# Complementarity in lepton-flavour violating muon decay experiments

PSI-PR-16-15

Complementarity in lepton-flavour violating muon decay experiments

Paul Scherrer Institut,

CH-5232 Villigen PSI, Switzerland

IPNL, CNRS/IN2P3, 4 rue E. Fermi,

69622 Villeurbanne cedex, France;

Université Lyon 1, Villeurbanne; Université de Lyon, F-69622, Lyon, France

Physik-Institut, Universität Zürich,

Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

This note presents an analysis of lepton-flavour-violating muon decays within the framework of a low-energy effective field theory that contains higher-dimensional operators allowed by QED and QCD symmetries. The decay modes and are investigated below the electroweak symmetry-breaking scale, down to energies at which such processes occur, i.e. the muon mass scale. The complete class of dimension-5 and dimension-6 operators is studied systematically at the tree level, and one-loop contributions to the renormalisation group equations are fully taken into account. Current experimental limits are used to extract bounds on the Wilson coefficients of some of the operators and, ultimately, on the effective couplings at any energy level below the electroweak symmetry-breaking scale. Correlations between two couplings relevant to both processes illustrate the complementarity of searches planned for the MEG II and Mu3e experiments.

## 1 Introduction

This note presents a specific example of a correlation that occurs in lepton-flavour-violating (LFV) muonic decays in the context of effective field theories (EFTs).

Whilst in the neutrino sector evidence for LFV is now established beyond doubt , the absence of experimental hints of LFV in the charged lepton sector, together with the smallness of the neutrino mass scale, indicate that a very incisive flavour conservation mechanism is at work. Although allowed in the Standard Model (SM) with right-handed neutrinos, the branching ratios (BRs) of such transitions are suppressed by , making them too small to be observable in any conceivable experiment. Consequently, any LFV production channel or decay mode offers a promising benchmark against which to search for physics beyond the SM.

Among charged LFV processes, muonic transition occurs in a relatively clean experimental environment, to the point that the MEG experiment has recently set a stringent limit on . This represents the strongest existing bound on ‘forbidden’ decays, while the SINDRUM result obtained almost three decades ago is still very competitive with regard to the current experimental status of other sectors. The well-known outcomes of these experiments are:

(1) | ||||

(2) |

Furthermore, there are good prospects for future MEG II and Mu3e experiments. The former is expected to reach a limit of , while the latter might even achieve a four-orders-of-magnitude improvement on the existing limit. All the aforementioned experiments are being carried out at the Paul Scherrer Institut’s experimental facilities. The present analysis does not consider LFV transitions in a nuclear environment (coherent and incoherent muon conversion in nuclei). See Refs and for extensive treatments of this topic.

From a theoretical perspective, LFV processes have been studied in
many specific extensions of the SM. In some cases the matching of such
extensions to a low-energy effective theory has also been
considered . However, this
analysis follows a bottom-up approach in which effective interactions
are included in a low-energy Lagrangian that
respects the and gauge symmetries. In exploiting the
Appelquist-Carazzone theorem , it is possible
to extend the QCD and QED Lagrangian^{b}^{b}bWithout the top quark field. with higher-dimensional operators

(3) |

Here, is the ultraviolet (UV) completion energy scale, which in this context is required not to exceed the electroweak symmetry-breaking (EWSB) scale, where the SM dynamic degrees of freedom and symmetries must be adequately restored and matched with those of the low-energy theory.

Having established the theoretical background, the main focus is on the interpretation of correlations between operators in the BRs of both and at the muon mass energy scale and beyond. Experimental limits are then applied to the parameter space in a search for allowed regions.

The popular parametrisation of dipole and four-fermion LFV operators

(4) |

where is an ad hoc parameter to be interpreted strictly at the muon mass energy scale, allows to switch from a pure dipole interaction () to a pure four-fermion interaction (). Although this approach ensures a descriptive phenomenological understanding of the contributions of different operators to different observables, a more consistent theoretical approach can be achieved without losing interpretive power.

The advantage of a systematic effective QFT approach lies in the fact that it can be used to link phenomenological observables at different energy scales unambiguously through the renormalisation-group evolution (RGE) of the Wilson coefficients. In this regard, the RGE between the muon mass energy scale and the EWSB scale is calculated at the leading order (up to the one-loop level) in QED and QCD for any operator contributing to LFV muon decays. This encompasses possible mixing effects among operators, which in this study are taken into account in a similar way to recent theoretical works . From this analysis, it is possible to extract limits both on the Wilson coefficients defined at the phenomenological energy scale and on the coefficients defined at the UV matching scale.

