Renormalisation-group improved analysis of μ → e processes in a systematic effective-field-theory approach

Received: February Renormalisation-group improved analysis of processes in a systematic e ective- eld-theory CH- 0 2 Villigen PSI 0 2 Switzerland 0 2 IPNL 0 2 CNRS/IN 0 2 P 0 2 Open Access 0 2 c The Authors. 0 2 0 4 rue E. Fermi , 69622 Villeurbanne cedex , France 1 Physik-Institut, Universitat Zurich 2 F-69622 Lyon , France 3 Paul Scherrer Institut In this article, a complete analysis of the three muonic lepton- avour violating processes processes; in; a; systematic; e; ective-; eld-theory; Beyond Standard Model; E ective Field Theories; Renormalization Group - approach ! 3e and coherent nuclear ! e conversion is performed in in the light of current data and future experimental prospects. Contents 1 Introduction 2 3 4 Low-energy Lagrangian LFV muon decays ! e conversion in nuclei Renormalisation-group evolution Phenomenological analysis Conclusions and outlook A Anomalous dimensions Introduction SINDRUM [3, 4] collaborations are Au ! e Au) The future experimental prospects for upgrade to MEG II [5], the sensitivity of Br( Mu3e will improve the sensitivity on ! e transitions are also promising. With the ! e ) will reach 10 14. Furthermore, ! 3e by up to 4 orders of magnitude [6]. Concerning Okada [13] reviewed ! e avour-changing processes and experiments, and the operator basis required to parameterise them. lepton operators [19]. In the quark avour sector, a long-time e ort allowed to establish proved that the self-renormalisation of the dipole reduces the coe cient at low energy. Also in the case of (g 2) (lepton- avour conserving e ective interactions), the The results of [25, 26, 28] were used in [33], to translate the current experimental scale to the new-physics scale, using QED QCD invariant operators below ! e bound from the mW , and the SU(2)-invariant operators above.2 ! 3e and coherent ! e conversion and include the lowest non-vanishing order in QED and QCD ( ! e conversion and ! 3e in the phenomenological analysis. In addition, we include the dimension-seven lepton-gluon operator that is relevant in ! e conversion [35]. QED invariant Low-energy Lagrangian that is valid below some scale with mW Therefore, it consists of all operators that are invariant under U(1)QED SU(3)QCD and contain the fermion 1For an analogous analysis in the quark sector see for example [29{32]. in [33, 34]. tiplied by dimensionless Wilson coe cients C. Having ! e transitions in mind, we respect to the other elds. Concretely, our Lagrangian reads Le = LQED + LQCD CLDOLD + CfVfLLOfVfLL + CfVfLROfVfLR + CfSfLLOfSfLL ChThLLOhThLL + ChShLR OhShLR + CgLgOgLg + L $ R with the explicit form of the operators given by OLD = e m (e OfVfLL = (e OfVfLR = (e OfSfLL = (ePL ) f PLf ; OhShLR = (ePL ) hPRh ; OhThLL = (e OgLg = s m GF (ePL ) Ga Ga ; where PL=R = I and Ga , respectively. Regarding the matter elds, f represents any fermion (hadron) or the and constitutes a minimal basis. (when considering the RGE). factor GF is included to resize the dimensionality down to 6. In the scenario where BSM physics is realised at a scale < mW , this NP directly gives EWSB scale, as it does not respect the SU(2) symmetry. important e ects. LFV muon decays In this section, the expressions for the processes muon-to-electron conversion in muonic atoms ! e+e e+, and coherent ! e N in terms of the coe cients at the scale of the process. is the LFV muonic process with the most stringent experimental ! e ) = is the width of the muon. The scale of the process is = m . Operators other than the dipole will enter this process only through the RGE. ! eee The bounds on Ci(m ) reads ! 3e) = where the interference term with the dipole operator is given by RehCLD CeVe RL +2CeVe RR i suppressed by powers of me. So in contrast to only impose two constraints, the upper bound on Br( ! e and coherent ! e conversion, which ! 3e) sets independent constraints on several four-lepton operators. Through the RGEs, this process is also sensitive to operators involving quarks or other leptons. ! e conversion in nuclei For this process, the Lagrangian Le as given in eq. (2.1) is not directly suitable. CgLg ! C~gLg = CgLg CqSqLL + CqSqLR GF m mq N ! e N ) can then be written as N!e = ( !e = 4 4 e CLD DN + 4 GF m mpC~(SpL)SN(p) + C~(Vp)R V (p) + p ! n N + L $ R; eq. (3.5) can be expressed in terms of our Wilson coe cients as C~(Vp=Rn) = C~(SpL=n) = CqVq RL + CqVq RR fV(qp)=n ; CqSqLL + CqSqLR m mqGF fS(qp)=n + C~gLg fGp=n be evaluated at the scale calculation [46].3 In summary, we use fS(up) = (20:8 fS(dp) = (41:1 fS(sp) = fS(sn) = (53 fS(un) = (18:9 fS(dn) = (45:1 isation we get fGp=n = fS(qp)=nA : of the target N and we use the numerical values [38] DAu = 0:189; eq. (3.5), divided by the capture rate. For the latter we use cAaupt = 8:7 10 15 MeV; capt = 4:6 10 16 MeV; taken from [49]. Renormalisation-group evolution The operators present in Le , eq. (2.1), will give rise to ! e transitions. Thus, experiWilson coe cients at the high scale. (...truncated)