IFT-10/2013
Lepton Flavor Violation in the Standard Model with general
arXiv:1312.0634v2 [hep-ph] 18 Feb 2014
Dimension-Six Operators
A. Crivellina, 1 , S. Najjarib,2 and J. Rosiekb,3 a
Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern. b
Institute of Theoretical Physics, Department of Physics, University of Warsaw.
Abstract We study lepton flavor observables in the Standard Model (SM) extended with all dimension-6 operators which are invariant under the SM gauge group. We calculate the complete one-loop predictions to the radiative lepton decays µ → eγ, τ → µγ and τ → eγ as well as to the closely related anomalous magnetic moments and electric dipole moments of charged leptons, taking into account all dimension-6 operators which can generate lepton flavor violation. Also the 3-body flavor violating charged lepton decays τ ± → µ± µ+ µ− , τ ± → e± e+ e− , τ ± → e± µ+ µ− , − τ ± → µ± e+ e− , τ ± → e∓ µ± µ± , τ ± → µ∓ e± e± and µ± → e± e+ e− and the Z 0 decays Z 0 → ℓ+ i ℓj are considered, taking into account all tree-level contributions.
1 2 3
e-mail:
[email protected] e-mail:
[email protected] e-mail:
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2 I.
INTRODUCTION
The Standard Model (SM) of strong and electroweak interactions has been successfully tested to a great precision [1]. Nevertheless, it is commonly accepted that it constitutes only an effective theory which is valid up an energy scale Λ where new physics (NP) enters and additional dynamic degrees of freedom become important. A renormalizable quantum field theory valid above this scale should satisfy the following requirements: (i) Its gauge group must contain the SM gauge group SU (3)C × SU (2)L × U (1)Y . (ii) All SM degrees of freedom should be incorporated either as fundamental or as composite fields. (iii) At low-energies it should reduce to the SM provided no undiscovered weakly coupled light particles exist (like axions or sterile neutrinos). In most theories of physics beyond the SM that have been considered, the SM is recovered at low energies via the decoupling of the heavy particles with masses of the order of Λ ≫ MZ . That such a decoupling at the perturbative level is possible in a renormalization quantum field theory is guaranteed by the Appelquist-Carazzone decoupling theorem [2]. This leads to the appearance of higher-dimensional operators which are suppressed by powers of Λ and are added to the SM Lagrangian: LSM =
(4) LSM
1 1 X (6) (6) 1 X (5) (5) Ck Q k + O . Ck Q k + 2 + Λ Λ Λ3
(I.1)
k
k
(4)
Here LSM is the usual renormalizable part of the SM Lagrangian which contains dimension-2 (5)
and dimension-4 operators only. Qk (6)
Qk
is the Weinberg operator giving rise to neutrino masses, (n)
denote dimension-6 operators, and Ck
stand for the corresponding dimensionless coupling
constants, i.e. the Wilson coefficients. Even if the ultimate theory of NP at some high energy scale is not a quantum field theory, at low energies the effective theory still reduces to a quantum field theory [3] and it is possible to parametrize its effects at the electroweak scale in terms of these operators and the associated Wilson coefficients. Thus, one can search for NP in a model independent way by studying the SM extended with gauge invariant effective higher dimensional operators. Later, once a specific model is chosen, the Wilson coefficients can be calculated as a function of model parameters by matching
3 the model of NP under consideration on the SM extended with such higher dimensional operators and one can calculate bounds on the specific model as well. Flavor observables, especially flavor changing neutral current (FCNC) processes are an excellent probe of new physics since they are suppressed in the SM and therefore sensitive even to small NP contributions. This also means that these processes can stringently constrain the Wilson coefficients of the dimension-6 operators induced by NP. Especially the search for lepton flavor violation (LFV) is very promising since in the SM (extended with massive neutrinos) all flavor violating effects in the charged lepton sector are proportional to the very small neutrino masses - e.g. the decay rates of heavy charged leptons into lighter 2 and thus are by far too small to be measurable in any ones are suppressed by the ratio m2ν /MW
foreseeable experiment. This in turn means that any observation of LFV would prove the existence of physics beyond the SM. In addition, LFV processes have the advantage of being “theoretically clean”, i.e. they can be computed precisely without problems with non-perturbative QCD effects affecting similar observables in the quark sector. Also the current experimental situation and prospects for the search for charged lepton flavor violation are very promising. In Tables I and II we list the experimental bounds on the radiative lepton decays ℓi → ℓf γ and on the three-body lepton decays ℓi → ℓj ℓk ℓl , respectively. Especially the limits on µ → e transitions are very stringent due to constraints from the MEG and SINDRUM collaborations at the PSI and will be even further improved in the future: MEG can measure Br[µ → eγ] down to 6 × 10−14 and a MEG upgrade [4] could increase the sensitivity by another order of magnitude. Furthermore, the electric dipole moments (EDM) and the anomalous magnetic moments of charged leptons are theoretically closely related to ℓi → ℓf γ transitions and also here the experimental accuracies are very good, leading to strong upper bounds for the EDMs (see Table III). In addition, there is a longstanding discrepancy between the SM prediction and the measurement of the anomalous magnetic moment of the muon, which might be a hint for physics beyond the SM. Lepton flavor violating processes have been studied in great detail in many specific extensions of the SM. For example in the MSSM non-vanishing decay widths for LFV processes are generated by flavor non-diagonal SUSY breaking terms [13, 14, 15, 16, 17]. Also extending the MSSM with righthanded neutrinos by the seesaw mechanism [18] gives rise to LFV [19, 20, 21, 22, 23, 24, 25, 26, 27], as well as allowing for R-parity violation [28, 29, 30]. Other models like the littlest Higgs Model
4
Process
Experimental bound
Br [τ → µγ]
4.4 × 10−8 [5, 6]
Br [τ → eγ]
3.3 × 10−8 [5]
Br [µ → eγ]
5.7 × 10−13 [7]
TABLE I: Experimental upper limits on the branching ratios of the radiative lepton decays.
Process
Experimental bound
Br [τ − → µ− µ+ µ− ]
2.1 × 10−8 [8]
Br [τ − → e− e+ e− ]
2.7 × 10−8 [8]
Br [τ − → e− µ+ µ− ]
2.7 × 10−8 [8]
Br [τ − → µ− e+ µ− ]
1.7 × 10−8 [8]
Br [µ− → e− e+ e− ]
1.0 × 10−12 [9]
TABLE II: Experimental upper limits on the branching ratios of the three body charged lepton decays.
with T-Parity [31], two-Higgs-doublet models with generic flavor structures [32, 33, 34, 35] or models with an extended fermion sector [36] have sources of lepton flavor violation, too. In order to make models of New Physics consistent with the non-observation of LFV processes in Nature, the assumption of Minimal Flavor Violation [37] has been extended to the lepton sector (see e.g. [38, 39]). Flavor changing τ decays have been studied in Ref. [40] in a model independent way taking into account a (reducible) set of four-lepton operators and the magnetic lepton operators. However, a detailed model independent analysis with all gauge invariant operators is still pending4 . In this article we perform such a model independent analysis by considering the SM extended with all dimension-6 operators giving rise to lepton flavor violation which are invariant under the SM gauge group. We study the radiative lepton decays ℓi → ℓf γ and three-body charged lepton decays ℓi → ℓj ℓk ℓl , as well as the anomalous magnetic moments and EDMs of charged leptons and
+ the flavor violating Z 0 → ℓ− i ℓj decays. 4
For a model independent analysis for the Higgs sector of the SM see Ref. [41, 42] and for anomalous top couplings Ref. [41, 43].
5
EDM
|de |
|dµ |
dτ
Bound [e cm] 8.7 × 10−29 [10] 1.9 × 10−19 [11] [−2.5, 0.8] × 10−17 [12] TABLE III: Experimental upper bounds (or allowed range for dτ ) on electric dipole moments of the charged leptons.
