Non-Abelian Gauge Fields as Pseudo-Goldstone Vector Bosons
J.L. Chkareuli and J.G. Jejelava
E. Andronikashvili Institute of Physics and
I. Chavchavadze State University, 0177 Tbilisi, Georgia
Abstract
We argue that non-Abelian gauge fields can be treated as the pseudo-Goldstone
vector bosons caused by spontaneous Lorentz invariance violation (SLIV). To this
end, the SLIV which evolves in a general Yang-Mills type theory with the nonlinear
vector field constraint Tr(AµA
µ) = ±M2 (M is a proposed SLIV scale) imposed
is considered in detail. With an internal symmetry group G having D generators
not only the pure Lorentz symmetry SO(1, 3), but the larger accidental symmetry
SO(D, 3D) of the SLIV constraint in itself appears to be spontaneously broken as
well. As a result, while the pure Lorentz violation still generates only one genuine
Goldstone vector boson, the accompanying pseudo-Goldstone vector bosons related
to the SO(D, 3D) breaking also come into play in the final arrangement of the entire
Goldstone vector field multiplet. Remarkably, they remain strictly massless, being
protected by gauge invariance of the Yang-Mills theory involved. We show that,
although this theory contains a plethora of Lorentz and CPT violating couplings,
they do not lead to physical SLIV effects which turn out to be strictly cancelled in
all the lowest order processes considered. However, the physical Lorentz violation
could appear if the internal gauge invariance were slightly broken at very small
distances influenced by gravity. For the SLIV scale comparable with the Planck one
the Lorentz violation could become directly observable at low energies.
http://arxiv.org/abs/0704.0553v5
1 Introduction
The old idea[1] that spontaneous Lorentz invariance violation (SLIV) may lead to
an alternative theory of quantum electrodynamics still remains extremely attractive
in numerous theoretical contexts[2] (for some later developments, see the papers[3]).
The SLIV could generally cause the appearance of massless vector Nambu-Goldstone
modes which are identified with photons and other gauge fields underlying the mod-
ern particle physics framework like as Standard Model and Grand Unified Theory.
At the same time, the Lorentz violation by itself has attracted a considerable at-
tention in recent years as an interesting phenomenological possibility appearing in
various quantum field and string theories[4-9].
Early models realizing the SLIV conjecture were based on the four fermion
(current-current) interaction, where the proposed gauge field may appear as a fermion-
antifermion pair composite state[1], in a complete analogy with a massless composite
scalar field in the original Nambu-Jona-Lazinio model[10]. Unfortunately, owing to
the lack of a starting gauge invariance in such models and composite nature of
Goldstone modes appeared it is hard to explicitly demonstrate that these modes
really form together a massless vector boson being a gauge field candidate. Actu-
ally, one must make a precise tuning of parameters, including a cancellation between
terms of different orders in the 1/N expansion (where N is the number of fermion
species involved), in order to achieve the massless photon case[11]. Rather, there are
in general three separate massless Goldstone modes, two of which may mimic the
transverse photons polarizations, while the third one must properly be suppressed.
In this connection, the more instructive laboratory for SLIV consideration proves
to be some simple class of the QED type models having from the outset a gauge
invariant form, whereas the Lorentz violation is realized through the nonlinear dy-
namical constraint imposed on the starting vector field Aµ
A2µ = n
2 (1)
where nµ is an properly oriented unit Lorentz vector, while M is a proposed SLIV
scale. This constraint means in essence that the vector field Aµ develops the vacuum
expectation value 〈Aµ(x)〉 = nµM and Lorentz symmetry SO(1, 3) breaks down to
SO(3) or SO(1, 2) depending on the time-like (n2µ = +1) or space-like (n
µ = −1)
SLIV. Such QED model was first studied by Nambu a long time ago[12], but only
for the time-like SLIV case and in the tree approximation. For this purpose he
applied the technique of nonlinear symmetry realizations which appeared successful
in the handling of the spontaneous breakdown of chiral symmetry in the nonlinear
σ model[13] and beyond1.
1Actually, the simplest possible way to obtain the above supplementary condition (1) could be
an inclusion the “standard” quartic vector field potential V (A) = −
A2µ +
(A2µ)
2 into the
QED type Lagrangian, as can be motivated to some extent[14] from the superstring theory. This
potential inevitably causes the spontaneous violation of Lorentz symmetry in a standard way, much
as an internal symmetry violation is caused in a linear σ model for pions[13]. As a result, one has a
In the present paper, we mainly address ourselves to the Yang-Mills gauge fields
as the possible vector Goldstone modes (Sec.3) once some basic ingredients of the
Goldstonic QED model are established in a general SLIV case (Sec.2). This prob-
lem has been discussed many times in the literature within quite different contexts,
such as the Yang-Mills gauge fields as the Goldstone modes for the spontaneous
breaking of general covariance in a higher-dimensional space[17] or for the nonlinear
realization of some special infinite parameter gauge group[18]. However, all these
considerations look rather speculative and optional. Specifically, they do not give
a correlation between the SLIV induced photon case, from the one hand, and the
Yang-Mills gauge field case, from the other. In contrast, our approach is solely
based on the spontaneous Lorentz violation thus properly generalizing the Nambu’s
QED model[12] to the non-Abelian internal symmetry case. Just in this approach
evolved the interrelation between both of cases appears most transparent. We will
see that in the Yang-Mills theory case with an internal symmetry group G having
D generators not only the pure Lorentz symmetry part SO(1, 3) in the symmetry
SO(1, 3) ×G of the Lagrangian, but the larger accidental symmetry SO(D, 3D) of
the SLIV constraint Tr(AµA
µ) = ±M2 in itself is spontaneously broken as well.
Because the starting non-Abelian theory proves to be expanded about the vacuum
which violates the much higher accidental symmetry appeared, many extra mass-
less modes, the pseudo-Goldstone vector bosons (PGB), have to arise. Actually,
while the spontaneous Lorentz violation on its own still generates only one genuine
Goldstone vector boson, the accompanying vector PGBs related to the SO(D, 3D)
breaking also come into play in the final arrangement of the entire Goldstone vec-
tor field multiplet. Remarkably, in contrast to the familiar scalar PGB case[13] the
vector PGBs remain strictly massless being protected by the non-Abelian gauge in-
variance of the Yang-Mills theory involved. Then in Sec.4 we show by some examples
of the lowest order SLIV processes that, while the Goldstonic non-Abelian theory
evolved contains a rich variety of Lorentz and CPT violating couplings, it proves
to be physically indistinguishable from a conventional Yang-Mills theory. Actually,
one of the goals of the present work is to explicitly demonstrate that a conventional
Yang-Mills theory (as well as QED) is in fact the spontaneously broken theory. The
Lorentz violation, due to the quadratic field constraint of the type (1), renders this
theory highly nonlinear in the Goldstone vector modes, while physically equivalent
to the usual one. So, as well as in the pure QED case, the SLIV only means the
noncovariant gauge choice in the otherwise gauge invariant and Lorentz invariant
Yang-Mills theory. However, even a tiny breaking of the starting gauge invariance at
massive Higgs mode (with mass
2mA) together with a massless Goldstone mode associated with
photon. Furthermore, just as in the pion model one can go from the linear model for the SLIV to
the non-linear one taking a limit λA → ∞, m
A → ∞ (while keeping the ratio m
A/λA to be finite).
This immediately leads to the constraint (1) for vector potential Aµ with n
2 = m2A/λA, as it
appears from a validity of its equation of motion. Another motivation for the nonlinear vector field
constraint (1) might be an attempt to avoid the infinite self-energy of the electron in a classical
electrodynamics, as was originally indicated by Dirac[15] and extended later to various vector field
theory cases[16].
very small distances influenced by gravity would render the SLIV physically signifi-
cant. For the SLIV scale comparable with the Planck one the spontaneous Lorentz
violation could become directly observable at low energies. We summarize the results
obtained in the final Sec.5.
2 Goldstonic quantum electrodynamics
The simplest SLIV model is given by a conventional QED Lagrangian for the charged
fermion field ψ
L(A,ψ) = −
µν + ψ(iγ · ∂ −m)ψ − eAµψγ
µψ (2)
where the nonlinear vector field constraint (1) is imposed[12]. For the resulting
Lorentz violation, one can rewrite the Lagrangian L(A,ψ) in terms of the standard
parametrization for the vector potential Aµ
Aµ = aµ +
(n ·A) (n2 ≡ n2µ) (3)
where the aµ is pure Goldstonic mode
n · a = 0 (4)
while the effective Higgs mode (or the Aµ component in the vacuum direction) is
given according to the above nonlinear constraint (1) by
n ·A = (M2 − n2a2ν)
2 =M −
n2a2ν
+O(1/M2) (5)
where, for definiteness, the positive sign for the above square root was taken when
expanding it in powers of a2ν/M
2. Putting the parametrization (3) with the SLIV
constraint (1, 5) into our basic gauge invariant Lagrangian (2) one comes to the
truly Goldstonic model for QED. This model might look unacceptable due to the
inappropriately large Lorentz violating fermion bilinear eMψ(γ ·n)ψ stemming from
the vector-fermion current interaction eAµψγ
µψ in the Lagrangian L (2) when the
expansion (5) is taken. However, thanks to a local invariance of the Lagrangian L
this term can be gauged away by a suitable redefinition of the fermion field
ψ → eieM(n·x)ψ (6)
after which the above fermion bilinear is exactly cancelled by an analogous term
stemming from the fermion kinetic term. So, one eventually comes to the essentially
nonlinear SLIV Lagrangian for the Goldstonic aµ field of the type (taken in the first
approximation in a2ν/M
L(a, ψ) = −
δ(n · a)2 −
n2a2ρ
+ (7)
+ψ(iγ · ∂ +m)ψ − eaµψγ
en2a2ρ
ψ(γ · n)ψ
We denoted its strength tensor by fµν = ∂µaν − ∂νaµ, while h
µν = nµ∂ν −nν∂µ is a
new SLIV oriented differential tensor. This tensor hµν acts on the infinite series in
a2ρ coming from the expansion of the effective Higgs mode (5) from which the first
order term −n2a2ν/2M was only taken in this expansion throughout the Lagrangian
L(a, ψ). Also, we explicitly included the orthogonality condition n · a = 0 into
Lagrangian through the term which can be treated as the gauge fixing term (taking
the limit δ → ∞) and retained the former notation for the fermion ψ.
