Lorentzian Goldstone modes shared among photons and gravitons
Eur. Phys. J. C
Lorentzian Goldstone modes shared among photons and gravitons
J. L. Chkareuli 0 1
J. Jejelava 0 1
Z. Kepuladze 0 1
0 E. Andronikashvili Institute of Physics , 0177 Tbilisi , Georgia
1 Center for Elementary Particle Physics, Ilia State University , 0162 Tbilisi , Georgia
It has long been known that photons and gravitons may appear as vector and tensor Goldstone modes caused by spontaneous Lorentz invariance violation (SLIV). Usually this approach is considered for photons and gravitons separately. We develop the emergent electrogravity theory consisting of the ordinary QED and the tensorfield gravity model which mimics the linearized general relativity in Minkowski spacetime. In this theory, Lorentz symmetry appears incorporated into higher global symmetries of the lengthfixing constraints put on the vector and tensor fields involved, A2μ = ±M A2 and Hμ2ν = ±M H2 (MA and MH are the proposed symmetry breaking scales). We show that such a SLIV pattern being related to breaking of global symmetries underlying these constraints induces the massless Goldstone and pseudoGoldstone modes shared by photon and graviton. While for a vector field case the symmetry of the constraint coincides with Lorentz symmetry S O(1, 3) of the electrogravity Lagrangian, the tensorfield constraint itself possesses much higher global symmetry S O(7, 3), whose spontaneous violation provides a sufficient number of zero modes collected in a graviton. Accordingly, while the photon may only contain true Goldstone modes, the graviton appears at least partially to be composed of pseudoGoldstone modes rather than of pure Goldstone ones. When expressed in terms of these modes, the theory looks essentially nonlinear and contains a variety of Lorentz and CPT violating couplings. However, all SLIV effects turn out to be strictly cancelled in the lowest order processes considered in some detail. How this emergent electrogravity theory could be observationally different from conventional QED and GR theories is also briefly discussed.

think that spontaneous violation of spacetime symmetries
and, particularly, spontaneous Lorentz invariance violation
(SLIV), could also provide some dynamical approach to
quantum electrodynamics [1], gravity [2] and Yang–Mills
theories [3] with photon, graviton and nonAbelian gauge
fields appearing as massless Nambu–Goldstone (NG) bosons
[4,5] (for some later developments, see [6–14]). In this
connection, we recently suggested [15,16] an alternative
approach to the emergent gravity theory in the framework
of nonlinearly realized Lorentz symmetry for the underlying
symmetric twoindex tensor field in a theory, which mimics
linearized general relativity in Minkowski spacetime. It was
shown that such a SLIV pattern, due to which a true vacuum
in the theory is chosen, induces massless tensor Goldstone
and pseudoGoldstone modes, some of which can naturally
be associated with the physical graviton.
This approach itself has had a long history, dating back
to the model of Nambu [17] for QED with a nonlinearly
realized Lorentz symmetry for the underlying vector field.
This may indeed appear through the “lengthfixing” vector
field constraint
A2μ = n2 M A2,
A2μ ≡ Aμ Aμ, n2 ≡ nν nν = ±1
(
1
)
(where nν is a properly oriented unit Lorentz vector, while
MA is the proposed scale for Lorentz violation) much as it
works in the nonlinear σ model [18] for pions, σ 2+π 2 = fπ2,
where fπ is the pion decay constant. Note that a
correspondence with the nonlinear σ model for pions may appear
rather suggestive in view of the fact that pions are the only
presently known Goldstone particles whose theory, chiral
dynamics [18], is given by the nonlinearly realized chiral
SU (
2
) × SU (
2
) symmetry rather than by an ordinary linear
σ model. The constraint (
1
) means in essence that the vector
field Aμ develops some constant background value,
and the Lorentz symmetry S O(
1, 3
) formally breaks down
to S O(3) or S O(
1, 2
) depending on the timelike (n2 > 0)
(2)
< Aμ(x ) > = nμ MA,
or spacelike (n2 < 0) nature of SLIV. However, in sharp
contrast to the nonlinear σ model for pions, the nonlinear
QED theory, due to the starting gauge invariance involved,
ensures that all the physical Lorentz violating effects turn out
to be nonobservable. It was shown [17], while only in the
tree approximation and for the timelike SLIV (n2 > 0), that
the nonlinear constraint (
1
) implemented into the standard
QED Lagrangian containing a charged fermion field ψ (x )
1
L Q E D = − 4 Fμν F μν + ψ (i γ ∂ + m)ψ − e Aμψ γ μψ
(
3
)
as a supplementary condition appears in fact as a possible
gauge choice for the vector field Aμ, while the Smatrix
remains unaltered under such a gauge convention. Really,
this nonlinear QED contains a plethora of Lorentz and CPT
violating couplings when it is expressed in terms of the pure
emergent photon modes (aμ) according to the constraint
condition (
1
)
Aμ=aμ + nμ(M A2 − n2a2) 21 , nμaμ = 0 (a2 ≡ aμaμ).
(
4
)
(
5
)
For definiteness, one takes the positive sign for the square
root (giving an effective Higgs mode) when expanding it in
powers of a2/M A2,
n2
Aμ = aμ + MAnμ − 2MA a2nμ + O(1/M A2).
However, the contributions of all these Lorentz violating
couplings to physical processes completely cancel out among
themselves. So, SLIV is shown to be superficial as it affects
only the gauge of the vector potential Aμ at least in the tree
approximation [17].
Some time ago, this result was extended to the oneloop
approximation and for both the timelike (n2 > 0) and
spacelike (n2 < 0) Lorentz violation [19]. All the
contributions to the photon–photon, photon–fermion and fermion–
fermion interactions violating physical Lorentz invariance
were shown to exactly cancel among themselves in the
manner observed long ago by Nambu for the simplest treeorder
diagrams. This means that the constraint (
1
), having been
treated as a nonlinear gauge choice at the tree (classical) level,
remains as a gauge condition when quantum effects are taken
into account as well. So, in accordance with Nambu’s original
conjecture, one can conclude that physical Lorentz invariance
is left intact at least in the oneloop approximation, provided
we consider the standard gauge invariant QED Lagrangian
(
3
) taken in flat Minkowski spacetime. Later this result was
confirmed for the spontaneously broken massive QED [20],
nonAbelian theories [21] and tensorfield gravity [15,16].
The point is, however, that all these calculations represent
somewhat “empirical” confirmation of gauge invariance of
the nonlinear QED and other emergent theories rather than a
theoretical one. Indeed, whether the constraint (
1
) amounts
in general to a special gauge choice for a vector field is an
open question unless the corresponding gauge function
satisfying the constraint condition is explicitly constructed. We
discuss this important issue in more detail in Sect. 4.
Let us note that, in principle, the vector field constraint (
1
)
may be formally obtained in some limit from a conventional
potential that could be included in the QED Lagrangian (
3
),
U ( A) = λA( A2μ − n2 M A2)2,
thus extending QED to the socalled bumblebee model [22].
Here λA > 0 stands for the coupling constant of the vector
field, while values of n2 = ± 1 determine again its
possible vacuum configurations. Indeed, one can readily see that
the potential (
6
) inevitably causes spontaneous violation of
Lorentz symmetry in an ordinary way, much as an internal
symmetry violation is caused in the linear σ model for pions
[18]. As a result, one has a massive “Higgs” mode (with mass
2√2λA MA) together with massless Goldstone modes
associated with the photon components. However, as was argued in
[23], the bumblebee model adding the potential terms (
6
) to
the standard QED Lagrangian is generally unstable. Indeed,
its Hamiltonian appears to be unbounded from below unless
the phase space is constrained just by the nonlinear condition
A2μ = n2 M A2. With this condition imposed, the Hamiltonian
becomes positive, the massive Higgs mode never emerges,
and the model is physically equivalent to the Nambu model
[17]. Remarkably, this pure Goldstone theory limit can be
reached when, just as in the σ model for pions, one goes
from the linear model for the SLIV to the nonlinear one by
taking the limit λA → ∞. This immediately fixes in (
6
)
the vector field square to its vacuum value, thus leading to
the above constraint (
1
). As a matter of fact, the vector field
theory turns out to be stable in this limit only.
