Anomalous magnetic and weak magnetic dipole moments of the \(\tau \) lepton in the simplest little Higgs model
Eur. Phys. J. C
Anomalous magnetic and weak magnetic dipole moments of the ? lepton in the simplest little Higgs model
M. A. ArroyoUre?a 1
G. Hern?ndezTom? 0 1
G. TavaresVelasco 1
0 Present address: Departamento de Fi?sica, Centro de Investigacio?n y de Estudios Avanzados del Instituto Polite?cnico Nacional , Apartado Postal 14740, 07000 Mexico City , Mexico
1 Facultad de Ciencias Fi?sicoMatema?ticas, Beneme?rita Universidad Auto?noma de Puebla , C.P. 72570 Puebla, PUE , Mexico
We obtain analytical expressions, both in terms of parametric integrals and PassarinoVeltman scalar functions, for the oneloop contributions to the anomalous weak magnetic dipole moment (AWMDM) of a charged lepton in the framework of the simplest little Higgs model (SLHM). Our results are general and can be useful to compute the weak properties of a charged lepton in other extensions of the standard model (SM). As a byproduct we obtain generic contributions to the anomalous magnetic dipole moment (AMDM), which agree with previous results. We then study numerically the potential contributions from this model to the ? lepton AMDM and AWMDM for values of the parameter space consistent with current experimental data. It is found that they depend mainly on the energy scale f at which the global symmetry is broken and the t? parameter, whereas there is little sensitivity to a mild change in the values of other parameters of the model. While the ? AMDM is of the order of 10?9, the real (imaginary) part of its AWMDM is of the order of 10?9 (10?10). These values seem to be out of the reach of the expected experimental sensitivity of future experiments. The most general dimensionfive effective interaction of a neutral V gauge boson (V = ? , Z ) to a charged lepton that respects Lorentz invariance can be written in terms of six independent form factors:

? V (q2) = i e[??(FVV ? FAV ?5) ? q?(i FSV + FPV ?5)
a email:
b email:
c email:
where q is the incoming transfer fourmomentum of the
gauge boson. The electromagnetic (weak) properties of the
lepton are determined by the photon (Z gauge boson) vertex
function. The C Pviolating terms define the static electric
dipole moment (EDM) and the static weak electric dipole
moment (WEDM):
Since the standard model (SM) predictions for these C
Pviolating dipole moments are highly suppressed, they may
serve to search for new sources of C P violation. In this work
we are interested instead in the static anomalous magnetic
dipole moment (AMDM) and the static anomalous weak
magnetic dipole moment (AWMDM), which are defined in
terms of the C Peven form factor as follows:
The measurement of the AMDM of a lepton has long been
considered a probe for the SM, which considers leptons as
pointlike objects. The AMDM of the electron ae, which
receives its main contributions from quantum
electrodynamics (QED), has been calculated up to order of ?5 [1] and the
agreement between the theoretical and the experimental
values has reached the level of ten significant digits [2], which
represents one of the greatest milestones of QED.
As far as the muon is concerned, the E821 experiment
at Brookhaven National Lab (BNL) measured its AMDM
a? with an unprecedent precision of 0.54 ppm. The current
average experimental measurement is [3]
a?Exp. = 11659209.1(5.4)(3.3) ? 10?10,
and further improvement is expected at future experiments by
the (g?2)? [4] and JPARC (g?2)/EDM [5] Collaborations,
which aim to reach a precision of ?0.2 ppm. As for the
theoretical prediction of the SM [6]
a?SM = 116591803(1)(42)(26) ? 10?11,
there is still a large uncertainty in the estimate of the hadronic
contribution, whereas the QED and electroweak
contributions have been determined with a great precision [7]. Thus a
more accurate evaluation of the leading order hadronic
contribution together with the future experimental measurement
are needed to settle down the discrepancy between the SM
prediction and the experimental value of a?, which currently
stands at the level of 3.6 standard deviations [6]:
a? = a?Exp. ? a?SM = 288(63)(49) ? 10?11.
Since the muon AMDM has become a powerful tool to test
the validity of the SM and searching for new physics (NP)
effects, a plethora of calculations within the framework of
several SM extensions has been reported in the literature in
order to explain the a? discrepancy [7,8].
As far as the ? lepton is concerned, the SM prediction is
a?SM = 117721(5) ? 10?8 [9,10]. The error of the order of
10?8 hints that SM extensions predicting values for a? above
this level could be worth studying. Since the SM prediction
for a? is far from the experimental sensitivity, which is one
order of magnitude below the leading QED contribution, a
more precise determination of the experimental value is
necessary. Due to its short lifetime (290.3 ? 0.5 ? 10?15 s) [11],
the ? lepton does not allow for a high precision
measurement of its AMDM via a spin precession method. The most
stringent current bound on a? is [12]
which was obtained using LEP1, SLD and LEP2 data for ?
lepton production. It also has been pointed out recently that
the ? electromagnetic moments can be probed in ? ? and
? e collisions at CLIC, which can lead to improved bounds
[13,14]. In this regard, it has been pointed out that super B
factories could allow for a precise determination of a? up to
the 10?6 level using unpolarized or polarized electron beams
[15?17]. Furthermore, due to its large mass, it is expected that
the ? AMDM can be very sensitive to NP effects [18] since
its electroweak contribution would be ten times larger than
the uncertainty of the hadronic contribution [10]. Therefore,
it is worth estimating the ? AMDM in any SM extension
as future measurements may allow us to search for NP in a
rather clean environment.
Unlike the electromagnetic dipole moments of leptons,
little attention has been paid to the study of their weak
properties. In the experimental arena, the current best limits on
the ? AWMDM and WEDM, with 95% C.L., are [19]
Re(d?W ) < 0.5 ? 10?17 ecm,
I m(d?W ) < 1.1 ? 10?17 ecm,
which were extracted from the data collected at the LEP from
1990 to 1995, corresponding to an integrated luminosity of
155 pb?1. Somewhat weaker bounds were obtained in [20]
via a study of the pp ? ? +? ? and pp ? Z h ? ? +? ?h
cross sections at the LHC. The current experimental bounds
on a?W are well above the SM theoretical prediction, which
was calculated in Ref. [21]:
It is thus interesting to analyze whether NP contributions can
give a significant enhancement and be at the reach of future
experimental detection.
In this work we evaluate the AMDM and AWMDM of a
lepton, with special focus on those of the ? lepton, predicted
by the simplest little Higgs model (SLHM) [22], which is an
appealing SM extension. This model is aimed to deal with
the hierarchy problem by conjecturing that the Higgs boson
is a pseudoGoldstone boson arising from a global symmetry
broken spontaneously. At the same scale, the local symmetry
is also broken by a collective symmetry breaking mechanism.
The top quark and the electroweak gauge bosons have heavy
partners that give rise to new contributions that exactly cancel
the quadratic divergences to the Higgs boson mass at the
oneloop level, thereby rendering a mass of about 100 GeV
without the need of fine tuning [23,24].
The rest of this presentation is organized as follows. A
brief review on the SLHM is presented in Sect. 2, whereas
Sect. 3 is devoted to the analytical results for the AWMDM.
As a byproduct we will obtain the corresponding expressions
for the AMDM. A brief discussion of the current constraints
on the parameters of the model, and the numerical analysis of
the ? electromagnetic and weak dipole moments is presented
in Sect. 4. Section 5 is devoted to the conclusions, whereas
the SLHM Feynman rules as well as explicit expressions for
the loop integrals are shown in the appendices.
