Treelevel equivalence between a Lorentzviolating extension of QED and its dual model in electron–electron scattering
Eur. Phys. J. C
Treelevel equivalence between a Lorentzviolating extension of QED and its dual model in electronelectron scattering
Giuliano R. Toniolo 1
H. G. Fargnoli 1
L. C. T. Brito 1
A. P. Baêta Scarpelli 0
0 Departamento de Polícia Federal , Setor TécnicoCientífico, Rua Hugo D'Antola, 95, Lapa, São Paulo , Brazil
1 Departamento de Física, Universidade Federal de Lavras , Caixa Postal 3037, Lavras, Minas Gerais 37.200000 , Brazil
Smatrix amplitudes for the electronelectron scattering are calculated in order to verify the physical equivalence between two Lorentzbreaking dual models. We begin with an extended Quantum Electrodynamics which incorporates CPTeven Lorentzviolating kinetic and mass terms. Then, in a process of gauge embedding, its gaugeinvariant dual model is obtained. The physical equivalence of the two models is established at tree level in the electronelectron scattering and the unpolarized cross section is calculated up to second order in the Lorentzviolating parameter.

In some situations, it is possible to establish relations between
models which are essentially different but are equivalent
in describing the physical behavior of a system. These are
called dual models. This concept of duality is very useful,
because there are some physical properties which are hidden
in one model but are explicit in its dual theory. We refer to
[1] in order to exemplify this particularly interesting
property of Quantum Field Theories. Different expansions for
the same Hamiltonian in a quantum model can be written, as
H = H0 + g H1 = H0 + g H1, where H0 and H0 allow
simple known solutions. Besides, H0 and H0 are expressed in a
simple form in terms of the fields ϕ and ϕ , respectively. On
the other hand, the relation between ϕ and ϕ is complicated
and nonlocal. Usually, the coupling constants obey a relation
of the type g ∼ 1/g , so that g becomes small when g is
large and vice versa. A very important fact is that if g and
g are not of the same order of magnitude, the description
in terms of one of the fields will be appropriate for a
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turbative analysis. As a good example, the relation between
electric and magnetic couplings is implemented by the dual
mapping of a weakly coupled theory into a strongly coupled
one.
We are interested in the kind of duality, investigated
in the seminal work of Deser and Jackiw, between the
threedimensional spacetime selfdual and Maxwell–Chern–
Simons models [2], which were discussed as a part of a wide
class of models in [3]. Since then, different techniques to
attain the duality between models have been elaborated [1,4].
Among the approaches to obtain physically equivalent
models, we can cite the master action method [5] and the gauge
embedding technique [6]. In the first approach, the socalled
master action, roughly speaking, is written in terms of two
vector fields. The dual models are then obtained by
eliminating one of the fields from the action in favor of the other with
the use of field equations. In the gauge embedding procedure
(also called Noether dualization method), on the other hand, a
gauge theory is obtained from a gaugebreaking model by the
use of iterative embedding Noether counterterms, which
vanish on mass shell. The Noether dualization method (NDM)
is based on the idea of local lifting a global symmetry. This
type of procedure is reminiscent of the earlier construction
of componentfield supergravity actions [7–9]. A particular
interesting feature of such kind of dual models is that one can
consider the noninvariant model as a gauge fixed version of
a gauge theory. In other words, one model would reduce to
the other under some gauge fixing conditions.
The gauging iterative Noether dualization method has
been shown to be effective in establishing dualities between
some models [10]. This method provides a strong suggestion
of duality, since it yields the expected result in the
paradigmatic duality between the selfdual and Maxwell–Chern–
Simons models in three dimensions. However, an intriguing
result has been shown to be general when NDM is applied to
Procalike models [11]. The gauge model obtained from the
dualization algorithm, although sharing the physical
spectrum with the original theory, acquires ghost modes. The
gauge model obtained by means of gauge embedding
encompasses the physical spectrum of the original Procalike theory
and, in addition, the spectrum of the corresponding massless
model. However, these new modes appear with the wrong
sign, characterizing ghosts, which may be dangerous for the
model. In [11], a relation between the propagators of dual
models was obtained, which shed some light on this fact.