Actually, for the tensorfield gravity we use a similar
nonlinear constraint for a symmetric twoindex tensor field,
Hμ2ν = n2 M H2 ,
Hμ2ν ≡ Hμν H μν , n2 ≡ nμν nμν = ±1
(where nμν is now a properly oriented unit Lorentz tensor,
while MH is the proposed scale for Lorentz violation in the
gravity sector) which fixes its length in the same manner as
it appears for the vector field (
1
). Again, the nonlinear
constraint (
7
) may in principle appear from the standard potential
terms added to the tensorfield Lagrangian
U (H ) = λH (Hμ2ν − n2 M H2 )2
in the nonlinear σ model type limit when the coupling
constant λH goes to infinity. Just in this limit the tensor field
theory appears stable, though, due to the corresponding Higgs
mode excluded, it does not lead to physical Lorentz violation
[15,16].
(
6
)
(
7
)
(
8
)
Usually, an emergent gauge field framework is
considered either regarding emergent photons or regarding
emergent gravitons. For the first time, we consider it
regarding them both in the socalled electrogravity theory where
together with the Nambu QED model [17] with its gauge
invariant Lagrangian (
3
) we propose the linearized Einstein–
Hilbert kinetic term for the tensor field preserving a
diffeomorphism (diff) invariance. We show that such a combined
SLIV pattern, conditioned by the constraints (
1
) and (
7
),
induces the massless Goldstone modes which appear shared
among photon and graviton. Note that one needs in common
nine zero modes both for photon (three modes) and graviton
(six modes) to provide all necessary (physical and auxiliary)
degrees of freedom. They actually appear in our
electrogravity theory due to spontaneous breaking of high symmetries
of the constraints involved. While for the vector field case the
symmetry of the constraint coincides with the Lorentz
symmetry S O(
1, 3
), the tensor field constraint itself possesses
a much higher global symmetry S O(
7, 3
), whose
spontaneous violation provides a sufficient number of zero modes
collected in a graviton. These modes are largely
pseudoGoldstone modes (PGMs) since S O(
7, 3
) is a symmetry of
the constraint (7) rather than the electrogravity Lagrangian
whose symmetry is only given by Lorentz invariance. The
electrogravity theory we start with becomes essentially
nonlinear, when expressed in terms of the Goldstone modes, and
contains a variety of Lorentz (and CPT) violating couplings.
However, as our calculations show, all SLIV effects turn out
to be strictly cancelled in the low order physical processes
involved once the tensorfield gravity part of the
electrogravity theory is properly extended to general relativity (GR).
This can be taken as an indication that in the
electrogravity theory physical Lorentz invariance is preserved in this
approximation. Thereby, the lengthfixing constraints (
1
) and
(
7
) put on the vector and tensor fields appear as the gauge
fixing conditions rather than sources of the actual Lorentz
violation just as it was in the pure nonlinear QED framework
[17]. From this viewpoint, if this cancellation were to work in
all orders, one could propose that emergent theories, like as
the electrogravity theory, are not different from conventional
gauge theories. We argue, however, that even in this case
some observational difference between them could
unavoidably appear, if gauge invariance were presumably broken by
quantum gravity at the Planck scale order distances.
The paper is organized in the following way. In Sect. 2
we formulate the model for the tensorfield gravity and find
corresponding massless Goldstone modes some of which are
collected in the graviton. Then in Sect. 3 we consider in
significant detail the combined electrogravity theory consisting
of QED and tensor field gravity. In Sect. 4 we derive general
Feynman rules for basic interactions in the emergent
framework. The model appears to be in essence threeparametric
containing the inverse Planck and SLIV scales, 1/MP , 1/MA
and 1/MH , respectively, as the perturbation parameters, so
that the SLIV interactions are always proportional to some
powers of them. Further, some lowest order SLIV processes,
such as an elastic photon–graviton scattering and photon–
graviton conversion are considered in detail. We show that
all these effects, taken in the tree approximation, appear in
fact to be vanishing so that the physical Lorentz invariance
is ultimately restored. Finally, in Sect. 5 we present our
conclusion.
2 Tensorfield gravity
+∂ν Htr∂μ Hμν ,
We propose here, closely following Refs. [15,16], the
tensorfield gravity theory which mimics linearized general
relativity in Minkowski spacetime. The corresponding Lagrangian
for one real vector field Aμ (still representing all sorts of
matter in the model)
L(H, A) = L(H ) + L( A) + Lint
consists of the tensor field kinetic terms of the form
1 1
L(H ) = 2 ∂λ H μν ∂λ Hμν − 2 ∂λ Htr∂λ Htr − ∂λ H λν ∂μ Hμν
(Htr stands for the trace of Hμν , Htr = ημν Hμν ), which is
invariant under the diff transformations
δ Hμν = ∂μξν + ∂ν ξμ, δx μ = ξ μ(x ),
and the interaction terms
1
Lint(H, A) = − MP Hμν T μν ( A).
1
L( A) = − 4 Fμν F μν , T μν ( A)
= −F μρ Fρν + 41 ημν Fαβ F αβ .
The L( A) and T μν ( A) are the conventional free Lagrangian
and energymomentum tensor for a vector field
(
9
)
(
10
)
(
11
)
(
12
)
(
13
)
It is clear that, in contrast to the tensor field kinetic terms,
the other terms in (
9
) are only approximately invariant under
the diff transformations (
11
). They become more and more
invariant when the tensorfield gravity Lagrangian (
9
) is
properly extended to GR with higher terms in H fields included.1
Following the nonlinear σ model for QED [17], we propose
the SLIV condition (
7
) as some tensor field lengthfixing
constraint which is supposed to be substituted into the total
1 Such an extension means that in all terms included in the GR action,
particularly in the QED Lagrangian term, (−g)1/2gμν gλρ Fμλ Fνρ , one
expands the metric tensors
gμν = ημν + Hμν /MP , gμν = ημν − H μν /MP + H μλ Hλν /MP2 + · · ·
taking into account the higher terms in H fields.
Lagrangian L(H, A) prior to the variation of the action. This
eliminates, as was mentioned above, a massive Higgs mode in
the final theory, thus leaving only massless Goldstone modes,
some of which are then collected in a graviton.
Let us first turn to the spontaneous Lorentz violation itself
in a gravity sector, which is caused by the constraint (
7
),
while such a violation in a QED sector is assumed to be
determined by the constraint (
1
). The latter leads, as was
mentioned above, only to two possible breaking channels of
the starting Lorentz symmetry, namely to S O(
3
) or S O(
1, 2
),
depending on the timelike (n2 > 0) or spacelike (n2 < 0)
nature of SLIV. For the tensorfield constraint (7) the choice
turns out to be wider. Indeed, this constraint can be written
in the more explicit form
Hμ2ν = H020 + Hi2= j + (√2Hi = j )2
√
−( 2H0i )2 = n2 M H2 = ± M H2
for the negative sign. These cases can be readily derived
taking an appropriate exponential parametrization for the tensor
field:
Hαβ = einμν J μσ ηντ hστ /MH γ δ nγ δ MH ,
αβ
J
μν βα = i δαμηνβ − δαν ημβ
determined by the matrix nμν . The initial Lorentz symmetry
S O(
1, 3
) of the Lagrangian L(H, A) given in (9) then
formally breaks down at a scale MH to one of its subgroups.