2 The simplest little Higgs model
We now present an overview of the SLHM focusing only
on the details relevant for our calculation. For a detailed
account of this model and the study of its phenomenology we
refer the reader to Refs. [22,25?28], which we will follow
closely in our discussion below. The SLHM is the most
economic version of simplegroup little Higgs models, which
have the feature that the SM gauge group is embedded into a
larger simple gauge group instead of a product gauge group.
the noself conjugate neutral gauge boson Y0, and the
neutral gauge boson Z0 (following Ref. [25] we will denote the
gauge eigenstates with the subindex 0)
3 ? t W2 A8 + tW B X
with masses of the order of f . Four gauge fields remain
massless at this stage. While A1, A2, A3 identify with the SU (2)L
gauge bosons W a , the charged gauge bosons W ? and the
hypercharge gauge boson are
3 ? t W2 B X
After the electroweak symmetry breaking (EWSB), the
weak gauge bosons W ? and Z0 acquire mass and the heavy
gauge boson masses get corrected. Up to order (v/ f )2 W , X
and Y0 coincide with the mass eigenstates and their masses
are [26]
The SLHM has a [SU (3) ? U (1)]2 global symmetry and a
SU (3)L ? U (1)X gauge symmetry, which requires the
introduction of nine gauge bosons. At the TeV scale, the global
symmetry is broken down spontaneously to [SU (2)?U (1)]2
via the vacuum expectation values (VEVs) f1 and f2 of two
sigma fields 1 and 2, giving rise to ten Goldstone bosons.
At the same scale the gauge group breaks to the SM gauge
group SU (2)L ? U (1)Y and five Goldstone bosons are eaten
by the heavy fields: a charged gauge boson X ?, a noself
conjugate neutral boson Y0 = Y0?, and an extra neutral gauge
boson Z , which thus acquire masses of the order of the scale
f1 ? f2. The remaining Goldstone bosons are
accommodated in a complex doublet (the SM one) and a real singlet
of SU (2). The Goldstone bosons can be parametrized by the
triplets
1 = e
where the pion matrix is
1
= f ??
here f = f12 + f22, h is the SU (2) complex doublet of
the SM, and ? is a real scalar field. The normalization is
chosen to produce canonical kinetic terms. The dynamics
of the Goldstone bosons is described by a nonlinear sigma
model
Lkin = D? 12 + D? 22,
with the SU (3)L ? U (1)X covariant derivative
D? = ?? + i gT a Aa? ? i gX Q X B?X ,
where T a (a = 1 . . . 8) are the SU (3)L generators in the
fundamental representation, Aa? are the SU (3)L gauge fields,
B?X is the U (1)X gauge field, and Q X = ?1/3 for 1 and 2.
The new gauge bosons accommodate in a complex SU (2) ?
U (1) doublet (X +, Y0) with hypercharge 21 and a neutral
singlet Z0. The matching of the gauge coupling constants
yields
gX =
1 ? t W2 /3
with tW = sW /cW the tangent of the Weinberg angle ?W .
As mentioned above, after the first stage of symmetry
breaking, there emerge the pair of charged gauge bosons X ?,
with t? = tan ? = f1/ f2. If higher order terms are
considered W and X need to be rotated to obtain the physical states
[26]. On the other hand, the photon and the light neutral Z0
gauge boson are given by
Finally, Z0 and Z0 need to be rotated to obtain the mass
eigenstates Z and Z , which are given by
with ?Z = ? (1?tW28)cW3?tW2 vf 22 . The respective masses, up to
order (v/ f )2, are [26]
m Z =
?2g f
The kinetic Lagrangian of the gauge bosons gives rise to
the trilinear gauge boson couplings necessary for our
calculation. It can be written as
1 1 Aa ?? Aa ?? ,
= ? 4 B X ?? B?X? ? 4
with the Abelian and nonAbelian gauge strength tensors
A?? = ?? A? ? ?? A? + g fabc Ab? Ac? ,
a a a
with fabc the structure constants of the SU (3) group. From
the relations between gauge eigenstates and mass eigenstates
(20)?(24) and (28)?(31) we can obtain after some lengthy
algebra the Feynman rules listed in ?Appendix A? for the
V V j?V j? vertices, namely AW ?W ?, A X ? X ?, Z W ?W ?,
and Z X ? X ?.
In the lepton sector of the SLHM, for each generation
there is a lefthanded triplet L mT = (?Lm , Lm , i NLm ), which
is completed with a new neutral lepton NLm , and two
righthanded singlets Rm and NRm . The Yukawa Lagrangian can
be written, in the basis where flavor and mass Nm eigenstates
coincide, as
?
LY = i ?mN N Rm 2 Lm +
i j k
Rm i jk 1 2 Ln +H.c., (37)
where = 4? f is the cutoff of the effective theory. Here
m and n are generation indices, whereas i , j , and k are
SU (3) indices. After EWSB this Lagrangian yields the
lepton masses and the heavy neutrino masses up to order (v/ f )2
[26],
where ?? = ?2vf t? represents the mixing between a heavy
neutrino and a SM neutrino of the same generation. Notice
also that the rotation that diagonalizes ?N does not
necessarily diagonalizes ? so there is mixing between the charged
leptons and the heavy neutrinos mediated by the charged
gauge bosons. The charged lepton mass eigenstates Lm are
thus related to the flavor eigenstates Lm0 by the rotation
Lm0 = V mi Li ,
where V mi is a CKMlike mixing matrix. Also, in each
generation, the SM and heavy neutrino mass eigenstates are
obtained through
NLm0 =
where again the 0 subindex stands for flavor eigenstates. The
lepton masses up to order (v/ f )2 are [26]
m = ?
The vertices of a gauge boson to a lepton pair are obtained
from the lepton kinetic Lagrangian, which can be written as
LF = L? m i ? ? D? Lm + ?Rm i ? ? D? Rm
where the covariant derivative was given in Eq. (18), with
Q X = ?1/3, 0 and 1 for Lm , Nm and m . We need to
introduce the mass eigenstates to obtain the interactions of the
physical gauge bosons Z , W , X , and Z to a lepton pair,
which are necessary for our calculation of the AMDM and
AWMDM of a lepton. They are given by
+ 21 ??2 N? Li ? ? NLi ? 2 (?? V im N? Lm ? ??Li + H.c.)
1
?sW2 ?Ri ? ? Ri ? cW2 N? Li ? ? NLi
??2
+ 1 ? 2
= cW2
The ? Z Z vertex vanish since ? is C Podd scalar, but
the H ? Z coupling does arise, though its contribution to the
Notice that there is lepton flavor violation mediated by the
charged gauge bosons. Finally, the interactions of the photon
with a charged lepton pair are dictated by QED,
The scalar Higgs bosons H and ? also contribute to a
lepton AMDM and AWMDM. From the Lagrangian (17)
we can obtain the respective interactions with the Z and Z
gauge bosons. After some algebra one can extract the vertices
Z Z H and Z Z H , which are given as follows to the leading
order in (v/ f ):
LZ Z H =
g2v(1 ? t W2 )
H Z ? Z?,
together with the Z Z H interaction
LZ Z H = gm Z
H Z ? Z?.
