#### A basis of dimension-eight operators for anomalous neutral triple gauge boson interactions

Celine Degrande
0
0
Department of Physics, University of Illinois at Urbana-Champaign
, 1110 W. Green Street, Urbana,
IL 61801, U.S.A
Four independent dimension-eight operators give rise to anomalous neutral triple gauge boson interactions, one CP-even and three CP-odd. Only the CP-even operator interferes with the Standard Model for the production of a pair of on-shell neutral bosons. However, the effects are found to be tiny due mainly to the mismatch of the Z boson polarization between the productions from the SM and the new operator.
1 Introduction
2 The operators
2.1 Without the Higgs field
2.2 With two Higgs fields
2.3 With four Higgs fields
3 Phenomenology 3.1 ZZ production 3.2 AZ production
4 Conclusion A Translation to off-shell anomalous couplings
1 Introduction
The recent discovery of the Higgs boson has increased the confidence in the validity of
the Standard Model (SM). On the other hand, the remaining issues of the SM like the
absence of a dark matter candidate claim for new physics. This dilemma can only be solved
experimentally by either directly searching for new particles or by looking for deviations
from the SM predictions. In this article, we use the well motivated effective field theory
(EFT) approach to pin down the expected first deviations from heavy new physics on the
neutral triple gauge couplings (nTGC).
Anomalous neutral gauge couplings have been actively searched for at LEP [13],
at the Tevatron [4, 5] and at the LHC [6, 7]. The constraints are given following the
parametrization of the anomalous vertices for the neutral gauge bosons [811]
ieZV (q1, q2, q3) = e(q23 m2V )
MZ2
h1V (q2g q2g) + Mh2VZ2 q3[(q3q2)g q2q3]
where V is a photon or a Z boson and is off-shell while the two other bosons are on-shell.
The parametrization of those vertices has been extended for off-shell bosons in ref. [10].
So far, the size of the fiV and hiV coefficients is unknown. Since the vertices contain three
momenta, they are expected to arise from at least dimension-six operators. Therefore the
contribution from heavy new physics is expected to be suppressed by at least the square of
the ratio of the weak and the new physics scale. They have be computed or estimated for
some extensions of the SM [10, 12]. Alternatively, their size as well as their dependence in
a smaller number of parameters can be obtained for any heavy new physics model using
EFT [13]. As a matter of fact, any extension the SM can be parametrized at low energy
by the effective Lagrangian
where d is the dimension of the operator Oid and is the new physics scale. The new
operators are built out of the SM fields and respect its gauge symmetries. In the limit
, this Lagrangian tends to the SM one. At energies well below , only the finite
set of operators with the lowest dimension are relevant. Therefore the Lagrangian (1.3) is
predictive even if the coefficients Ci are kept as free parameters and can be used to search
for heavy new physics in a model independent way.
Since no dimension-six operator induces nTGC [14, 15], the effects of heavy new physics
on nTGC is expected to be rather small. Whether or not they can be observed or
constrained depends on the presence of dimension-eight operators with nTGC and the size of
their contributions. Two dimension-eight operators invariant under the SM gauge group
have been given in [16]. Those operators contain only the neutral gauge bosons and the
Higgs field but they can be transformed using the equations of motion to operators without
nTGC. In the following, we allow the operators to contain the charged gauge bosons as
well since many dimension-six and dimension-eight operators can change they interactions
and may cancel the effects of the operators with nTGC.
The list of independent operators is derived in section 2 while their effects on
neutral diboson production are discussed in section 3. Finally, a summary is given in the
last section.
The operators
In this section, we go through all the possible dimension-eight operators invariant under the
SM gauge group with neutral triple gauge vertices to find a set of independent operators.