If one assumes a “minimal” vacuum configuration in the
S O(
1, 3
) space with the VEV (15) developed on a single
Hμν component, there are in fact the following three
breaking channels:
(where the summation over all indices (i, j = 1, 2, 3) is
imposed) and means in essence that the tensor field Hμν
develops the vacuum expectation value (VEV) configuration
(a) n00 = 0, S O(
1, 3
) → S O(3),
(b) ni= j = 0, S O(
1, 3
) → S O(
1, 2
),
(c) ni = j = 0, S O(
1, 3
) → S O(
1, 1
),
for the positive sign in (
14
), and
(d)n0i = 0, S O(
1, 3
) → S O(2),
(
14
)
(
15
)
(
16
)
(
17
)
(
18
)
where the Lorentz generators J μν are explicitly included.2
Accordingly, there are only three Goldstone modes in the
cases (a, b) and five modes in the cases (c)–(d). In order to
associate at least one of the two transverse polarization states
of the physical graviton with these modes, one could have any
of the abovementioned SLIV channels except for the case
(a) where the only nonzero Goldstone modes are given by the
tensor components h0i (i = 1, 2, 3). Indeed, it is impossible
for a graviton to have all vanishing spatial components, as in
the case (a). However, these components may be provided by
some accompanying pseudoGoldstone modes, as we argue
below. Apart from the minimal VEV configuration, there are
many others as well. A particular case of interest is that of
the traceless VEV tensor nμν
nμν ημν = 0,
(
19
)
in terms of which the emergent gravity Lagrangian acquires
an especially simple form (see below). It is clear that the
VEV in this case can be developed on several Hμν
components simultaneously, which in general may lead to total
Lorentz violation with all six Goldstone modes generated.
For simplicity, we will use sometimes this form of vacuum
configuration in what follows, while our arguments can be
applied to any type of VEV tensor nμν .
Aside from the pure Lorentz Goldstone modes, the
question of the other components of the symmetric
twoindex tensor Hμν naturally arises. Remarkably, they turn
out to be pseudoGoldstone modes (PGMs) in the
theory. Indeed, although we only propose Lorentz invariance
of the Lagrangian L(H, A), the SLIV constraint (
7
)
formally possesses the much higher global accidental
symmetry S O(
7, 3
) of the constrained bilinear form (14), which
manifests itself when considering the Hμν components as
the “vector” ones under S O(
7, 3
). This symmetry is in
fact spontaneously broken, side by side with Lorentz
symmetry, at the scale MH . Assuming again a minimal
vacuum configuration in the S O(
7, 3
) space, with the VEV
(
15
) developed on a single Hμν component, we have either
timelike (S O(
7, 3
) → S O(
6, 3
)) or spacelike (S O(
7, 3
)
→ S O(
7, 2
)) violations of the accidental symmetry
depending on the sign of n2 = ±1 in (
14
). According to the number
of broken S O(
7, 3
) generators, just nine massless Goldstone
modes appear in both cases. Together with an effective Higgs
2 One may alternatively argue starting from the vector representation
of the higher S O(
7, 3
)symmetry determined by the constraint equation
(
14
) itself (see below). Thereby, one has a standard parametrization
HA = einM J MN hN /MH BA nB MH
where the “big” indices A, B, M, N correspond to the pairs of different
values of the old indices (μν) appearing in (
14
). Consequently, one
has the equality nM J M N h N = nμν J μσ ηντ hστ when going to the
standard Lorentz indices so that antisymmetry in the indices (M, N )
goes to antisymmetry in the index pairs (μν, σ τ ).
1
− 2MH MP
h2 nμν F μρ Fρν ,
written in the O(h2/M H2 ) approximation in which, besides
the conventional graviton bilinear kinetic terms, there are also
three and fourlinear interaction terms in powers of hμν in
the Lagrangian. Some of the notations used are collected
here:
h2 ≡ hμν hμν , htr ≡ ημν hμν .
The bilinear vector field term
MH
MP
nμν F μρ F ν
ρ
in the third line in the Lagrangian (
22
) merits special notice.
This term arises from the interaction Lagrangian Lint (
12
)
after application of the tracelessness condition (
19
) for the
VEV tensor nμν . It could significantly affect the dispersion
relation for the vector field A (and any other matter as well)
thus leading to an unacceptably large Lorentz violation if
the SLIV scale MH were comparable with the Planck mass
MP . However, this term can be gauged away [15,16] by an
appropriate redefinition of the vector field by going to the
new coordinates
(
22
)
(
23
)
(
24
)
(25)
(26)
component, on which the VEV is developed, they complete
the whole tencomponent symmetric tensor field Hμν of the
basic Lorentz group as is presented in its parametrization
(
20
). Some of them are true Goldstone modes of the
spontaneous Lorentz violation, others are PGMs since, as was
mentioned, an accidental S O(
7, 3
) symmetry is not shared
by the whole Lagrangian L(H, A) given in (9). Notably, in
contrast to the scalar PGM case [18], they remain strictly
massless being protected by the starting diff invariance which
becomes exact when the tensorfield gravity Lagrangian (
9
) is
properly extended to GR1. Owing to this invariance, some of
the Lorentz Goldstone modes and PGMs can then be gauged
away from the theory, as usual.
Now, one can rewrite the Lagrangian L(H, A) in terms of
the tensor Goldstone modes explicitly using the SLIV
constraint (
7
). For this purpose, let us take the following handy
parameterization for the tensor field Hμν :
+∂ν htr∂μhμν +
n2 1
− MH h2nμλ ∂λ∂ν hμν − 2 ∂μ∂λhtr
n2 nμλnνλ
+ 8M H2
MH
+L( A) + MP
ημν −
n2
∂μh2∂ν h2
1
nμν F μρ F ν
ρ − MP
hμν T μν
Hμν = hμν + nμν (M H2 − n2h2) 21 ,
n · h = 0 (n · h ≡ nμν hμν ).
Here hμν corresponds to the pure emergent modes
satisfying the orthogonality condition, while the effective “Higgs”
mode (or the Hμν component in the vacuum direction) is
given by the square root for which we take again the positive
sign when expanding it in powers of h2/M H2 (h2 ≡ hμν hμν )
2 2
n h
Hμν = hμν + nμν MH − 2MH + O(1/M H2 ).
It should be particularly emphasized that the modes
collected in the hμν are generally the Goldstone modes of the
broken accidental S O(
7, 3
) symmetry of the constraint (7),
thus containing the Lorentz Goldstone modes and PGMs put
together. If Lorentz symmetry is completely broken then the
pure Goldstone modes appear enough to be solely collected
in a physical graviton. On the other hand, when one has a
partial Lorentz violation, some PGMs should be added.
Putting then the parameterization (
21
) into the total
Lagrangian L(H, A) given in (
9
), (
10
), and 12), one arrives at
the truly emergent tensorfield gravity Lagrangian L(h, A),
containing an infinite series in powers of the hμν modes. For
the traceless VEV tensor nμν (
19
) it takes, without loss of
generality, the especially simple form
1 1
L(h, A) = 2 ∂λhμν ∂λhμν − 2 ∂λhtr∂λhtr − ∂λhλν ∂μhμν
(
20
)
(
21
)
x μ → x μ + ξ μ.