From the Yukawa Lagrangian (37) we can also obtain the
interactions of the scalar Higgs bosons H and ? to leptons,
which we need for our calculation. They are diagonal and are
given to leading order in (v/ f )2 by [25]
LH ? = i g 2mW
AMDM and AWMDM of a lepton vanishes. A similar result
was found in Ref. [29], where the contributions of twoHiggs
doublet models (THDMs) to the AWMDM of a fermion were
calculated.
Other details of this model are irrelevant for our
calculation, so we refrain from presenting a discussion of the quark
sector and the Coleman?Weinberg scalar potential.
3 Anomalous magnetic and weak magnetic dipole
moments in the SLHM
We now turn to a presentation of our results. All the Feynman
rules necessary for our calculation follow straightforwardly
from the above interaction Lagrangians and are presented in
?Appendix A?. Since we are interested in the ? V ? vertex
with all the particles on their mass shell, the loop amplitudes
will be gauge independent. We used the unitary gauge as it
is best suited for our calculation method. In order to solve
the loop integrals, we used both Feynman parametrization
and the Passarino?Veltman reduction scheme. We will first
present the results for the AWMDM, from which the results
for the AMDM will follow easily. As far as the C Pviolating
properties are concerned, we will assume that there is no new
sources of C Pviolation in this model?s version, so both the
EDM and the WEDM will vanish. In the most general
scenario, C Pviolation could arise from new additional phases
in the extended Yukawa sector of the SLHM.
3.1 Anomalous weak magnetic dipole moment
In the SLHM, in addition to the pure SM contributions, the
AWMDM receives new physics contributions arising from
the loops carrying only new particles, but also from loops
involving only SM particles. The latter are due to
corrections to the SM vertices and appear as a series of powers of
v/ f , so it is enough to consider the leading order terms. The
AWMDM of lepton i can thus be written as
aW ?SL H M = aW ?SM + aW ?N P ,
i i i
where aWi ?SM stands for the SM contributions and aWi ?N P
for the new physics ones, which can be written as
aW ?N P = aW ?Gauge + aW ?Scalar,
i i i
with aW ?Gauge (aW ?Scalar) the contributions arising from the
i i
gauge (scalar) sector of the SLHM.
In the gauge sector, the NP contributions to the AWMDM
arise from the Feynman diagrams of Fig. 1, where V stands
for the charged gauge bosons W and X , ?i is a SM
neutrino, and Nk is a heavy neutrino. According to our discussion
Fig. 1 Feynman diagrams that contribute to the AWMDM of charged
lepton i at the oneloop level in the gauge sector of the SLHM. Here V
can be either the W gauge boson or the new charged X gauge boson, ?i
and N j stand for a SM neutrino and a new heavy one predicted by the
SLHM, respectively. Notice that diagram (4) involves the nondiagonal
vertex Z ??i N j
above, for the loops involving only W gauge bosons and SM
neutrinos we consider the leading order v/ f contributions
arising from the corrections to the W ?? vertex. The
contributions to the AWMDM of each Feynman diagram will be
written as follows:
where the threeletter superscript stands for the particles
circulating in each loop diagram ( A2 and A3 are the particles
attached to the external Z gauge boson, whereas A1 is the
particle attached to the external leptons). Here f ZA1 A2 A3 are
coefficients involving all the couplings appearing in each
amplitude, whereas I A1 A2 A3 stand for the loop integrals, which
Z
depend on the masses of the virtual particles. We present in
?Appendix B? both the f A1 A2 A3 coefficients and the loop
Z
integrals in terms of parametric integrals and Passarino?
Veltman scalar functions. We have verified that the
contribution of each diagram to the AWMDM is free of ultraviolet
divergences. The full contribution of the gauge sector can be
written as
+aWi ?Z i i ,
where the index j runs over the three lepton families.
In the scalar sector of the SLHM there are contributions
to the AWMDM of a lepton arising from both the SM scalar
boson H and the new pseudoscalar boson ? via the
Feynman diagrams of Fig. 2. The contributions of the SM Higgs
Fig. 2 Feynman diagrams that contributes to the AWMDM of a lepton
in the scalar sector of the SLHM at the oneloop level. We do not show
the diagram obtained by exchanging the Z (Z ) gauge boson and the
Higgs boson in diagram (2). The new contributions of the SM Higgs
boson contributions are due to the diagram with the Z gauge boson and
also to corrections to the SM vertices H and H Z Z
boson arise from corrections of the order of (v/ f )2 to the
SM vertices H and H Z Z . The respective contributions to
the AWMDM can also be written as in Eq. (56), where the
f A1 A2 A3 coefficients and the IZA1 A2 A3 functions are presented
Z
in ?Appendix B?. Again we have verified that the contribution
of each diagram to the AWMDM is ultraviolet finite.
The full scalar contribution is thus
V =Z,Z
3.2 Anomalous magnetic dipole moment
In the gauge sector of the SLHM, the AMDM of a lepton
arises from the Feynman diagrams (1) and (2) of Fig. 1, with
the Z gauge boson replaced by the photon. There are also
contributions arising from the scalar sector, which are induced
by the scalar bosons H and ? via a Feynman diagram similar
to diagram (1) of Fig. 2. The corresponding contributions to
the AMDM can be obtained straightforwardly from those to
the AWMDM by considering the limit m Z ? 0 and
substituting the Z coupling constants by those of the photon. We
can write the contributions to a arising from each diagram
as
where again the threeletter superscript corresponds to the
three particles circulating in the loop. Explicit expressions
for the f?A1 A2 A3 constants and the I?A1 A2 A3 functions are
presented in ?Appendix B? in terms of parametric integral and
Passarino?Veltman scalar functions.
The overall NP contribution of the SLHM to a i is thus
a Ni P =
where the index j runs over the three lepton families.
Table 1 Bound on the symmetry breaking scale f of the SLHM
obtained from several observables
Lower bound on f (TeV)
It is worth mentioning that our results for the ? AMDM in
terms of parametric integrals, obtained by a limiting
procedure from our results for the AWMDM, agree with previous
calculations presented in the literature [7,30]. This serves as
a crosscheck for our calculation.
4 Numerical analysis
We now present our numerical results for the AMDM and
AWMDM of the ? lepton in the context of the SLHM. We will
briefly review the existing bounds on the free parameters of
the model and afterwards analyze the potential contributions
to the AMDM and the AWMDM of the ? lepton for parameter
values consistent with these bounds.
4.1 Bounds on the parameter space of the SLHM
The SLHM parameters involved in our calculation are f , t? ,
m Nk , m?, ?? , and the matrix elements Vlmi . We will discuss
the current bounds on these parameters obtained from the
study of experimental data of several observables as reported
in the literature.
Symmetry breaking scale f : bounds on this parameter arise
from several observables. We list the most relevant in Table
1. We can observe that the most stringent bound f ? 5.6 TeV
arises from electroweak precision data (EWPD), whereas the
weakest limit f ? 1.7 TeV arises from parity violation in
cesium.
f1 t o f2 r ati o t? : a fit on 21 electroweak precision
observables from LEP, SLC, Tevatron, and the Higgs boson data
reported by the LHC ATLAS Collaboration and CMS
Collaboration, allowed the authors of Refs. [33,34] to find the
allowed region in the t? ? f plane, which we will take into
account for our numerical analysis. For the strongest bound
f ? 5.6 TeV, the allowed interval of t? values is 1?9. We
will analyze below the dependence on t? of the ? AMDM
and AWMDM in the allowed interval.
