Anomalous moments of the top quark arises from one loop corrections to the
vertices $\bar t t g$ and $\bar t t \gamma$. We study these anomalous couplings
in different frameworks: effective theories, Standard Model and 2HDM. We use
available experimental results in order to get bounds on these anomalous
couplings.Comment: 8 pages, 2 figures, talk presented by R. Martinez at the X Mexican
  School of Particles and Fields, Playa del Carmen, Mexico, 200

Martinez, R.

Poveda, N.

Rodriguez, J. -Alexis

English

arXiv

R. Martinez

Crossref

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Effective theories and constraints on new physics
R. Martinez, N. Poveda and J-Alexis Rodriguez
Departamento de Física, Universidad Nacional de Colombia
Bogota, Colombia
Abstract.
Anomalous moments of the top quark arises from one loop corretions to the vertices̄ttg andt̄tγ.
We study these anomalous couplings in different frameworks: effective theories, Standard Model
and 2HDM. We use available experimental results in order to get bounds on these anomalous
couplings.
INTRODUCTION
The top quark is the heaviest fermion in the standard model (SM) with a mass of
174.3±5.1 GeV1. In the framework of the SM, the couplings of the top quark arefix d
by the gauge symmetry. Anomalous couplings between the top quark and gauge bosons
might affect the top quark production at high energies and also its decay rate. Precisely
measured quantities with virtual top quark contributions will yield further information
regarding these couplings.
Since the top quark mass is so heavy, it is expected that its phy ics may be different
from the lighter fermions and that the top quark might couplequite strongly to the
electroweak symmetry breaking sector. This suggests the SMis just an effective theory
and that the physics beyond the SM may be manifested through an effective Lagrangian
involving the top quark.
The framework of effective theories, as a mean to parametrize physics beyond the SM
in a model independent way, has been used recently. Two casesh v been consider in the
literature, the decoupling case, which includes the Higgs boson, and the non-decoupling
case, where there is not any Higgs boson. We shall consider only the first case, in which
the SM is a low-energy limit of a renormalizable theory. In this approach, the effective
theory parametrizes the effects at low energy of the full reno malizable theory by means
of high order dimensional non-renormalizable operators2.
The effective Lagrangian approach is a convenient model independent parametriza-
tion of the low-energy effects of the new physics that may show up at high energies.
Effective Lagrangians, employed to study processes at a typical energy scaleE can be
written as a power series in 1/Λ, where the scaleΛ is associated with the heavy particles
masses of the underlying theory3. The coefficients of the different terms in the effective
Lagrangian arise from integrating out the heavy degrees of freedom. In order to define
an effective Lagrangian it is necessary to specify the symmetry and the particle content
of the low-energy theory. In our case, we require the effectiv Lagrangian to beCP-
conserving, invariant under SM symmetrySU(2)L⊗U(1)Y, and to have as fundamental
fields the same ones appearing in the SM spectrum. Therefore we consider a Lagrangian
in the form
Le f f = LSM+∑
n
αnOn (1)
where the operatorsOn are of dimension greater than four.
THE ANOMALOUS MAGNETIC MOMENT
The aim now is to extract indirect information on the magnetic dipole moment of the top
quark from LEP data, specifically we use the ratiosRb andRl defined by
Rb =
Γ(Z → bb̄)
Γ(Z → hadron) , Rl =
Γ(Z → hadron)
Γ(Z → l l̄) , ΓZ (2)
in the context of an effective Lagrangian approach. The oblique and QCD corrections to
theb quark and hadronicZ decay widths cancel off in the ratioRb. This property makes