Alternatives to avoid the emergence of ghosts in the process
of dualization were studied [12–15]. In some cases, the price
to be paid is the loss of locality [12]. For a model with a
spin2 selfdual field in three spacetime dimensions, it was shown
that the dual theory constructed with gauge embedding does
not suffer from the presence of ghosts [15].
In some cases, such as the threedimensional selfdual
model, it is simple to see that these extra nonphysical modes
are not harmful to the theory. This is because, in these cases,
it is evident that the nonphysical particles have no dynamics
or decouple from the rest of the model, since they do not
contribute to the propagator saturated by conserved currents.
Nevertheless, in some cases, it is not simple to check if the
new modes spoil the gauge theory obtained with the process
of dualization. Examples are the dualized Lorentzviolating
models treated in Refs. [16–19]. It is not obvious that the
ghosts and the physical particles decouple in these models.
So, a deeper analysis is required.
In this paper, we will focus on the model of Ref. [18].
The model is a modified QED, which incorporates two
CPTeven Lorentzbreaking terms: a mass part of the type
−(1/2)m2(gμν − βbμbν ) and the kinetic aetherlike term
ρ (bμ F μν )2, in which β and ρ are dimensionless
of [20], − 2
parameters and bμ is a background vector. The model can
be entirely accommodated in the Standard Model Extension
(SME) [21,22], which provides a description of Lorentz and
CPT violation in Quantum Field Theories, controlled by a
set of coefficients whose small magnitudes are, in principle,
fixed by experiments. The aether term is a particular case of
the more general CPTeven Lorentzviolating kinetic part of
SME. On the other hand, the Lorentzviolating mass term
can be generated by spontaneous gauge symmetry breaking
[23], coming from the symmetric part of the secondrank
background tensor which couples to the kinetic part of the
Higgs field. Models with Lorentzviolating mass terms [24–
27] may present lots of interesting aspects, like superluminal
modes or even instantaneous longrange interactions. The
present model has been studied in many aspects in [18],
and the aforementioned properties were shown to appear
for some values of the parameters β and ρ. However, the
dual gauge theory obtained in the process of dualization, as
commented on above, presents ghost modes whose role is
still not clear. Since only the gauge sector has been studied,
an analysis including the fermionic and interaction terms is
missing.
In this paper, we reassess the model of [18] with the
inclusion of the fermionic and the interaction sectors. The
dualization by gauge embedding is carried out and the dual
gaugeinvariant theory is obtained. In addition to the action achieved
in the previous work, new interaction terms, which are
nonminimal, are generated. We perform a practical calculation,
the electron–electron scattering at tree level, in order to check
the decoupling of the nonphysical modes (calculations of
scattering processes in Lorentzviolating models have been
performed, for example, in Refs. [28,29]). It is shown that
nontrivial cancelations occur in the calculation with the
dualized action, so that the two models yield identical results.
Moreover, the unpolarized cross section was calculated up to
second order in the Lorentzviolating parameter.
The paper is organized as follows. In Sect. 2, we describe
and justify the model under analysis and, afterwards, we
use the gauge embedding procedure to obtain its
gaugeinvariant dual theory. In Sect. 3, the treelevel calculation
of the electron–electron scattering is performed using the
two models. We also obtain the unpolarized cross section
up to second order in the Lorentzviolating parameter. The
concluding comments are in Sect. 4.
2 Description of the model and dualization
In the present work, we consider the CPTeven
Lorentzbreaking model of [18], but now with the vector field Aμ
minimally coupled to a Dirac fermion. Thus, we have an
extended QED model defined by the Lagrangian density
where hμν = gμν − βbμbν and bμ is a constant background
fourvector. We should notice that the magnitude of bμ is
small compared to the other parameters of the theory. Here,
m and M are the masses of the gauge field Aμ and the
electron, respectively, while ρ and β are dimensionless
parameters introduced simply to make the contributions from
distinct Lorentzviolating terms, which appear in (1), explicit.