Mixing between light and heavy neutrinos ?? : this
parameter is experimentally constrained to be small [26], with the
corresponding bound being flavor dependent: ??e ? 0.03,
??? ? 0.05, and ??? ? 0.09 with 95% C.L. Since we are
interested in the study of the ? lepton, we need to make sure
that we use values of f and t? consistent with the bound
?? = v/(?2 f t? ) ? 0.09, which in turn translates into the
bound f t? 1932 GeV. Such a constraint is fulfilled for the
values of f and t? chosen in our analysis.
Pseudoscalar mass m?: this parameter is basically dependent
on the ? parameter (m? ? ?) appearing in the scalar potential
via the term ??2( 1? 2 + H.c.). In our analysis we will
explore values consistent with the lower bound m? ? 7 GeV,
which arises from the nonobservation of the ? ? ?? decay
[35]. Although the dominant contribution to the AMDM can
arise from a very light pseudoscalar, with mass of the order of
10 GeV, this requires relatively large values of t? , which are
already excluded according to the above discussion [33,34].
Mixing matrix elements V mi : previous studies on LFV within
the SLHM [26?28,36,37] have parametrized the mixing
matrix V and considered bounds on the mixing angles from
experimental data on LFV processes. A simple approach was
taken by the authors of Ref. [26] in which a scenario with
mixing between the first and second families only was
considered. It was found that the respective angle is tightly
constrained: the limit on ? ? e conversion yields an upper bound
on sin 2?12 of the order of 0.005. As we are interested in the
? AMDM and AWMDM, we will assume a scenario with
mixing between the second and third families only, namely
we will consider the following mixing matrix:
We will analyze whether current experimental bounds
on processes such as the muon AMDM and the ? ? ??
decay can be helpful to find a bound on the mixing angle ? .
We already have presented the results for the AMDM and
AWMDM of a lepton, we will now present the SLHM
contribution to the ? ? ?? decay in terms of both parametric
integrals and Passarino?Veltman scalar functions. The
Feynman diagrams inducing this decay at the oneloop level are
shown in Fig. 3. The corresponding amplitude can be written,
in the limit of massless j , as
with q = pi ? p j and FL given by
e? 2
?N N V V ?k j V ki fL (xk )
V =W,X
where xk = (m Nk /mV )2 (V = W, X ), ?N N V = ?? for the
2
?? ) for the X gauge
W gauge boson, and ?N N V = (1 ? 2
Fig. 3 SLHM contribution to the i ? j ? decay at the oneloop
level. In the loop circulate the charged W and X gauge bosons and a
heavy neutrino Nk
boson. The f (xk ) function is presented in ?Appendix C?.
We have verified that our results are in agreement with [26].
The i ? j ? decay width is given by
The current experimental limit BR(? ? ?? ) < 4.4 ? 10?8
[6] can translate into a bound on sin 2? . For this purpose,
we introduce the mass splitting ?23 ? z3 ? z2 with zk =
(m Nk /m X )2 and show in Fig. 4 the contours of the branching
ratio of the ? ? ?? decay in the ?23 vs. t? plane for z2 =
1 and f = 2000 GeV. We conclude that even for a large
splitting ?23, the branching ratio BR(? ? ?? ) would hardly
reach a level above 10?8 for sin 2? of the order of unity,
thereby yielding a very weak constraint on this parameter. On
the other hand, the muon AMDM also does not yields a useful
bound on ? as the heavy neutrino contribution is negative and
cannot account for the muon AMDM discrepancy of Eq. (8).
As we will see below, the only relevant contribution to
the AMDM and WAMDM involving the mixing matrix
elements is the contribution of the heavy neutrino, which has
the generic form
V ?i3V i3 F (xk )
(F (x? ) + sin2 ? (F (x?) ? F (x? ))).
V =W,X i
V =W,X
V =W,X
It turns out that the second term is subdominant for a small
mixing angle ? and a small splitting ?23. Following the
authors of Ref. [27] in their study of LFV hadronic ? decays,
Fig. 4 Contours of the SLHM contribution to the ? ? ?? branching
ratio in the ?23 vs. t? plane. We set z2 = 1 and f = 2000 GeV
we will consider values of sin 2? of the order of 10?1. Under
this assumption, the term proportional to sin2 ? becomes
subdominant, which is equivalent to consider an approximately
diagonal mixing matrix. In addition, we have found that there
is little sensitivity of our results to a small change in the value
of sin 2? . A larger value of this parameter would increase
slightly the ? AMDM and AWMDM, but an enhancement
larger than one order of magnitude would hardly be attained.
Heavy neutrino mass: we will follow the approach of Ref.
[26], in which m Nk is parametrized through the ratio zk =
(m Nk /m X )2. As observed in Fig. 5, for the most stringent
limit f = 5.6 TeV, m Nk ? 0.836 TeV, which corresponds to
the value zk = 0.1, whereas m Nk ? 2.644 TeV for zk = 1.
Again, the ? AMDM and AWMDM contributions arising
from the heavy neutrinos show little sensitivity to moderate
changes in the value of z3 and the mass splitting ?23, so we
will use as reference values z3 1 and ?23 ? 0.1.
In conclusion, we will use the set of values shown in
Table 2 for the SLHM parameters involved in our
numerical analysis. We have found that except for f and t? there
is little sensitivity of the ? AMDM and AWMDM to a mild
change in the values of the remaining parameters as far as
they lie between the allowed intervals.
In order to estimate the ? AMDM we used the
Mathematica numerical routines to evaluate the parametric
integrals involved in our calculation. A crosscheck was done
by evaluating the results expressed in terms of Passarino?
Veltman scalar functions via the numerical FF/LoopTools
routines [38,39]. As already mentioned, it is convenient to
Fig. 5 Heavy neutrino mass as a function of f for t? = 9 and three
values of zk = (m Nk /m X )2. The vertical lines represent the lower
bounds on f arising from several observables (see Table 1)
Table 2 Values used in our
numerical analysis for the
parameters involved in the
AMDM and AWMDM of the ?
lepton
analyze the behavior of the AMDM and AWMDM as
functions of the symmetry breaking scale f since the mass of the
new particles and the corrections to the SM couplings and
particle masses depend on it. Also, since the mixing angle ??
depends on t? , it is worth examining the dependence on this
parameter in the allowed interval.
f
t?
z3
?23
m?
??
sin 2?
In the left plot of Fig. 6 we show the absolute values of the
main partial contributions to a?N P along with the total sum
as a function of f for t? = 9, whereas in the right plot we
set f = 4 TeV and show the dependence of a?N P on t? . For
the remaining parameters we use the values shown in Table
2. We have refrained from showing the curves for the most
suppressed contributions.
We first discuss the behavior observed in Fig. 6a. Notice
that the magnitude of each contribution depends highly on
the respective f?A1 A2 A3 coefficient and to a lesser extent on
the magnitude of the loop integral, which in turn dictates its
behavior. Therefore, the ?? X X , N? W W , and ?? W W
contributions, which are not shown in the plot, are the most
suppressed ones, with values below the 10?10 level. This stems
from the fact that the f?A1 A2 A3 coefficients associated with
these contributions include two powers of the coupling
constants gLV n , which are of the order of ?? ? v/ f , thereby being
considerably suppressed for large f . Although the H ? ? and
Z ? ? contributions are less suppressed, they are below the
10?9 level, whereas the ?? ? and N? X X contributions are
the largest ones and can reach values up to the order of 10?8
for f around 2 TeV, which is a result of the fact that the
respective f?A1 A2 A3 coefficients have no (v/ f )2 suppression
factor. Another point worth to mention is that all the partial
contributions are negative except for the H ? ? and N? W W
ones. Since these contributions are relatively small, they will
not interfere with the dominant contributions, which will add
up constructively. In conclusion both the ?? ? and the N? X X
contributions will represent the bulk of the total contribution
to a?N P , which is of the order of 10?8 for f = 2 TeV, but has
a decrease of about one order of magnitude as f increases
up to 6 TeV, as observed in the plot.