The Lagrangian (7) completes the Goldstonic QED construction for the charged
fermion field ψ. The model, as one can see, contains the massless Goldstone modes
given by the tree broken generators of the Lorentz group, while keeping the massive
Higgs mode frozen. These modes, lumped together, constitute a single Goldstone
vector boson associated with photon2. In the limit M → ∞ the model is indistin-
guishable from a conventional QED taken in the general axial (temporal or pure
axial) gauge. So, for this part of the Lagrangian L(a, ψ) given by the zero-order
terms in 1/M the spontaneous Lorentz violation only means the noncovariant gauge
choice in otherwise the gauge invariant (and Lorentz invariant) theory. Remarkably,
furthermore, also all the other (first and higher order in 1/M) terms in the L(a, ψ)
(7), though being by themselves the Lorentz and CPT violating ones, do not lead to
the physical SLIV effects which turn out to be strictly cancelled in all the physical
processes involved. So, the nonlinear constraint (1) imposed on the standard QED
Lagrangian (2) appears, in fact, as a possible gauge choice, while the S-matrix re-
mains unaltered under such a gauge convention. This conclusion was first reached
at tree level[12] and recently extended to the one-loop approximation[19]. All the
one-loop contributions to the photon-photon, photon-fermion and fermion-fermion
interactions violating the physical Lorentz invariance were shown to be exactly can-
celled as well. This means that the vector field constraint A2µ = n
2 which has
been treated as the nonlinear gauge choice at a tree (classical) level, remains just
as a pure gauge condition when quantum effects are also taken into account. Re-
markably, this conclusion appears to work also for a general Abelian theory case[20],
particularly, when the internal U(1) charge symmetry is spontaneously broken hand
in hand with the Lorentz one. As a result, the massless photon being first generated
by the Lorentz violation become then massive due to the standard Higgs mechanism,
while the SLIV condition in itself remains to be a gauge choice3.
2Strictly speaking one can no longer use the standard definition of photon as a state being
the spin-1 representation of the (now spontaneously broken) Poincare group. However, due to
gauge symmetry of the starting QED Lagrangian (2) the separate SLIV Goldstone modes appear
combined in such a way that a standard photon (taken in an axial gauge (4)) emerges.
3Note in this connection that there was discussed[12] a possibility of an explicit construction of
the gauge function corresponding to the nonlinear gauge constraint (1) that would eliminate the
need for all the kinds of checks of gauge invariance mentioned above. Remarkably, the equation
for this gauge function appears to be mathematically equivalent to the classical Hamilton-Jacobi
equation of motion for a charged particle. Thus, this gauge function should in principle exist
because there is a solution to the classical problem. However, this formal analogy only works for
the time-like SLIV (n2µ = +1) in the pure QED leaving aside a general Abelian theory when the
gauge invariance can spontaneously be broken. Apart from that, it does not generally extend to
3 Goldstonic Yang-Mills theory
In this section, we extend our discussion to the non-Abelian internal symmetry case
given by a general group G with generators ti([ti, tj ] = icijktk and Tr(titj) = δij
where cijk are structure constants and i, j, k = 0, 1, ...,D − 1). The corresponding
vector fields which transform according to its adjoint representation are given in the
proper matrix form Aµ = A
i, while the matter fields (fermions, for definiteness)
are presented in the fundamental representation column ψr (r = 0, 1, ..., d− 1) of G.
By analogy with the above Goldstonic QED case we take for them a conventional
Yang-Mills type Lagrangian
L(A, ψ) = −
Tr(F µνF
µν) + ψ(iγ · ∂ −m)ψ + gψAµγ
µψ (8)
(where F µν = ∂µAν − ∂νAµ − ig[Aµ,Aν ] and g stands for the universal coupling
constant in the theory) with the nonlinear SLIV constraint
Tr(AµA
µ) = n2µM
2, n2µ = ±1 (9)
imposed4. One can easily see that, although we propose only the SO(1, 3) × G
invariance in the theory, the SLIV constraint taken (9) possesses, in fact, the much
higher accidental symmetry SO(D, 3D) determined by the dimensionality D of the
G group adjoint representation to which the vector fields Aiµ are belonged. This
symmetry is indeed spontaneously broken at a scale M
Aiµ(x)
= niµM (10)
with the vacuum direction given now by the ‘unit’ rectangular matrix niµ which
describes both of the generalized SLIV cases at once, time-like (SO(D, 3D) →
SO(D−1, 3D)) or space-like (SO(D, 3D) → SO(D, 3D−1)), respectively, depending
on the sign of the n2µ ≡ n
µ,i = ±1. This matrix has only one non-zero element for
both of cases determined by the proper SO(D, 3D) rotation. They are, particularly,
0 or n
3 provided that the vacuum expectation value (10) is developed along the
i = 0 direction in the internal space and along the µ = 0 or µ = 3 direction, respec-
tively, in the Minkowskian space-time. In response to each of these two breakings,
side by side with one true vector Goldstone boson and the D − 1 scalar Goldstone
bosons corresponding to the spontaneous violation of actual SO(1, 3) ⊗ G symme-
try of the total Lagrangian L, the D − 1 vector pseudo-Goldstone bosons related
to breaking of the accidental SO(D, 3D) symmetry of the SLIV constraint taken
(9) are also produced. Remarkably, in contrast to the familiar scalar PGB case[13]
the non-Abelian case (see next Section).
4As in the Abelian case, the existence of such a constraint could be related with some non-
linear σ type SLIV model proposed for the vector field multiplet Aiµ in the Yang-Mills theory
(8). Note in this connection that, due to its generic antisymmetry, the familiar quadrilinear terms
g2Tr([Aµ, Aν ])
2 in the Lagrangian (8) do not contribute into the SLIV since they identically
vanish for any single-valued vacuum configuration
the vector PGBs remain strictly massless being protected by the non-Abelian gauge
invariance of the starting Lagrangian (8). Together with the aforementioned true
vector Goldstone boson they complete the entire Goldstonic vector field multiplet
of the internal symmetry group G.
As in the Abelian case, upon an explicit use of the corresponding SLIV constraint
(9) being so far the only supplementary condition for vector field multiplet Aiµ, one
comes to the pure Goldstone field modes aiµ identified in a similar way
Aiµ = a
(n · A) , n · a ≡ niµa
µ,i = 0 (n2 ≡ n2µ) , (11)
At the same time, an effective Higgs mode (i.e., the Aiµ component in the vacuum
direction niµ) is given by the product n · A ≡ n
µ,i determined by the SLIV con-
straint
n · A =
M2 − n2(aiν)
2 =M −
2(aiν)
+O(1/M2) (12)
where, as earlier in the Abelian case, we took the positive sign for the square
root when expanding it in powers of (aiν)
2/M2. Note that the general Goldstonic
modes aiµ, apart from pure vector fields, contain the D − 1 scalar ones, a
0 and a
(i′ = 1...D − 1), for the time-like (niµ = n
0gµ0δ
i0) and space-like (niµ = n
3gµ3δ
SLIV, respectively. They can be eliminated from the theory if one puts the proper
supplementary conditions on the aiµ fields which were still the constraint free. Using
their overall orthogonality (11) to the physical vacuum direction niµ one can formu-
late these supplementary conditions in terms of a general axial gauge for the entire
aiµ multiplet
n · ai ≡ nµa
µ,i = 0, i = 0...D − 1 (13)
where nµ is the unit Lorentz vector introduced in the Abelian case which is now
oriented in Minkowskian space-time so as to be parallel to the vacuum matrix niµ.
For such a choice the simple equation holds
µ = s
inµ (s
n · ni
) (14)
which shows that the rectangular vacuum matrix niµ has the factorized ”two-vector”
form. As a result, apart from the Higgs mode excluded earlier by the orthogonality
condition (11), all the scalar fields also appear eliminated, and only pure vector fields,
(µ′ = 1, 2, 3) or ai
(µ′′ = 0, 1, 2) for time-like or space-like SLIV, respectively,
are only left in the theory.