It is worth noting that analyzing the LFV processes using the gauge-invariant basis of dimension6 operators automatically assures that the final results are also gauge invariant and contain all relevant contributions. Otherwise, one risks including just subset of diagrams contributing to a given process. For example it is quite common in the literature to calculate in a model of NP only the effective flavor changing Z 0 -boson coupling to charged leptons and neglect the corrections to W couplings, as the latter do not contribute at the tree-level to neutral current processes. However, both Z 0 and W (and also Goldstone boson) couplings come from the same set of gauge-invariant higher-order operators, and are thus of the same size. In fact, (as our calculation shows explicitly) their contributions at least to some processes, like e.g. ℓi → ℓf γ, are equally important and should be always considered together. The outline of this article is as follows: after recalling the relevant dimension-6 operators in the next Section we will consider radiative lepton decays in Sec. III (including the related anomalous magnetic moments and electric dipole moments of charged leptons), three-body charged lepton + decays in Sec. IV and the flavor changing Z 0 decays, Z 0 → ℓ− i ℓj , in Sec. V. We calculate the
full one-loop predictions for the ℓi → ℓf γ decays and all tree-level contributions for ℓi → ℓj ℓk ℓl decays in terms of the Wilson coefficients of the dimension-6 operators. Sec. VI deals with the numerical evaluation of our results and finally we conclude in Sec. VII. An Appendix summarizes the Feynman rules arising from the dimension-6 operators after electroweak symmetry breaking and the additional form-factors for ℓi → ℓf γ ∗ amplitude for the case of an off-shell photon.
II.
THE LEPTON FLAVOR VIOLATING OPERATORS OF DIMENSION-6
The complete (but still reducible) list of independent operators of dimension-5 and dimension-6 which can be constructed out of SM fields and which are invariant under the SM gauge group fields was first derived in Ref. [44]. In this article we follow the notation Ref. [45] where the operator basis of Ref. [44] was reduced to a minimal set. For completeness, we list below again the operators
6
fermions
scalars
field
ℓaLi
eRi
a qLi
uRi
dRi
ϕa
hypercharge Y
− 12
−1
1 6
2 3
− 31
1 2
TABLE IV: Our conventions for the hypercharges of the SM fields.
relevant for our discussion. We use the following indices and symbols: • a, b = 1, 2 label the components of the weak isospin doublets. • i, j, k, l are flavor indices running from 1 to 3. • L and R stand for the chiralities. νLi uLi and qi = stand for the lepton and the quark doublets. • ℓi = ℓLi dLi
• ei = ℓRi , ui = uRi and di = dRi are the right-handed isospin singlets. • ϕa is the SM Higgs doublet where ϕ2 is the neutral component.
The hypercharges of the SM fields are summarized in Table IV. The sign convention for the covariant derivatives is a
(Dµ ℓ) =
1 I ′ I δab ∂µ + igτab Wµ + ig Yℓ δab Bµ ℓb . 2
(II.1) ←
with τ I being the Pauli matrices. The hermitian derivative terms are (ϕ†D µ ϕ ≡ (Dµ ϕ)† ϕ): ← ↔ ← ↔ (II.2) and ϕ† i DµI ϕ ≡ iϕ† τ I Dµ − D µ τ I ϕ . ϕ† i Dµ ϕ ≡ iϕ† Dµ − D µ ϕ The gauge field strength tensors read I Wµν = ∂µ WνI − ∂ν WµI − gεIJK WµJ WνK ,
(II.3)
Bµν = ∂µ Bν − ∂ν Bµ .
(II.4)
In general, the SM can be extended by higher dimensional operators starting from dimension5. However, there is just a single dimension-5 term respecting the SM gauge symmetry which, after electroweak symmetry breaking, generates neutrino masses and mixing angles - the Weinberg operator (C is the charge conjugation matrix and ε12 = +1): Qνν = εab εcd ϕa ϕc (ℓbi )T Cℓdj .
(II.5)
7 This operator does not contribute directly (other then modifying the UP M N S matrix) to LFV processes in the charged lepton sector, consequently we do not consider it in the rest of the paper. In Table V we collect the independent dimension-6 operators relevant for our discussion, i.e. all operators which can contribute to LFV processes in the charged lepton sector at the treelevel or at the 1-loop level. We neglect the operators which could give LFV effects only via the interference with the dimension-4 SM vertices containing the PMNS matrix, since such effects are suppressed by the small neutrino masses which we assume to be zero. The names of operators in the left column of each block should be supplemented with generation indices of the fermion (1)
(1)ijkl
fields whenever necessary, e.g. Qℓq → Qℓq
. Dirac and color indices (not displayed) are always (1)
(3)
contracted within the brackets. The same is true for the isospin indices, except for Qℓequ and Qℓequ .
ℓℓℓℓ
ℓℓϕ2 D and ℓℓϕ3
ℓℓXϕ
(ϕ† iDµ ϕ)(ℓ¯i γ µ ℓj )
Qϕℓ
(3)
(ϕ† iDµI ϕ)(ℓ¯i τ I γ µ ℓj )
Qϕe
(ϕ† iDµ ϕ)(¯ ei γ µ ej )
Qeϕ3
(ϕ† ϕ)(ℓ¯i ej ϕ)
(ℓ¯i γµ ℓj )(ℓ¯k γ µ ℓl )
I QeW (ℓ¯o σ µν ej )τ I ϕWµν Qϕℓ
Qee
(¯ ei γµ ej )(¯ ek γ µ el )
QeB
Qℓe
(ℓ¯i γµ ℓj )(¯ ek γ µ el )
(ℓ¯i σ µν ej )ϕBµν
↔
(1)
Qℓℓ
↔
↔
ℓℓqq (1)
Qℓd
(ℓ¯i γµ ℓj )(d¯k γ µ dl )
Qℓu
(ℓ¯i γµ lj )(¯ uk γ µ ul )
Qℓq (ℓ¯i γµ τ I ℓj )(¯ qk γ µ τ I ql ) Qed
(¯ ei γµ ej )(d¯k γ µ dl )
Qeu
(¯ ei γµ ej )(¯ uk γ µ ul )
(¯ ei γ µ ej )(¯ qk γµ ql )
(ℓ¯ai ej )(d¯k qla )
Qℓequ
(1)
(ℓ¯ai ej )εab (¯ qkb ul )
Qℓq
(ℓ¯i γµ ℓj )(¯ qk γ µ ql )
(3)
Qeq
Qℓedq
(3) Qℓequ (ℓ¯ai σµν ea )εab (¯ qkb σ µν ul )
TABLE V: Complete list of the dimension-6 operators (invariant under the SM gauge group) which contribute to the LFV observables under consideration at the tree or at the one-loop level.
Note that different flavor index combinations of the 4-lepton operators can correspond to the ilkj kjil klij same operator (for example Qijkl ℓℓ = Qℓℓ = Qℓℓ = Qℓℓ ). For this reason, in the following we will
only consider one of these combinations which avoids the introduction of combinatorial factors. This can be achieved by the requirement i ≥ k, j ≥ l for Qijkl ℓℓ,ee , so that the relevant part of the
8 Lagrangian can be written as: L=
1 Λ2
X
ijkl,i≥k,j≥l
1 X ijkl ijkl ijkl ijkl ijkl ijkl Cℓℓ Qℓℓ + Cee Qee + 2 Cℓe Qℓe . Λ
(II.6)
ijkl
ijkl Note that for Cℓe all possible flavor index permutations correspond to different operators. Due ijkl jilk⋆ to the hermiticity of the Lagrangian we find the additional relations like Cℓℓ = Cℓℓ . Similar
ones hold for all four-fermion operators. The dominant contributions to the processes considered in this article are given by diagrams with flavor changing gauge boson vertices or contact 4-fermion vertices. However, to preserve gauge-invariance, also Goldstone boson exchanges has to be taken into account even if, with few exceptions of mixed W ± G∓ diagrams, they are suppressed by additional powers of light lepton masses over v, the Higgs field VEV. In general, the operators listed in Table V give rise also to flavor violating physical Higgs boson couplings. We neglect them in our analysis as they are again of the higher order in mℓ /mh0 . The (ϕ† ϕ)(ℓ¯i ej ϕ) operator does not contain gauge boson fields and modifies only Higgs and Goldstone boson couplings, which in principle could affect our results. However, it gives also new O(1/Λ2 ) contribution to the charged lepton mass matrix: v3 v fi Ceϕ3 . mℓf i = √ Yfℓ δf i + √ 2 2 2 2Λ
(II.7)
The necessary rediagonalization of lepton masses has the effect of modifying the relation between the Yukawa coupling and the charged lepton masses (and the PMNS matrix). However, one can see that in the triple Goldstone boson couplings to leptons still the physical lepton masses and i the physical PMNS matrix enter so the Qfeφ3 does not generate flavor violation in these couplings.