Fig. 6 Absolute values of the main partial contributions from the
SLHM to a?N P as functions of f for t? = 9 (left plot) and as
functions of t? for f = 4 TeV (right plot). For the remaining parameters of
the model we use the values shown in Table 2. The partial contributions
below the 10?10 level are not shown. The threeletter tags denote the
virtual particles circulating in each type of Feynman diagram. All the
contributions are negative except the H ? ? one. The absolute value of
the sum all of the contributions is also shown (solid lines with squares)
Fig. 7 Contours of the SLHM contribution to a?N P  in the f vs. t?
plane. For the remaining parameters of the model we use the values
shown in Table 2
We now turn to a discussion of the dependence of a?N P on
the t? parameter as depicted in Fig. 6b. We observe that the
N? X X contribution, which has a very slight dependence on
t? indeed, is the dominant one, with marginal contributions
arising from other diagrams. In the allowed t? interval, the
H ? ? contribution is negligible and is not shown in the plot.
For low t? , the ?? W W and N? W W contributions can be as
large as the N? X X one, but the ?? ? contribution is the one
that becomes important when t? increases. Since these
contributions are directly proportional to the square of the
mixing parameter ?? = v/(?2t? f ), they get suppressed by two
orders of magnitude as t? goes from 1 to 9. We observe that
the total contribution of the SLHM to a?N P remains almost
constant in this t? interval as it is dominated by the N? X X
contribution.
The behavior of the SLHM contribution to a?N P is best
illustrated in Fig. 7, where we plot the contours of a?N P  in
the f vs. t? plane for the parameter values of Table 2. We
observe that the SLHM contribution to a?N P is of the order of
10?8 for f between 2 and 5 TeV, but it is below 10?9 for f
above 5 TeV and decreases rapidly as f increases. So, if we
consider the most stringent constraint on f , namely 5.6 TeV,
we can expect values of a?N P of the order of 10?9. We also
observe that there is little dependence of a?N P on the value
of t? , but such dependence is more pronounced for large f .
It is interesting to make a comparison with the typical
predictions of some popular extension models as reported in the
literature. In this respect, several extension models predict
values for a?N P lying in the interval between 10?9 and 10?6
[40?43]. We note that although the SLHM contribution is of
the same order than the potential contribution of leptoquark
models (LQM) [40], it is disfavored with respect to the
contributions of THDMs [29,44], the minimal supersymmetric
standard model (MSSM) [45,46], and unparticles (UP) [41],
which can reach values as high as 10?6.
4.3 Anomalous weak magnetic dipole moment of the ?
lepton
We now present the analysis of the NP contribution of the
SLHM to the AWMDM of the ? lepton. There are some
differences with respect to the behavior of the AMDM: apart
that the AWMDM receives extra contributions, it can develop
an imaginary part, which arises from the diagrams where the
external Z gauge boson is attached to a couple of particles
whose total mass is lower than m Z . This occurs when there
are two internal SM neutrinos or ? leptons in the loop. Again,
the magnitude of each partial contribution to the AWMDM is
highly dependent on the corresponding f A1 A2 A3 coefficient,
Z
while its behavior is dictated by the loop integral. We show
in Fig. 8 the real part of the dominant partial contributions of
the SLHM to a?W ?N P as well as the total sum as functions
of f for t? = 9 (left plot) and as functions of t? for f = 4
TeV (right plot). For the remaining parameters we use the
same set of values used in our analysis of the AMDM. All
the contributions not shown in the plots are negligible.
We first discuss the behavior of Re[a?W ?N P ] as a function
of f as shown Fig. 8a. Numerical evaluation shows that for
f around 2 TeV the magnitude of the partial contributions
ranges from 10?14 to 10?9, with an additional suppression of
at least one order of magnitude for f around 5 TeV. The most
suppressed contributions are the ?? X X , X ?? ?? and W N? N?
ones, which are proportional to the (v/ f )2 factor and are
below the 10?13 level. Other contributions such as ? H Z ,
?? W W , W N? ?? , N? W W , ?? ? , H ? ? , and X X N? are less
suppressed but are also below the 10?10 level. In fact, only the
contributions shown in the plot, ? H Z , N? X X , and W ?? ?? ,
are relevant for the total sum. While the ? H Z contribution
is the dominant one, the N? X X contribution plays a
subdominant role, which again is due to the fact that the f N? X X
Z
coefficient is not suppressed by the (v/ f )2 factor. We also
observe that the W ?? ?? contribution, which together with
the ? H Z contribution are absent in the AMDM, is below the
10?9 level. Very interestingly, the H ? ? and ?? ?
contributions, which were not very suppressed in the AMDM case,
are now negligible as they are proportional to the small gVZ
coupling. Note also that all the contributions shown in the
plot are positive except for the ? H Z one. As a result the total
contribution will have an additional suppression as the main
contributions will have a large cancelation due to their
opposite signs. This becomes evident in the curve for the total
Fig. 8 Absolute value of the real part of the main partial contributions
to a?W ?N P and the total sum as a function of f for t? = 9 (left plot)
and as a function of t? for f = 4 TeV (right plot). For the remaining
parameters of the model we use the values of Table 2. The contributions
below the 10?10 level are not shown. All the contributions are positive
except the ? H Z one. The absolute value of the sum all of the
contributions is also shown (solid lines with squares). In the bottomright corner
of the right plot we zoom into the region where a?W ?N P changes from
positive to negative due to the cancellation between the distinct partial
contributions
contribution, which appears below the curve for the ? H Z
contribution.
As far as the behavior of the real part of a?W ?N P as a
function of t? is concerned, we observe in Fig. 8b that the
? H Z and W ?? ?? contributions are highly dependent on t? :
in the interval where this parameter increases from 1 to 10,
the ? H Z contribution is negative and increases by one order
of magnitude, whereas the W ?? ?? contribution is positive
and decreases by one order of magnitude. On the other hand,
the N? X X contribution is positive and remains almost
constant throughout this t? interval. It is interesting that due
to the opposite signs of the partial contributions, there is a
flip of sign of the total contribution around t? = 6.8. For
low t? , the total contribution is positive as it arises mainly
from the N? X X and W ?? ?? contributions, with the ? H Z
contribution being subdominant. As t? increases, the
magnitude of the ? H Z contribution increases, whereas that of
the W ?? ?? contribution decreases. Therefore, the total sum
cancels out around t? = 6.8 and becomes negative above this
value since the ? H Z contribution becomes the dominant one.
This effect is evident in the large dip of the total
contribution curve, which is due to the flip of sign of the AWMDM.
This behavior of the partial and total contributions around
t? = 6.8 is best illustrated in the zoomed region displayed
at the bottomright corner of Fig. 8b, where we show the ?
AWMDM contributions without taking their absolute values.