We now show that the such constrained Goldstone vector fields aiµ (with the
supplementary conditions (13) taken) appear truly massless when the starting non-
Abelian Lagrangian L (8) is rewritten in the form determined by the SLIV. Actually,
putting the parametrization (11) with the SLIV constraint (12) into the Lagrangian
(8) one is led to the highly nonlinear Yang-Mills theory in terms of the pure Gold-
stonic gauge field modes aiµ. However, as in the above Abelian case, one should
first gauge away (using the local invariance of the Lagrangian L) the enormously
large, while false, Lorentz violating terms appearing in the theory in the form of
the fermion and vector field bilinears. As one can readily see, they stem from the
couplings gψAµγ
µψ and −1
g2Tr([Aµ, Aν ])
2, respectively, when the effective Higgs
mode expansion (12) is taken in the Lagrangian (8). Making the appropriate redef-
initions of the fermion (ψ ) and vector (aµ ≡ a
i) field multiplets
ψ → U(ω)ψ , aµ → U(ω)aµU(ω)
†, U(ω) = eigM(n
i·x)ti (15)
and using the evident equalities for the linear (in coordinate) transformations U(ω)
with the single-valued vacuum matrix niµ (n
0 or n
3 for the particular SLIV cases)
∂µU(ω) = ign
iU(ω) = igU(ω)niµt
i (16)
one can confirm that the abovementioned Lorentz violating terms are exactly can-
celled with the analogous bilinears stemming from their kinetic terms. So, the final
Lagrangian for the Goldstonic Yang-Mills theory takes the form (in the first approx-
imation in (aiν)
2/M2)
L(a,ψ) = −
Tr(fµνf
δ(n · ai)2 +
Tr(fµνh
2(aiν)
+ψ(iγ · ∂ −m)ψ + gψaµγ
gn2(aiν)
ψ(γ · nk)tkψ (17)
where the tensor fµν is, as usual, fµν = ∂µaν − ∂νaµ − ig[aµ,aν ], while hµν is a
new SLIV oriented tensor of the type
hµν = nµ∂ν − nν∂µ + ig([nµ,aν ]− [nν ,aµ]), nµ ≡ n
k (18)
This tensor hµν acts on the infinite series in (a
2 coming from the expansion of the
effective Higgs mode (12) from which only the first order term −n2(aiν)
2/2M was
taken throughout the Lagrangian L(a,ψ). We also retained the former notations for
the fermion and vector field multiplets after transformations (15), and explicitly in-
cluded the (axial) gauge fixing term into Lagrangian according to the supplementary
conditions taken (13).
The theory derived gives a proper generalization of the nonlinear QED model[12]
for the non-Abelian case. It contains the massless vector boson multiplet aiµ (con-
sisting of one Goldstone and D − 1 pseudo-Goldstone vector states) which gauges
the starting internal symmetry G. In the limit M → ∞ it is indistinguishable from
a conventional Yang-Mills theory taken in the general axial gauge. So, for this part
of the Lagrangian L(a,ψ) given by the zero-order in 1/M terms the spontaneous
Lorentz violation only means the noncovariant gauge choice in the otherwise gauge
invariant (and Lorentz invariant) theory. However, one may expect that, just as
it appears in the nonlinear QED model, also all the first and higher order in 1/M
terms in the L (17), though being by themselves the Lorentz and CPT violating
ones, do not lead to the physical SLIV effects due to the mutual cancellation of their
contributions into all the physical processes appeared.
4 The lowest order SLIV processes
Let us now show that the simple tree level calculations related to the Lagrangian
L(a,ψ) confirms in essence this proposition. As an illustration, we consider SLIV
processes in the lowest order in g and 1/M being the fundamental parameters of the
Lagrangian (17). They are, as one can readily see, the vector-fermion and vector-
vector elastic scattering going in the order g/M , which we turn to once the Feynman
rules in the Goldstonic Yang-Mills theory are established.
4.1 Feynman rules
The corresponding Feynman rules, apart from the ordinary Yang-Mills theory rules
(i) the vector-fermion vertex
− ig γµ t
i (19)
(ii) the vector field propagator (taken in a general axial gauge nµaiµ = 0)
Dijµν (k) = −
gµν −
nµkν + kµnν
n · k
n2kµkν
(n · k)2
which automatically satisfies the orthogonality condition nµD
µν(k) = 0 and on-shell
transversality kµD
µν(k) = 0 (k
2 = 0); the latter means that free vector fields with
polarization vector ǫiµ(k, k
2 = 0) are always appeared transverse kµǫiµ(k) = 0;
(iii) the 3-vector vertex (with vector field 4-momenta k1, k2 and k3; all 4-momenta
in vertexes are taken ingoing throughout)
gcijk[(k1 − k2)γgαβ + (k2 − k3)αgβγ + (k3 − k1)βgαγ ] (21)
include the new ones, violating Lorentz and CPT invariance, for
(iv) the contact 2-vector-fermion vertex
(γ · nk)τkgµν δ
ij (22)
(v) another 3-vector vertex
(k1 · n
i)k1,αgβγδ
jk + (k2 · n
j)k2,βgαγδ
ki + (k3 · n
k)k3,γgαβδ
where the second index in the vector field 4-momenta k1, k2 and k3 denotes their
Lorentz components;
(vi) the extra 4-vector vertex (with the vector field 4-momenta k1,2,3,4 and their
proper differences k12 ≡ k1 − k2 etc.)
[cijpδklgαβgγδ(n
p · k12) + c
klpδijgαβgγδ(n
p · k34) +
+cikpδjlgαγgβδ(n
p · k13) + c
jlpδikgαγgβδ(n
p · k24) + (24)
+cilpδjkgαδgβγ(n
p · k14) + c
jkpδilgαδgβγ(n
p · k23)]
where only the terms which can not lead to contractions of the rectangular vacuum
matrix n
µ with vector field polarization vectors ǫ
µ(k) are presented. These contrac-
tions are in fact vanished due to the gauge taken (13), np · ǫi = sp(n · ǫi) = 0 (with
a factorized two-vector form for the matrix n
µ (14) used).
Just the rules (i-vi) are needed to calculate the lowest order amplitudes of the
processes we have mentioned in the above.
4.2 Vector boson scattering on fermion
This process is directly related to two SLIV diagrams one of which is given by the
contact a2-fermion vertex (22), while another corresponds to the pole diagram with
the longitudinal a-boson exchange between Lorentz violating a3 vertex (23) and
ordinary a-boson-fermion one (19). Since ingoing and outgoing a-bosons appear
transverse (k1 · ǫ
i(k1) = 0, k2 · ǫ
j(k2) = 0) only the third term in this a
3 coupling
(23) contributes to the pole diagram so that one comes to a simple matrix element
iM for both of diagrams
iM = i
ū(p2)τ
(γ · nl) + i(k · nl)γµkνDµν(k)
u(p1)[ǫ(k1) · ǫ(k2)] (25)
where the spinors u(p1,2) and polarization vectors ǫ
µ(k1) and ǫ
µ(k2) stand for the in-
going and outgoing fermions and a-bosons, respectively, while k is the 4-momentum
transfer k = p2 − p1 = k1 − k2. Upon the further simplifications in the square
bracket related to the explicit form of the a boson propagator Dµν(k) (20) and ma-
trix niµ (14), and using the fermion current conservation ū(p2)(p̂2 − p̂1)u(p1) = 0,
one is finally led to the total cancellation of the Lorentz violating contributions to
the a-boson-fermion scattering in the g/M approximation.
Note, however, that such a result may be in some sense expected since from the
SLIV point of view the lowest order a-boson-fermion scattering discussed here is
hardly distinct from the photon-fermion scattering considered in the nonlinear QED
case[12]. Actually, the fermion current conservation which happens to be crucial for
the above cancellation works in both of cases, whereas the couplings being peculiar
to the Yang-Mills theory have not yet touched on. In this connection the next
example seems to be more instructive.
4.3 Vector-vector scattering
The matrix element for this process in the lowest order g/M is given by the contact
SLIV a4 vertex (24) and the pole diagrams with the longitudinal a-boson exchange
between the ordinary a3 vertex (21) and Lorentz violating a3 one (23), and vice
versa. There are six pole diagrams in total describing the elastic a − a scattering
in the s- and t-channels, respectively, including also those with an interchange of
identical a-bosons. Remarkably, the contribution of each of them is exactly canceled
with one of six terms appeared in the contact vertex (24). Actually, writing down the
matrix element for one of the pole diagrams with ingoing a-bosons (with momenta k1
and k2) interacting through the vertex (21) and outgoing a-bosons (with momenta
k3 and k4) interacting through the vertex (23) one has
cijpδkl[(k1 − k2)µgαβ + (k2 − k)αgβµ + (k − k1)βgαµ] ·
·Dpqµν(k)gγδkν(n
q · k)[ǫi,α(k1)ǫ
j,β(k2)ǫ
k,γ(k3)ǫ
l,δ(k4)] (26)
where polarization vectors ǫi,α(k1), ǫ
j,β(k2), ǫ
k,γ(k3) and ǫ
l,δ(k4) belong, respectively,
to ingoing and outgoing a-bosons, while k = −(k1 + k2) = k3 + k4 according to the
momentum running in the diagrams taken above. Again, as in the previous case of
vector-fermion scattering, due to the fact that outgoing a-bosons appear transverse
(k3 · ǫ
k(k3) = 0 and k4 · ǫ
l(k4) = 0), only the third term in the Lorentz violating a
coupling (23) contributes to this pole diagram. Upon evident simplifications related
to the a-boson propagator Dµν(k) (20) and matrix n
µ (14) one comes to the expres-
sion which is exactly cancelled with the first term in the contact SLIV vertex (24)
when it is properly contracted with a-boson polarization vectors. Likewise, other
terms in this vertex provide the further one-to-one cancellation with the remaining
pole matrix elements iM
(2−6)
. So, again, the Lorentz violating contribution to the
vector-vector scattering is absent in Goldstonic Yang-Mills theory in the lowest g/M
approximation.