This paper is organised as follows. Section 2 introduces the LFV effective Lagrangian, and in Section 3 the observables connected with the and searches are briefly discussed. Section 4 provides a brief phenomenological analysis, and in Section LABEL:sec:5 conclusions are drawn. Formulae relevant to the RGE of the Wilson coefficients are provided in the appendix.

## 2 LFV effective Lagrangian at the muon energy scale

The Appelquist-Carazzone theorem is exploited to construct an effective Lagrangians valid below the EWSB scale, with higher-dimensional operators that respect the QCD and QED symmetries. This allows for an interpretation of BSM effects at high energy scales in terms of new, non-renormalisable interactions at the low energy scale.

In this respect, all possible QCD and QED invariant operators relevant to transitions are considered up to dimension 6. These can be arranged in the following effective Lagrangian with dimensionless Wilson coefficients and the decoupling energy scale :

(5) |

where and specify that sums run over the quark and lepton flavours, respectively. The explicit structure of the operators is given by

(6) | ||||

(7) | ||||

(8) | ||||

(9) | ||||

(10) | ||||

(11) |

and an analogous notation is assumed for cases in which the exchange is applied. In the previous equations, the convention is understood. Apart from being multiplied by the QED coupling , the operator in Eq.6 is also rearranged into a dimension-6 operator with an appropriate normalisation factor . The reason is that this operator is directly related to a dimension-6 operator in the SMEFT .

Direct comparison of Eq. 5 and Eq. 4 reveals that the latter assumes a tree-level correlation between independent operators. This assumption is manifestly inconsistent when quantum fluctuations are considered. Notably, an analysis of LFV transitions in nuclei calls for a further dimension-7 operator relating to the leading-order muon-electron-gluon interaction, which is generated by threshold corrections induced by the heavy quark operators (see Ref. for details).

## 3 Lepton-flavour-violating muonic observables

This section describes two of the most relevant LFV muon decay processes, and . Since the following analysis does not include a study of angular distributions (as in Ref. for the case of polarised -lepton decays), the charges of the external states need not be specified. The following partial widths should be divided by the total muon decay width, i.e. , in order to obtain the corresponding BRs.

### 3.1

The simplest and most investigated LFV muonic process is . On the one hand, the serious experimental bounds on this kinematically simple transition clearly indicate that there is an indisputable conservation mechanism at work. On the other hand, any observation of a non-zero in current or future experiments would indicate the existence of BSM physics. The Lagrangian in Eq. 5 results in a branching ratio

(12) |

from which it is clear that, with the Wilson coefficients defined at the muon energy scale, the associated BR is related only to the dipole operators . According to the RGEs presented in Eq. LABEL:rgedipole, these operators will receive contributions from scalar ( with ) and tensor ( and with ) operators, with non-vanishing coefficients at higher scales.

### 3.2

The second representative channel for muonic LFV decays is . Prospects for future experimental developments in this rare muon process are very promising: the current experimental limit is expected to be improved considerably by the Mu3e experiment. Again, any signal of such a rare decay would be a clear signal for BSM physics.

## 4 Limits on Wilson coefficients and correlations

In this section, the present experimental limits together with anticipated updates are applied to the observables of Eqs. 12 and 3.2 defined at a UV-completion energy scale.

Closer examination of Eqs. 12 and 3.2 together with the RGE equations in the appendix reveals that only two classes of operators – the dipole () and the scalar () – are manifestly correlated at the one-loop level in two self-consistent systems (separate by chirality) of ordinary differential equations (ODE). In principle, more complicated relations occur if non-zero tensorial quark or -lepton operators are considered. In addition, at the two-loop level, even the vectorial operators mix with the dipole. However, a complete quantitative treatment of all possible correlations is beyond the scope of this analysis.

For illustrative purposes, in the following discussion, we consider a scenario where an underlying UV-complete theory produces non-vanishing SMEFT coefficients. We assume that matching this SMEFT to the low-energy Lagrangian of Eq. 5, only two categories of non-vanishing coefficients are produced, namely and .

According to the RGE described by
Eqs. LABEL:rgedipole and LABEL:rgescalar, if the RGE effects are neglected for the EM coupling and
fermion masses^{c}^{c}cIf the running
of the electromagnetic (EM) coupling and the fermion masses is taken
into account, then the evolution of the couplings is more involved,
but at the same time the qualitative conclusion of this note will
remain unchanged., then the running of these two operators can be
described by a relatively simple system of two ODE. The solutions are

(14) | ||||

(15) |

where is the phenomenological energy scale at which the coefficients should be evaluated, and is the normalised EM coupling.

By combining these results with the BRs of Section 3 and applying the experimental limits, at the muon mass scale , we obtain the constraints on the coefficients and shown in Figure 1 (right-chirality ones give the same result). Note that the evolution of the EM coupling and fermion masses is taken into account in these numerical results.