The triple coupling of the physical Higgs boson h0 to charged leptons, as well as all quadruple and i quintuple vertices derived from Qfeφ3 can still be flavor violating. Nonetheless, their contributions
to the processes discussed below vanish or are small due to an additional suppression of mℓ /mh0 , (1)
(3)
compared to the dominant contributions from Qϕe , Qϕℓ and Qϕℓ operators5 . Thus, we neglect this operator (and thus the entire ℓℓϕ3 class) in our analysis, provided that the rediagonalization of the lepton mass matrix has been performed. The operators of the ℓℓXϕ class (as defined in Table V) can give rise to both radiative lepton decays and to three-body neutral current lepton decays already at the tree-level. The 4-lepton 5
fi Oeϕ3 generates flavour-changing couplings of the SM-Higgs. The resulting effects have been studied in Refs. [46, 47, 48]
9 ℓℓℓℓ operators and the operators of the ℓℓϕ2 D class can contribute to ℓi → ℓj ℓk ℓl decays at the tree-level and to ℓi → ℓf γ decays at the 1-loop level. Finally, the operators of the ℓℓqq class can contribute to both types of decays only at the 1-loop level. However, for 3-body decays we are only interested in the tree-level contributions and concerning the radiative lepton decays, it turns (3)
out that only Qℓequ gives a non-zero contribution. In the Appendix we list the Feynman rules arising from the operators given in Table V which are necessary in order to calculate the flavor observables discussed in the next Sections.
III.
OBSERVABLES RELATED TO THE EFFECTIVE LEPTON-PHOTON COUPLING
As outlined in the introduction, observables related to effective lepton-photon coupling: radiative lepton decays (especially µ → eγ), EDMs of charged leptons and their anomalous magnetic moments are very sensitive to NP and allow to constrain stringently the relevant Wilson coefficients. The general form of the flavor violating photon-lepton vertex can be written as: fi µ Vℓℓγ =
i i h µ fi fi fi fi fi fi µ µν µν . γ (F P + F P ) + (F P + F P )q + (F iσ P + F iσ P )q L R L R L R ν VL VR SL SR TL TR Λ2 (III.1)
In this Section we calculate the expressions for the formfactors in Eq. (III.1) necessary to calculate the branching ratio for the ℓi → ℓf γ decays (with i > f ) at the 1-loop level up the order 1/Λ2 . In addition, the obtained results are directly related to the anomalous magnetic moments and the electric dipole moments (EDM) of leptons after setting f = i.
A.
Radiative lepton decays
Gauge-invariance requires that FV L and FV R must vanish for on-shell external particles. The form-factors FSL and FSR do not contribute to the ℓi → ℓf γ decay amplitude and the branching
i ratio can be expressed in terms of FTf Li and FTf R only: m3ℓi f i 2 f i 2 . Br [ℓi → ℓf γ] = FT R + FT L 16πΛ4 Γℓi
The total decay width of the muon is given by Γµ =
G2F m5µ 192π 3
(III.2)
and for the tau lepton Γτ includes the
leptonic and hadronic decay channels. Only the operators QeW (here W denotes the neutral gauge boson of the SU (2)L gauge group)
10 i at the tree-level. If their coefficients are comparable to other and QeB can contribute to FTf L,R
Wilson coefficient of the dimension 6 operators, they dominate the effective photon-lepton vertex, with the form-factors simply given by (v =
2MW g2
):
√ √ i fi fi ≡ v 2Cγf i . = FTifL⋆ = v 2 cW CeB FTf R − sW CeW
(III.3)
However, in a renormalizable theory of NP the operators QeW and QeB can only be generated at the loop-level while other operators, like the effective four-lepton couplings, can already be generated at the tree-level. In some extensions of the SM CeW and CeB may even not be generated at all [49]. Thus, comparable (or even dominant) contributions to the flavor violating lepton-lepton-photon vertex can come from other dimension-6 operators, which for consistency should be included at the 1-loop level. The generic topologies of the diagrams which could contribute to ℓi → ℓf γ at the 1-loop level in the order 1/Λ2 and the relevant momenta assignments are shown in Fig. 1. γµ
γµ
γµ
γµ
q = pi − pf pi ℓ
i
pf ℓ
f
ℓ
i
ℓf
ℓ
f
ℓ
i
ℓi ℓ
f
ℓ
i
Z 0 , γ 0 , G0 ℓf
FIG. 1: Topologies of diagrams contributing to radiative decay ℓi → ℓf γ. .
The list of all 1-loop diagrams contributing to the effective lepton-photon vertex is given in Fig. 2 (lepton self-energy contributions) and Fig. 3 (1-particle irreducible vertex corrections). The diagrams contributing to photon-photon and Z 0 -photon self-energies are the same as in the SM (with W bosons, charged ghosts, charged Goldstone bosons and charged fermion as virtual particles). In our loop calculations we do not take into account flavor violating photon and Z 0 couplings generated at the one-loop level by the operators QeW and QeB because if their coefficients are nonnegligible, than already the tree-level contribution of Eq. (III.3) would dominate the whole process anyway. Our final 1-loop results for the form-factors FT L and FT R are given in Table VI. We group them into subsets; within these subsets the vector form-factors FV L and FV R vanish separately in the on-shell limit. We kept only the leading term in 1/Λ2 and we expand all diagrams involving Z 0 and W bosons (or the associated Goldstone bosons) in the charged lepton masses, keeping only the leading terms in mℓ /mW , mℓ /mZ . For this expansion we used two independent approaches
11
Z 0 (G0 )
ℓi
ℓj ℓ
f
ℓi
a)
νj
ℓf
ℓi
b)
ℓj (ν j , dj , uj )
ℓi
G−
W − (G− )
ℓf c)
ℓj
ℓf
ℓi
d)
ℓf e)
FIG. 2: Diagrams contributing to LFV self-energy of charged leptons. .
for calculating the diagrams. In the first approach the exact calculation of all loop integrals is performed, followed by their expansion in the external momenta. In the second approach we used asymptotic expansion [50] and expanded the diagrams in external momenta before performing the loop integrals, finding the same result as with the first approach. The final expressions collected in Table VI are compact and simple. As mentioned above, if the external particles in the flavor violating lepton-photon vertex are onshell, gauge invariance requires FVf iL = FVf iR = 0 for i 6= f . As the diagrams involving dimension-6 vertices may have complicated tensor structures, the vanishing of the FV L and FV R is an important check of our calculation. As an additional check we performed the whole calculation in a general Rξ gauge finding that the ξ dependence cancels for all form-factors. Here one should keep in mind that taking into account only 1PI irreducible diagrams is sufficient for the calculation of FT L , FT R - however, taking into account also lepton, photon and mixing photon-Z 0 self-energies diagrams is obligatory to cancel completely the vector form-factors and to get a gauge-independent renormalization constant for the electric charge. We see that in the final result, at the 1-loop level and in the first order of expansions in 1/Λ2 (3)f ijj
f jji and ml , only the five Wilson coefficients Cℓe , Cℓequ
(3)f i
, Cϕℓ
(1)f i
fi , Cϕe and Cϕℓ
enter, while the
contribution of all other Wilson coefficients is zero. (3)
It is interesting to note that the term proportional to Cℓequ is the only one containing a diver-
12
Group (diagrams of Figs. 2, 3)
Z 0 (3a, 2a(Z 0 ))
Tensor form-factors FTZLf i = FTZRf i =
G0 (3b, 2a(G0 ))
4e
h
4e
h
(1)f i
Cϕℓ
(1)f i Cϕℓ
(3)f i
+ Cϕℓ +
(3)f i Cϕl
i fi mf (1 + s2W ) − Cϕe mi ( 23 − s2W )
2 3(4π) i fi mi (1 + s2W ) − Cϕe mf ( 32 − s2W )
3(4π)2
0
FTGL f i = 0 0
FTGR f i = 0 (3)f i
W (3c,d,e,j,k, 2b(W ))
FTWL f i = − FTWR f i = −
G± (3f,g,h, 2b(G± ))
±
FTGL
±
FTGR W G “bubble” (3i,l, 2c)
fi
=0
fi
=0
10emf Cϕℓ
3(4π)2 (3)f i 10emi Cϕℓ 3(4π)2
FTWLG f i = 0 FTWRG f i = 0
contact 4-fermion (3m, 2d)
FT4fL f i FT4fR f i
contact 4-lepton (3n, 2e)
16e P3 (3)f ijj⋆ =− C muj 3(4π)2 j=1 ℓequ 16e P3 (3)f ijj =− C muj 3(4π)2 j=1 ℓequ
∆ − log ∆ − log
m2uj µ2
m2uj µ2
!