4.4 Other processes
Other tree level Lorentz violating processes, related to a bosons and fermions, appear
in higher orders in the basic SLIV parameter 1/M . They come from the subsequent
expansion of the effective Higgs mode (12) in the Lagrangian (17). Again, their
amplitudes are essentially determined by an interrelation between the longitudinal
a-boson exchange diagrams and the corresponding contact a-boson interaction dia-
grams which appear to cancel each other thus eliminating physical Lorentz violation
in theory.
Most likely, the same conclusion can be derived for SLIV loop contributions
as well. Actually, as in the massless QED case considered earlier [19], the corre-
sponding one-loop matrix elements in Goldstonic Yang-Mills theory either vanish
by themselves or amount to the differences between pairs of the similar integrals
whose integration variables are shifted relative to each other by some constants (be-
ing in general arbitrary functions of external four-momenta of the particles involved)
that in the framework of dimensional regularization leads to their total cancellation.
So, the Goldstonic vector field theory (17) for a non-Abelian charge-carrying
matter is likely to be physically indistinguishable from a conventional Yang-Mills
theory.
5 Conclusion
The spontaneous Lorentz violation in 4-dimensonal flat Minkowskian space-time was
shown to generate vector Goldstone bosons both in Abelian and non-Abelian theo-
ries with the corresponding nonlinear vector field constraint (1) or (9) imposed. In
the Abelian case such a massless vector boson is naturally associated with photon.
In non-Abelian case, although the pure Lorentz violation still generates only one
genuine Goldstone vector boson, the accompanying vector PGBs related to a vio-
lation of the larger accidental symmetry SO(D, 3D) of the SLIV constraint (9) in
itself come also into play in the final arrangement of the entire Goldstone vector
field multiplet of the internal symmetry group G. Remarkably, they remain strictly
massless being protected by the gauge invariance of the Yang-Mills theory involved.
These theories, both Abelian and non-Abelian, while being essentially nonlinear in
the Goldstone vector modes, are physically indistinguishable from conventional QED
and Yang-Mills theory. One could actually see that just the gauge invariance not
only provides these theories to be free from the unreasonably large Lorentz violation
stemming from the fermion and vector field bilinears (see Sections 2 and 3), but also
render all the other physical SLIV effects (including those which are suppressed by
the Lorentz violation scale M) non-observable (Section 4). As a result, Abelian and
non-Abelian SLIV theory appear, respectively, as standard QED and Yang-Mills
theory taken in the nonlinear gauge (to which the vector field constraints (1) and
(9) are virtually reduced), while the S-matrix remains unaltered under such a gauge
convention.
So, while at present the Goldstonic nature of gauge fields, both Abelian and non-
Abelian, seems to be highly plausible, the most fundamental question of physical
Lorentz violation in itself, that only could uniquely point toward such a possibility,
is still an open question. Note, that here we are not dealing with direct (and quite
arbitrary in essence) Lorentz non-invariant extensions of QED or Standard Model
which were intensively discussed on their own in recent years [6-8]. Rather, the case
in point is a construction of genuine SLIV models which would generate gauge fields
as the proper vector Goldstone bosons, from one hand, and could lead to observed
Lorentz violating effects, from the other. In this connection, somewhat natural
framework for physical Lorentz violation to occur would be a model where the
internal gauge invariance were slightly broken at very small distances through some
high-order operators stemming from the gravity-influenced area. Such physical SLIV
effects would be seen in terms of powers of ratio M/MP l (where MP l is the Planck
mass). So, for the SLIV scale comparable with the Planck one they would become
directly observable. Remarkably enough, if one has such internal gauge symmetry
breaking in an ordinary Lorentz invariant theory this breaking appears vanishingly
small at laboratory being properly suppressed by the Planck scale. However, the
spontaneous Lorentz violation would render it physically significant: the higher
Lorentz scale, the greater SLIV effects observed. If true, it would be of particular
interest to have a better understanding of the internal gauge symmetry breaking
mechanism that brings out the spontaneous Lorentz violation at low energies. We
return to this basic question elsewhere.
Acknowledgments
We would like to thank Colin Froggatt, Rabi Mohapatra and Holger Nielsen for useful
discussions and comments. One of us (J.L.C.) is grateful for the warm hospitality
shown to him during a visit to Center for Particle and String Theory at University
of Maryland where part of this work was carried out.
References
[1] W. Heisenberg, Rev. Mod. Phys. 29 (1957) 269;
J.D. Bjorken, Ann. Phys. (N.Y.) 24 (1963) 174;
I. Bialynicki-Birula, Phys. Rev. 130 (1963) 465 ;
G. Guralnik, Phys. Rev. 136 (1964) B1404;
T. Eguchi, Phys.Rev. D 14 (1976) 2755;
H. Terazava, Y. Chikashige and K. Akama, Phys. Rev. D 15 (1977) 480 .
[2] C.D. Froggatt and H.B. Nielsen, Origin of Symmetries (World Scientific, Sin-
gapore, 1991).
[3] J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen, Phys. Rev. Lett. 87 (2001)
091601;
J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen Nucl. Phys. B 609 (2001) 46;
J.D. Bjorken, hep-th/0111196;
Per Kraus and E.T. Tomboulis, Phys. Rev. D 66 (2002) 045015;
A. Jenkins, Phys. Rev. D 69 (2004) 105007;
J.L. Chkareuli, C.D. Froggatt, R.N. Mohapatra and H.B. Nielsen,
hep-th/0412225;
J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen, hep-th/0610186.
[4] D. Colladay and V.A. Kostelecky, Phys. Rev. D58 (1998) 116002 ;
V.A. Kostelecky, Phys. Rev. D69 (2004) 105009 ;
R. Bluhm and V.A. Kostelecky, Phys. Rev. D 71(2005) 065008;
CPT and Lorentz Symmetry, ed. A. Kostelecky (World Scientific, Singapore,
1999, 2002, 2005).
[5] S.M. Carroll, G.B. Field and R. Jackiw, Phys. Rev. D 41 (1990) 1231;
R. Jackiw and V.A. Kostelecky, Phys. Rev. Lett. 82 (1999) 3572.
[6] S. Coleman and S.L. Glashow, Phys. Rev. D 59 (1999) 116008.
http://arxiv.org/abs/hep-th/0111196
http://arxiv.org/abs/hep-th/0412225
http://arxiv.org/abs/hep-th/0610186
[7] J. W. Moffat, Int. J. Mod.Phys. D2 (1993) 351;
J.W. Moffat, Int. J. Mod.Phys. D12 (2003) 1279.
[8] O. Bertolami and D.F. Mota, Phys. Lett. B 455 (1999) 96.
[9] T. Jacobson, S. Liberati and D. Mattingly, Ann. Phys. (N.Y.) 321 (2006) 150.
[10] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345.
[11] M. Suzuki, Phys. Rev. D 37 (1988) 210 .
[12] Y. Nambu, Progr. Theor. Phys. Suppl. Extra 190 (1968).
[13] S. Weinberg, The Quantum Theory of Fields, v.2, Cambridge University Press,
2000.
[14] V.A. Kostelecky and S. Samuel, Phys. Rev. D 39 (1989) 683;
V.A. Kostelecky and R. Potting, Nucl. Phys. B 359 (1991) 545.
[15] P.A.M. Dirac, Proc. Roy. Soc. 209A (1951) 292;
P.A.M. Dirac, Proc. Roy. Soc. 212A (1952) 330.
[16] R. Righi and G. Venturi, Lett. Nuovo Cim. 19 (1977) 633;
R. Righi, G. Venturi and V. Zamiralov, Nuovo Cim. A47 (1978) 518.
[17] Y.M. Cho and P.G.O. Freund, Phys. Rev. D 12 (1975) 1711.
[18] E. A. Ivanov and V.I. Ogievetsky, Lett. Math. Phys. 1 (1976) 309 .
[19] A.T. Azatov and J.L. Chkareuli, Phys. Rev. D 73 (2006) 065026.
