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)