!
2e P3 C f jji mj (4π)2 j=1 ℓe 2e P3 = C jif j mj (4π)2 j=1 ℓe
FT4ℓLf i = FT4ℓRf i
i i TABLE VI: One-loop contributions to form factors FTf L and FTf L giving rise to ℓi → ℓf γ up to order 1/Λ2 .
gence. This divergence must be canceled by a counter-term to QeW and/or QeB . The appearance of this divergence can be understood by looking at a UV complete theory of NP. Consider as an example a theory with a heavy scalar particle. Directly calculating the contributions to FT L and FT R in the full theory one would obtain a finite result. However, when matching to full theory on the SM extended with dimension-6 operators the situation is more complicated: integrating out (3)
the heavy particle at the matching scale Λ gives rise to CeW and CeB at the loop-level and Cℓequ at the tree-level. However, as all Wilson coefficients, CeW and CeB can only contain the hard part of the corresponding loop-contribution while the soft part must be canceled by the loop-contribution (3)
of Qℓequ to CeW and CeB in an effective theory. It turns out that the hard part which contributes
13
γµ
γµ
ℓj
ℓj
ℓj
Z0
ℓi
ℓf
G0
ℓi
ℓ
G− ℓ
f
c)
ℓ
W− ℓ
d)
e)
G−
G−
νj ℓ
f
ℓ
G−
νj
i
g)
f
ℓ
νj
i
γµ
γµ ℓf
ℓi
ℓf i)
γµ
G−
G−
ℓi
ℓf
γµ
W−
ℓi
G− ℓf
j)
k)
γµ
ℓf
G−
h)
γµ
W−
G−
f)
νj
i
ℓf
γµ
G−
νj
i
νj
ℓi
γµ
W−
γµ
ℓf
W−
b)
γµ
ℓ
W−
ℓj
a)
i
γµ
G−
ℓi
ℓf l)
γµ
ℓj (dj , uj ) ℓi
ℓf
ℓj ℓi
ℓf
m)
n)
FIG. 3: 3-point 1PI diagrams contributing to the radiative charged lepton decay ℓi → ℓf γ at the 1-loop level. .
14 to CeW and CeB has a infrared divergence which is canceled by the UV divergence of the soft part (as can be best seen using asymptotic expansion). Comparing this result with the one in the full (3)
theory we see that the µ-dependence in the contribution of Cℓequ to FT L and FT R must be replaced by the mass of the heavy scalar, i.e. Λ. In our numerical analysis we neglect (possible but rather (3)
exotic in the lepton sector) contributions from Qℓequ operator - coefficients of such lepton-quark contact terms can be independently constrained using the LHC measurements [51].
B.
Anomalous magnetic moments and electric dipole moments
The form-factors listed in Table VI for f = i can directly be used to calculate also the electric dipole moments of charged leptons and the contribution (in addition to the SM) to their anomalous magnetic moments: dℓi =
aℓi =
−1 Im FTiiR , 2 Λ 2mℓi Re FTiiR . 2 eΛ
(III.4)
(III.5)
The experimental bounds on the EDM of charged leptons are given in Table III. The anomalous magnetic moment of the electron is usually used to determine the fine structure constant, but determining αem from rubidium atom experiments [52], one can still use it for obtaining bounds on NP [17, 53, 54]. For the anomalous magnetic moment of the muon there is the long known discrepancy between experiment and the SM prediction for aµ = (g − 2)/2 [55, 56, 57, 58, 59]: SM ≈ (2.7 ± 0.8) × 10−9 . ∆aµ = aexp µ − aµ
(III.6)
This discrepancy could point towards physics beyond the SM and, if verified, could make the search for ℓi → ℓf γ decay even more promising, as both processes depend on the operators with formally the same field and Dirac structure, differing only by the choice of flavor indices. The current experimental limit on the anomalous magnetic moment of the tau lepton is rather weak, but it can be improved in the future [60]: − 0.052 ≤ aτ ≤ 0.013 .
(III.7)
15 IV.
ℓi → ℓj ℓk ℓ¯l DECAY RATE
LFV operators of dimension-6 also give contributions to another set of experimentally strongly constrained decays, namely decays of heavy charged lepton into three lighter charged leptons6 . Such decays can be generated already at the tree-level by Z 0 and neutral Goldstone boson exchange, flavor violating photon couplings generated by QeW and QeB operators, or even directly by the 4−lepton operators. In this Section we list the general expressions for the lowest order contributions to all such 3-body charged lepton decays. Since all operators enter already at the tree-level we choose not to consider loop-diagrams for these processes. We split the expressions for the ℓi → ℓj ℓk ℓ¯l decays into 3 groups, depending on composition of the final state leptons: (A) Three leptons of the same flavor: µ± → e± e+ e− , τ ± → e± e+ e− and τ ± → µ± µ+ µ− . (B) Three distinguishable leptons: τ ± → e± µ+ µ− and τ ± → µ± e+ e− . (C) Two lepton of the same flavor and charge and one with different flavor and opposite charge: τ ± → e∓ µ± µ± and τ ± → µ∓ e± e± . lk
pk pj
θ
pi
θ′ ¯ll
lj
pl
FIG. 4: Kinematics of ℓi → ℓj ℓk ℓ¯l decay in the CMS frame. .
We decompose the amplitude A for the decay ℓi → ℓj ℓk ℓ¯l as A = A0 + Aγ ,
(IV.1)
where A0 contains all operators for which one can neglect the momenta of the external leptons and Aγ is the photon contribution generated in our approximation by QeW and QeB only. The 6
Experimental bounds are usually given on positively charged muon decays, as they do not form bound state with atoms what would decrease the accuracy of measurements [61].
16 amplitude A0 can without loss of generality be written as7 : A0 =
1 X CI [¯ u(pj )QI u(pi )][¯ u(pk )Q′I v(pl )] Λ2
(IV.2)
I
with the momenta assignments shown in Fig. 4. The basis of quadrilinears QI × Q′I is given by: OV XY = γ µ PX × γ µ PY , OSXY = PX × PY , OT X = σ µν × σ µν PX , where X, Y stands for the chiralities L and R.
(IV.3)
For processes with two identical leptons in
the final state one needs to include crossed diagrams in which the different spinor ordering [¯ u(pj )QI u(pl )][¯ u(pk )Q′I v(pi )] appears. However, one can always reduced these contributions to form given in Eq. (IV.2) by the appropriate Fierz transformations (see e.g. [62]). The contributions from photon exchange for various types of decays (A), (B), (C) read (retaining only 1/Λ2 terms): ev 1 (A) µν Aγ = 2 [¯ u(pj )iσ (CγL PL + CγR PR )(pi − pj )ν u(pi )][¯ u(pk )γµ v(pl )] − (pj ↔ pk ) Λ (pi − pj )2 1 ev [¯ u(pj )iσ µν (CγL PL + CγR PR )(pi − pj )ν u(pi )][¯ u(pk )γµ v(pl )] Aγ(B) = 2 Λ (pi − pj )2 A(C) = 0 γ
(IV.4)
In Eq. (IV.2) and Eq. (IV.4) we did not write explicitly flavor indices of CI , Cγ but we specify them later in the next Section. The general expression for the spin averaged square matrix element M =
1 2
P
pol
|A|2 is com-
plicated, but due to the hierarchy of the charged lepton masses, in most cases it is sufficient to assume mi ≡ M ≫ mj , mk , ml and neglect the lighter lepton masses (which also eliminates the contribution of Goldstone bosons). Only the contribution from the photon penguin requires more care due to singularity of photon propagator for small momenta. For the photon penguin, in order to get the correct final result one needs to expand matrix element and phase space kinematics at least up to the order of m2 /M 2 . Then, using standard expressions for 3-particle phase space one 7
We define the amplitude in such a way that calculating a diagram equals iA, which means that the Wilson coefficients are purely real in the absence of CP violation.