ξ μ(x ) =
MH nμν xν ,
MP
In fact, with a simple choice of the parameter function ξ μ(x )
being linear in the 4coordinate,
the term (
24
) is cancelled by an analogous term stemming
from the vector field kinetic term in L( A) given in (
12
). On
the other hand, since the diff invariance is an approximate
symmetry of the Lagrangian L(H, A) we started with (
9
),
this cancellation will only be accurate up to the linear order
corresponding to the tensor field theory. Indeed, a proper
extension of this theory to GR1 with its exact diff invariance
will ultimately restore the usual dispersion relation for the
vector (and other matter) fields. We will consider all that in
significant detail in the next section.
Together with the Lagrangian one must also specify other
supplementary conditions for the tensor field hμν (appearing
eventually as possible gauge fixing terms in the emergent
tensorfield gravity) in addition to the basic emergent “gauge”
condition nμν hμν = 0 given above (
20
). The point is that the
spin 1 states are still left in the theory being described by some
of the components of the new tensor hμν . This is certainly
inadmissible.3 Usually, the spin 1 states (and one of the spin
3 Indeed, the spin1 component must be necessarily excluded in the
tensor hμν , since the sign of the energy for the spin1 component is
always opposite to that for the spin2 and spin0 ones.
0 states) are excluded by the conventional Hilbert–Lorentz
condition,
So, the total starting electrogravity Lagrangian is, therefore,
proposed to be
(27)
Ltot = L( A) + L(H ) + L(ϕ) + Lint(H, A, ϕ)
∂μhμν + q∂ν htr = 0
(q is an arbitrary constant, giving for q = −1/2 the standard
harmonic gauge condition). However, as we have already
imposed the emergent constraint (
20
), we cannot use the
full Hilbert–Lorentz condition (27) eliminating four more
degrees of freedom in hμν . Otherwise, we would have an
“overgauged” theory with a nonpropagating graviton. In
fact, the simplest set of conditions which conform with the
emergent condition n · h = 0 in (
20
) turns out to be
∂ρ (∂μhνρ − ∂ν hμρ ) = 0.
(28)
This set excludes only three degrees of freedom4 in hμν
and, besides, it automatically satisfies the Hilbert–Lorentz
spin condition as well. So, with the Lagrangian (
22
) and
the supplementary conditions (
20
) and (28) lumped together,
one eventually arrives at a working model for the emergent
tensorfield gravity [15,16]. Generally, from ten components
of the symmetric twoindex tensor hμν four components are
excluded by the supplementary conditions (
20
) and (28). For
a plane gravitational wave propagating in, say, the z direction
another four components are also eliminated, due to the fact
that the above supplementary conditions still leave freedom
in the choice of a coordinate system, x μ → x μ+ξ μ(t −z/c),
much as in standard GR. Depending on the form of the
VEV tensor nμν , caused by SLIV, the two remaining
transverse modes of the physical graviton may consist solely of
Lorentzian Goldstone modes or of pseudoGoldstone modes,
or include both of them. This theory, similar to the nonlinear
QED [17], while suggesting an emergent description for the
graviton, does not lead to physical Lorentz violation [15,16].
3 Electrogravity theory
3.1 Emergent photons and gravitons together
So far we considered the vector field Aμ as an unconstrained
material field which the emergent gravitons interact with.
Now, we propose that the vector field also develops the VEV
through the SLIV constraint (
1
), thus generating the massless
vector Goldstone modes associated with a photon. We also
include the complex scalar field ϕ (taken to be massless, for
simplicity) as an actual matter in the theory
L(ϕ) = Dμϕ Dμϕ ∗ ,
Dμ = ∂μ + i e Aμ.
(29)
4 The solution for a gauge function ξμ(x) satisfying the condition (28)
can generally be chosen as ξμ = −1(∂ρ hμρ ) + ∂μθ, where θ(x) is an
arbitrary scalar function, so that only three degrees of freedom in hμν
are actually eliminated.
where the Lagrangians L( A) and L(H ) were given above in
Eqs. (
13
) and (
10
), while the gravity interaction part,
1
Lint(H, A, ϕ) = − MP Hμν [T μν ( A) + T μν (ϕ)],
contains the tensor field couplings with canonical
energymomentum tensors of vector and scalar fields.
In the symmetry broken phase one goes to the pure
Goldstone vector and tensor modes, aμ and hμν , respectively.
Whereas the tensor modes hμν , including their kinetic and
interaction terms, have been thoroughly discussed in terms of
Eq. (
22
), the vector modes aμ are not yet properly exposed.
Putting the parametrization (
4
) into the Lagrangian (
13
) one
has from the vector field kinetic term (taken to the first order
in a2/M A2)
1 1
L( A) → L(a) = − 4 fμν f μν − 2 δ(n · a)2
1 n2
− 4 MA
fμν (∂μν a2).
We have denoted the aμ strength tensor by fμν = ∂μaν −
∂ν aμ, while ∂μν = nμ∂ν − nν ∂μ is a new SLIV oriented
differential tensor acting on the infinite series in a2 coming from
the expansion of the effective “Higgs” mode (M A2 −n2a2) 21 in
(
4
), from which we have only included the lowest order term
−n2a2/2MA throughout the Lagrangian L(a). We have also
explicitly introduced in the Lagrangian the emergent
orthogonality condition n · a = 0, which can be treated as the gauge
fixing term (when taking the limit δ → ∞). At the same time,
the scalar field Lagrangian L(ϕ) in (30) is going now to
L(ϕ) =
∂μ + i eaμ + i e MAnμ − i e
n2
2MA
2
a2nμ ϕ ,
(30)
(31)
(32)
(33)
while the tensor field interacting terms (31) in Lint(H, A, ϕ)
convert to
1
Lint = − MP
n2
hμν + MH nμν − 2MH h2nμν
n2
× T μν aμ − 2MA a2nμ
+ T μν (ϕ) ,
(34)
where the vector field energymomentum tensor is now solely
a function of the Goldstone aμ modes.
3.2 Constraints and zero mode spectrum Before going any further, let us make some necessary comments. Note first of all that, apart from dynamics described
by the total Lagrangian Ltot, the vector and tensor field
constraints (
1
) and (
7
) are also proposed. In principle, these
constraints could be formally obtained from the conventional
potential terms included in the total Lagrangian Ltot, as was
discussed in Sect. 1. The most general potential, where the
vector and tensor field couplings possess the Lorentz and
S O(
7, 3
) symmetry, respectively, must be solely a function of
A2μ ≡ Aμ Aμ and Hμ2ν ≡ Hμν H μν . Indeed, it cannot include
any contracted and intersecting terms like Htr, H μν Aμ Aν
and others, which would immediately reduce the above
symmetries to the common Lorentz one. So, one may only write
U ( A, H ) = λA( A2μ − n2 M A2)2 + λH (Hμ2ν − n2 M H2 )2
where λA,H,AH stand for the coupling constants of the vector
and tensor fields, while the values of n2 = ±1 and n2 = ±1
determine their possible vacuum configurations. As a
consequence, an absolute minimum of the potential (35) might
appear for the couplings satisfying the conditions
+λAH A2μ Hρ2ν
λA,H > 0, λAλH > λAH /4.
However, as in the pure vector field case discussed in Sect. 1,
this theory is generally unstable with the Hamiltonian being
unbounded from below unless the phase space is constrained
just by the above nonlinear conditions (
1
) and (
7
). They in
turn follow from the potential (35 ) when going to the
nonlinear σ model type limit λA,H → ∞. In this limit, the
massive Higgs mode disappears from the theory, the Hamiltonian
becomes positive, and one arrives at the pure emergent
electrogravity theory considered here.
