contribution (S Al p >> S Al n ) we find from Ref. [38] S Al (m µ ))/S Al (0) ≃ 0.29.
Loop effects and the RGEs -QED and QCD loops change the magnitude of some operator coefficients, and QED loops can transform one operator into another. Such Standard Model loops are neccessarily present, and their dominant (log-enhanced) effects are included in the evolution with scale of the operator coefficients, as described by the Renormalisation Group Equations (RGEs) of QED and QCD (see [5] for an introduction to the RG running of operators with the scale µ). If the New Physics scale is well above m W , loops involving the W, Z, and h could also be relevant. However, we focus here on the RGE evolution from the experimental scale µ N up to the weak scale m W . Since any UV model can be mapped into a set of operator coefficients at µ = m W , our calculation does not lose generality while remaining quite simple.
We consider the one-loop RGEs of QED and QCD for µ ↔ e flavour-changing operators [27,28]. Defining λ = αs(mW ) αs(µN ) , their solution can be approximated as where I, J represent the super-and subscripts which label operator coefficients. The a I describe the QCD running and are only non-zero for scalars and tensors: for N f = 5 one has a I = Γ s II 2β0 = {− 12 23 , 4 23 } for I = S, T,. We use this scaling to always give results in terms of coefficients at the low scale µ N = 2 GeV, where we match quarks to nucleons. Γ e is the one-loop QED anomalous dimension matrix, rescaled [39,40] for J, I ∈ T, S to account for the QCD running: In the estimates presented here, we focus on the effects of the off-diagonal elements of Γ e JI , which mix one operator into another, and neglect the QED running of individual coefficients.
In RG evolution down to µ N , photon exchange between the external legs of a tensor operator can mix it to a scalar operator. This contribution to the scalar coefficient is where f T S is from Eq. (12). The tensor operator also mixes to the dipole, when the quark lines are closed and an external photon is attached. This gives a contribution to the dipole coefficient (14) which is suppressed by m q /m µ , due to a mass insertion on the quark line. For tensor operators involving u, d or s quark bilinears, the mixing to the scalar operator described in Eq. (13) gives a larger contribution to SI µ → e conversion than this mixing to the dipole. So for the remainder of this letter, we do not discuss the contribution of Eq. (14) to µ → e conversion. We will discuss heavier quarks 5 in a later publication [41]. Curiously, one-loop QED corrections to the axial operator generate the vector [28] 6 . If a New Physics model induces a non-zero coefficient C qq A,Y (m W ), then photon exchange between the external legs induces a contribution to the vector coefficient at the experimental scale: As a result, the SI and SD processes will have comparable sensitivities to axial vector operators.
Results -To interpret our results, we first estimate the sensitivity of SD and SI µ → e conversion to the coefficients of the tensor and axial operators of eqn (2). We allow a single operator coefficient to be non-zero at m W , and consider its various contributions to SD and SI µ → e conversion (sometimes refered to as setting bounds "one-operator-at-a-time").
Suppose first that only the tensor coefficient C uu T,L is present at m W . Recall that C uu T,L (m W ) can contribute to µ → e conversion in three ways: to the SI rate via the finite momentum transfer effects of eqn (7), to the SI rate via the RG mixing to the scalar given in eqn (13), and directly to the SD rate as given in eqn (10). It is easy to check that the RG mixing contribution to C N N S (µ N ) is an order of magnitude larger than the finite recoil contribution. Furthermore, the RG mixing effect is dominant contribution of C uu T,L (m W ) to µ → e conversion, as can been seen numerically by calculating the SD and SI contributions to the branching ratio: where the coefficients are at the experimental scale, and the second term is the A 2 -enhanced SI contribution. The RG mixing is the largest contribution of C uu T,L (m W ) to µ → e conversion due to three enhancements: first, the anomalous dimension Γ e T S is large, and second, the G N,q S coefficients of eqn (3) are an order of magnitude larger than G N,q T . The combination of these gives ∆ C N N S,X (µ N ) > ∼ C pp T,X (µ N ), which respectively contribute to the SI and SD rates. Finally, the scalar coefficient benefits from a further A 2 enhancement in the SI conversion rate. This shows that including the RG effects can change the branching ratio by orders of magnitude.
A similar estimate for the axial operator O uu A,L gives We see that the RG mixing of O uu A,L into O uu V,L , whose coefficient contributes to SI µ → e conversion, also gives the best sensitivity to C uu A,L . However, the ratio of SI to SD contributions is smaller than in the tensor case, due to the smaller anomalous dimension in eqn (15).
SI µ → e conversion will also give the best sensitivity to tensor and axial operators involving d quarks. However, in the case of strange quarks, the vector current vanishes in the nucleon, so O ss A,Y only contributes to SD µ → e conversion. The largest contribution of the strange tensor operator is via its mixing to the scalar, with a sensitivity to C ss T,X reduced by a factor ∼ G N s T /2G N u T with respect to C uu T,X . The strange tensor also mixes significantly to the dipole (see eqn (14)) which contributes to µ → eγ; we estimate that the sensitivity to C ss T,Y of the MEG experiment with BR ∼ 2×10 −14 (as expected after their upgrade), would be comparable to that of COMET or Mu2e with BR ∼few×10 −16 .
Let us now focus on the complementarity of SD and SI contributions to the µ → e conversion rate, which depend on different combinations of operator coefficients. So once a signal is observed, measuring µ → e conversion in targets with and without spin could assist in differentiating among operators or models. To illustrate this complementarity, we restrict to scalar and tensor operators involving u quarks, whose coefficients we would like to determine. Figure 1 represents the allowed parameter space for C uu T,L and C uu S,L evaluated at µ N (dotted blue) and m W (solid red). We see that, irrespective of the operator scale, SD µ → e conversion always gives an independent constraint. In its absence, there would be an unconstrained direction in parameter space, corresponding to C uu T,Y at the experimental scale, or the diagonal red band at m W . The figure also shows that the enhanced sensitivity of SI conversion illustrated in eqn (16) requires the (model-dependent) assumption that the model does not induce a scalar contribution which cancels the mixing of the tensor into the scalar, which would correspond to venturing along the red ellipse in the plot.
Prospects -In this letter, we followed the pragmatic low-energy perspective of parametrising charged Lepton Flavour Violating interactions with effective operators, and considered the contribution of axial vectors and tensors to µ → e conversion. To our knowledge, this has not been studied previously. We found that the Spin- Dependent process depends on different operator coefficients from the Spin-Independent case, so comparing µ → e conversion rates in targets with and without spin would give additional constraints, and could allow to identify axial or tensor operators coefficients. In future work [41], we plan to give rates for a complete set of operators, estimate their uncertainties due to higher order terms and neglected effects, and explore realistic prospects for distinguishing models/operators using targets with and without spin, such as different isotopes of T i, a nucleus used for the past µ → e conversion conversion searches. VC and SD thank Martin Hoferichter for discussions, and acknowledge the partial support and hospitality of the Mainz Institute for Theoretical Physics (MITP) during the completion of this work. The work of Y.K. is supported in part by the Japan Society of the Promotion of Science (JSPS) KAKENHI Grant No. 25000004.