[20] J.L. Chkareuli and Z.R. Kepuladze, Phys. Lett. B 644 (2007) 212;
J.L. Chkareuli and Z.R. Kepuladze, Proc. of XIV Int. Seminar “Quarks-2006”,
eds. S.V. Demidov at al (Moscow, INR, 2006); hep-th/0610227.
http://arxiv.org/abs/hep-th/0610227
	Introduction
	Goldstonic quantum electrodynamics
	Goldstonic Yang-Mills theory
	The lowest order SLIV processes
	Feynman rules
	Vector boson scattering on fermion
	Vector-vector scattering
	Other processes
	Conclusion

  We argue that non-Abelian gauge fields can be treated as the pseudo-Goldstone
vector bosons caused by spontaneous Lorentz invariance violation (SLIV). To
this end, the SLIV which evolves in a general Yang-Mills type theory with the
nonlinear vector field constraint $Tr(% \boldsymbol{A}_{\mu
}\boldsymbol{A}^{\mu})=\pm M^{2}$ ($M$ is a proposed SLIV scale) imposed is
considered in detail. With an internal symmetry group $G$ having $D$ generators
not only the pure Lorentz symmetry SO(1,3), but the larger accidental symmetry
$SO(D,3D)$ of the SLIV constraint in itself appears to be spontaneously broken
as well. As a result, while the pure Lorentz violation still generates only one
genuine Goldstone vector boson, the accompanying pseudo-Goldstone vector bosons
related to the $SO(D,3D)$ breaking also come into play in the final arrangement
of the entire Goldstone vector field multiplet. Remarkably, they remain
strictly massless, being protected by gauge invariance of the Yang-Mills theory
involved. We show that, although this theory contains a plethora of Lorentz and
$CPT$ violating couplings, they do not lead to physical SLIV effects which turn
out to be strictly cancelled in all the lowest order processes considered.
However, the physical Lorentz violation could appear if the internal gauge
invariance were slightly broken at very small distances influenced by gravity.
For the SLIV scale comparable with the Planck one the Lorentz violation could
become directly observable at low energies.

Introduction
The old idea[1] that spontaneous Lorentz invariance violation (SLIV) may lead to
an alternative theory of quantum electrodynamics still remains extremely attractive
in numerous theoretical contexts[2] (for some later developments, see the papers[3]).
The SLIV could generally cause the appearance of massless vector Nambu-Goldstone
modes which are identified with photons and other gauge fields underlying the mod-
ern particle physics framework like as Standard Model and Grand Unified Theory.
At the same time, the Lorentz violation by itself has attracted a considerable at-
tention in recent years as an interesting phenomenological possibility appearing in
various quantum field and string theories[4-9].
Early models realizing the SLIV conjecture were based on the four fermion
(current-current) interaction, where the proposed gauge field may appear as a fermion-
antifermion pair composite state[1], in a complete analogy with a massless composite
scalar field in the original Nambu-Jona-Lazinio model[10]. Unfortunately, owing to
the lack of a starting gauge invariance in such models and composite nature of
Goldstone modes appeared it is hard to explicitly demonstrate that these modes
really form together a massless vector boson being a gauge field candidate. Actu-
ally, one must make a precise tuning of parameters, including a cancellation between
terms of different orders in the 1/N expansion (where N is the number of fermion
species involved), in order to achieve the massless photon case[11]. Rather, there are
in general three separate massless Goldstone modes, two of which may mimic the
transverse photons polarizations, while the third one must properly be suppressed.
In this connection, the more instructive laboratory for SLIV consideration proves
to be some simple class of the QED type models having from the outset a gauge
invariant form, whereas the Lorentz violation is realized through the nonlinear dy-
namical constraint imposed on the starting vector field Aµ
A2µ = n
2 (1)
where nµ is an properly oriented unit Lorentz vector, while M is a proposed SLIV
scale. This constraint means in essence that the vector field Aµ develops the vacuum
expectation value 〈Aµ(x)〉 = nµM and Lorentz symmetry SO(1, 3) breaks down to
SO(3) or SO(1, 2) depending on the time-like (n2µ = +1) or space-like (n
µ = −1)
SLIV. Such QED model was first studied by Nambu a long time ago[12], but only
for the time-like SLIV case and in the tree approximation. For this purpose he
applied the technique of nonlinear symmetry realizations which appeared successful
in the handling of the spontaneous breakdown of chiral symmetry in the nonlinear
σ model[13] and beyond1.
1Actually, the simplest possible way to obtain the above supplementary condition (1) could be
an inclusion the “standard” quartic vector field potential V (A) = −
A2µ +
(A2µ)
2 into the
QED type Lagrangian, as can be motivated to some extent[14] from the superstring theory. This
potential inevitably causes the spontaneous violation of Lorentz symmetry in a standard way, much
as an internal symmetry violation is caused in a linear σ model for pions[13]. As a result, one has a
In the present paper, we mainly address ourselves to the Yang-Mills gauge fields
as the possible vector Goldstone modes (Sec.3) once some basic ingredients of the
Goldstonic QED model are established in a general SLIV case (Sec.2). This prob-
lem has been discussed many times in the literature within quite different contexts,
such as the Yang-Mills gauge fields as the Goldstone modes for the spontaneous
breaking of general covariance in a higher-dimensional space[17] or for the nonlinear
realization of some special infinite parameter gauge group[18]. However, all these
considerations look rather speculative and optional. Specifically, they do not give
a correlation between the SLIV induced photon case, from the one hand, and the
Yang-Mills gauge field case, from the other. In contrast, our approach is solely
based on the spontaneous Lorentz violation thus properly generalizing the Nambu’s
QED model[12] to the non-Abelian internal symmetry case. Just in this approach
evolved the interrelation between both of cases appears most transparent. We will
see that in the Yang-Mills theory case with an internal symmetry group G having
D generators not only the pure Lorentz symmetry part SO(1, 3) in the symmetry
SO(1, 3) ×G of the Lagrangian, but the larger accidental symmetry SO(D, 3D) of
the SLIV constraint Tr(AµA
µ) = ±M2 in itself is spontaneously broken as well.
Because the starting non-Abelian theory proves to be expanded about the vacuum
which violates the much higher accidental symmetry appeared, many extra mass-
less modes, the pseudo-Goldstone vector bosons (PGB), have to arise. Actually,
while the spontaneous Lorentz violation on its own still generates only one genuine
Goldstone vector boson, the accompanying vector PGBs related to the SO(D, 3D)
breaking also come into play in the final arrangement of the entire Goldstone vec-
tor field multiplet. Remarkably, in contrast to the familiar scalar PGB case[13] the
vector PGBs remain strictly massless being protected by the non-Abelian gauge in-
variance of the Yang-Mills theory involved. Then in Sec.4 we show by some examples
of the lowest order SLIV processes that, while the Goldstonic non-Abelian theory
evolved contains a rich variety of Lorentz and CPT violating couplings, it proves
to be physically indistinguishable from a conventional Yang-Mills theory. Actually,
one of the goals of the present work is to explicitly demonstrate that a conventional
Yang-Mills theory (as well as QED) is in fact the spontaneously broken theory. The
Lorentz violation, due to the quadratic field constraint of the type (1), renders this
theory highly nonlinear in the Goldstone vector modes, while physically equivalent
to the usual one. So, as well as in the pure QED case, the SLIV only means the
noncovariant gauge choice in the otherwise gauge invariant and Lorentz invariant
Yang-Mills theory. However, even a tiny breaking of the starting gauge invariance at
massive Higgs mode (with mass
2mA) together with a massless Goldstone mode associated with
photon. Furthermore, just as in the pion model one can go from the linear model for the SLIV to
the non-linear one taking a limit λA → ∞, m
A → ∞ (while keeping the ratio m
A/λA to be finite).
This immediately leads to the constraint (1) for vector potential Aµ with n
2 = m2A/λA, as it
appears from a validity of its equation of motion. Another motivation for the nonlinear vector field
constraint (1) might be an attempt to avoid the infinite self-energy of the electron in a classical
electrodynamics, as was originally indicated by Dirac[15] and extended later to various vector field
theory cases[16].
very small distances influenced by gravity would render the SLIV physically signifi-
cant. For the SLIV scale comparable with the Planck one the spontaneous Lorentz
violation could become directly observable at low energies. We summarize the results
obtained in the final Sec.5.
2 Goldstonic quantum electrodynamics
The simplest SLIV model is given by a conventional QED Lagrangian for the charged
fermion field ψ
L(A,ψ) = −
µν + ψ(iγ · ∂ −m)ψ − eAµψγ
µψ (2)
where the nonlinear vector field constraint (1) is imposed[12]. For the resulting
Lorentz violation, one can rewrite the Lagrangian L(A,ψ) in terms of the standard
parametrization for the vector potential Aµ
Aµ = aµ +
(n ·A) (n2 ≡ n2µ) (3)
where the aµ is pure Goldstonic mode
n · a = 0 (4)
while the effective Higgs mode (or the Aµ component in the vacuum direction) is
given according to the above nonlinear constraint (1) by
n ·A = (M2 − n2a2ν)
2 =M −
n2a2ν
+O(1/M2) (5)
where, for definiteness, the positive sign for the above square root was taken when
expanding it in powers of a2ν/M
2. Putting the parametrization (3) with the SLIV
constraint (1, 5) into our basic gauge invariant Lagrangian (2) one comes to the
truly Goldstonic model for QED. This model might look unacceptable due to the
inappropriately large Lorentz violating fermion bilinear eMψ(γ ·n)ψ stemming from
the vector-fermion current interaction eAµψγ
µψ in the Lagrangian L (2) when the
expansion (5) is taken. However, thanks to a local invariance of the Lagrangian L
this term can be gauged away by a suitable redefinition of the fermion field
ψ → eieM(n·x)ψ (6)
after which the above fermion bilinear is exactly cancelled by an analogous term
stemming from the fermion kinetic term. So, one eventually comes to the essentially
nonlinear SLIV Lagrangian for the Goldstonic aµ field of the type (taken in the first
approximation in a2ν/M
L(a, ψ) = −
δ(n · a)2 −
n2a2ρ
+ (7)
+ψ(iγ · ∂ +m)ψ − eaµψγ
en2a2ρ
ψ(γ · n)ψ
We denoted its strength tensor by fµν = ∂µaν − ∂νaµ, while h
µν = nµ∂ν −nν∂µ is a
new SLIV oriented differential tensor. This tensor hµν acts on the infinite series in
a2ρ coming from the expansion of the effective Higgs mode (5) from which the first
order term −n2a2ν/2M was only taken in this expansion throughout the Lagrangian
L(a, ψ). Also, we explicitly included the orthogonality condition n · a = 0 into
Lagrangian through the term which can be treated as the gauge fixing term (taking
the limit δ → ∞) and retained the former notation for the fermion ψ.
