Eur. Phys. J. C (2016) 76:370 DOI 10.1140/epjc/s10052-016-4207-5 Regular Article - Theoretical Physics μ → eγ and matching at mW Sacha Davidson1,2,3,a 1 IPNL, CNRS/IN2P3, 4 rue E. Fermi, 69622 Villeurbanne Cedex, France 2 Université Lyon 1, Villeurbanne, France 3 Université de Lyon, 69622 Lyon, France Received: 18 March 2016 / Accepted: 14 June 2016 / Published online: 4 July 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract Several experiments search for μ ↔ e flavour attempt to reconstruct the fundamental Lagrangian of New change, for instance in μ → e conversion, μ → eγ and Physics from the operator coefficients. This is probably not μ → eee¯ . This paper studies how to translate these experi- feasible, but could give interesting perspectives. A first step mental constraints from low energy to a New Physics scale in this “bottom-up” approach, explored in this paper, is to M mW . A basis of QCD × QED-invariant operators (as use Effective Field Theory (EFT) [31] to translate the exper- appropriate below mW ) is reviewed, then run to mW with imental bounds to the coefficients of effective operators at one-loop Renormalisation Group Equations (RGEs) of QCD the New Physics scale M > mW . and QED. At mW , these operators are matched onto SU(2)- The goal would be to start from experimental constraints invariant dimension-six operators, which can continue to run on μ–e flavour change, and obtain at M the best bound up with electroweak RGEs. As an example, the μ → eγ on each coefficient from each observable. These constraints bound is translated to the scale M, where it constrains two should be of the correct order of magnitude, but not precise sums of operators. The constraints differ from those obtained beyond one significant figure. This preliminary study restricts in previous EFT analyses of μ → eγ , but they reproduce the the experimental input to the bound on BR(μ → eγ), and expected bounds on flavour-changing interactions of the Z makes several simplifications in the translation up to the New and the Higgs, because the matching at mW is pragmatically Physics scale M. Firstly, the EFT has three scales: a low scale performed to the loop order required to get the “leading” mμ ∼ mb, the intermediate weak scale mW , and the high contribution. scale M. Secondly, at a given scale, the EFT contains lighter Standard Model particles and dimension-six, gauge-invariant operators (one dimension-seven operator is listed; however, 1 Introduction dimension-eight operators are neglected). The final simplifi- cation might have been to match at tree level, and run with Neutrino masses and mixing angles imply that “New” one-loop Renormalisation Group Equations (RGEs). How- Physics from beyond the Standard Model(SM) must be ever, a bottom-up EFT should reproduce the results of top- present in the lepton sector, and must induce charged Lepton down model calculations, and it is straightforward to check Flavour Violation (LFV; for a review, see [1]). However, nei- that one- and two-loop matching is required at mW to obtain ther LFV nor the origin of neutrino masses has yet been dis- the correct bounds from μ → eγ on LFV interactions of covered. This study assumes that the required new particles the Z and Higgs. So the matching at mW is performed to the are heavy, with masses at or beyond M > mW . In addition, order required to get the known bounds. between mW and M, there should be no other new particles The paper is organised in two parts: the Sects. 2–4 con- or interactions which affect the LFV sector. One approach struct some of the framework required to obtain experimental to identifying this New LFV Physics, is to construct a moti- constraints on SU(2)-invariant operator coefficients at mW , vated model, and identify its signature in observables [2–30]. then Sect. 5 focusses on using, checking and improving this A more pragmatic approach, which requires optimism but formalism to obtain bounds from μ → eγ on operator coef- no model-building skills, is to parametrise the New Physics ficients at M. The formalism can be organised in four steps: at low energy with non-renormalisable operators, map the matching at mμ, running to mW , matching at mW , then run- experimental constraints onto the operator coefficients, and ning up to the New Physics scale M. Section 2 reviews the basis of QCD×QED-invariant operators, as appropriate a e-mail: [email protected] 123 370 Page 2 of 19 Eur. Phys. J. C (2016) 76 :370 below mW . These operators, of dimension