17 can integrate the matrix element and obtain the branching ratios (for comparison see [26]): Nc M 5 4 |CV LL |2 + |CV RR |2 + |CV LR |2 + |CV RL |2 3 4 6144π Λ Γℓi + |CSLL |2 + |CSRR |2 + |CSLR |2 + |CSRL |2 + 48 |CT L |2 + |CT R |2 + Xγ
Br(ℓi → ℓj ℓk ℓ¯l ) =
(IV.5)
where Nc = 1/2 if two of the final state leptons are identical, Nc = 1 in all other cases and Γℓi is the total decay width of the initial lepton. The photon penguin contribution reads: 1 1 16ev ⋆ ⋆ (A) Re 2CV LL + CV LR − CSLR CγR + 2CV RR + CV RL − CSRL CγL Xγ = − M 2 2 2 2 2 64e v 11 M + log 2 − (|CγL |2 + |CγR |2 ) 2 M m 4 16ev ⋆ ⋆ Re (CV LL + CV LR ) CγR + (CV RR + CV RL ) CγL Xγ(B) = − M M2 32e2 v 2 log 2 − 3 (|CγL |2 + |CγR |2 ) + M2 m Xγ(C) = 0
(IV.6)
A.
Decay ℓi → ℓj ℓj ℓ¯j
This option responds to the physical decays µ → 3e, τ → 3e and τ → 3µ. In general, at the
tree-level diagrams mediated by photon, Z 0 , the neutral Goldstone boson and 4-lepton contact
terms can contribute to the matrix element. The quantities CX in Eq. (IV.5) can be expressed in terms of Wilson coefficients of operators in Table V as (with Cγji defined in Eq. (III.3)): (3)ji (1)ji jijj + Cℓℓ CV LL = 2 (2s2W − 1) Cϕℓ + Cϕℓ ji jijj CV RR = 2 2s2W Cϕe + Cee 1 (1)ji (3)ji jijj + Cℓe CV LR = − CSRL = 2s2W Cϕℓ + Cϕℓ 2 1 jjji ji CV RL = − CSLR = (2s2W − 1)Cϕe + Cℓe 2 CSLL = CSRR = CT L = CT R = 0 √ CγL = 2Cγij⋆ √ CγR = 2Cγji
(IV.7)
18 B.
Decay ℓi → ℓj ℓk ℓ¯k
Such a decay can be realized as τ ± → e± µ+ µ− e or τ ± → µ± e+ e− . The coefficients CX read: (3)ji (1)ji jikk + Cℓℓ CV LL = (2s2W − 1) Cϕℓ + Cϕl ji jikk + Cee CV RR = 2s2W Cϕe (1)ji (3)ji jikk + Cℓe CV LR = 2s2W Cϕℓ + Cϕℓ jkki ji CV RL = (2s2W − 1)Cϕe + Cℓe jkki CSLR = −2Cℓe jikk CSRL = −2Cℓe
CSLL = CSRR = CT L = CT R = 0 √ CγL = 2Cγij⋆ √ CγR = 2Cγji
C.
(IV.8)
¯∓ ± ± Decay ℓ± i → ℓj ℓk ℓk
Again, only τ lepton can decay into such channels, τ ± → e∓ µ± µ± or τ ± → µ∓ e∓ e∓ . In this
case photon and Z 0 -mediated diagrams are suppressed by 1/Λ4 and only contact 4−lepton diagram can contribute to these (rather exotic) process. The coefficients CX are given by: kikj CV LL = 2Cℓℓ kikj CV RR = 2Cee 1 kikj CV LR = − CSRL = Cℓe 2 1 kjki CV RL = − CSLR = Cℓe 2 CSLL = CSRR = CT L = CT R = 0
CγL = CγR = 0
V.
(IV.9)
LEPTON FLAVOR VIOLATING Z 0 DECAYS
+ The branching ratio for the lepton flavor violating decays of a Z 0 boson Z 0 → ℓ− f ℓi is given by: 2 i h mZ ZR 2 ZL 2 ZL 2 ZR 2 mZ 0 ± ∓ + Γf i + Γf i + Cf i Cf i , (V.1) Br Z → ℓf ℓi = 24πΓZ 2
19 where ΓZ ≈ 2.495 GeV is the total decay width of the Z 0 boson. We included all tree-level contributions and ΓZL fi ΓZR fi CfZR i
v 2 (1)f i (3)f i 2 + 1 − 2sW δf i , Cϕl + Cϕl = 2sW cW Λ2 2 e v fi 2 = C − 2sW δf i , 2sW cW Λ2 ϕe v ZL⋆ = Cif = −√ Cfi 2Λ2 Z
e
(V.2) (V.3) (V.4)
where CZf i is defined as fi fi . CZf i = sW CeB + cW CeW
(V.5)
The experimental bounds on these decays are given in Table VII. Their current sensitivities are not as good as for the other lepton flavor violating decays but a future linear collider could significantly + improve them [63]. Note that theoretical prediction in Eq. (V.1) is for the decay Z 0 → ℓ− f ℓi or
− − + − + 0 Z 0 → ℓ+ f ℓi while the experimental values are for the sum Z → ℓf ℓi + ℓi ℓf . Therefore, Eq. (V.1)
must be multiplied by a factor of 2 in order to compare it to the experimental values.
Process
Experimental bound
Br Z 0 → µ± e∓ Br Z 0 → τ ± e∓ Br Z 0 → τ ± µ∓
1.7 × 10−6 [64] 9.8 × 10−6 [64] 1.2 × 10−5 [64]
TABLE VII: Experimental upper limits (95 % CL) on the lepton flavor violating Z 0 decay rates.
VI.