The behavior of the real part of a?W ?N P as a function of f
and t? is best illustrated in Fig. 9, where we show the contours
of the real part of the total SLHM contribution to a?W ?N P in
the f vs t? plane. It can be observed that the real part of
a?W ?N P reaches its largest values, of the order of 10?9, for
low and high t? , irrespective of the value of f . There is a band
centered around t? ? 7 where the lowest values of real part
?
t
Fig. 9 Contours of the SLHM contribution to the absolute value of the
real part of a?W ?N P in the f vs. t? plane. For the remaining parameters
of the model we use the values shown in Table 2
of a?W ?N P are reached. Such a band (darkest region) widens
as f increases. We observe that, for t? around 10, there is a
slow decrease of Re[aWW ?N P ] as f increases but in general
its magnitude is of the order of 10?10 ? 10?9. A comparison
of the SLHM contribution with typical predictions of some
SM extensions allow us to conclude that, for f up to 4 TeV,
the SLHM contribution can be above the 10?9 level and can
be larger than the contributions predicted by THDMs, the
MSSM and UP, which are of the order of 10?10. For f 4
TeV, the SLHM contribution decreases and it is expected
to be smaller than the values predicted by other extension
models.
We now turn to an examination of the behavior of the
imaginary part of the partial contributions of the SLHM to
a?W ?N P . We will follow the same approach as that used for
the analysis of the real part. There are only four contributions
to a?W ?N P that can develop an imaginary part. In Fig. 10a
we show the behavior of such contributions as a function of
f for t? = 9. Contrary to what happens with the real part,
the imaginary part of the W ?? ?? contribution is the dominant
one by far, at the level of 10?10, with the imaginary parts of
the ?? ? , H ? ? , and Z ? ? contributions suppressed by more
than one order of magnitude. Therefore, the imaginary part
of a?W ?N P will be completely dominated by the W ?? ??
contribution. In fact the curves for the W ?? ?? contribution and
the total contribution overlap. In this region of the parameter
space of the SLHM, the imaginary part of a?W ?N P is
positive, with a magnitude of the order of 10?10, which slightly
decreases as f increases. As far as the behavior of the
imaginary part of a?W ?N P as a function of t? is concerned (Fig.
10b), there is no considerable change in the analysis as in the
allowed t? interval the imaginary part of the W ?? ??
contribution is dominant, whereas the remaining contributions are
negligibly small. For low t? , the imaginary part of the total
contribution to a?W ?N P can be of the order of 10?9, but it
decreases by almost one order of magnitude as t? goes up to
10.
Finally we present the contours of the imaginary part of
a?W ?N P in the f vs. t? plane in Fig. 11. We observe that
the imaginary part of a?W ?N P can be of the order of 10?9
for low values of t? , irrespective of the value of f . This
is also true for t? ? 10 and f ? 3 TeV, but there is a
pronounced decrease of about one order of magnitude for
larger values of f . We also can observe that Im[a?W ?N P ]
decreases mildly as f increases. As far as the values predicted
by other extension models, although the imaginary part of the
total contribution of the SLHM is larger than that predicted
by typeI and typeII LQMs (of the order of 10?10) it is well
below the contributions predicted by THDMs and the MSSM
(of the order of 10?7).
5 Conclusions
In this work we have calculated analytical expressions,
both in terms of parametric integrals and Passarino?Veltman
scalar functions, for the oneloop contributions to the static
anomalous magnetic and weak magnetic dipole moments of
a charged lepton in the context of the SLHM. We have
considered the scenario in which there is no C P violation in
the model and thereby there are no electric nor weak
electric dipole moments. The expressions presented for the weak
properties are very general and can be useful to compute the
weak properties of a charged lepton in other extension
models. For the numerical analysis we have focused on the case of
the ? lepton since their electromagnetic and weak properties
are the least studied in the literature and also because they
have great potential to be experimentally tested in the future.
For values of the parameters of the model allowed by current
experimental data we find that the respective contribution to
the ? AMDM is of the order of 10?9, whereas the real
(imaginary) part of the ? AWMDM is of the order of 10?9 (10?10).
The SLHM contribution to the ? AMDM could have some
enhancement in the scenario in which there is a very light
pseudoscalar boson ?, with a mass of the order of about 10
GeV, and t? is of the order of 20. However, such a value of t?
are already excluded according to the bounds obtained from
experimental data. Proposed future experiments are expected
to reach a sensitivity to the ? AMDM and the real part of
the AWMDM of the order of 10?6 [15?17] and 10?4 [21].
Therefore, the values predicted for these observables by the
SLHM would be out of the reach of the experimental
detection.
One further remark is in order here. Little Higgs
models are effective theories valid up to the cutoff scale
= 4? f . Below this scale a good prediction is
provided by the effective theory, but at higher energies the
physics would become strongly coupled and the effective
theory must be replaced by its ultraviolet (UV)
completion, which would be a QCDlike gauge theory with a
confinement scale around 10 TeV (see for instance [47]). This
gives rise to the possibility that the EWSB is driven by
strong dynamics such as occurs in technicolor theories. It
is thus possible that the ? AMDM and AWMDM can receive
some enhancement from the UV completion, but an
analysis along these lines is beyond the scope of the present
work.
Acknowledgements We acknowledge financial support from Sistema
Nacional de Investigadores (Mexico), Consejo Nacional de Ciencia y
Tecnolog?a (Mexico) and Vicerrector?a de Investigaci?n y Estudios de
Posgrado (BUAP).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Feynman rules in the SLHM
We now present the Feynman rules necessary for our
calculation (see [26] for a complete set of SLHM Feynman
rules). We note that the coupling constant associated with the
A1 A2 A3 vertex will be written as i eg A1 A2 A3 , so the f A1 A2 A3
V
(V = Z , ? ) coefficients of Eqs. (56) and (59) will be given in
terms of the g A1 A2 A3 constants. In Table 3 we show the
Feynman rules for the trilinear gauge boson couplings Vi V j+V j?,
whereas in Table 4 we show the ones for the vertices of a
gauge boson to a fermion pair V f?i f j . Finally, Table 5
gathers all the Feynman rules for the scalar interactions.
Table 3 Feynman rules for the trilinear gauge boson couplings
V?( p1)V?+( p2)V??( p3). The Feynman rule for all these couplings have
the generic form iegVi Vj Vj (g?? ( p2 ? p1)? +g?? ( p3 ? p2)? +g?? ( p1 ?
p3)? ), where the gVi Vj Vj constants are shown in the right column. All
the fourmomenta are taken incoming
Table 4 Feynman rules for the vertices of a gauge boson to a lepton
pair V? f?i f j . The Feynman rule for this class of vertices has the generic
form ie(gLV f?i f j PL + gRV f?i f j PR)??, with PL and PR the left and
rightchirality projectors, where the gLV f?i f j and gVR f?i f j constants are shown
in the second and third columns. The vector and axial couplings are
2gVV f?i f j = gLV f?i f j + gRV f?i f j and 2gVA f?i f j = gLV f?i f j ? gRV f?i f j
Vi Vj Vj vertex
Z X + X ?
Z W +W ?
Table 5 Feynman rules for the
vertices involving the scalar
bosons H and ? at the leading
order in v/ f in the SLHM
A1 A2 A3 vertex
H Z? Z?
Appendix B: Loop integrals
Appendix B.1.1: Anomalous weak magnetic dipole moment
The AWMDM and AWMDM of a lepton are given by Eqs.
(56) and (59). The f A1 A2 A3 coefficients are shown in Table
V
6 and the loop functions IVA1 A2 A?3 (V = Z , ? ) will be
presented below. The loop integration was performed via
both Feynman parametrization and the Passarino?Veltman
method [48]. For the Dirac algebra and the Passarino?