We note again that the Goldstone modes appearing in the
theory are caused by breaking of global symmetries related
to the constraints (
1
) and (
7
) rather than directly to Lorentz
violation. Meanwhile, for the vector field case symmetry of
the constraint (
1
) coincides in fact with Lorentz symmetry
whose breaking causes the Goldstone modes depending on
the vacuum orientation vector nμ, as can be clearly seen from
an appropriate exponential parametrization for the starting
vector field,
Aα = einμJ μνaν /MA
β
α
nβ MA
where nμJ μν just corresponds to the broken Lorentz
generators. However, in the tensor field case, due to the higher
symmetry S O(
7, 3
) of the constraint (7), there are much more
tensor zero modes than would appear from SLIV itself. In
fact, they complete the whole tensor multiplet hμν in the
parametrization (
20
). However, as was discussed in the
previous section, only a part of them are true Goldstone modes;
the others are pseudoGoldstone ones. In the minimal VEV
configuration case, when these VEVs are developed only on
the single Aμ and Hμν components, one has several
possibilities determined by the vacuum orientations nμ and nμν in
(35)
(36)
(37)
Eqs. (37), (
16
), (
17
) and (
18
), respectively. There appear 12
zero modes in total, three from Lorentz violation itself and
nine from a violation of the S O(
7, 3
) symmetry, which is
more than enough to have the necessary three photon modes
(two physical and one auxiliary ones) and six graviton modes
(two physical and four auxiliary ones). We list below all
interesting cases classifying them according to the corresponding
n − n values.
(
1
) For the timelike–timelike SLIV, when both n0 = 0
and n00 = 0, the photon is determined by the space
Goldstone components ai (i = 1, 2, 3) of the partially broken
Lorentz symmetry S O(
1, 3
) → S O(3), while the space–
space components hi j needed for physical graviton and its
auxiliary components can be only provided by the
pseudoGoldstone modes following from the timelike symmetry
breaking S O(
7, 3
) → S O(
6, 3
) related to the tensorfield
constraint (
7
).
(
2
) Another interesting case seems to be the timelike–
spacelike SLIV, when n0 = 0 and ni= j = 0 (one of the
diagonal space components of the unit tensor nμν is nonzero).
Now, Lorentz symmetry is broken up to the plane rotations
S O(
1, 3
) → S O(2), so that the five true Goldstone bosons
appear shared among photon and graviton in the following
way. The photon is given again by three space components ai ,
while the graviton is determined by two space–space
components, h12 and h13 (if the VEV was developed along the
direction n11), as directly follows from the parametrization
Eqs. (37) and (
18
). Thus, again one necessary component h23
for physical graviton, as well as its gauge degrees of freedom,
should be provided by the proper pseudoGoldstone modes
following from the spacelike symmetry breaking S O(
7, 3
)
→ S O(
7, 2
) related to the tensorfield constraint (
7
).
(
3
) For the similar timelike–spacelike SLIV case, when
n0 = 0 and ni = j = 0 (one of the nondiagonal space
components of the unit tensor nμν is nonzero), the Lorentz symmetry
appears to be fully broken so that the photon has the same
three space components ai , while the graviton physical
components are given by the tensor field space components hi j .
This is the only case when all physical components of both
photon and graviton are provided by the true SLIV
Goldstone modes, whereas some gauge degrees of freedom for
a graviton are given by the PGM states stemming from the
spacelike symmetry breaking S O(
7, 3
) → S O(
7, 1
) related
to the tensor field constraint (
7
).
(
4
) Using the parametrization equations (37) and (
18
) one
can readily consider all other possibilities as well;
particularly, the spacelike–timelike (nonzero ni and n00), spacelike–
spacelike diagonal (nonzero ni and ni= j ) and spacelike–
spacelike nondiagonal (nonzero ni and ni = j ) cases. In all
these cases, while the photon may only contain true
Goldstone modes, some pseudoGoldstone modes appear to be
necessary so as to be collected in the graviton together with
some true Goldstone modes.
3.3 Emergent electrogravity interactions
To proceed, one should eliminate, first of all, the large terms
of the false Lorentz violation proportional to the SLIV scales
MA and MH in the interaction Lagrangians (33) and (34).
Arranging the phase transformation for the scalar field in the
following way:
ϕ → ϕ exp[−i eMAnμx μ]
one can simply cancel that large term in the scalar field
Lagrangian (33), thus arriving at
where the covariant derivative Dμ is read from now on as
Dμ = ∂μ + i eaμ. Another unphysical set of terms, like the
already discussed term (
24
), may appear from the gravity
interaction Lagrangian Lint (34) where the large SLIV entity
MH nμν couples to the energymomentum tensor. They also
can be eliminated by going to the new coordinates (25), as
was mentioned in the previous section.
For infinitesimal translations ξμ(x ) the tensor field
transforms according to (
11
), while the scalar and vector fields
transform as
δϕ = ξμ∂μϕ, δaμ = ξλ∂λaμ + ∂μξν aν ,
respectively. One can see, therefore, that the scalar field
transformation has only the translation part, while the vector one
has an extra term related to its nontrivial Lorentz structure.
For the constant unit vector nμ this transformation looks as
δnμ = ∂μξν nν ,
having no translation part. Using all that and also expecting
that the phase parameter ξλ is in fact linear in coordinate xμ
(which allows one to drop its highderivative terms), we can
easily calculate all scalar and vector field variations, such as
δ Dμϕ
= ξλ∂λ(Dμϕ) + ∂μξλ Dλϕ, δ fμν = ξλ∂λ fμν
+∂μξ λ fλν + ∂ν ξ λ fμλ
(38)
(39)
(40)
(41)
(42)
and others. This finally leads to the total variations of the
above Lagrangians. Whereas the pure tensorfield Lagrangian
L(H ) (
10
) is invariant under diff transformations, δL(H ) =
0, the interaction Lagrangian Lint in (30) is only
approximately invariant, being compensated (in the lowest order in
the transformation parameter ξμ) by kinetic terms of all the
fields involved. However, this Lagrangian becomes
increasingly invariant once our theory is extending to GR1.
In contrast, the vector and scalar field Lagrangians acquire
some nontrivial additions,
n2
× f μν fνλ + MA
nλ Jλ ,
where Jμ stands for the conventional vector field source
current
Jμ = i e[ϕ∗ Dμϕ − ϕ Dμϕ ∗],
while Dν ϕ is the SLIV extended covariant derivative for the
scalar field
Dν ϕ = Dν ϕ − i e
n2
2MA
a2nν ϕ.
The first terms in the variations (43) are unessential, since
they simply show that these Lagrangians transform, as usual,
like scalar densities under diff transformations.
Combining these variations with Lint (34) in the total
Lagrangian (30) one finds after simple, though long,
calculations that the largest Lorentz violating terms it contains,
(43)
(44)
(45)
(47)
(48)
−
MH ∂μξλ + ∂λξμ
MP nμν − 2
n2
× − f μν f λ
ν − MA
fλν ∂μλa2 + 2Dν ϕ Dμϕ ∗ , (46)
will immediately cancel if the transformation parameter is
chosen exactly as in Eq. (26). So, with this choice we finally
have for the modified interaction Lagrangian
1 1
Lint(h, a, ϕ) = − MP hμν T μν (a, ϕ) + MP MA L1
1 MH
+ MP MH L2 + MP MA L3
where
1
− 4 fλρ ∂λρ a2 + nλ Jλ
L1 = n2hμν fλν ∂μλa2 − nμ J ν + ημν
L2 = 21 n2h2nμν − f μλ fλν + 2Dν ϕ Dμϕ ∗ ,
1
L3 = n2nμλ 2 fρν ∂ρν aμaλ
− (aμaλ)nν Jν .