NUMERICAL ANALYSIS
In the absence of fine-tuning and accidental cancellations the Wilson coefficients of the flavor changing 4-lepton operators and of the flavor changing Z 0 -lepton-lepton vertex are most strinfi fi is best − sW CeW gently constrained by the three-body charged lepton decays, while Cγf i = cW CeB
restricted by the radiative lepton decays. Henceforth, as a first approximation one can obtain the approximate bounds on Cγf i from the experimental upper limits on Br[ℓi → ℓf γ], assuming that all
20 other Wilson coefficients are negligible: 2 r Λ Br [µ → eγ] , 1 TeV 5.7 × 10−13 2 r q Λ Br [τ → eγ] Cγ13 2 + Cγ31 2 ≤ 2.35 × 10−6 , 1 TeV 3.3 × 10−8 2 r q Λ Br [τ → µγ] C 23 2 + C 32 2 ≤ 2.71 × 10−6 . γ γ 1 TeV 4.4 × 10−8
q Cγ12 2 + Cγ21 2 ≤ 2.45 × 10−10
(VI.1)
Here, the numbers dividing the branching ratios are the current experimental bounds given in Table I. We see that the resulting bounds are very strong, of the order of 10−10 for µ → e transitions and of the order of 10−6 for τ → µ, e transitions for NP at the TeV scale. This means that, even though in a renormalizable theory of NP Cγf i can only be induced at the loop level, an
additional suppression mechanism is needed (especially for µ → eγ) in order the make TeV-scale NP compatible with experiment. Knowing that Cγf i must be tiny one can set them to zero in order to constrain other Wilson coefficients using the bounds from the ℓi → ℓf ℓf ℓ¯f decay rates. Here we find (again normalizing the branching ratios to current limits listed in Table II): −5
Cµeee ≤ 3.29 × 10
−2
Cτ eee ≤ 1.28 × 10
−2
Cτ µµµ ≤ 1.13 × 10
Λ 1 TeV Λ 1 TeV Λ 1 TeV
2 r 2 r 2 r
Br [µ → eee] , 1 × 10−12 Br [τ → eee] , 2.7 × 10−8
(VI.2)
Br [τ → µµµ] , 2.1 × 10−8
with Cℓi ℓf ℓf ℓf given by 2 f if f (3)f i 2 (1)f i f if f fi + 0.46 Cϕe 2 | Cℓℓ − 0.54 Cϕℓ + Cϕℓ + 2 Cee 2 12 fffi f if f (3)f i 2 (1)f i fi . + Cℓe + 0.46 Cϕℓ + Cϕℓ + Cℓe − 0.54 Cϕe
Cℓ i ℓ f ℓ f ℓ f =
(VI.3)
From Eq. (VI.2) and Eq. (VI.3) we see that also the Wilson coefficient of the flavor changing 4lepton and the Z 0 -lepton-lepton vertices must be small for Λ ∼ O(1) TeV: of the order of 10−5 for µ → e transitions and on the order of 10−2 for τ → µ and τ → e transitions. These constraints are less stringent then the ones derived from radiative photon decays in Eq. (VI.1) but one should keep in mind that unlike OeB and OeW , the other operators are not necessarily induced at the loop-level but can already be generated at tree-level. ∓ Also the constraints from Z 0 → ℓ± f ℓi can be brought into a form in which one can directly read
21 off the bounds on the Wilson coefficients:
r (1)13 (3)13 2 13 2 + C 13 2 + C 31 2 Cϕℓ + Cϕℓ + Cϕe Z Z r (1)23 (3)23 2 23 2 + C 23 2 + C 32 2 Cϕℓ + Cϕℓ + Cϕe Z Z
2
s
Br Z 0 → µ± e∓ , 1.7 × 10−6 s 2 Br Z 0 → τ ± e∓ Λ ≤ 0.14 , 1 TeV 9.8 × 10−6 s 2 Br Z 0 → τ ± µ∓ Λ ≤ 0.16 . 1 TeV 1.2 × 10−5
r 2 2 2 (1)12 (3)12 2 12 21 12 Cϕℓ + Cϕℓ + Cϕe + CZ + CZ ≤ 0.06
Λ 1 TeV
(VI.4)
These constraints are less stringent than the ones from ℓi → ℓf ℓf ℓ¯f and ℓi → ℓf γ but they put
bounds on the linear combination CZf i which is orthogonal to Cγf i (see Eq. (III.3) and Eq. (V.5)),
fi fi so that using both Eq. (VI.1) and Eq. (VI.4) one can independently constrain both CeW and CeB .
Finally, one can give similar simplified expressions for the bounds resulting from the anomalous magnetic moments of charged leptons and from the EDMs. Neglecting small lepton mass ratios and taking into account that some of the Wilson coefficients of the 4-lepton and the Z 0 -lepton vertices are real in the flavor conserving case we find for the EDMs: 1 TeV 2 11 −18 −5 3113 de = −2.08 × 10 Im 2 × 10 Cℓe + Cγ e cm , Λ 1 TeV 2 22 −18 −5 3223 e cm , dµ = −2.08 × 10 Im 2 × 10 Cℓe + Cγ Λ 33 1 TeV 2 −18 e cm , dτ = −2.08 × 10 Im Cγ Λ
(VI.5)
and for the anomalous magnetic moments:
3113 ae = 1.17 × 10−6 Re 2 × 10−5 Cℓe + Cγ11 −4
aµ = 2.43 × 10
−3
aτ = 4.1 × 10
−5
Re 2 × 10 h
−5
Re 10
3223 Cℓe
+
Cγ22
1 TeV Λ 1 TeV Λ
2 2
, ,
(VI.6)
i 1 TeV 2 (3)33 (1)33 33 33 3333 . × 1.6 Cϕℓ + 2.0 Cℓe − 1.7 Cϕℓ + Cϕe + Cγ Λ
Here we kept the loop induced contributions from the Qℓe and Qϕe since they are not (or weakly) constrained from other processes. In order to illustrate the interplay between different Wilson coefficients in ℓi → ℓf γ and ℓi →
ℓf ℓf ℓ¯f decays let us consider as an example the dependence of both decays on the Wilson coefficients
fi fi of the operators Oϕe and OeW , as shown in Fig. 5. We see that the regions which respect both the
bound from ℓi → ℓf γ and ℓi → ℓf ℓf ℓ¯f are very small, especially for µ → e transitions. We also
show the predicted branching ratios for Z 0 → ℓf ℓi to illustrate that in this plane indirect limits
22 Μ®eee, Μ®eΓ and Z®Μe
Τ®eee, Τ®eΓ and Z®Τe
15
-11 0.02 2´10
-6
8´10
8´10-6
10 4´10-6
10 4´10-6 5
-12
CΦe
13
1´10
0.00
2´10-6
5
0
1´10-12
0
5´10-7
-5
-0.01 5´10-12 1´10-11
5´10-7
-5
2´10-6
-6 -10 4´10
-0.02 2´10-11
-10
8´10-6
-0.5
0.0
0.5
1.0
2´10-6 4´10-6 8´10-6
-15
-1.0
2´10-6 5´10-7
5´10-7 23
1´10
0.01 5´10-12
CΦe
-11
12 CΦe
Τ®ΜΜΜ, Τ®ΜΓ and Z®ΤΜ
15
-4
0
-2
6
12 CeW ´10
2
-15 -10
4
0
-5
3
13
5
10
3
23
CeW ´10
CeW ´10
fi fi FIG. 5: Allowed regions in the CeW –Cϕe plane for Λ = 10 TeV. Yellow (lightest): ℓi → ℓf ℓf ℓ¯f , red (gray):
ℓi → ℓf γ. The blue region is allowed by both decay modes simultaneously. The contour lines show the
predicted branching ratio for Z 0 → ℓf ℓi . Note that in the parameter space plotted, the dependence of fi Br[Z 0 → ℓf ℓi ] on CeW is very weak. Br@Μ ® eΓD
Br@Τ ® eΓD
Br@Τ ® ΜΓD
Br@Μ ® eeeD
Br@Τ ® eeeD
Br@Τ ® ΜΜΜD
4000 120
110
400
200
50 60
250
130
-2000
300 350
CΓ23
80
0
CΦ{
H1L 13
160
100
H1L 23
CΓ13
150
0
70
200
400
0
CΦ{
140
CΦ{
H1L 12
CΓ12
2000
90
450
-200
170
-100
500
-400
-4000 -4000 -2000
0
2000
4000
0
-400 -200
12 CΦe CΓ12
200
400
fi Cϕe
Cγf i
-100
0
100
200
23
13
CΦe CΓ23
CΦe CΓ13
FIG. 6: Ratios Br[ℓi → ℓf γ]/Br[ℓi → ℓf ℓf ℓ¯f ] in the
-200 -200
(1) f i
–
Cϕℓ
Cγf i
plane (independent of the scale Λ of NP).
from the other two processes are currently stronger then the directly measured upper bounds given in Table VII. Another interesting aspect is the correlation between the radiative lepton decays and the three-body charged lepton decays. In Fig. 6 we show as an example the ratios (1) f i Cϕℓ Cfi Br [ℓi → ℓf γ] /Br ℓi → ℓf ℓf ℓ¯f as a function of ϕe f i and f i . Note that such ratios are indeCγ
Cγ
pendent of the scale Λ of NP and depend only on the ratios of Wilson coefficients. Thus, given
a specific model, one can determine the branching ratio for one process in terms of the other one independently of the scale of new physics and also of other possible cancellations of NP model parameters which can occur in the ratios
fi Cϕe Cγf i
(1) f i
and
Cϕℓ
Cγf i
. As known in the literature, the ratio
23 Μ®eΓ
aΤ and Z®ΤΤ 0.04
Τ®ΜΓ and Z®ΤΜ
10
3
0.4
2 5
0.2
1 ´10
3213
0
0
C{e
33
0.0
5
0.02
CeB
23
CeB
0
-1
-0.2
-5 -2
-0.04
-0.4 -0.4
-0.2
0.0
0.2
0.4
-10 -0.02 -10 -5
0 33
23
CeW
CeW
5
10
-3 -3
-2
0
-1
1332
C{e
1
2
3
5
´10
23 23 FIG. 7: Left: Allowed regions from Br[Z 0 → τ µ] (yellow) and Br[τ → µγ] (blue) in the CeW –CeB plane for
Λ = 1 TeV. Middle: Correlations between the anomalous magnetic moment of the τ lepton and Z 0 → τ τ .