Rb very sensitive to direct corrections to theZbb vertex, specially those involving the
heavy top quark , whileΓZ andRl are more sensitive to the oblique corrections.
In the present work, we consider the following dimension sixandCP-conserving
operators,
OabuW = Q̄
a
Lσ
µνWiµντ
i φ̃UbR , O
ab
uB = Q̄
a
Lσ
µνYBµν φ̃UbR , (3)
whereQaL is the quark isodoublet,U
b
R is the up quark isosinglet,a , b are the fam-
ily indices, Bµν andWµν are theU(1)Y andSU(2)L field strengths, respectively, and
φ̃ = iτ2φ∗. We use the notation introduced by Buchmüller and Wyler4. After spon-
taneous symmetry breaking, these fermionic operators generate also effective vertices
proportional to the anomalous magnetic moments of quarks. The above operators for
the third family give rise to the anomaloustt̄γ vertex and the unknown coefficientsεabuB
andεabuW are related with the anomalous magnetic moment of the top quark through
δκt =−
√
2
mt
mW
g
eQt
(sWε33uW+cWε
33
uB) , (4)
wheresW denotes the sine of the weak mixing angle.
The expression forRb is given by
Rb = R
SM
b (1+(1−RSMb )δb) , (5)
whereRSMb is the value predicted by the SM andδb is the factor which contains the new
physics contribution, and it is defined as follows
δb =
2
(
gSMV g
NP
V +g
SM
A g
NP
A
)
+
(
gNPV
)2
+
(
gNPA
)2
(gSMV )
2+(gSMA )
2
(6)
and gSMV and g
SM
A are the vector and axial vector couplings of theZbb̄ vertex. The
contributions from new physics, eq. (3), toRl and ΓZ are of two classes. One from
vertex correction toZbb̄ in theΓhadr and the other from the oblique correction through
∆κ in the sin2θW. These can be written as
Rl = R
SM
l (1−0.1851∆κ +0.2157δb) ,
Rb = R
SM
b (1−0.03∆κ +0.7843δb) ,
ΓZ = ΓSMZ (1−0.2351∆κ +0.1506δb) (7)
where∆ρ is equal to zero for the operators that we are considering.
We will consider the one loop contribution of the above effective operators to theZbb
vertex. After evaluating the Feynman diagrams, with insertions of the effective operators
OabuB andO
ab
uW we obtain
gNPV = 4
√
2ε33uWGFm
3
Wmt
{
3cW(C̃12−C̃11)−
m2t√
2m2W
(C12−C11+C0)
+
(1+a)
8cW
(C11+C12+C0)+
1√
2
(C12−C11−C0)
− 3a
4cWm2Z
(B1−B0)
}
,
gNPA = 4
√
2ε33uWGf m
3
Wmt
{
− a
2cW
(C0+C12−C11)
− 1√
2
(C12−C0−C11)+
m2t√
2m2W
(C12−C11+C0)
−2mts
2
W
mW
(C̃0+C̃12−C̃11)−
3
4cWmZ2
(B1−B0)
}
(8)
for the operatorOuW and,
gNPV = g
NP
A =
4
√
2
3
ε33uBGFm
3
Wmt
sW
cW
[
m2t√
2m2W
(C12−C11+C0)
− 1√
2
(−C11+C12−C0)] , (9)
for the OuB. In the above equationsa = 1− 83s2W while Ci j = Ci j (mW,mt ,mt), C̃i j =
C̃i j (mt ,mW,mW) andBi = Bi(0,mt,mW) are the Passarino-Veltman scalar integral func-
tions5. The combinationB0−B1 has a pole ind = 4 dimensions that is identified with
the logarithmic dependence and can be replaced by lnΛ2/m2Z.
Now we have various posibilities to explore the space of the parametersε33uW, ε
33
uB and
δκt . Using experimental values forΓZ, Rb and Rl we find the allowed region in the
planeε33uW− ε33uB. If we do not neglect the term of the order
(
gNP
)2
, we get the following
expressions
−0.057εB+0.058ε2B+0.053εW +0.0015ε2W+0.01εBεW = 1− (Γ
exp
Z /Γ
SM
Z ),
−0.023εB+0.026ε2B+0.016εW+0.0007ε2W+0.0048εBεW = 1− (R
exp
b /R
SM
b ),
−0.656εB+0.686ε2B+0.536εW +0.018ε2W+0.1264εBεW = 1− (R
exp
l /R
SM
l ),
where we have omitted the superindex 33.
The SM values for the parameters that we have used are
ΓZ = 2.4963GeV, Rl = 20.743,
Rb = 0.21572,Γhadr = 17427MeV, Γl = 84.018MeV,
with the input parameters:
mt = 174.3GeV, αs(mZ) = 0.118,
mZ = 91.1861GeV, mH = 100GeV, Λ = 1TeV.
And the experimental values are
ΓZ = 2.4952±0.0023GeV, Rl = 20.804±0.050,
Rb = 0.21653±0.00069.
After doing aχ2 analysis at 95% C.L. we find the allow region forε33uW− ε33uB pa-
rameters. In this kind of scenarios new physics is explored assuming that its effects are
smaller than the SM effects, consequently one expect that
∣
∣
∣
gNPV,A/g