Namely, we are using
I. The Higgs field equation of motion
II. The W field equation of motion
DDH = 2H 2 HH H + fermion densities;
and the B field equation of motion
DWI = ig HI DH DHI H
+ fermion currents
+ fermion currents;
III. Bianchi identities
IV. Jacobi identity
V. The commutator of two covariant derivatives is a linear combination of strength field
tensors;
VI. Integration by part.
Rules I and II allow to transform an operator to operators with more Higgs fields or fermion
fields or dimension-six operators with at least two Higgs fields. No dimension-six with two
Higgs fields have nTGC even without using the equations of motion. As a matter of fact,
only two independent (using rule VI only) operators with four derivatives and two Higgs
fields cannot be written as operators with at least one strength tensor,
Since their parts with three neutral gauge bosons have an imaginary coefficient they cancel
each other in those hermitian operators. If there is a strength tensor, the two derivatives
can be chosen to be applied on the Higgs fields only. If they are acting on two different fields,
they both give a Z and ZZ gives zero when contracted with a strength tensor. If they
are applied on the same field, they can be replaced by strength tensors and there is again
no nTGC. Consequently, we will start with the operators with no Higgs and use those two
rules to remove some operators with a low number of Higgs fields in favor of operators with
more Higgs fields. Similarly, rule V allows to change the order of the covariant derivatives
while generating operators with more strength tensors and less covariant derivatives. In
particular, the operator is replaced by operators with more Higgs fields or more strength
tensors
Using the equation of motion in this way, we only keep the operators with nTGC that are
not equivalent to operators without nTGC (See for example ref. [17] for operators involving
fermions currents). Finally, we assume that the gauge bosons are produced by the collision
of massless fermions and they are produced either on-shell or decay into massless fermions
consistently with the traditional searches for nTGC. Therefore operators with vertices
proportional to the scalar product of the polarization vector and the momentum of the
same boson can be removed as well.
Without the Higgs field
Operators with only one strength tensor or with two strength tensors can be removed
using rules VIII and VI as long as the two Lorentz indices of the strength tensors are not
contracted together. Thoses particular cases can be removed using rule IV.
Only the normal derivatives in the covariant derivatives can contribute to TGC in the
operators with three strength tensors and two covariant derivatives. Furthermore, there
is no neutral TGC if the three strength tensors are SU(2) triplet contracted with IJK
and therefore there is at least one B . This B can always be chosen without covariant
derivatives acting on it by VI. Consequently, operators with two derivatives and three
strength tensors can be written as
where X is an antisymmetric tensor made of two covariant derivatives and two identical
strength tensors. The covariant derivatives can be forbidden to have the same Lorentz
index as the strength field on which they are applied thanks to rule VIII. The cases where
the two derivatives are adjacent and contracted together can be transformed into operators
with one more strength tensors or with a current by using Jacobi identity. Consequently,
the various possible expression for X when it contains two W strength tensors can be
written as
X = DDDW W E ,
X = DDD W W E ,
X = DDD W W E ,
X = DD W DW E ,
X = DDW DW E ,
X = DDW DW E ,
X = DDW DW E
where h. . . i denotes the trace, or equivalent operators up to a sign or the exchange of and
.The operators in eqs.(2.11), (2.12) and (2.14) are symmetric. In eqs. (2.8) and (2.10),
the derivatives can be replaced by a strength tensor. The operators in eqs. (2.9) and (2.13)
give equivalent operators when contracted with B up to operators with a current through
integration by part. The operator in eq. (2.13) is related to the operators in eqs. (2.11)
and (2.12) through jacobi identity. The arguments remain if the W strength tensor are
replaced by B strength tensors or if B is replaced by its dual tensor. Only the case with
one dual W strength tensor need to be further discussed. The operators from eqs. (2.8)
and (2.10) can still be replaced by operators with four strength tensors and operators from
eq. (2.9) can still be replaced by the one from eq.(2.13) independently of which W got the
tilde. The operators from eq. (2.12) are equivalent to a linear combination of operators
with two derivatives contracted together acting on one of the strength tensors including
B. The operators from eq. (2.14) are equivalent to the operators from eq. (2.8) and some
operators with a current by VI. The derivative on the W strength tensor on the operators
from eq. (2.11) can be moved to the two other strength tensors by VI. The term with this
derivative acting on the B strength tensor can be transformed into a operator with a dual
B strength tensor using
1
DBWf = 2 DBeW. (2.15)
This relation is obtained by multiplying Jacobi identity by the dual tensor. The remaining
piece is removed by VIII. The operators in eq. (2.13) with one dual strength tensor are
still related to the operators from eqs. (2.11) and (2.12) through jacobi identity. To sum
up, there are no operators with three strength tensors and two covariant derivatives that
contribute to nTGC and cannot be written as operators with two or more Higgs fields or
more strength tensors.