Yellow (light grey): region allowed by the aτ , blue (dark grey): region allowed by Z 0 → τ τ . The contour lines
1332 3213 indicate the value of aτ for Λ = 1 TeV. Right: Allowed regions from Br[µ → eγ] in the Cℓe –Cℓe plane for 12 12 12 12 Λ = 1 TeV and different values of CeW . Yellow: CeW = 0, red: CeW = 6 × 10−8 , green: CeW = −6 × 10−8.
of both decay rates in case in which only Cγf i is non-zero depends solely on SM parameters and α (log is given by 1/( 3π
m2f m2i
−
11 4 ))
(which corresponds to points (0, 0) in Fig. 6). From Fig. 6 one (1) f i
fi can see that contributions from Cϕe and Cϕℓ
can only slightly enhance but more significantly
suppress this ratio. This is important from the point of view of planned new experiments searching for µ → eee with increased sensitivity. As observed in Sec. V, processes involving photon and Z 0 couplings to leptons constrain “orthogonal” combinations of the Wilson coefficients of the operators OeB and OeW . Thus, using a suitable pair of measurements, one can obtain absolute upper bounds on each of CeB and CeW . An example of such an exclusion is shown in the left panel of Fig. 7: the bound on the radiative decay τ → µγ strongly correlates the allowed values for CeB and CeW values to a thin straight
belt, while Z 0 → τ µ bound cuts the length of this belt to a wider but finite compartment.
Concerning flavor diagonal transitions we can correlate the anomalous magnetic moments to the corresponding Z 0 → ℓℓ decays. For the electron and the muon the constraints from the
anomalous magnetic moments are so strong that no sizable effects of NP in Z 0 → ee or Z 0 → µµ are possible. However, for the tau lepton the constraints on NP generated terms from Z 0 → τ τ
and from the anomalous magnetic moment are not that different. The allowed region in the 33 –C 33 plane is shown in the middle plot of Fig. 7. In order to obtain these constraint we used CeW eB 0 Br Z → τ τ = (3.370±0.008)% [1] and included radiative corrections into our tree-level expression
24 for Z 0 → ℓf ℓi , Eq. (V.1), multiplying it by a correction factor Br Z 0 → τ τ SM /Br Z 0 → τ τ tree where Br Z 0 → τ τ SM includes radiative corrections and can be found in Ref. [1]. We also find (1) jj
that the precision of Z 0 → ℓj ℓ¯j decay width measurements limit the sizes of Cϕℓ
(3) jj
, Cϕℓ
jj and Cϕe
Wilson coefficients so stringently that no sizable effects in the corresponding anomalous magnetic moments are possible for any lepton flavor. Another interesting aspect is that one can constrain some of the 4-lepton contact terms by using only the radiative lepton decays. This is possible because the 4-lepton operator Oℓe affects the ℓi → ℓf γ amplitude at the 1-loop level, as calculated in Sec. III A. Once the values of Wilson coefficients defining the photon coupling Cγ are fixed, the bounds on the 4-lepton couplings can be fairly strong - as illustrated in example in the right panel of Fig. 7. There we see that the bounds 1332 and C 3213 from µ → eγ for Λ = 1 TeV are O(10−5 ). Note that these coefficients (with on Cℓe ℓe
double τ flavor index) cannot be constrained from any other process considered in this article.
VII.
CONCLUSIONS
In these article we calculated the expressions for several theoretically important and experimentally well constrained lepton flavor violating processes within the Standard Model extended with the most general set of effective LFV operators of dimension-6 invariant under the SM gauge group. We computed the complete set of 1-loop contributions (to the leading order in mℓ /mW ) to the radiative lepton decays ℓi → ℓf γ and to the related electric dipole moments and anomalous magnetic moments of charged leptons (see Table VI). We also obtained the full expression for the 3-body charged lepton decay rates ℓi → ℓj ℓk ℓl (Eq. (IV.5)–Eq. (IV.9)) and for the flavor violating Z 0 → lf ¯li decays taking into account all possible tree-level contributions.
The predictions for all processes are given in terms of Wilson coefficients of the effective operators, automatically assuring that the final results are gauge-invariant (which we confirmed explicitly in our calculation) and that all relevant contributions are included. The derived expressions allow to obtain model-independent bounds on the Wilson coefficients of LFV operators, which can be later easily compared to their values calculated within specific UV complete extensions of the SM. To facilitate the comparison, we included in Sec. VI approximate numerical formulae directly relating the Wilson coefficients to current experimental upper bounds on the discussed processes (Eq. (VI.1)–Eq. (VI.6)). We show that bounds on the effective LFV couplings are already very strong if the scale of NP is low, O(1) TeV, and weaken proportionally to the square of NP scale. We
25 also illustrated possible correlations between Wilson coefficients of various dimension-6 operators and showed that the loop contributions to ℓi → ℓf γ decays are capable to constrain 4-lepton operators which would be unbounded otherwise.
Acknowledgments The authors would like to thank Stefan Pokorski for suggesting the subject of this article and helpful discussions. A.C. is grateful to Christoph Greub for his help and proofreading the article. We also thank Marco Giovanni Pruna and Adrian Signer for useful discussions. A.C. is supported by the Swiss National Science Foundation (SNF). The work of S.N. has been suported by the Marie Curie Initial Training Network of the EU Seventh Framework Programme under contract number PITN-GA-2009-237920-UNILHC. She thanks the CERN Theory Division for the hospitality during the stay there. The work of J.R. is supported by the National Science Center in Poland under the research grants DEC-2011/01/M/ST2/02466 and DEC-2012/05/B/ST2/02597.
APPENDIX
We summarize below the Feynman rules arising from the dimension-6 operators after the electroweak symmetry breaking. i, i1 , i2 and f, f1 , f2 denote the flavor indices of incoming and outgoing leptons, respectively. We list only the vertices actually used in our tree level or 1-loop calculations. For completeness we also include few necessary purely SM couplings.
A.
Feynman rules involving gauge and Goldstone bosons
. ℓi
γµ
q→
ℓf
i h fi fi PL + CγR PR qν i eγ µ δf i + iσ µν CγL CfγR i
=
CfγL⋆ i
√ v 2 fi fi = 2 cW CeB − sW CeW Λ
26
ℓi
Zµ
q→
ℓf
ZL ZR µν i γ µ ΓZL Cf i PL + CfZR f i PL + Γf i PR + iσ i PR qν v 2 (1)f i (3)f i 2 Cφℓ + Cφℓ + 1 − 2sW δf i = 2sW cW Λ2 2 v fi e C − 2s2W δf i ΓZR fi = 2sW cW Λ2√ φe v 2 fi fi ZL⋆ CfZR = C = − s C + c C W eB W eW i if Λ2 e
ΓZL fi
νi L P MNS µ iΓW γ PL f j Vji
ℓf
W −µ
L ΓW fj
ℓf
p
v 2 (3)f j C + δf j Λ2 φℓ
1 + δf i mℓi PL − v 1 G0 R + pΓ ✁ f i − v δf i mℓi PR
ℓi
G0
e = −√ 2sW
0
G0 L pΓ ✁ fi
v (1)f i (3)f i + C C φℓ φℓ Λ2 v fi = 2 Cφe Λ
L = ΓG fi 0
R ΓG fi
νi i G−
p
−L ΓG fj p ✁
√
2 δf j mℓf − v
ℓf −L ΓG fj
√ v 2 (3)f j = − 2 Cφl Λ
!
VjiP M N S PL
27
G− p2
ie (pµ1 + pµ2 )
γµ
p1
G−
γµ p3 ↓
Wν
← p2
p1 →
ie[gνλ (p1 − p2 )µ + gλµ (p2 − p3 )ν + gµν (p3 − p1 )λ ] Wλ
νi P MNS µ iΓGγL γ PL f j Vji
γµ
ℓf p
ΓGγL fj
√ ev 2 (3)f j = − 2 Cφl Λ
G− ℓi
ℓf
W +µ p G−
R L PR PL + ΓGW iγ µ ΓGW fi fi L =− ΓGW fi
ev (1)f i C Λ2 sW φℓ
R =− ΓGW fi
ev Cfi Λ2 sW φe
ℓi
G−
ℓf
p2 p1
GGR PR i (p✁1 + p✁2 ) ΓGGL f i PL + Γf i ΓGGL =− fi
G−
=− ΓGGR fi
1 (1)f i (3)f i C − C φl φℓ Λ2 1 fi C Λ2 φe
28 γµ
G−
ℓi
ℓf
p2 p1
GGγR P + Γ P iγ µ ΓGGγL L R fi fi ΓGGγL =− fi
G−
ΓGGγR =− fi B.