SM
V,A
∣
∣
∣
≪ 1 and, then we
get the inequalities
∣
∣ε33uW
∣
∣≤ 0.11,
∣
∣ε33uB
∣
∣≤ 0.48. By using the eq.(4) and the bounds got
in the numerical analysis, we obtain forδκt the following values
−2.94≤ δκt ≤ 1.3, −0.76≤ δκt ≤ 1.9,
−1.3≤ δκt ≤ 1.7 (10)
which correspond toΓZ, Rb andRl , respectively. Therefore for these observablesΓZ,
Rb andRl , we get the allowed region
−2.94≤ δκt ≤ 1.9. (11)
ANOMALOUS CHROMOMAGNETIC DIPOLE MOMENT I
We are interested in studying possible deviations from the SM on the decayb → sγ
within the context of the effective Lagrangian approach. Several authors have been used
the CLEO results on radiativeB decays to set bounds on the anomalous coupling of the
t-quark. We will use dimension-six operator which are full strong and electroweak gauge
invariant and contribute tob→ sγ in order to bound the chromomagnetic dipole moment
of the top quark.
Since the anomalous chromomagnetic dipole moment of the topquark appears in the
top quark cross section, it is possible, due to uncertainties, o estimate the constraints
that it would impose on the∆κ tg. For the LHC, the anomalous coupling is constrained to
lie in the range−0.09≤ ∆κ tg ≤ 0.16. Similar range is obtained for the future NLC. The
influence of an anomalous∆κ on the cross section and associated gluon jet energy for
tt̄g has been also analyzed. Events produced at 500 GeV in+e− colliders, with a cut
on the gluon energy of 500 GeV and integrated luminosity of 30f b−1, lead to a bound
of −2.1≤ ∆κ tg ≤ 0.66.
>From the experimental information it is possible to get a limit on the∆κ tg from
Tevatron. Following the reference by F. del Aguila7 nd assuming that the only non-
zero coupling is precisely the chromomagnetic dipole moment of the top quark, we find
from the collected data that the allowed region is|∆κ tg| ≤ 0.45.
We consider the following dimension-six, CP-conserving operators, which are
SU(3)C⊗S(2)L⊗U(1)Y gauge invariant
O
ab
uG = Q̄
a
LσµνG
µν i λ i
2
φ̃ubR , (12)
whereGµν i is the gluon field strength tensor anda,b are the family indices. The above
operator gives rise to the anomaloustt̄g vertex and its respective unknown coefficient
εuGab is related with the anomalous chromomagnetic moment of the top quark by
δκ tg =
√
2
g
gs
mt
MW
ε33uG . (13)
where by means of the dimension 5 coupling to an on-shell gluon, the anomalous
chromomagnetic dipole moment of the top quark is defined as
L5 = i(
∆κ tg
2
)
gs
2mt
ū(t)σµνqνTau(t)Gµ,a (14)
with gs andTa are theSU(3)c coupling and generators, respectively.
The effective Hamiltonian used to compute theb→ s transition is given by8
He f f =−
4GF√
2
V∗tsVtb
8
∑
i=1
ci(µ)Oi(µ) , (15)
whereµ is the energy scale at whichHe f f is applied. Fori = 1−6, Oi(µ) correspond
to four-quark operators,O7(µ) is the electromagnetic dipole moment andO8(µ) is
the chromomagnetic dipole operator. At low energy,µ ≈ mb, the only operator that
contributes tob→ s transition isO7(µ) which results from a mixing among theO2(MW),
O7(MW) andO8(MW) operators.
The total contribution of the effective operator (12) to theO8(MW) operator can be
written as:
c8(MW) = c8(MW)
SM+δκ tg∆c8(MW) , (16)
where
∆c8(MW) =
1
4Vts
ln
(
Λ2
M2W
)
+
1
Vts
x−x2+x(2−x) ln(x)
8(1−x)2
+
3x−4x2+x3+2xln(x)
8(1−x)3 (17)
andx= m2t /M
2
W.
Using the recent data from CLEO collaboration for the branchi g fraction of the
processB(b → sγ) = (3.21± 0.43± 0.27)×10−4 9, we get an allowed region for the
anomalous chromomagnetic dipole moment of the top quark to be10
−0.03≤ ∆κ tg ≤ 0.01 (18)
ANOMALOUS CHROMOMAGNETIC DIPOLE MOMENT II
Our next objective is to evaluate the contribution at the onel op-level to the anomalous
chromomagnetic dipole moment of the top quark in different scenarios with the gluon
boson on-shell. Beginning with the SM, the typical QCD correction through gluon