The Lagrangian (7) completes the Goldstonic QED construction for the charged
fermion field ψ. The model, as one can see, contains the massless Goldstone modes
given by the tree broken generators of the Lorentz group, while keeping the massive
Higgs mode frozen. These modes, lumped together, constitute a single Goldstone
vector boson associated with photon2. In the limit M → ∞ the model is indistin-
guishable from a conventional QED taken in the general axial (temporal or pure
axial) gauge. So, for this part of the Lagrangian L(a, ψ) given by the zero-order
terms in 1/M the spontaneous Lorentz violation only means the noncovariant gauge
choice in otherwise the gauge invariant (and Lorentz invariant) theory. Remarkably,
furthermore, also all the other (first and higher order in 1/M) terms in the L(a, ψ)
(7), though being by themselves the Lorentz and CPT violating ones, do not lead to
the physical SLIV effects which turn out to be strictly cancelled in all the physical
processes involved. So, the nonlinear constraint (1) imposed on the standard QED
Lagrangian (2) appears, in fact, as a possible gauge choice, while the S-matrix re-
mains unaltered under such a gauge convention. This conclusion was first reached
at tree level[12] and recently extended to the one-loop approximation[19]. All the
one-loop contributions to the photon-photon, photon-fermion and fermion-fermion
interactions violating the physical Lorentz invariance were shown to be exactly can-
celled as well. This means that the vector field constraint A2µ = n
2 which has
been treated as the nonlinear gauge choice at a tree (classical) level, remains just
as a pure gauge condition when quantum effects are also taken into account. Re-
markably, this conclusion appears to work also for a general Abelian theory case[20],
particularly, when the internal U(1) charge symmetry is spontaneously broken hand
in hand with the Lorentz one. As a result, the massless photon being first generated
by the Lorentz violation become then massive due to the standard Higgs mechanism,
while the SLIV condition in itself remains to be a gauge choice3.
2Strictly speaking one can no longer use the standard definition of photon as a state being
the spin-1 representation of the (now spontaneously broken) Poincare group. However, due to
gauge symmetry of the starting QED Lagrangian (2) the separate SLIV Goldstone modes appear
combined in such a way that a standard photon (taken in an axial gauge (4)) emerges.
3Note in this connection that there was discussed[12] a possibility of an explicit construction of
the gauge function corresponding to the nonlinear gauge constraint (1) that would eliminate the
need for all the kinds of checks of gauge invariance mentioned above. Remarkably, the equation
for this gauge function appears to be mathematically equivalent to the classical Hamilton-Jacobi
equation of motion for a charged particle. Thus, this gauge function should in principle exist
because there is a solution to the classical problem. However, this formal analogy only works for
the time-like SLIV (n2µ = +1) in the pure QED leaving aside a general Abelian theory when the
gauge invariance can spontaneously be broken. Apart from that, it does not generally extend to
3 Goldstonic Yang-Mills theory
In this section, we extend our discussion to the non-Abelian internal symmetry case
given by a general group G with generators ti([ti, tj ] = icijktk and Tr(titj) = δij
where cijk are structure constants and i, j, k = 0, 1, ...,D − 1). The corresponding
vector fields which transform according to its adjoint representation are given in the
proper matrix form Aµ = A
i, while the matter fields (fermions, for definiteness)
are presented in the fundamental representation column ψr (r = 0, 1, ..., d− 1) of G.
By analogy with the above Goldstonic QED case we take for them a conventional
Yang-Mills type Lagrangian
L(A, ψ) = −
Tr(F µνF
µν) + ψ(iγ · ∂ −m)ψ + gψAµγ
µψ (8)
(where F µν = ∂µAν − ∂νAµ − ig[Aµ,Aν ] and g stands for the universal coupling
constant in the theory) with the nonlinear SLIV constraint
Tr(AµA
µ) = n2µM
2, n2µ = ±1 (9)
imposed4. One can easily see that, although we propose only the SO(1, 3) × G
invariance in the theory, the SLIV constraint taken (9) possesses, in fact, the much
higher accidental symmetry SO(D, 3D) determined by the dimensionality D of the
G group adjoint representation to which the vector fields Aiµ are belonged. This
symmetry is indeed spontaneously broken at a scale M
Aiµ(x)
= niµM (10)
with the vacuum direction given now by the ‘unit’ rectangular matrix niµ which
describes both of the generalized SLIV cases at once, time-like (SO(D, 3D) →
SO(D−1, 3D)) or space-like (SO(D, 3D) → SO(D, 3D−1)), respectively, depending
on the sign of the n2µ ≡ n
µ,i = ±1. This matrix has only one non-zero element for
both of cases determined by the proper SO(D, 3D) rotation. They are, particularly,
0 or n
3 provided that the vacuum expectation value (10) is developed along the
i = 0 direction in the internal space and along the µ = 0 or µ = 3 direction, respec-
tively, in the Minkowskian space-time. In response to each of these two breakings,
side by side with one true vector Goldstone boson and the D − 1 scalar Goldstone
bosons corresponding to the spontaneous violation of actual SO(1, 3) ⊗ G symme-
try of the total Lagrangian L, the D − 1 vector pseudo-Goldstone bosons related
to breaking of the accidental SO(D, 3D) symmetry of the SLIV constraint taken
(9) are also produced. Remarkably, in contrast to the familiar scalar PGB case[13]
the non-Abelian case (see next Section).
4As in the Abelian case, the existence of such a constraint could be related with some non-
linear σ type SLIV model proposed for the vector field multiplet Aiµ in the Yang-Mills theory
(8). Note in this connection that, due to its generic antisymmetry, the familiar quadrilinear terms
g2Tr([Aµ, Aν ])
2 in the Lagrangian (8) do not contribute into the SLIV since they identically
vanish for any single-valued vacuum configuration
the vector PGBs remain strictly massless being protected by the non-Abelian gauge
invariance of the starting Lagrangian (8). Together with the aforementioned true
vector Goldstone boson they complete the entire Goldstonic vector field multiplet
of the internal symmetry group G.
As in the Abelian case, upon an explicit use of the corresponding SLIV constraint
(9) being so far the only supplementary condition for vector field multiplet Aiµ, one
comes to the pure Goldstone field modes aiµ identified in a similar way
Aiµ = a
(n · A) , n · a ≡ niµa
µ,i = 0 (n2 ≡ n2µ) , (11)
At the same time, an effective Higgs mode (i.e., the Aiµ component in the vacuum
direction niµ) is given by the product n · A ≡ n
µ,i determined by the SLIV con-
straint
n · A =
M2 − n2(aiν)
2 =M −
2(aiν)
+O(1/M2) (12)
where, as earlier in the Abelian case, we took the positive sign for the square
root when expanding it in powers of (aiν)
2/M2. Note that the general Goldstonic
modes aiµ, apart from pure vector fields, contain the D − 1 scalar ones, a
0 and a
(i′ = 1...D − 1), for the time-like (niµ = n
0gµ0δ
i0) and space-like (niµ = n
3gµ3δ
SLIV, respectively. They can be eliminated from the theory if one puts the proper
supplementary conditions on the aiµ fields which were still the constraint free. Using
their overall orthogonality (11) to the physical vacuum direction niµ one can formu-
late these supplementary conditions in terms of a general axial gauge for the entire
aiµ multiplet
n · ai ≡ nµa
µ,i = 0, i = 0...D − 1 (13)
where nµ is the unit Lorentz vector introduced in the Abelian case which is now
oriented in Minkowskian space-time so as to be parallel to the vacuum matrix niµ.
For such a choice the simple equation holds
µ = s
inµ (s
n · ni
) (14)
which shows that the rectangular vacuum matrix niµ has the factorized ”two-vector”
form. As a result, apart from the Higgs mode excluded earlier by the orthogonality
condition (11), all the scalar fields also appear eliminated, and only pure vector fields,
(µ′ = 1, 2, 3) or ai
(µ′′ = 0, 1, 2) for time-like or space-like SLIV, respectively,
are only left in the theory.