Finally, the operators with four strength tensors induce vertices with at least four fields
and are therefore irrelevant.
With two Higgs fields
In this section, we will remove the derivatives on the H field by integration by part. The
operators with six covariant derivatives can be replaced by operators with at least one
strength tensor or by operators with four Higgs fields using rule VII.
If there is one strength tensor and four derivatives, exactly one derivative should be
applied to the strength tensor due to rules VII and VIII. Those operators should therefore
be written as
or with a different order of the Lorentz indices for the derivatives acting on the Higgs field.
If the derivatives carrying the same index as the strength tensor are adjacent, they can be
replaced by a sum of strength tensors. If they are not, they can be brought again close to
each other using the strength tensors. Consequently, we can forgot also the operators with
four derivatives.
Operators with two derivatives and two strength tensors can be written as
The cases with the indices of the two derivatives summed together have been removed
using rules I and IV. The operators in eq. (2.18) can be transformed into the previous
operators by integration by part and one operator with a derivative on each Higgs fields
which contains vertices with at least four vector bosons and eventually an operator with an
extra current. Moreover, the two derivatives can be replaced here by they anticommutator
as the commutator will be treated later. The operator in eq. (2.19) do not contain nTGC
because the indices of three vector fields can only be contracted with IJK . Three operators
Four extra operators with nTGC can be obtained with the dual strength tensors,
OBW = i HB W {D, D } H,
OW W = i HW W {D, D } H,
OBB = i HB B {D, D } H.
OBeW = i HBe W {D, D } H,
OBWf = i HB Wf {D, D } H,
OWfW = i HWf W {D, D } H,
OBeB = i HBe B {D, D } H.
B Be = 14 BBe,
and its equivalent for W transform the two last operators to operators with more Higgs
or dimension-six operators. The operators from eq. (2.24) is equivalent to the one from
eq. (2.23) up to operators with more currents, more strength tensors or with quartic gauge
boson couplings only due to eq. (2.15).
If there are only three strength tensors in addition to the two Higgs fields, the operator
is antisymmetric under the exchange of the three strength tensors and no nTGC can be
generated. There are also no operators with one dual strength tensor and two strength
tensors due to eq. (2.27).
With four Higgs fields
If the four derivatives are applied each on one Higgs doublet, only quartic interactions are
generated. Two derivatives should then be applied on one Higgs field and they should have
different Lorentz indices due to the Higgs equation of motion. Only the Z field can be
selected with the Higgs vacuum expectation value. Consequently, the part of the operators
with four Higgs contributing to the nTGC can be written as ZZ Z and have all the
same nTCG vertex proportional to
This vertex does not contribute when the vector bosons are on-shell or attached to a
massless fermion line. There are also no operator with one or two strength tensors with nTGC.
Phenomenology
Once added to the Lagrangian with their hermitian conjugate, the four operators found in
section 2.2 produces nTCG vertices independent of the imaginary part of their coefficients.
Namely, none of the combinations Oi Oi contains vertices with three neutral gauge boson.
The Lagrangian for nTGC can therefore be written as
where i run over the label of the four operators from equations (2.20) to (2.23). In the
definitions of the operators we use the following convention:
W = I (WI WI + gIJK WJ WK )
with I J = IJ /2.
The CP-conserving anomalous couplings for the production of two on-shell Z bosons
(see eq. (1.1)) are given by
f5Z = 0
h3Z =
h4Z = 0
and the CP-violating by
f4Z =
MZ2 v2 cw2 CB4B + 2cwsw CB4W + 4sw2 CW4W
cwsw CB4B + CB4W cw2 sw2 + 4cwsw CW4W
For one on-shell Z boson and one on-shell photon (see eq. (1.2)), the CP conserving
couplings are
and the CP-violating couplings are
h1Z =
h2Z = 0
h1Z =
cwsw CB4B + CB4W cw2 sw2 + 4cwsw CW4W
MZ2 v2 sw2 CBB
4 2cwsw CB4W + 4cw2 CW4W
The anomalous couplings for off-shell bosons are given in appendix A. Many couplings are
zero because the CP-conserving ZAA and ZZZ vertices vanish. Those expressions implies
two relations between the non-vanishing couplings,
CBeW /4 can be constrained using the mearsurements of f5 and h3Z . The limits on f5 [6]
give the strongest constraint, i.e.