Veltman reduction we used the Feyncalc routines [49], and
a further simplification was done with the help of the
Mathematica symbolic algebra routines. We will first present the
results in terms of parametric integrals.
Appendix B.1: Parametric integrals
After introducing Feynman parameters and integrating over
the fourmomentum space, the IVA1 A2 A3 loop integrals can be
cast in the following form after one Feynman parameter is
integrated out:
F A1 A2 A3 (x )dx .
We will first present the F A1 A2 A3 (x ) functions necessary to
Z
calculate the AWMDM of a lepton.
Table 6 Coefficients fZA1 A2 A3 and f?A1 A2 A3 required in Eqs. (56) and
(59), respectively, to compute the AWMDM and AMDM of a charged
lepton arising from loops carrying the A1 A2 A3 particles. Here n is the
SM neutrino ? or a heavy one N . The coupling constants g A1 A2 A3 are
shown in ?Appendix A?
H V (V = Z , Z )
We start by introducing the functions
f1A1 A2 A3 (x ) =
f2A1 A2 A3 (x ) = log[Y A1 A2 A3 (x )].
For diagram (1) we obtain
F nV V (x ) = x x ((3x (2 ? x ) ? 2)xZ + 6x 2 ? 8x + 2)
Z
?4) f2nV V (x ) + (2(x 3(xZ ? 2)(xn + x ? 1)
+x 2(2xn(3 ? 2xZ ) ? 2(xZ ? 1)x + 3xZ ? 8)
+x (xn (5xZ ? 4) + xZ x ? 2) ? 2xn xZ )
+Z nV V (x )(4x 2(xZ ? 2) + x (12 ? 9xZ )
+4(xZ ? 1))) ? f1nV V (x ) ,
X nV V (x ) = x xZ ,
Y nV V (x ) = (1 ? x ) (xn ? x x ) + x ,
Z nV V (x ) = xZ (4Y nV V (x ) ? x 2xZ ),
where we have introduced the dimensionless variable xa =
(ma /mV )2.
As for the contribution of diagram (2), the FZZ function
can be written as
= (gLZ
? x (3x ? 2)gLZ
X Z Y Z Z Z
F Z Z?L (x )(gLZ Z?R (x ) = F Z
+x (3x ?) f2Z
+ gRZ )[2(1 ? 2x )x
(x ) + 2(2x 2(2y + 1) ? 14x
(x )(3x ? 2) + 12) f1Z
with ya = (ma /m Z )2 and
We now present the contribution of diagram (3):
F V nn(x ) = x [2x (2x ? 1) + (2 ? 3x )x f2V nn(x )
Z
+(4x 2(xn ? 2x ? 3) + 4x (2(x ? xn) + 9)
+2Z V nn(x ) (2 ? 3x ) ? 24) f1V nn(x )], (B.15)
X V nn(x ) = x xZ ,
Y V nn(x ) = x (xn ? x ? 1) + x 2x + 1,
Z V nn(x ) = xZ (Y V nn(x ) ? x 2xZ ).
Finally, the loop function arising from diagram (4) is given
by
F V ? N (x ) = xZ [4x (2x ? 1)xZ + 2x xN ( f2V1? N (x )
? f2V2? N (x )) + x (2 ? 3x )xZ ( f2V1? N (x )
+ f2V2? N (x )) + 2?xZ ((x ? 2)x xN xZ
?(x ? 2)x N2 ? (3x ? 2)Z V nn(x )
?2(x ? 1)xZ (2x x + 3x ? 6))
where the functions f1Aa1 A2 A3 (x ) and f2Aa1 A2 A3 (x ) are
obtained from f1A1 A2 A3 (x ) and f2A1 A2 A3 (x ) after the
replacements X A1 A2 A3 (x ) ? XaA1 A2 A3 (x ) and Y A1 A2 A3 (x ) ?
YaA1 A2 A3 (x ), respectively. In addition
X V ? N (x ) = x xZ ? xn,
X V ? N (x ) = x xZ + xn,
We now turn to the parametric integrals for the diagrams
of Fig. 2. For diagram (1) we obtain
FZH (x ) = ?8x (2 ? x )w f1H (x ),
with wa = (ma /m H )2 and
X H (x ) = x wZ ,
FZ? (x ) = 8x 2z f1? (x ),
with za = (ma /m?)2, X ? (x ) = X H (x ) [wa ? za ], and
Z ? (x ) = Z H (x ) [wa ? za ].
As for the contribution of diagram (2) and the one obtained
by exchanging the internal gauge boson with the scalar boson,
it is as follows:
?x
FZH V (x ) = ?xZ [4x (1 ? 2x )xZ + ((1 ? 2x )x H + 2x ? 3)
( f21H V (x ) ? f22H V (x ))
+(3x ? 2)x xZ ( f21H V (x ) + f22H V (x ))
?2?xZ (x 2xZ (5x H + 2xZ ? 28x + 5)
?x (x H (3xZ ? 4) + 2x H2 ? 20xZ x + 5xZ + 2)
+x H (x H ? 4) + 3x 3xZ (4x ? xZ )
?4xZ x + 3) ? ( f11H V (x ) ? f12H V (x ))],
(B.29)
+x xZ (2 ? 8x ) + 4xZ x ? 1.
Appendix B.1.2: Anomalous magnetic dipole moment
For completeness we present the contributions to the AMDM,
which can be obtained from the AWMDM results after taking
the limit m Z ? 0 and substituting the Z coupling constants
by the photon ones. The f?A1 A2 A3 coefficients of Eq. (59) are
presented in Table 6, whereas the respective loop integrals
are of the form of (B.1).
As far as Fig. 1 is concerned, there are only contributions
from diagrams (1) and (2), but with the external Z gauge
boson replaced by the photon. The contribution of diagram
(1) can be written as
x x
F?nV V (x ) = ? Y nnV (x ) (x 2 (xn + x + 2)
?x (3xn + x ? 2) + 2xn).
As for the contribution of diagram (2), the F?Z function
is given by an analogous expression to Eq. (B.8) but now
F Z ? ?L (x ), with ? ?R (x ) = F Z
F Z 4x y
(x 2 (y + 1) ? 3x + 2),
(x 2 y ? 2(x ? 1)).
?45))) + 18xn3 log(xn)),
I?Z?L R = ?3I?Z?L = 4y .
Finally, there are also contributions of the scalar bosons H
and ? arising from the diagram (1) of Fig. 2, with the Z gauge
boson replaced by the photon. The respective F?A1 A2 A3 (x )
functions can be written as
All of the above results agree with previous calculations
of the AMDM of a lepton (see for instance [7,30]).
Appendix B.2: Passarino?Veltman scalar functions
We now present the results for the AWMDM and AMDM of
a lepton in terms of Passarino?Veltman scalar functions.
Appendix B.2.1: Anomalous weak magnetic dipole moment
We first introduce the following ultraviolet finite functions
given in terms of twopoint Passarino?Veltman scalar
integrals,
1 = B0(0, m2N , m2V ) ? B0(m2, m2N , m2V ),
2 = B0(m2Z , m2V , m2V ) ? B0(m2, m2N , m2V ),
3 = B0(0, m2, m2Z ) ? B0(m2, m2, m2Z ),
4 = B0(m2Z , m2, m2) ? B0(m2, m2, m2Z ),
5 = B0(m2Z , m2N , m2N ) ? B0(m2, m2N , m2V ),
7 = B0(m2, 0, m2V ) ? B0(m2Z , 0, m2N ),
8 = B0(m2Z , 0, m2N ) ? B0(0, 0, m2V ),
and we use a shorthand notation for the following threepoint
scalar functions:
C1 = m2V C0(m2, m2, m2Z , m2V , m2N , m2V ),
C2 = m2Z C0(m2, m2, m2Z , m2, m2Z , m2),
C3 = m2V C0(m2, m2, m2Z , m2N , m2V , m2N ),
C4 = m2V C 0(m2, m2, m2Z , m2N , m2V , 0).