exchange implies two different Feynman diagrams. After theexplicit calculation of the
loops, we find that
∆κ tg =−
1
6
αs(mt)
π
. (19)
We note that its natural size is of the order ofαs/π similar to the QED anomalous
coupling, but now in combination with a factor−1/6 coming from the color structure in
the diagram i.e.TaTbTa =−Tb/6 with Ta being the generators ofSU(3)C.
The other possible contribution in the framework of the SM comes from electroweak
interactions. The relevant contributions occur when neutral Higgs boson and the would-
be Goldstone boson ofZ are involved in the loop. This contribution reads
∆κ tg =−
√
2GFm2t
8π2
[H1(mh)+H2(mZ)], (20)
where
H1(m) =
∫ 1
0
dx
x−x3
x2− (2−m2/m2t )x+1
,
H2(m) =
∫ 1
0
dx
−x+2x2−x3
x2− (2−m2/m2t )x+1
.
The SM contribution is showed in figure 1 where we have added thQCD contribution.
It is worth noting that the behaviour of the curve for a large Higgs boson mass indicates
decoupling and that the values of∆κ tg lie within the allowed region for∆κ tg coming from
b→ sγ.
FIGURE 1. Standar Model contribution to the anomalous cromomagneticdipole moment of the top
quark vs the higgs boson mass
The contributions within a general 2HDM will be different from the SM contributions
because of the presence of the virtual five physical Higgs bosons which appear in any
two Higgs doublet model after spontaneous symmetry breaking: H0, A0, h0, H± 11.
Therefore, 2HDM predictions depend on their massses and on the two mixing angles
α andβ . For smallβ , the charged Higgs boson contribution is suppressed due to its
large mass and the small bottom quark mass.
The expression for the contribution of the neutral Higgs bosons is given by
∆κ tg =
√
2GF
8π2
[λ 2H0ttH1(M
2
H)+λ
2
h0ttH1(M
2
h)
+ λ 2A0ttH2(M
2
A)+λ
2
G0ttH2(M
2
Z)] (21)
whereλitt are the Yukawa couplings in the so-called models of type I, IIand III . Table
I shows the couplings in the usual convention.
TABLE 1. Couplings of the Higgs eigenstates with the top
quark in 2HDM. We omit the factorg/2mW and in the model
type III we use the Sher-Cheng approach for the flavour
changing couplings.
λitt model type I model type II model type III
λH0tt
mt sinα
sinβ
mt cosα
cosβ (1+
ηUtt√
2
)mt sinα
λh0tt
mt cosα
sinβ
mt sinα
cosβ (1+
ηUtt√
2
)mt cosα
λA0tt cotβmt tanβmt
ηUtt mt√
2
λG0tt mt mt mt
The Yukawa couplings of a given fermion to the Higgs scalars are proportional to
the mass of the fermion and they are therefore naturally enhanced in this case. In the
model type III appears flavour changing neutral couplings attree level which can be
parametrized in the Sher-Cheng approach where a natural value for the flavour changing
couplings from different families should be of the order of the geometric average of their
Yukawa couplings,hi j = gηi j
√
mimj/(2mW) with ηi j of the order of one.
In order to show the behaviour of the contribution of the 2HDMto the anomalous
chromomagnetic dipole moment of the top quark, we evaluate explicitly the contribution
FIGURE 2. Contour plot for the contribution of 2HDM to the anomalous cromomagnetic dipole
moment of the top quark in the plane tanβ −mH for MA = 200 GeV(solid line) andMA = 400 GeV
(dashed line) using the bound from theb→ sγ process. The allowed region is above the curve
for couplings type II. We show in figure 2 the allowed region (above the curve) for the
plane tanβ vs mH using equation (21) and assuming that−0.03≤ ∆κ tg ≤ 0.01 from
b → sγ. We fix the following parameters:mH = mh, mA = 200(400) GeV solid line
(dashed line). The solid line for the scalar Higgs mass smaller (bigger) than 240 GeV
corresponds to the cut between equation (21) and the upper(low r) limit from b→ sγ.
ACKNOWLEDGMENTS
We thank COLCIENCIAS for finantial support
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Effective theories and constraints on new physics

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