at 95% C.L. The coefficients on the CP-violating operators can be constrained from the
measurements of f4Z , f4 [6] and h1 [18],
47 TeV4 < CBe4W < 47 TeV4
96 TeV4 < CB4B < 102 TeV4
114 TeV4 < CB4W < 114 TeV4
47 TeV4 < CW4W < 46 TeV4.
Only the constraints on anomalous couplings obtained without form factor have been used
to extract the limits on the coefficients of the dimension-eight operators. For order one
coefficients, the new physics scale is constrained to be above a few hundred GeV. Such
large coefficients require a strongly interacting new sector since the anomalous operators
arise at the loop level in a UV complete theory. Consequently, the coefficients are expected
to be smaller in a weakly coupled model and the mass limit lower. For example, fermions
with a non-zero axial coupling to the Z boson can generate the Levi-Civita tensor of the
CP-conserving operator at one-loop. The contribution such a heavy fermion implies [10]
CBeW =
where Q, gA, gV , MF are respectively the charge, the axial and vector couplings to the Z
boson and the mass of the fermion (see eq. 42 in [10] for the couplings convention). The
limit in eq. (3.18) forces the fermion mass to be of a few tens of GeV depending on its
charge and couplings to the Z boson. It should be noted that the dependence in the inverse
fermion mass in quadratic and not quartic as expected from the dimension of the operator.
However, the fermion mass is related to the EWSB scale since the left and right-handed
fermions have to transform differently under the weak gauge group to generate the axial
coupling. It the new physics is due to new fermion or scalar multiplets, the CP-violating
nTCG requires more than one loop contribution as their interaction to the photon or the
Z-boson do not break CP. The CP breaking term can come either from the interaction to
the charged vector bosons or from the scalar potential. In these cases, the CP-violating
coefficients can be estimated by
Ci
where gCP and gCP are CP-conserving and CP-violating couplings of the new sector. Again,
the experimental bounds imply that the new physics scale as to be of the order a few tens
of GeV. In both case, the limits are only rough estimates as the effective approach is not
expected to be reliable for such low masses.
If the new physics is heavy, the largest new physics contribution to f f ZZ/ZA is
expected from the interference between the SM and the dimension-eight operators,
The first and second dots represent the interference between the SM and the dimension-ten
and dimension-twelve operators respectively. In the following, the given partial O 8
contribution is the squared of the amplitude with dimension-eight operators. This term
is given to understand the suppression of the interference but do not induce a leading
contribution from the heavy new physics unless the interferences between the SM and the
dimension-eight and dimension-ten operators are both strongly suppressed. Although the
dimension-six operators do not induce nTGC at the tree-level, they can have a effect on
nTGC at one-loop. Those contributions would be of the order 4EM s2 while the tree-level
contribution from the dimension-eight operators are of the order sv42 . Consequently, the
contribution of the dimension-eight operators dominates the one-loop contribution of the
dimension-six operator for
s 10 TeV.
The three first CP-odd operators do not contribute to the on-shell production of two
neutral gauge bosons at the order 4. On the contrary, the CP-even operator contributes
to ZZ and ZA productions thanks to the ZZA vertex. As a consequence, only the CP-even
operator will be studied in the following.
2 (MSMMdim8) (f f ZZ)
= 2 (MSMMdim8) (f f ZT ZL)
= e4v2NcQ Q(Q + 1)s4w 2Qs2wT3 + T32 CBeW 4MZ2 s s 2 c2 + 1 MZ2 s2s
4 2c3ws3w 4MZ4 4s2MZ2 s + s2s2
where the average over the spin is done but not over the color, Q and T3 are the fermion
charge and weak isospin, s the center of mass energy, the scattering angle, Nc is the
number of color and should be set to one for leptons. The dependence in Q and T3 is due
to the fact that only the new physics diagram with the photon in the s-channel has an
effect. By comparison, the SM contribution is
ZZ production
The new operator only contributes to the production of one longitudinally polarized boson
and one transversally polarized boson,
The 1/8 term from the new physics amplitude squared should be neglected for
phenomenological purpose. However, it has been kept only to show that the interference is
c4ws4w 4MZ4 4s2MZ2 s + s2s2 2
+1.4 104CB2eW
1 TeV 4
1 TeV 8
and becomes constant at high energy while the new physics contribution grows like s.