Thereby, apart from a conventional gravity interaction part
given by the first term in (47), there are Lorentz violating
couplings in L1,2,3 being properly suppressed by corresponding
mass scales. Note that the coupling presented in L3 between
the vector and scalar fields is solely induced by the
tensorfield SLIV. Remarkably, this coupling may be in principle
of the order of a normal gravity coupling or even stronger,
if MH > MA. However, appropriately simplifying this
coupling (and using also a full derivative identity) one arrives
at
L3 ∼ n2 nμλaμaλ nρ ∂ν fνρ − Jρ ,
which, after applying of the vector field equation of motion,
turns it into zero. We consider it in more detail in the next
section where we calculate some tree level processes.
(49)
4 The lowest order SLIV processes
4.1 Preamble
The emergent gravity Lagrangian in (
22
) taken alone or
considered together with the material vector and scalar fields
presents in fact a highly nonlinear theory which contains
lots of Lorentz and CPT violating couplings. Nevertheless,
as shown in [15,16] in the lowest order calculations, they
all are cancelled and do not manifest themselves in physical
processes. This may mean that the lengthfixing constraints
(
7
) put on the tensor fields appear as gauge fixing conditions
rather than a source of an actual Lorentz violation.
However, as was mentioned in Sect. 1, one cannot be sure
that these calculations, as well as the similar calculations in
the Nambu model itself, fully confirm gauge invariance of
the emergent theory considered. Indeed, whether the
constraint (
1
) in QED amounts in general to a special gauge
choice for a vector field Aμ(x ) is an open question unless the
corresponding gauge function ω(x ) satisfying the constraint
condition
[ Aμ(x ) + ∂μω(x )]2 = n2 M A2
is explicitly constructed for an arbitrary Aμ(x ). The
original Nambu argument [17] was related to the observation that
for the positive n2 the constraint equation (50) is
mathematically equivalent to a classical Hamilton–Jacobi equation for
a massive charged particle,
[∂μ S(x ) + e Aμ(x )]2 = m2,
where S(x ) is an action of a system, while e and m stand
for the particle charge and mass, respectively. Comparison of
Eqs. (50 ) and (51) shows the correspondence ω(x ) = S(x )/e
and n2 M A2 = m2/e2. Thus, the constraint equation (50)
should have a solution inasmuch as there is a solution to
the classical problem described by Eq. (51). This
conclusion was actually confirmed by Nambu for the timelike SLIV
(n2 = +1) in the lowest order calculation of the physical
processes in [17] and then was extended to the oneloop
approximation and for both the timelike (n2 > 0) and spacelike
(n2 < 0) Lorentz violation in [19]. Thus, the status of the
constraint (
1
) as a special gauge choice in QED is only
par(50)
(51)
tially confirmed by some low order calculations rather than
having a serious theoretical reason.
The same may be said of the emergent tensorfield gravity.
Its diff gauge invariance could only be fully approved if the
corresponding gauge function ξμ(x ) satisfying the constraint
condition (
7
)
(52)
[Hμν (x ) + ∂μξν (x ) + ∂ν ξμ]2 = n2 M H2
is explicitly constructed for an arbitrary Hμν (x ). However, in
contrast to the above nonlinear QED case where at least some
heuristic argument could be applied, one cannot be sure that
there exists a solution to Eq. (52) in a general case. So, the
only way to answer this question is to explicitly check it in
physical processes that in the lowest approximation has been
done in [15,16]. Again, though the result appears positive,
one cannot be sure that this will work in all orders.
The present electrogravity theory, in contrast to the pure
QED and tensorfield gravity theories, contains both the
photon and the graviton as the emergent gauge fields. This adds
new variety of Lorentz and CPT violating couplings (47),
being expressed in terms of tensor and vector Goldstone
modes. In general, one cannot be sure that, even though
both the emergent QED and the tensorfield gravity taken
separately preserve Lorentz invariance (in the low order
processes), the combined electrogravity theory does not lead to
physical Lorentz violation as well. However, as shown by our
calculations given below, just this appears to be the case. All
Lorentz violation effects turn out again to be strictly cancelled
among themselves at least in the lowest order SLIV processes
in the electrogravity theory. Thus, similar to emergent
vector field theories, both Abelian [17,19,20] and nonAbelian
[21], as well as in the pure tensorfield gravity [15,16], such a
cancellation may only mean that at least in the lowest
approximation the SLIV constraints (
1, 7
) amount to a special gauge
choice in the otherwise diff and Lorentz invariant emergent
electrogravity theory presented here.
We will consider the lowest order SLIV processes, once
the corresponding Feynman rules are properly established.
For simplicity, both in the above Lagrangians and in
forthcoming calculations, we continue to use the tracelessness of
the VEV tensor nμν (
19
), while our results remain true for
any type of vacuum configuration caused by SLIV.
4.2 Feynman rules
Though the Feynman rules and processes related to the
nonlinear QED, as well as with emergent gravity with the matter
scalar fields, are thoroughly discussed in our previous works
[15,20], there are many new Lorentz and CPT breaking
interactions in the total interaction Lagrangian (47). We present
below some basic Feynman rules which are needed for
calculations of different SLIV processes just appearing in the
emergent electrogravity.
(i) The first and most important is the graviton propagator
which only conforms with the emergent gravity Lagrangian
(
22
) and the gauge conditions (
20
) and (28),
1
− i Dμναβ (k) = 2k2 ηβμηαν + ηβν ηαμ − ηαβ ημν
+ nαρ kρ kν kμkβ + nβρ kρ kν kαkμ
(53)
(where (nkk) ≡ nμν kμkν and (knnk) ≡ kμnμν nνλkλ).
It automatically satisfies the orthogonality condition nμν
Dμναβ (k) = 0 and onshell transversality kμkν Dμναβ (k, k2
= 0) = 0. This is consistent with the corresponding
polarization tensor μν (k, k2 = 0) of the free tensor fields, being
symmetric, transverse (kμ μν = 0), traceless (ημν μν (k) = 0)
and also orthogonal to the vacuum direction, nμν μν (k) = 0.
As one can see, only standard terms given by the first
bracket in (53) contribute when the propagator is sandwiched
between the conserved energymomentum tensors of matter
fields, and the result is always Lorentz invariant.
We will also need the photon propagator,
i k2 Dμν = ημν − nμkνn+· knν kμ + (n n· 2k)2 kμkν , (54)
which in accordance with the vector field Lagrangian
(32) possesses the following properties: nμ Dμν = 0 and
kμ Dμν (k2 = 0) = 0.
(ii) Next is the threegraviton vertex hhh, again from the
Lagrangian (
22
), with graviton polarization tensors (and
4momenta) given by αα (k1), ββ (k2) and γ γ (k3) we have
αα ββ γ γ
3h
i
= 2MH
[ ηβγ ηβ γ
+ ηβγ ηβ γ
Pαα (k1)
+ ηαγ ηα γ
+ ηαγ ηα γ
+ ηβαηβ α + ηβα ηβ α
Pββ (k2)
Pγ γ (k3)
where the momentum tensor Pμν (k) is
Pμν (k) = nνρ kρ kμ + nμρ kρ kν − ημν (nkk).
Note that all 4momenta at the vertices are taken ingoing
throughout.