The kinetic Lorentzviolating term, which we call aether term
[20], is a particular version of the more general CPTeven part
of the gauge sector of the Standard Model Extension [21,22],
and it can be radiatively induced [30–33] when nonminimal
couplings to fermions [34–37] are considered. On the other
hand, the Lorentzbreaking mass term in the gauge sector
may, for example, be generated by spontaneous gauge
symmetry breaking in a Lorentzviolating gaugeHiggs model
[23], emerging from the symmetric part of the secondrank
background tensor which couples to the kinetic part of the
Higgs field.
The gauge sector of this model was investigated in detail
in [18] and it was shown that it incorporates very
interesting features. For example, it presents physical massive poles
which, depending on the choice of the coefficients ρ and β,
have their degrees of freedom changed. For this class of
models, uncommon physical aspects can be accommodated for
particular values of ρ and β; for example, the presence of
propagating superluminal modes.
We now proceed to the gauge embedding procedure. First,
we calculate the variation of (1) with respect to an
infinitesimal change δ Aμ in the gauge field:
δL(0) = {∂β F βμ + ρbβ bα∂β F αμ − ρbμbα∂β F αβ
It should be noticed that it is an offshell method, since we
have J μ = 0 in the space of solutions. The current J μ in (2)
can be used to construct a second Lagrangian density,
= L
in which Bμ is an auxiliary vector field, chosen such that
δ Bμ = δ Aμ. We calculate the variation of (3) with respect
to δ Aμ and get
with δ Jμ = m2hμν δ Aν . Knowing the result (4), we use a
compensatory quadratic term in the auxiliary field Bμ in
order to build a gaugeinvariant Lagrangian density, given
by
= L
in which it is simple to check that δL(2) = 0. Finally, we
calculate the variation of L(2) with respect to Bμ to obtain
β
Lμν = gμν + 1 − βb2 bμbν .
Equation (6) is used to write L(2) in terms of the field Aμ. The
resulting dual gaugeinvariant Lagrangian associated with
the original model (1) reads
+ ρ2 (bμ Fμν )2 − 21α (∂μ Aμ)2
in which we have inserted a gauge fixing term and
+ ρ[−Lμν (b · ∂)2 + Lαν (b · ∂)(bμ∂α + bα∂μ)
By construction, the gauge embedding method gives rise
to a gaugeinvariant Lagrangian, whereas the original model
(1) does not have this symmetry. It is believed that the
noninvariant model can be considered as the gauge fixed version
of a gauge theory. Note that now the Dirac fermions are
nonminimally coupled to the gauge field Aμ. Besides, the dual
Lagrangian has a contribution of a fourfermion
nonrenormalizable vertex, which is similar to the result obtained in the
duality between the selfdual and Maxwell–Chern–Simons
models coupled to fermions [38].
It is also interesting to comment on the change of sign of
the Maxwell kinetic term when compared with the original
model. We refer to [18], in which a relation between the gauge
propagators of the two models is obtained. The new
gaugeinvariant model acquire new massless ghost modes which
combine with the physical sector to produce this “wrong”
sign. However, these nonphysical modes, as we will see in
the sections below, decouple from the physical sector and
produce no effect in phenomenological calculations.
3 Electron–electron scattering
We now proceed to perturbative calculations in order to
check, in a practical calculation at tree level, the physical
equivalence of the models. We first write the two photon
propagators, which were obtained in [18]. From the quadratic
terms in Aμ, the propagators for the gauge field in
momentum space for the original (1) and the dual (8) models are
given, respectively, by
Table 1 Multiplicative table
fulfilled by θ, ω, and
Fig. 1 Feynman diagrams for the electron–electron scattering at order
e2. In the original model, only these two diagrams contribute. The gauge
propagator is DμOν and the vertex is the same as in ordinary QED. In the
dual model, these diagrams must be accounted with the replacements
γ μ → μ and DμOν → DμDν . The external lines represent onshell
Dirac electrons, where p1 and p2 are the momenta of the incoming
particles, whereas p1 and p2 are the momenta of the outgoing particles.