Futhermore, the SM contribution to one longitudinally polarized boson and one
transversally polarized boson,
2s 4 c2 + 1 MZ4 4s2MZ2 s + s4s2
c4ws4w 4MZ4 4s2MZ2 s + s2s2 2
goes like 1/s at high energy. As expected, the SM produces mainly transversally
polarized bosons at high energy. Those analytical expressions also show that the NP and the
Numerically, the interference between new operator and the SM is small. For example,
the the cross-section for an electron positron collider of 200 GeV is
(ee ZZ)/f b = 1252 3.2 103CBeW
highly suppressed as already mentioned. The fact that the new physics only contributes to
one longitudinally polarized boson and one transversally polarized boson partially explains
this suppression. As a matter of fact, the SM contribution to those polarizations is only
one third of its total rate at this energy. If a cut on the scattering angle, c2 < ss+22MMZZ22 ,
is added,
cut(ee ZT ZL)/f b = 233 2.6 102CBeW
1 TeV
+7.9 105CB2eW
the interference suppression is further reduced. In fact, the two parts of the interference
with an opposite sign nearly cancel each other close to threshold. The new physics
contribution is expected to be larger compared to the SM at higher energies but remains
challenging since the new physics scale should also be pushed to higher energy for
consistency. For a collision of 1 TeV, the new physics contribution is well below the fb for a new
physics scale at a few TeV, i.e.
1 TeV
1 TeV
+ . . . . (3.31)
The suppression of the interference remains at high energy but the cut in the scattering
angle does not have such a large effect anymore,
1 TeV
1 TeV
+ . . . (3.32)
The suppression is rather explained by the fact that the polarization from the SM
contribution are mainly transverse far from threshold. Only slightly less than a percent of the SM
production has one longitudinal Z boson at 1 TeV. The remaining part of the suppression
of the interference at both energies comes from the large phase difference between the SM
and new physics amplitudes. Since the interference between the SM and the
dimensioneight operators is quite suppressed, the next order in the 1/ expansion may have a larger
contribution for a new physics scale not very far from the energy probed by the experiment.
AZ production
The dimension-eight operator contributes both to the production of longitudinally and
transversally polarized Z boson,
2 (MSMMdim8) (f f AT ZL)/f b = e4NcQv2 CBeW
4
2 (MSMMdim8) (f f AT ZT )/f b = e4NcQv2 CBeW
4
However, there are no new physics contributions to all the possible combination of
transversally polarized Z boson. In fact, the polarization vectors of the Z boson and the photon
have to be orthogonal to each other. At high energy, the production longitudinal Z boson
dominates. On the contrary, the SM contribution,
cw2sw2 s MZ2 2
contains mainly transversally polarized Z bosons at high energy. Contrary to the new
physics contribution, the SM one depends on the scattering angle. Namely, the bosons are
produced mainly at small scattering angle in the SM. Finally, the dependence in the center
of mass energy is the same as for Z boson pair production for both the SM and the new
operator.
For the following cross-sections, we have only applied a cut on the pT of the photon,
i.e. pT > 10 GeV, to remove the divergence. For a center of mass energy of 200 GeV, the
new physics effects are of the order of a few fb if the new physics scale is at 600 GeV,
(ee AT ZT )/f b = 15620 + 7.96 102CBeW
1 TeV 4
1 TeV 8
+2.11 103CB2eW
+4.53 104CB2eW
+2.17 102CB2eW
(ee AT ZT )/f b = 866 + 4.02 103CBeW
1 TeV 4
1 TeV 8
1 TeV 4
We remind once more that the 1/8 terms are only given to understand the suppression
of the interference. At 1 TeV and for a new physics scale around 3 TeV, the effects of the
new physics are slightly below 0.01 fb,
At that energy, the secondary longitudinal polarizations are at the level of the percent and
the interference is highly suppressed. Like for ZZ production, there is a mismatch between
the polarization states produced at high energy by the SM and by the new operator. On
the one hand, it allows to disentangle the new physics contribution from the SM one. On
the other hand, it suppress the interference. In fact, this mismatch is also present for
charged bosons pair production from the SM and the dimension-six operators with the
Higgs field [19]. Finally a cut on the scattering angle would improve the ratio of the new
physics contribution over the SM one but will this time reduce the already small new
physics contribution.