(iii) Next, we address the contact tensor–tensor–vector–
vector interaction coupling hhaa coming from the Lagrangian
L2 in (48). However, it would be useful to give first the
standard tensor–vector–vector vertex haa with tensor and vector
(55)
(56)
field polarizations, αα and ξ μ,ν , respectively,
αα μν
st
i
= − MP
T αα μν (aλ)
where T αα μν (aλ) stands for the conserved
energymomentum tensor of the Goldstone vector field aλ,
T αα μν
ηαμ ηα ν + ηαν ηα μ
One can see that this tensor is symmetric both in the (α, α )
and (μ, ν) indices, though these pairs are not interchangeable.
Using all that we are ready now to give the contact tensor–
tensor–vector–vector interaction vertex,
with the corresponding tensor field (ββ and γ γ ) and vector
field (μ, ν) polarization indices.
(iv) We have also to derive the fourlinear tensor–vector
interaction vertex haaa coming from the Lagrangian L1 in
(48). Note that the last term in it which is proportional to htr
will not contribute in the processes with graviton on external
lines, since its polarization tensor is traceless. For the other
terms one has the vertex
αα μνλ
h3a
i n2
= MP MA
ηαμ ηα ν + ηαν ηα μ
× ηνλ (n · k1) ( p + k1)μ ην μ
+nμ ( p + k1)ρ k1ν ηρμ − k1ρ ην μ
+ημλ (n · k2) ( p + k2)μ ην ν
+nμ ( p + k2)ρ k2ν ηρν − k2ρ ην ν
+ηνμ (n · k3) ( p + k3)μ ην λ
+nμ ( p + k3)ρ k3ν ηρλ − k3ρ ην λ
where the polarization αα ( p) stands interacting tensor field,
while polarizations ξ1μ(k1), ξ2ν (k2), ξ3λ(k3) for interacting
vector fields.
(v) For the threevector Goldstone mode interaction aaa
we have the wellknown vertex [20] following for the pure
vector field Lagrangian (32),
μνλ
3a
= −i
n2
MA
+(n · k3)ημν k3λ ,
(n · k1)ηνλk1μ + (n · k2)ημλk2ν
(57)
,
(58)
(59)
(60)
(61)
(62)
(63)
(64)
and the new one coming from the Lagrangian L3 in (48),
(n · k1)nνλk1μ + (n · k2)nμλk2ν
3μaνλ = i
where Jν is the conserved scalar field current discussed in
the previous section.
These are rules that are actually needed to calculate the
lowest order SLIV processes mentioned above. Note also
that some of these processes could in principle appear in
the pure nonlinear QED [20] or in the nonlinear tensorfield
gravity [15,16] where, as is well known, all the physical
Lorentz violation effects are eventually vanishing. Therefore,
we consider the SLIV contributions which only appear in
the combined nonlinear vector–tensor electrogravity theory
presented here.
4.3 Elastic photon–graviton scattering
This SLIV part of this process, γ + g → γ + g, may only
be of order of 1/MP MH due to the emergent nature of the
graviton. There are in fact two matrix elements: the first one
is related to the contact diagram with the hhaa vertex (60),
i n2
Mcon = MP MH ( 1 · 2) nαα T αα μν ξ3μξ4ν ,
while the second one is related to the pole diagram with the
longitudinal graviton exchange between the Lorentz
violating h3 (55) and the standard haa (57) vertices,
M pole = 1γ γ 2ββ 3αhα ββ γ γ Dαα λρ (q) sλtρμν ξ3μξ4ν , (66)
with the graviton and photon polarizations 1,2 and ξ3,4,
respectively (q is the momentum of the propagating
graviton).
Note now that all the terms in the propagator Dαα λρ which
are proportional to the propagating momentum will render
the energymomentum tensor to zero, thus there are left only
a few terms in the pole matrix element M2. Using also the
fact that in the vertex 3h there survives one term only when
the transversality and tracelessness of a graviton is used we
arrive at
i n2 i
M pole = MH ( 1 · 2) 2nαρ qρ qα − ηαα (nqq) q2 ·
which after evident simplifications is exactly cancelled with
the contact matrix element M1 given above in (65),
Mtot = Mcon + Mpole = 0.
Thereby, physical Lorentz invariance is left intact in the
emergent graviton–photon scattering.
(68)
4.4 Photon–graviton conversion
This SLIV process γ + g → γ + γ appears in the order
of 1/MA MP (now, due to the emergent nature of photon).
Again, this process in the tree approximation is basically
related to the interplay between the contact and pole
diagrams.
The contact haaa diagram being determined by the
interaction vertex (61) has a matrix element
Mcon = αα ( p) hα3αaμνλξ1μ(k1)ξ2ν (k2)ξ3λ(k3)
(69)
where the polarization αα belongs to the graviton, while the
polarizations ξ1μ(k1), ξ2ν (k2), ξ3λ(k3) belong to the photons
( p and k1 are incoming and k2 and k3 outgoing momenta).
In turn, the pole diagrams with the longitudinal photon
exchange between the Lorentz violating a3 (62) and the
standard haa (57) vertices consist in fact of three diagrams
differing from each other by the interchangeable external photon
legs. Their total matrix element is
where q is the propagating momentum, while the momenta
of the graviton and photons, p and k1,2,3, refer to the
polarizations αα and ξ1,2,3.
Using again the orthogonality properties and mass shell
conditions for polarizations of the photons and graviton one
can split the contact amplitude (69) into three terms which
exactly cancel the corresponding terms in the pole amplitude
(70). So, we will not have any physical Lorentz violation in
this process as well.5
4.5 Elastic photon–scalar scattering
One can also consider a new type of vector field scattering
process on the charged scalar, γ + s −→ γ + s, appearing at
the e MH /MP MA order due to the emergent nature of both
5 Note that together with the pure QED a3 vertex (62) we could also use
the new a3 vertex (63) in the above pole diagrams. This would give some
new contribution into this process with the lesser order MH /MA MP2 .
One may expect, however, that such a contribution will be cancelled by
the corresponding contact term appearing in the same order when going
to GR (see the footnote1).
(65)
αα μν Dνλ(q) 3λaρσ (ξ1μξ2ρ ξ3σ
M pole = αα st
+ξ2μξ1ρ ξ3σ + ξ3μξ1ρ ξ2σ ),
(70)
×
ηαληα ρ + ηα ληαρ − ηαα ηλρ
2
qα qαnλρ
− (nqq)
−i T λρμν ξ3μξ4ν ,
MP
(67)
photon and graviton.6 Again, there are contact and pole
diagrams for this process which cancel each other. The contact
diagram corresponds to the vertex aϕϕ∗ (64) appearing from
the Lagrangian L3 in (48) and leads to the matrix element
Mcon = i
Meanwhile, for the pole diagram with the longitudinal photon
exchange between the Lorentz violating aaa vertex (63) and
the standard scalar field current (44) in L3 one has, using the
mass shell properties of the vector field polarization,
Mpole = i
n2 MH
This amplitude, when applying an explicit form of propagator
and the current conservation qρ Jρ = 0, is exactly cancelled
with the contact one (71). Therefore, we show once again that
there is no real physical SLIV effect in the theory considered.
4.6 Other processes
Many other tree level Lorentz violating processes related to
gravitons and vector fields (interacting with each other and
the matter scalar field in the theory) appear in higher orders
in the basic SLIV parameters 1/MH and 1/MA, by
iteration of the couplings presented in our basic Lagrangians (
22
)
and (47) or from further expansions of the effective vector
and tensorfield Higgs modes (
5
) and (
21
) inserted into the
starting total Lagrangian (30). Again, their amplitudes are
essentially determined by an interrelation between the
longitudinal graviton and photon exchange diagrams and the
corresponding contact interaction diagrams, which appear to
cancel each other, thus eliminating physical Lorentz
violation in the theory.