We now show that the such constrained Goldstone vector fields aiµ (with the
supplementary conditions (13) taken) appear truly massless when the starting non-
Abelian Lagrangian L (8) is rewritten in the form determined by the SLIV. Actually,
putting the parametrization (11) with the SLIV constraint (12) into the Lagrangian
(8) one is led to the highly nonlinear Yang-Mills theory in terms of the pure Gold-
stonic gauge field modes aiµ. However, as in the above Abelian case, one should
first gauge away (using the local invariance of the Lagrangian L) the enormously
large, while false, Lorentz violating terms appearing in the theory in the form of
the fermion and vector field bilinears. As one can readily see, they stem from the
couplings gψAµγ
µψ and −1
g2Tr([Aµ, Aν ])
2, respectively, when the effective Higgs
mode expansion (12) is taken in the Lagrangian (8). Making the appropriate redef-
initions of the fermion (ψ ) and vector (aµ ≡ a
i) field multiplets
ψ → U(ω)ψ , aµ → U(ω)aµU(ω)
†, U(ω) = eigM(n
i·x)ti (15)
and using the evident equalities for the linear (in coordinate) transformations U(ω)
with the single-valued vacuum matrix niµ (n
0 or n
3 for the particular SLIV cases)
∂µU(ω) = ign
iU(ω) = igU(ω)niµt
i (16)
one can confirm that the abovementioned Lorentz violating terms are exactly can-
celled with the analogous bilinears stemming from their kinetic terms. So, the final
Lagrangian for the Goldstonic Yang-Mills theory takes the form (in the first approx-
imation in (aiν)
2/M2)
L(a,ψ) = −
Tr(fµνf
δ(n · ai)2 +
Tr(fµνh
2(aiν)
+ψ(iγ · ∂ −m)ψ + gψaµγ
gn2(aiν)
ψ(γ · nk)tkψ (17)
where the tensor fµν is, as usual, fµν = ∂µaν − ∂νaµ − ig[aµ,aν ], while hµν is a
new SLIV oriented tensor of the type
hµν = nµ∂ν − nν∂µ + ig([nµ,aν ]− [nν ,aµ]), nµ ≡ n
k (18)
This tensor hµν acts on the infinite series in (a
2 coming from the expansion of the
effective Higgs mode (12) from which only the first order term −n2(aiν)
2/2M was
taken throughout the Lagrangian L(a,ψ). We also retained the former notations for
the fermion and vector field multiplets after transformations (15), and explicitly in-
cluded the (axial) gauge fixing term into Lagrangian according to the supplementary
conditions taken (13).
The theory derived gives a proper generalization of the nonlinear QED model[12]
for the non-Abelian case. It contains the massless vector boson multiplet aiµ (con-
sisting of one Goldstone and D − 1 pseudo-Goldstone vector states) which gauges
the starting internal symmetry G. In the limit M → ∞ it is indistinguishable from
a conventional Yang-Mills theory taken in the general axial gauge. So, for this part
of the Lagrangian L(a,ψ) given by the zero-order in 1/M terms the spontaneous
Lorentz violation only means the noncovariant gauge choice in the otherwise gauge
invariant (and Lorentz invariant) theory. However, one may expect that, just as
it appears in the nonlinear QED model, also all the first and higher order in 1/M
terms in the L (17), though being by themselves the Lorentz and CPT violating
ones, do not lead to the physical SLIV effects due to the mutual cancellation of their
contributions into all the physical processes appeared.
4 The lowest order SLIV processes
Let us now show that the simple tree level calculations related to the Lagrangian
L(a,ψ) confirms in essence this proposition. As an illustration, we consider SLIV
processes in the lowest order in g and 1/M being the fundamental parameters of the
Lagrangian (17). They are, as one can readily see, the vector-fermion and vector-
vector elastic scattering going in the order g/M , which we turn to once the Feynman
rules in the Goldstonic Yang-Mills theory are established.
4.1 Feynman rules
The corresponding Feynman rules, apart from the ordinary Yang-Mills theory rules
(i) the vector-fermion vertex
− ig γµ t
i (19)
(ii) the vector field propagator (taken in a general axial gauge nµaiµ = 0)
Dijµν (k) = −
gµν −
nµkν + kµnν
n · k
n2kµkν
(n · k)2
which automatically satisfies the orthogonality condition nµD
µν(k) = 0 and on-shell
transversality kµD
µν(k) = 0 (k
2 = 0); the latter means that free vector fields with
polarization vector ǫiµ(k, k
2 = 0) are always appeared transverse kµǫiµ(k) = 0;
(iii) the 3-vector vertex (with vector field 4-momenta k1, k2 and k3; all 4-momenta
in vertexes are taken ingoing throughout)
gcijk[(k1 − k2)γgαβ + (k2 − k3)αgβγ + (k3 − k1)βgαγ ] (21)
include the new ones, violating Lorentz and CPT invariance, for
(iv) the contact 2-vector-fermion vertex
(γ · nk)τkgµν δ
ij (22)
(v) another 3-vector vertex
(k1 · n
i)k1,αgβγδ
jk + (k2 · n
j)k2,βgαγδ
ki + (k3 · n
k)k3,γgαβδ
where the second index in the vector field 4-momenta k1, k2 and k3 denotes their
Lorentz components;
(vi) the extra 4-vector vertex (with the vector field 4-momenta k1,2,3,4 and their
proper differences k12 ≡ k1 − k2 etc.)
[cijpδklgαβgγδ(n
p · k12) + c
klpδijgαβgγδ(n
p · k34) +
+cikpδjlgαγgβδ(n
p · k13) + c
jlpδikgαγgβδ(n
p · k24) + (24)
+cilpδjkgαδgβγ(n
p · k14) + c
jkpδilgαδgβγ(n
p · k23)]
where only the terms which can not lead to contractions of the rectangular vacuum
matrix n
µ with vector field polarization vectors ǫ
µ(k) are presented. These contrac-
tions are in fact vanished due to the gauge taken (13), np · ǫi = sp(n · ǫi) = 0 (with
a factorized two-vector form for the matrix n
µ (14) used).
Just the rules (i-vi) are needed to calculate the lowest order amplitudes of the
processes we have mentioned in the above.
4.2 Vector boson scattering on fermion
This process is directly related to two SLIV diagrams one of which is given by the
contact a2-fermion vertex (22), while another corresponds to the pole diagram with
the longitudinal a-boson exchange between Lorentz violating a3 vertex (23) and
ordinary a-boson-fermion one (19). Since ingoing and outgoing a-bosons appear
transverse (k1 · ǫ
i(k1) = 0, k2 · ǫ
j(k2) = 0) only the third term in this a
3 coupling
(23) contributes to the pole diagram so that one comes to a simple matrix element
iM for both of diagrams
iM = i
ū(p2)τ
(γ · nl) + i(k · nl)γµkνDµν(k)
u(p1)[ǫ(k1) · ǫ(k2)] (25)
where the spinors u(p1,2) and polarization vectors ǫ
µ(k1) and ǫ
µ(k2) stand for the in-
going and outgoing fermions and a-bosons, respectively, while k is the 4-momentum
transfer k = p2 − p1 = k1 − k2. Upon the further simplifications in the square
bracket related to the explicit form of the a boson propagator Dµν(k) (20) and ma-
trix niµ (14), and using the fermion current conservation ū(p2)(p̂2 − p̂1)u(p1) = 0,
one is finally led to the total cancellation of the Lorentz violating contributions to
the a-boson-fermion scattering in the g/M approximation.
Note, however, that such a result may be in some sense expected since from the
SLIV point of view the lowest order a-boson-fermion scattering discussed here is
hardly distinct from the photon-fermion scattering considered in the nonlinear QED
case[12]. Actually, the fermion current conservation which happens to be crucial for
the above cancellation works in both of cases, whereas the couplings being peculiar
to the Yang-Mills theory have not yet touched on. In this connection the next
example seems to be more instructive.
4.3 Vector-vector scattering
The matrix element for this process in the lowest order g/M is given by the contact
SLIV a4 vertex (24) and the pole diagrams with the longitudinal a-boson exchange
between the ordinary a3 vertex (21) and Lorentz violating a3 one (23), and vice
versa. There are six pole diagrams in total describing the elastic a − a scattering
in the s- and t-channels, respectively, including also those with an interchange of
identical a-bosons. Remarkably, the contribution of each of them is exactly canceled
with one of six terms appeared in the contact vertex (24). Actually, writing down the
matrix element for one of the pole diagrams with ingoing a-bosons (with momenta k1
and k2) interacting through the vertex (21) and outgoing a-bosons (with momenta
k3 and k4) interacting through the vertex (23) one has
cijpδkl[(k1 − k2)µgαβ + (k2 − k)αgβµ + (k − k1)βgαµ] ·
·Dpqµν(k)gγδkν(n
q · k)[ǫi,α(k1)ǫ
j,β(k2)ǫ
k,γ(k3)ǫ
l,δ(k4)] (26)
where polarization vectors ǫi,α(k1), ǫ
j,β(k2), ǫ
k,γ(k3) and ǫ
l,δ(k4) belong, respectively,
to ingoing and outgoing a-bosons, while k = −(k1 + k2) = k3 + k4 according to the
momentum running in the diagrams taken above. Again, as in the previous case of
vector-fermion scattering, due to the fact that outgoing a-bosons appear transverse
(k3 · ǫ
k(k3) = 0 and k4 · ǫ
l(k4) = 0), only the third term in the Lorentz violating a
coupling (23) contributes to this pole diagram. Upon evident simplifications related
to the a-boson propagator Dµν(k) (20) and matrix n
µ (14) one comes to the expres-
sion which is exactly cancelled with the first term in the contact SLIV vertex (24)
when it is properly contracted with a-boson polarization vectors. Likewise, other
terms in this vertex provide the further one-to-one cancellation with the remaining
pole matrix elements iM
(2−6)
. So, again, the Lorentz violating contribution to the
vector-vector scattering is absent in Goldstonic Yang-Mills theory in the lowest g/M
approximation.