The IZA1 A2 A3 loop functions arising from the diagrams of
Fig. 1 are given as follows. For diagram (1) we obtain
x
2 [ 4x ? xZ
xZ ? 2 ? 4
?4
?4x
xn ? 1 4x ? xZ
xZ ? 2
1 ?
x xZ xZ + 8 ? 60 + 2 xn 3xn
?4 xZ xZ ? 3 ? 6
2 + 2 x 2 24 ? 5xn xZ
?14xZ + 24 + 3x Z2 ? xZ ? 34
+x 3 xZ ? 2 ? 2x Z2 + 4xZ + 12 C1],
whereas the loop function arising from diagram (2) is given
by a similar expression to Eq. (B.8):
IZZ
= (gLZ
IZZ?R = IZZ?L (gLZ
IZZ?L R =
IZZ?L = ?
? gLZ
y ? 1 4y ? yZ gLZ
y 4y ? yZ + 34
2 ? 7y + 2yZ
[y 4y ? yZ + y ? 1 4y
?yZ
3 ?
y 4y ? yZ + 10
z Z ? 4z
+ z Z ? 10z
2 [z z Z ? 4z
? z ? 1 4z ?z Z
where 11, 12, and C6 are obtained from 9, 10 and C5,
respectively, after the replacement m H ? m?.
As for diagram (2) of Fig. 2, it yields
2?x
+ x ? 1 x
14 ? 4x
15 ? 2xZ 16
+2 2x x H + xZ ? 1 ? x H xZ C7],
As for diagram (3), the respective loop integral is
IZV nn = xZ ?x 4x 2 [ x + 2 xZ ? 4x
xn ? 1 x + 2 xZ ? 4x
+ x 2xn + xZ ? 22 ? 2 xn xZ + 6
?5xZ ? 6 + 2x2
?x 2xn xZ ? 5 + xn + 8xZ + 17 + xn2 xZ
2
Finally, the loop function arising from diagram (4) obeys
xZ ? 4x
[2x xZ x + 2 xZ ? 4x
+xZ xn ? 1 x + 2 xZ ? 4x
? 2x 3 6xn + xZ + x 2 xZ ? 18 xZ
?2xn 3xZ + 4
+ x xZ 20 ? 4xn + 9xZ
?2x Z2
2x 3 xZ ? 6xn + x 2 8xn xZ + 1
+ xZ ? 26 xZ + xZ x 4 ? 2xn xZ + 4
7 ? x 2x 2 xZ ? 6xn
+4 ? 5xZ ? 6
8 + 2x x 2 2xn xZ ? 3xn
?2xZ xn xZ ? 5 + 8xZ + 17
?2x 3xZ C4].
We now present the loop functions for the diagrams of Fig.
2. We will use the following additional Passarino?Veltman
scalar functions:
9 = B0(0, m2H , m2) ? B0(m2Z , m2, m2),
10 = B0(m2, m2H , m2) ? B0(m2Z , m2, m2),
C5 = m2H C0(m2, m2, m2Z , m2, m2H , m2).
For diagram (1) we obtain for the contribution of the SM
Higgs boson H
? (w ? 1) (4w ? wZ ) 9
+(16w 2 ? 2w (2wZ + 5) + wZ ) 10
?6w (4w ? wZ ? 1) C5],
13 = B0(0, m2H , m2) ? B0(0, m2H , m2V ),
14 = B0(0, m2H , m2V ) ? B0(0, m2, m2V ),
15 = B0(m2, m2H , m2) ? B0(m2, m2, m2V ),
16 = B0(m2, m2, m2V ) ? B0(m2Z , m2H , m2V ),
C7 = m2V C0(m2, m2, m2Z , m2H , m2, m2V ).
Appendix B.3: Anomalous magnetic dipole moment
For the loop integrals of the contributions to the AMDM of
the diagrams analogue to those of Fig. 1, but with the external
Z boson replaced by the photon, we have for diagram (1)
I?nV V = ? 81x [ xn x 5x + 12 ? 7xn2x ? 9xn + 3xn3
?x
x ? 12 x + 17 + 6 C1
15 ? 4xn x
where the primed scalar functions i and Ci are obtained
from the unprimed ones by setting m Z = 0. We note that
all the threepoint functions Ci appearing in the AMDM are
of the generic type C0(m2A, m2A, 0, m2B , m2 , m2B ), which can
C
be written in terms of twopoint scalar functions as follows
[50]:
?( m2A ? m2B ? mC2
+2 m2A ? m2B + mC2 ),
B ? 2mC2 C
with ?(x , y, z) = (x ? y ? z)2 ? 4yz and
B0(0, m2X , m2X ) ? B0(m2A, m2B , mC2 ).
fL (xk ) =
2(1 + xk ) ? x i xk
As for diagram (2) we obtain a similar expression to Eq.
(B.54), where
where x i = (m i /mV )2 (V = W, X ), whereas the fL
function is given as
I?Z?R = I?Z?L = ? 41y [ 3y ? 1 4y ? 3 C2
+2 1 ? y 2
?2y y + 1 ]
3 + y ? 3 2y ? 1
I?Z?L R = ? 21 [ 5 ? 12y C2 + 2 y ? 1
+ 5 ? 2y
Finally, the diagram (1) of Fig. 2 with the Z replaced by the
photon yields the loop functions
+ 5 ? 8w
? 1 C5]
11 + 5 12 + 4x ? 3 C6].
Again the primed scalar functions i and Ci are obtained
from the unprimed ones by setting m Z = 0.
Appendix C: Decay i ?
In this appendix we present the amplitude for the i ? j ?
decay both in terms of parametric integrals and Passarino?
Veltman scalar functions. The contributions arise from the
Feynman diagrams of Fig. 3. The decay width is given
in Eq. (64). We have obtained the fL (xk ) function [xk =
(m Nk /mV )2] appearing in Eq. (63) via the Feynman
parameters technique using the approximation of massless final
lepton m j = 0. We first define the function
1 + x (x ? 1)x i + z ? 1
1 + x(z ? 1)
x i (x ? 1)x2 (2x ? 1)x i + z ? 1
x (x ? 1)x i + z ? 1 + 1
?x i z ? 2
+(x ? 1)
?2 x + 1 z
(1 ? x) (z + 2)x2 + (z ? 6)x + 4
x(z ? 1) + 1
For the sake of completeness we also include the amplitude
in terms of Passarino?Veltman scalar functions. The result
reads
2xk (1 + xk )xk ? 2 ? x i xk ? 2
x i xk ? 1
? B i Nk V + 2 2(1 ? x i ) + xk C i Nk V ,
where we have defined
B i Nk V =
1 ? xk B0(m2i , m2Nk , m2Nk )
+xk B0(0, m2Nk , m2Nk ) ? B0(0, m2V , m2V ),
C i Nk V = m2V C0(m2i , 0, 0, m2Nk , m2V , m2V ).
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