Most likely, the same conclusion could be expected for
SLIV loop contributions as well. Actually, as in the massless
QED case considered earlier [19], the corresponding
oneloop matrix elements in our emergent electrogravity theory
could either vanish by themselves or amount to the
differences between pairs of similar integrals whose integration
variables are shifted relative to each other by some
constants (being in general arbitrary functions of the external
4momenta of the particles involved) which, in the
framework of dimensional regularization, could lead to their total
cancellation.
So, the emergent electrogravity theory considered here
is likely to eventually possess physical Lorentz invariance
6 Note that a similar SLIV process can independently appear in the
pure nonlinear scalar QED including the Lagrangians (32) and (33).
However, it was shown [20] that the corresponding Lorentz violation
terms are strictly cancelled in this scattering process.
provided that the underlying gauge and diff invariance in the
theory remains unbroken.
5 Conclusion
We have developed an emergent electrogravity theory
consisting of the ordinary QED and the tensorfield gravity model
(which mimics the linearized general relativity in Minkowski
spacetime) where both photons and gravitons emerge as
states solely consisting of massless Goldstone and
pseudoGoldstone modes. This appears due to spontaneous
violation of Lorentz symmetry incorporated into global
symmetries of the lengthfixing constraints put on the starting vector
and tensor fields, A2μ = ±M A2 and Hμ2ν = ±M H2 (MA and
MH are the proposed symmetry breaking scales). While for
the vector field case the symmetry of the constraint
coincides with Lorentz symmetry S O(
1, 3
) of the
electrogravity Lagrangian, the tensor field constraint itself possesses
the much higher global symmetry S O(
7, 3
), whose
spontaneous violation provides a sufficient number of zero modes
collected in a graviton. Accordingly, while the photon may
only contain true Goldstone modes, the graviton appears
at least partially composed from pseudoGoldstone modes
rather than from pure Goldstone ones. Thereby, the SLIV
pattern related to breaking of the constraint symmetries, due
to which the true vacuum in the theory is chosen, induces a
variety of zero modes shared among photon and graviton.
This theory looks essentially nonlinear and contains a
variety of Lorentz and CPT violating couplings, when expressed
in terms of the pure tensor Goldstone modes. Nonetheless, all
the SLIV effects turn out to be strictly cancelled in the lowest
order processes considered. This can be taken as an
indication that in the electrogravity theory physical Lorentz
invariance is preserved in this approximation. Thereby, the
lengthfixing constraints (
1
) and (
7
) put on the vector and tensor
fields appear as gauge fixing conditions rather than sources
of the actual Lorentz violation in the gauge and diff invariant
Lagrangian (30) we started with. In fact, some Lorentz
violation through deformed dispersion relations for the material
fields involved would appear in the interaction sector (34),
which only possesses an approximate diff invariance.
However, a proper extension of the tensorfield theory to GR,
with its exact diff invariance, ultimately restores the normal
dispersion relations and, therefore, the SLIV effects are
cancelled at least in the lowest order considered. If this
cancellation were to work in all orders, one could propose that
emergent theories, like as the electrogravity theory, are not differed
from conventional gauge theories. Accordingly, spontaneous
Lorentz violation caused by the vector and tensorfield
constraints (
1
) and (
7
) appear hidden in the gauge degrees of
freedom, and only results in a noncovariant gauge choice in
an otherwise gauge invariant emergent electrogravity theory.
From this standpoint, the only way for physical Lorentz
violation to occur would be if the above gauge invariance
were slightly broken at distances of the order of the Planck
scale, which could be presumably caused by quantum gravity.
This is in fact a place where the emergent vector and
tensorfield theories may drastically differ from conventional QED,
Yang–Mills and GR theories where gauge symmetry
breaking could hardly induce physical Lorentz violation. In
contrast, in emergent electrogravity such breaking could readily
lead to many violation effects including deformed dispersion
relations for all matter fields involved. Another basic
distinction of emergent theories with nonexact gauge invariance
is a possible origin of a mass for graviton and other gauge
fields (namely, for the nonAbelian ones, see [21]), if they,
in contrast to the photon, are partially composed of
pseudoGoldstone modes rather than of pure Goldstone ones. Indeed,
these PGMs are no longer protected by gauge invariance and
may properly acquire tiny masses, which still do not
contradict experiment. This may lead to a massive gravity theory
where the graviton mass emerges dynamically, thus avoiding
the notorious discontinuity problem [24].
So, while emergent theories with an exact local invariance
are physically indistinguishable from conventional gauge
theories, there are some principal distinctions when this local
symmetry is slightly broken, which could eventually allow us
to differentiate between the two types of theory in an
observational way. We may return to a more detailed consideration
of this interesting point elsewhere.
Acknowledgements We would like to thank Colin Froggatt, Archil
Kobakhidze, Rabi Mohapatra and Holger Nielsen for useful
discussions and comments. Z.K. acknowledges financial support from Shota
Rustaveli National Science Foundation (Grant # YS201681).
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.
1. J.D. Bjorken , Ann. Phys. (N.Y.) 24 , 174 ( 1963 )
2. P.R. Phillips , Phys. Rev . 146 , 966 ( 1966 )
3. T. Eguchi , Phys. Rev. D 14 , 2755 ( 1976 )
4. Y. Nambu , G. JonaLasinio , Phys. Rev . 122 , 345 ( 1961 )
5. J. Goldstone , Nuovo Cimento 19 , 154 ( 1961 )
6. J.L. Chkareuli , C.D. Froggatt , H.B. Nielsen , Phys. Rev. Lett . 87 , 091601 ( 2001 )
7. J.L. Chkareuli , C.D. Froggatt , H.B. Nielsen , Nucl. Phys. B 609 , 46 ( 2001 )
8. J.D. Bjorken . arXiv:hepth/0111196
9. Per Kraus , E.T. Tomboulis . Phys. Rev. D 66 , 045015 ( 2002 )
10. A. Jenkins, Phys. Rev. D 69 , 105007 ( 2004 )
11. V.A. Kostelecky, Phys. Rev. D 69 , 105009 ( 2004 )
12. Z. Berezhiani , O.V. Kancheli . arXiv: 0808 . 3181
13. V.A. Kostelecky , R. Potting , Phys. Rev. D 79 , 065018 ( 2009 )
14. S.M. Carroll , H. Tam , I.K. Wehus , Phys. Rev. D 80 , 025020 ( 2009 )
15. J.L. Chkareuli , J.G. Jejelava , G. Tatishvili, Phys. Lett. B 696 , 126 ( 2011 )
16. J.L. Chkareuli , C.D. Froggatt , H.B. Nielsen , Nucl. Phys. B 848 , 498 ( 2011 )
17. Y. Nambu , Progr. Theor. Phys. Suppl. Extra 190 ( 1968 )
18. S. Weinberg, The Quantum Theory of Fields, v.2 , Cambridge University Press ( 2000 )
19. A.T. Azatov , J.L. Chkareuli , Phys. Rev. D 73 , 065026 ( 2006 )
20. J.L. Chkareuli , Z.R. Kepuladze , Phys. Lett. B 644 , 212 ( 2007 )
21. J.L. Chkareuli , J.G. Jejelava , Phys. Lett. B 659 , 754 ( 2008 )
22. V.A. Kostelecky , S. Samuel, Phys. Rev. D 40 , 1886 ( 1989 )
23. R. Bluhm , N.L. Cage , R. Potting , A. Vrublevskis , Phys. Rev. D 77 , 125007 ( 2008 )
24. H. van Dam, M.J.G. Veltman , Nucl. Phys. B 22 , 397 ( 1970 )