We have defined k = p1 − p1 and k = p1 − p2
+ (ρ + β)λ2m2]ωμν + [(ρ + β)k2 − β A1] μν
We have written the propagators in terms of spin operators,
θμν = gμν − kμk2kν and ωμν = kμk2kν being the transversal
and the longitudinal operators, respectively. The operators
μν = bμbν and μν = bμkν emerged from the inclusion
of the external vector bμ (λ stands for μ μ = bμkμ). The
Lorentz algebra of these operators is shown in Table 1.
As carefully studied in [18], the propagator DμDν has,
besides the physical poles of DμOν , new nonphysical ones.
One way to proceed is to study the saturated propagator,
which makes use of the current conservation to discard the
nondynamical poles. Here, we intend to go further in a
practical calculation of the Smatrix contribution at order e2 for
the electron–electron scattering, which is the main purpose of
this letter, and establish, for this process, the physical
equivalence between the models.
3.1 Calculation with the original model
First, we consider the original model (1). Since the model has
only the usual Dirac fermion minimally coupled to the gauge
field Aμ, the two diagrams in Fig. 1 contribute at tree level
with the same vertex as ordinary QED. However, in this case,
with the Aμ propagator given by Eq. (10). The contribution
to the Smatrix amplitude is given by −(2π )4δ( p1 + p2 −
p1 − p2)e2τO , where
τO = u¯( p1)γμu( p1)DμOν (k)u¯( p2)γν u( p2)
− u¯( p2)γμu( p1)DμOν (k )u¯( p1)γν u( p2).
We have used p1 and p2 for external momentum of the free
electrons in the initial states described by the spinors u( p1)
and u( p2), and p1 and p2 for the free electrons in the final
states u¯( p1) and u¯( p1). Equation (13) can be obtained from
the direct application of the LSZ reduction formula.
Since the fermions are onshell, all terms from DμOν which
are dependent on the external momentum can be neglected
in the calculation of the amplitude. Thus, we are left with
τO = u¯( p1)γμu( p1) − A11(k) θ μν
Fig. 2 Fourfermion vertex diagrams which contribute for the
electron–electron scattering in the dual model. Again, p1 and p2 are
the momenta of the incoming electrons, whereas p1 and p2 are the
momenta of the outgoing electrons. These diagrams must be summed
to the diagrams of Fig. 1
− u¯( p2)γμu( p1) − A11(k ) θ μν
3.2 Calculation with the dual model
We now proceed to the calculation of the treelevel electron–
electron scattering by using the Feynman rules from the
gauge model of (8). Besides the two diagrams in Fig. 1, in
the dual model we must take into account the diagrams of
Fig. 2. Now, for the calculation of the diagrams of Fig. 1, we
must perform the replacements γ μ → μ and DμOν → DμDν .
In momentum space, we have
μ = m12 γν (k2 + ρλ2)θ μν + ρλ2ωμν
and the calculation is greatly simplified if we note that
By making these modifications in (13), using Eq. (16) and
the fact that fermions are onshell, after lengthy but
straightforward algebra, we obtain
τ1 = u¯( p1)γμu( p1) Qμν (k)
− u¯( p2)γμu( p1) Qμmν (2k )
1
Qμν (k) = − A1 A2
× (ρ + β)(ρ − β − 2ρβb2)λ2
+(ρ + β)(2 − βb2 + ρb2)k2
F
−(1 + ρb2)(1 − βb2) H k2 (k2 − βλ2)2
In addition, the contributions of the fourvertex diagrams of
Fig. 2 read
Finally, putting together contributions (17) and (19), we
obtain the total treelevel amplitude for the gaugeinvariant
theory:
1
τD = u¯( p1)γμu( p1) m2 [Qμν (k) + Lμν ]u¯( p2)γν u( p2)
1
−u¯( p2)γμu( p1) m2 [Qμν (k ) + Lμν ]u¯( p1)γν u( p2).
Using Eqs. (7) and (18) for Lμν and Qμν , respectively, and
after very lengthy algebra, we obtain
= A1
With this identity, we check that τO = τD and the equivalence
for this process is proved.