Analytical results presented above for both ZZ and ZA productions have been found
in agreement for the total rate and the matrix element of one phase space point using
FeynRules UFO output [20, 21], ALOHA [22] and Madgraph5 [23].
Conclusion
If the new physics is heavier than the scale probed by the experiments, it can be described
in a model independent way by an effective Lagrangian. Since no dimension-six
operator induces nTGC, the first new physics contribution to those interactions comes from
the dimension-eight operators. We have found that only one CP-even and three CP-odd
independent dimension-eight operators give rise to nTGC for on-shell bosons or bosons
decaying to massless fermions. Since the CP even one only induces the ZZA anomalous
vertex, only the f5 and h3Z CP-even anomalous couplings do not vanish. Similarly, two
of the CP-odd anomalous couplings vanish despite that the three anomalous vertex are
generated by CP-odd dimension-eight operators. The two relations from eq. (3.17) further
reduce the number independent parameters in the anomalous vertex approach. Only the
CP-even operator contributes to the production of a pair of on-shell neutral electroweak
bosons. This operator affects both to ZZ and ZA productions. As expected, the
contributions from the higher dimensional operator become larger compared to those of the
SM at high energy. They also differ by the polarization of the produced bosons. At high
energy, this operator produces almost exclusively one of the Z boson with a longitudinal
polarization contrary to the SM. The new physics contributions also differ from the SM
ones by the dependence in the scattering angle. In particular, the interference between the
SM and the dimension-eight operators change of sign at c2 = ss+22MMZZ22 for ZZ production
and the two pieces nearly cancel each other near threshold. The new physics contributions
are therefore quite distinguishable from the pure SM contribution. Although we have only
used the tree-level expression for the SM contribution, the SM behavior at high energy
should not be modified by loop corrections. Unfortunately, the difference between the SM
and new physics contributions also implies that the interference is strongly suppressed.
Consequently, advanced analysis like matrix element method would be welcome to observe
or constrain the coefficients of the effective operators if the new physics scale is at or above
the TeV. The present constraints on the anomalous couplings barely constrain the new
physics scale to be above the weak scale. Finally, the four dimension-eight operators are
implemented in FeynRules model [24] and can therefore be used in MadGraph5 [25].
Acknowledgments
We are grateful for discussion with J.-M. Gerard. This material is based upon
work supported in part by the U. S. Department of Energy under Contract
No. DE-FG02-13ER42001.
for the CP-conserving couplings and by
v2 q21 q22 CBeW
The contributions of the dimension-eight operators to the off-shell anomalous couplings [10]
are given by
q12 q22 v2 cw2 CB4B + 2cwsw CB4W + 4sw2 CW4W
q21 + 2q1.q2 v2 cw2 CB4B + 2cwsw 4
CB4W + 4sw2 CWW
2q1.q2 + q22 v2 cw2 CB4B + 2cwsw 4
CB4W + 4sw2 CWW
q12 q22 v2 sw2 CB4B 2cwsw CB4W + 4cw2 CW4W
q21 + q22 v2 sw2 CB4B 2cwsw CB4W + 4cw2 CW4W
q12 q22 v2 sw2 CB4B 2cwsw CB4W + 4cw2 CW4W
q23v2 cwsw CB4B + CB4W cw2 sw2 + 4cwsw CW4W
q12 q22 v2 cwsw CB4B + CB4W cw2 sw2 + 4cwsw CW4W
for the CP-violating couplings. We have dropped the star in the exponent of the coupling
and the dependence on the momenta to shorten the notation. The qi are the momenta of
the bosons as defined in [10]. We do not give the g couplings since we assume that the
bosons are on-shell or connected to a massless fermion line. Those couplings have been
translated to the parametrization of eqs. (1.1) and (1.2) using the relation from ref. [10].
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