ℓ i2
2e f i C Λ2 φe
Feynman rules for 4-fermion operators
i h f 1 i1 f 2 i2 µ Cℓℓ (γ PL )f1 i1 (γµ PL )f2 i2 Λ2
ℓ f1
ℓ i1
2e (1)f i (3)f i − C C φℓ φℓ Λ2
f 1 i1 f 2 i2 µ +Cee (γ PR )f1 i1 (γµ PR )f2 i2
f 1 i1 f 2 i2 µ +Cℓe (γ PL )f1 i1 (γµ PR )f2 i2
i
ℓ f2 ν i2
ℓ f1
ℓ i1
i h f 1 i1 f 2 i2 µ Cℓℓ (γ PL )i1 f1 (γµ PL )f2 i2 Λ2
f 1 i1 f 2 i2 +2Re(Cℓe )(γ µ PL )f1 i1 (γµ PR )f2 i2
i
ν f2
i h (1)f1 i1 f2 i2 (3)f i f i (Cℓq − Cℓq 1 1 2 2 )(γ µ PL )i1 f1 (γµ PL )f2 i2 2 Λ
ui 2
f 1 i1 f 2 i2 µ f 1 i1 f 2 i2 µ +Cℓu (γ PL )f1 i1 (γµ PR )f2 i2 + Ceq (γ PR )f1 i1 (γµ PL )f2 i2
ℓ f1
ℓ i1
f 1 i1 f 2 i2 µ +Ceu (γ PR )f1 i1 (γµ PR )f2 i2 (1)f i1 f2 i2
(PR )f1 i1 (PR )f2 i2 − Cℓequ1
(3)f i1 f2 i2
(σ µν PR )f1 i1 (σµν PR )f2 i2
−Cℓequ1 uf 2
−Cℓequ1
(1)i f1 i2 f2 ⋆
(3)i f1 i2 f2 ⋆
− Cℓequ1
(σ µν PL )f1 i1 (σµν PL )f2 i2
i
(PL )f1 i1 (PL )f2 i2
29 i h (1)f1 i1 f2 i2 (3)f i f i (Cℓq + Cℓq 1 1 2 2 )(γ µ PL )i1 f1 (γµ PL )f2 i2 Λ2
di2
f 1 i1 f 2 i2 µ f 1 i1 f 2 i2 µ +Cℓd (γ PL )f1 i1 (γµ PR )f2 i2 + Ceq (γ PR )f1 i1 (γµ PL )f2 i2
ℓ f1
ℓ i1
f 1 i1 f 2 i2 µ +Ced (γ PR )f1 i1 (γµ PR )f2 i2 f 1 i1 f 2 i2 i1 f 1 i2 f 2 ⋆ + Cℓedq (PR )f1 i1 (PL )f2 i2 + Cℓedq (PL )f1 i1 (PR )f2 i2
df2
C.
i
Vector and scalar form-factors contributing to off-shell ℓi → ℓf γ ∗ amplitude.
Gauge invariance requires that FV L and FV R (“vector”) form-factors vanish for the on-shell external particles. Thus, expressions for them must be proportional to the momentum of the outgoing photon and they do not contribute to ℓi → ℓf γ decay rate. The “scalar” form-factors FSL and FSR does not need to vanish on-shell, but they also cancel out from this amplitude after contracting with the photon polarization vector. Still, those form-factors can enter the expressions for the more complicated processes. Thus, we list them below, again splitted into groups of contributions within which the vector form-factors vanish in the on-shell limit. Note that some of them are infinite and require renormalization. We give only expressions for left scalar form-factor FSL - the right one can be obtained from FSL by changing the sign and exchanging the external fermion masses, i.e.: FSR = −FSL (mi ↔ mf )
(C.1)
Z 0 group - diagrams 3a, 2a(Z 0 ): mi mf 2e(1 − 2s2W )Q2 (1)f i (3)f i C + C 1 − 6 log ϕℓ ϕl 9(4π)2 MZ2 mi mf 4es2W Q2 f i = − (C.2) C 1 − 6 log 9(4π)2 ϕe MZ2 i mi mf 2e h (3)f i (1)f i 2 fi 2 + 2mi sW Cϕe 1 − 6 log mf (1 − 2sW ) Cϕℓ + Cϕl = 9(4π)2 MZ2
FVZLf i = FVZRf i Z fi FSL
30 W G group - diagrams 3c,d,e,i,j,k,l, 2b(W ),c and photon-Goldstone boson self-energy: 2eQ2 h (3)f i (1)f i (3)f i 2 + C C + 6c 16C W ϕl ϕℓ ϕl 9(4π)2 2 MW (1)f i (3)f i 2 3cW 15Cϕℓ + 16Cϕl ∆ − log 2 µ 2 2 2 2ecW Q f i M − Cϕe 2 + 15 ∆ − log W (C.3) 2 3(4π) µ2 e h 2 (3)f i (3)f i (1)f i fi 12cW (mi Cϕe − mf (Cϕℓ + Cϕl )) − 32mf Cϕl 9(4π)2 2 MW (3)f i (3)f i (1)f i 2 fi ∆ − log 2 3 15cW (mi Cϕe − mf (Cϕℓ + Cϕl )) − 2mf Cϕl µ
FVWLG f i = − + FVWRG f i = W G fi FSL =
+
G0 group - diagrams 3b, 2a(G0 ): 0
0
0
0
G fi G fi FVGL f i = FVGR f i = FSL = FSR =0
(C.4)
G± group - diagrams 3f,g,h, 2b(G± ): ±
FVGL
fi
±
= FVGR
fi
±
G = FSL
fi
±
G = FSR
fi
=0
(C.5)
4l group - contact 4-lepton diagrams 3n, 2e: FV4ℓLf i
3 2eQ2 X f ijj f ijj = − 2Cℓℓ + Cℓe 3(4π)2 j=1
FV4ℓRf i
µ2
3 m2ℓj 2eQ2 X jjf i f ijj 2Cee + Cℓe ∆ − log 2 = − 3(4π)2 µ j=1
4ℓ f i FSL
∆ − log
m2ℓj
!! !!
3 m2ℓj 2e X f ijj jjf i f ijj f ijj 2Cℓℓ mf − 2Cee mi − (Cℓe mi − Cℓe mf ) ∆ − log 2 = − 3(4π)2 µ j=1
(C.6) !!
31 4f group - contact 4-lepton and 2-lepton-2-quark diagrams 3m, 2d: FV4fL f i
3 m2uj 4eQ2 X (1)f ijj (3)f ijj f ijj ∆ − log + C − C C = ℓq ℓu ℓq 9(4π)2 µ2 j=1
3 X m2dj (1)f ijj (3)f ijj f ijj C + C + C ∆ − log − ℓq ℓq ℓd 9(4π)2 µ2 j=1 ! 3 m2uj 4eQ2 X f ijj (3)f ijj ∆ − log 2 Ceq + Ceu = 9(4π)2 µ j=1 ! 2 3 m 2eQ2 X f ijj d (3)f ijj ∆ − log 2j Ceq + Ced − 9(4π)2 µ
2eQ2
FV4fRf i
! ! (C.7)
j=1
4f f i FSL
3 m2uj 4e X (3)f ijj (1)f ijj f ijj f ijj (3)f ijj ∆ − log C + C − m + C − C m C = i f eq eu ℓq ℓu ℓq 9(4π)2 µ2 j=1
3 m2dj 2e X (3)f ijj (3)f ijj (1)f ijj f ijj f ijj ∆ − log 2 − mi Ceq + Ced + Cℓd + Cℓq mf Cℓq − 9(4π)2 µ j=1
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