3.3 The cross section
Finally, we present the treelevel unpolarized cross section
for the electron–electron scattering at second order in the
Lorentzviolating background vector bμ. In the
centerofmass reference frame, the cross section is given by
where τ = τO = τD and EC M is the energy in the center of
mass. Using the approximations m/M 1 and  b 2 1,
we obtain the following expansion:
in which the dots represent higher order terms in bμ. The
first term in (23) is just the wellknown result from ordinary
QED:
QED = 64π 2 (E C2 M − M 2)
e4 (2E C2 M − M 2)2
(E C2 M − M 2)2
sin4 θ − sin2 θ + (2E C2 M − M 2)2
The second term is the contribution due to the Proca term,
which reads1
μν = bμbν . Since we have calculated the unpolarized cross
section, we expect that only the coefficient of the isotropic
part contributes.
Finally, it is easy to check that the results of the
Lorentzinvariant limit of our models match the ones known from
the literature (without the Proca term; see, for example [39]).
3E C2 M − M 2 2 1 + cos2 θ 2
+ M 4(4 + sin2 θ ) − 4E C4 M cos4 θ − 8E C2 M M 2
64π 2 E C2 M (E C2 M − M 2)3 sin6 θ
where θ is the scattering angle between the direction of the
incident and the outgoing particles. The last term of (23)
is the correction introduced by the Lorentz violation. Just
to illustrate, we explicitly show the expressions for timelike
and spacelike bμ. For timelike bμ, we take bμ = (δ, 0, 0, 0),
such that the result reads
+(E C2 M − M 2)(4E C2 M − M 2) cos3 θ
+(5E C2 M − 3M 2)(6E C2 M + 3M 2) cos2 θ
+(12E C4 M − 3E C2 M M 2 − M 4) cos θ + 15E C4 M
−15E C2 M M 2 + 6M 4}.
For the spacelike case, we use bμ = (0, 0, 0, δ) and the
Lorentzviolating contribution reads
+ (E C2 M − M 2)(4E C2 M + M 2) cos5 θ
− 3(4E C4 M + 3M 4) cos θ + 3(7E C4 M + 4M 4)}.
To carry out these calculations we have chosen the
spatial part of bμ in the same direction of the outgoing particle
with momentum p1. Note that at second order in bμ only
the aether term, ρ2 (bμ Fμν )2, contributes to the cross section,
since only the parameter ρ appears in (23). This is because in
the expression for τ , the coefficient of the transversal
operator θ μν = gμν − kμk2kν does not depend on the β
parameter, which appears only in the coefficient of the operator
1 In these computations, we have used the FeynCalc Mathematica pack
The dimensionless parameters ρ and β can be used to “turn
off” the Lorentzbreaking terms, by simply taking the limits
ρ , β → 0.
4 Conclusion
Dual models are constructed with the aim of having
different descriptions of the same physical system; it is
appropriate to apply them in distinct situations. Therefore, for
some calculation one of the models may furnish an
obvious and simple result which is difficult to infer from the
other one. One of the artifacts of the dualization procedure
by gauge embedding is the production, besides the original
spectrum, of new nonphysical modes which, in some cases,
may turn the model meaningless. These ghosts, in some cases
as in the famous duality between the threedimensional
selfdual and Maxwell–Chern–Simons models, are easily seen
to have no dynamics. However, in most cases this is not an
obvious issue, like in the Lorentzbreaking models studied
in [18].
In this paper, we showed that, sometimes, the physical
equivalence of dual models is subtle. We carried out a
practical calculation of a physical process, more precisely the
cross section of the electron–electron scattering, using the
dual model studied in detail in [18]. The equivalence of the
models was shown at tree level through nontrivial
cancelations. For this purpose, an essential role was played by the
new fermionic couplings which emerged in the dualization
process. Although these new modes apparently couple to the
other sectors of the theory, these contributions are canceled
by other terms which come from new graphs due to this
nonrenormalizable quartic vertex. Finally, the unpolarized
cross section for this process was obtained. Besides the result
from ordinary QED with a Proca term, new contributions
were obtained up to second order in the background vector
bμ.
Acknowledgements A. P. B. S. acknowledges research grants from
CNPq. Giuliano Toniolo thanks CAPES for financial support.
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