five, six and seven, contribution of the LFV Higgs operator to LFV Z couplings describe three- and four-point functions involving a μ,an was beautifully studied in [45]. There are also many closely e and any other combination of flavour-diagonal light par- related works in the quark sector, reviewed in [46,47]. For ticles. To complete the first step, the experimental bounds instance, the QED anomalous dimension matrix for various should be matched onto these operator coefficients; how- vector four-quark operators is given in [48], and matching ever, this is delayed till Sect. 5, where only the bound on at mW of flavour-changing quark operators is discussed in μ → eγ is imposed on the dipole coefficients (the bounds [49]. However, colour makes the quarks different, so it is not from μ → e conversion and μ → eee¯ are neglected for always immediate to translate the quark results to leptons. simplicity; the strong interaction subtleties of matching to μ → e conversion are discussed in [33–36]). Section 3 dis- cusses the second step, which is to run the coefficients up to 2 A basis of μ–e interactions at low energy mW with the RGEs of QED and QCD. Appendix B gives the anomalous dimension matrix mixing the scalar and tensor 2.1 Interactions probed in muon experiments operators to the dipole (which is responsable for μ → eγ ). The anomalous dimension matrix for vector operators is Experiments searching for lepton-flavour change from μ to neglected for two reasons: although vectors contribute at tree e, probe three- and four-point functions involving a muon, level to μ → e conversion and μ → eee¯ , these experimen- an electron and one or two other SM particles. I focus here tal bounds are not included, and the leading order mixing on interactions that can be probed in μ → eγ , μ → eee¯ of vectors to the dipole is at two-loop in QED, whereas the and μ → e conversion, meaning that the interactions are running here is only performed at one-loop. The next step is otherwise flavour diagonal, and there is only one muon (so to match these operators at mW onto the Buchmuller–Wyler K →¯μe and other meson decays is not considered). [37] basis of SU(2)-invariant operators as pruned in [38], These “new physics” interactions can be written as non- which is referred to as the BWP basis. The tree-level match- renormalisable operators involving a single μ, and some ing for all operators is given in Sect. 4; if this is the leading combination of e,γ,g, u, d, or s. The operators should be contribution to the coefficients, then imposing SU(2) invari- QED and QCD invariant (because we are intested in LFV, ance above mW predicts some ratios of coefficients below not departures from the SM-gauge symmetries), and can be mW , as discussed in Sect. 4.2. Section 5 uses the formalism of any dimension (because the aim is to list the three-point of the previous sections to translates the experimental bound and four-point interactions that the data constrains). The list, on BR(μ → eγ)to sums of SU(2)-invariant operator coeffi- which can be found in [1,33–35] but with different names, cients at mW . Then a few finite-loop contributions are added, is: and the coefficients are run up to M, using a simplified ver- eμ αβ dipole O = mμ(eσ P μ)Fαβ sion of the one-loop QCD and electroweak RGEs [39,40,43]. D,Y Y Finally, Sect. 6 discusses various questions arising from this Oeμee = 1( γ α μ)( γ α ), 4 lepton YY e PY e PY e study, such as the loop order required in matching at mW , 2 μ 1 α α whether the non-SU(2)-invariant basis is required below mW , Oe ee = ( γ μ)( γ ), YX e PY e PX e and the importance of QED running below mW . 2 Oeμee = ( μ)( ) Many parts or this analysis can be found in previous lit- S,YY ePY ePY e erature. Czarnecki and Jankowski [41] emphasized the one- eμuu 1 α α loop QED running of the dipole operator (neglected in the 2 lepton 2 quark O = (eγ PY μ)(uγ PY u), YY 2 estimates here), which shrinks the coefficient at low energy. eμuu 1 α α Degrassi and Giudice [42] give the leading order QED mixing O = (eγ PY μ)(uγ PX u), YX 2 of vector operators to the dipole, which is also neglected here, eμuu O = (eP μ)(uP u), because it arises at two loop. In an early top-down analysis, S,YY Y Y Oeμuu = ( μ)( ), Brignole and Rossi [15] calculated a wide variety of LFV pro- S,YX ePY uPX u cesses as a function of operator coefficients above m , with- Oeμuu = ( σ μ)( σ ) W T,YY e PY u PY u out explicit Renormalisation Group running and a slightly eμdd 1 α α redundant basis.