4.4 Other processes
Other tree level Lorentz violating processes, related to a bosons and fermions, appear
in higher orders in the basic SLIV parameter 1/M . They come from the subsequent
expansion of the effective Higgs mode (12) in the Lagrangian (17). Again, their
amplitudes are essentially determined by an interrelation between the longitudinal
a-boson exchange diagrams and the corresponding contact a-boson interaction dia-
grams which appear to cancel each other thus eliminating physical Lorentz violation
in theory.
Most likely, the same conclusion can be derived for SLIV loop contributions
as well. Actually, as in the massless QED case considered earlier [19], the corre-
sponding one-loop matrix elements in Goldstonic Yang-Mills theory either vanish
by themselves or amount to the differences between pairs of the similar integrals
whose integration variables are shifted relative to each other by some constants (be-
ing in general arbitrary functions of external four-momenta of the particles involved)
that in the framework of dimensional regularization leads to their total cancellation.
So, the Goldstonic vector field theory (17) for a non-Abelian charge-carrying
matter is likely to be physically indistinguishable from a conventional Yang-Mills
theory.
5 Conclusion
The spontaneous Lorentz violation in 4-dimensonal flat Minkowskian space-time was
shown to generate vector Goldstone bosons both in Abelian and non-Abelian theo-
ries with the corresponding nonlinear vector field constraint (1) or (9) imposed. In
the Abelian case such a massless vector boson is naturally associated with photon.
In non-Abelian case, although the pure Lorentz violation still generates only one
genuine Goldstone vector boson, the accompanying vector PGBs related to a vio-
lation of the larger accidental symmetry SO(D, 3D) of the SLIV constraint (9) in
itself come also into play in the final arrangement of the entire Goldstone vector
field multiplet of the internal symmetry group G. Remarkably, they remain strictly
massless being protected by the gauge invariance of the Yang-Mills theory involved.
These theories, both Abelian and non-Abelian, while being essentially nonlinear in
the Goldstone vector modes, are physically indistinguishable from conventional QED
and Yang-Mills theory. One could actually see that just the gauge invariance not
only provides these theories to be free from the unreasonably large Lorentz violation
stemming from the fermion and vector field bilinears (see Sections 2 and 3), but also
render all the other physical SLIV effects (including those which are suppressed by
the Lorentz violation scale M) non-observable (Section 4). As a result, Abelian and
non-Abelian SLIV theory appear, respectively, as standard QED and Yang-Mills
theory taken in the nonlinear gauge (to which the vector field constraints (1) and
(9) are virtually reduced), while the S-matrix remains unaltered under such a gauge
convention.
So, while at present the Goldstonic nature of gauge fields, both Abelian and non-
Abelian, seems to be highly plausible, the most fundamental question of physical
Lorentz violation in itself, that only could uniquely point toward such a possibility,
is still an open question. Note, that here we are not dealing with direct (and quite
arbitrary in essence) Lorentz non-invariant extensions of QED or Standard Model
which were intensively discussed on their own in recent years [6-8]. Rather, the case
in point is a construction of genuine SLIV models which would generate gauge fields
as the proper vector Goldstone bosons, from one hand, and could lead to observed
Lorentz violating effects, from the other. In this connection, somewhat natural
framework for physical Lorentz violation to occur would be a model where the
internal gauge invariance were slightly broken at very small distances through some
high-order operators stemming from the gravity-influenced area. Such physical SLIV
effects would be seen in terms of powers of ratio M/MP l (where MP l is the Planck
mass). So, for the SLIV scale comparable with the Planck one they would become
directly observable. Remarkably enough, if one has such internal gauge symmetry
breaking in an ordinary Lorentz invariant theory this breaking appears vanishingly
small at laboratory being properly suppressed by the Planck scale. However, the
spontaneous Lorentz violation would render it physically significant: the higher
Lorentz scale, the greater SLIV effects observed. If true, it would be of particular
interest to have a better understanding of the internal gauge symmetry breaking
mechanism that brings out the spontaneous Lorentz violation at low energies. We
return to this basic question elsewhere.
Acknowledgments
We would like to thank Colin Froggatt, Rabi Mohapatra and Holger Nielsen for useful
discussions and comments. One of us (J.L.C.) is grateful for the warm hospitality
shown to him during a visit to Center for Particle and String Theory at University
of Maryland where part of this work was carried out.
References
[1] W. Heisenberg, Rev. Mod. Phys. 29 (1957) 269;
J.D. Bjorken, Ann. Phys. (N.Y.) 24 (1963) 174;
I. Bialynicki-Birula, Phys. Rev. 130 (1963) 465 ;
G. Guralnik, Phys. Rev. 136 (1964) B1404;
T. Eguchi, Phys.Rev. D 14 (1976) 2755;
H. Terazava, Y. Chikashige and K. Akama, Phys. Rev. D 15 (1977) 480 .
[2] C.D. Froggatt and H.B. Nielsen, Origin of Symmetries (World Scientific, Sin-
gapore, 1991).
[3] J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen, Phys. Rev. Lett. 87 (2001)
091601;
J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen Nucl. Phys. B 609 (2001) 46;
J.D. Bjorken, hep-th/0111196;
Per Kraus and E.T. Tomboulis, Phys. Rev. D 66 (2002) 045015;
A. Jenkins, Phys. Rev. D 69 (2004) 105007;
J.L. Chkareuli, C.D. Froggatt, R.N. Mohapatra and H.B. Nielsen,
hep-th/0412225;
J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen, hep-th/0610186.
[4] D. Colladay and V.A. Kostelecky, Phys. Rev. D58 (1998) 116002 ;
V.A. Kostelecky, Phys. Rev. D69 (2004) 105009 ;
R. Bluhm and V.A. Kostelecky, Phys. Rev. D 71(2005) 065008;
CPT and Lorentz Symmetry, ed. A. Kostelecky (World Scientific, Singapore,
1999, 2002, 2005).
[5] S.M. Carroll, G.B. Field and R. Jackiw, Phys. Rev. D 41 (1990) 1231;
R. Jackiw and V.A. Kostelecky, Phys. Rev. Lett. 82 (1999) 3572.
[6] S. Coleman and S.L. Glashow, Phys. Rev. D 59 (1999) 116008.
http://arxiv.org/abs/hep-th/0111196
http://arxiv.org/abs/hep-th/0412225
http://arxiv.org/abs/hep-th/0610186
[7] J. W. Moffat, Int. J. Mod.Phys. D2 (1993) 351;
J.W. Moffat, Int. J. Mod.Phys. D12 (2003) 1279.
[8] O. Bertolami and D.F. Mota, Phys. Lett. B 455 (1999) 96.
[9] T. Jacobson, S. Liberati and D. Mattingly, Ann. Phys. (N.Y.) 321 (2006) 150.
[10] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345.
[11] M. Suzuki, Phys. Rev. D 37 (1988) 210 .
[12] Y. Nambu, Progr. Theor. Phys. Suppl. Extra 190 (1968).
[13] S. Weinberg, The Quantum Theory of Fields, v.2, Cambridge University Press,
2000.
[14] V.A. Kostelecky and S. Samuel, Phys. Rev. D 39 (1989) 683;
V.A. Kostelecky and R. Potting, Nucl. Phys. B 359 (1991) 545.
[15] P.A.M. Dirac, Proc. Roy. Soc. 209A (1951) 292;
P.A.M. Dirac, Proc. Roy. Soc. 212A (1952) 330.
[16] R. Righi and G. Venturi, Lett. Nuovo Cim. 19 (1977) 633;
R. Righi, G. Venturi and V. Zamiralov, Nuovo Cim. A47 (1978) 518.
[17] Y.M. Cho and P.G.O. Freund, Phys. Rev. D 12 (1975) 1711.
[18] E. A. Ivanov and V.I. Ogievetsky, Lett. Math. Phys. 1 (1976) 309 .
[19] A.T. Azatov and J.L. Chkareuli, Phys. Rev. D 73 (2006) 065026.
[20] J.L. Chkareuli and Z.R. Kepuladze, Phys. Lett. B 644 (2007) 212;
J.L. Chkareuli and Z.R. Kepuladze, Proc. of XIV Int. Seminar “Quarks-2006”,
eds. S.V. Demidov at al (Moscow, INR, 2006); hep-th/0610227.
http://arxiv.org/abs/hep-th/0610227
	Introduction
	Goldstonic quantum electrodynamics
	Goldstonic Yang-Mills theory
	The lowest order SLIV processes
	Feynman rules
	Vector boson scattering on fermion
	Vector-vector scattering
	Other processes
	Conclusion
