Completeness and complementarity for μ → eγ, μ → $$ e\overline{e}e $$ and μA → eA

Abstract

Lepton Flavour Violation (LFV) is New Physics that must occur, but is stringently constrained by experiments searching for μ ↔ e flavour change, such as μ → eγ, μ →$$ e\overline{e}e $$ e e ¯ e or μ → e conversion. However, in an Effective Field Theory(EFT) parametrisation, there are many more μ ↔ e operators than restrictive constraints, so determining operator coefficients from data is a remote dream. It is nonetheless interesting to learn about New Physics from data, so this manuscript introduces “observable-vectors” in the space of operator coefficients, which identify at any scale the combination of coefficients probed by the observable. These vectors have an overlap ≳ 10−3 with most of the coefficients, and are used to study whether μ → eγ, μ →$$ e\overline{e}e $$ e e ¯ e and μ → e conversion give complementary information about New Physics. The appendix gives updated sensitivities of these processes, (and a subset of τ → ℓ decays), to operator coefficients at the weak scale in the SMEFT and in the EFT below mW.

What is Leading Order for LFV in SMEFT?

“…so there is sensitivity to vector coefficients for scales below 10 TeV (which is consistent with the bound in [10,11]). Approximating physical predictions in terms of powers of some parameter is always somewhat arbitrary and erroneous (Indeed, although we count in λ, we allow for √ λ in table 3).…”

“…(2.7) For µ → e flavour change, the MEG bound [26] on BR(µ → eγ) could probe couplings as small as [10] v…”

“…We first consider µ → e processes because the most restrictive experimental constraints on LFV arise in this sector, and upcoming experiments aim to improve the sensitivities by several orders of magnitude (see table 3; indeed, there are plans to reach a conversion ratio 10 −18 for µA → eA [43]). The Branching Ratios can be expressed (see eg [3,10,42]) in terms of the coefficients, evaluated at the experimental scale, of operators which contribute at tree level. For instance, the low-energy operators…”

“…Approximating physical predictions in terms of powers of some parameter is always somewhat arbitrary and erroneous (Indeed, although we count in λ, we allow for √ λ in table 3). In order to test our recipe, in table 4 we compare our power-counting estimates to the solutions of the RGEs; this estimate is obtained in the EFT below m W , for which the solution of the "leading order" RGEs is given in [10,11]. The table shows that our estimate of the scale Λ NP where C f f T would be ∼ 1, (obtained by combining eq.…”

“…Below the weak scale, a powercounting recipe for flavour is sufficient to organise a calculation, because the mass scales for the EFT are known, and the remaining couplings are few: in the RGEs for four-quark operators, QED effects can be included at appropriate subleading order in the expansion in α s log [7,9]. For LFV below the weak scale, the "leading order" operators and RGEs have been assembled: observables can be parametrised with three and four-point functions, which correspond to operators of dimensions five to eight (see eg [10] for a list), and the "leading order" RGEs, which include two-loop vector to dipole mixing, are given in [11]. However, above the weak scale, the situation is complicated by the dynamical Higgs and SU(2) gauge bosons, which introduce more particle mixing in the RGEs, and also by our ignorance of the mass scale of new particles, Λ NP .…”