Gauge invariance in gravity-like descriptions of massive gauge field theories
Dennis D. Dietrich
Institut für Theoretische Physik, Universität Heidelberg, Heidelberg, Germany
(Dated: January 13, 2019)
We discuss gravity-like formulations of massive Abelian and non-Abelian gauge field theories in
four space-time dimensions with particular emphasis on the issue of gauge invariance. Alternative
descriptions in terms of antisymmetric tensor fields and geometric variables, respectively, are anal-
ysed. In both approaches Stückelberg degrees of freedom factor out. We also demonstrate, in the
Abelian case, that the massless limit for the gauge propagator, which does not exist in the vector
potential formulation, is well-defined for the antisymmetric tensor fields.
PACS numbers: 11.15.-q, 11.15.Ex, 11.30.Qc, 12.15.-y
I. INTRODUCTION
Massive gauge bosons belong to the fundamental con-
cepts we use for picturing nature. Prominent exam-
ples are found in the physics of electroweak interac-
tions, superconductivity, and confinement. Even more
than in the massless case, gauge invariance is a severe
constraint for the construction of massive gauge field
theories. Usually additional fields beyond the original
gauge field have to be included in order to obtain gauge
invariant expressions.[26] Technically, this is linked to
the fact that the aforementioned gauge field—the Yang–
Mills connection—changes inhomogeneously under gauge
transformations and encodes also spurious degrees of
freedom arising from the construction principle of gauge
invariance. This complicates the extraction of physical
quantities. A variety of approaches has been developed
in order to deal with this situation. Wilson loops [1]
represent gauge invariant but non-local variables.[27] Al-
ternatively, there exist decomposition techniques like the
one due to Cho, Faddeev, and Niemi [2]. Here we first
pursue a reformulation of massive Yang–Mills theories in
terms of antisymmetric gauge algebra valued tensor fields
Baµν (Sect. II) and subsequently continue with a represen-
tation in terms of geometric variables (Sect. III).
In Sect. II A we review the massless case. It is re-
lated to gravity [3] formulated as BF gravity [4] and
thus linked to quantum gravity. The antisymmetric ten-
sor field can be seen as dual field strength and transforms
homogeneously under gauge transformations. This fact
already makes it simpler to keep track of gauge invari-
ance. In Sect. II B the generalisation to the massive case
is presented. In the Baµν field representation the (non-
Abelian) Stückelberg fields, which are commonly present
in massive gauge field theories and needed there in or-
der to keep track of gauge invariance, factor out com-
pletely. In other words, no scalar fields are needed for a
gauge invariant formulation of massive gauge field theo-
ries in terms of antisymmetric tensor fields. The case of
a constant mass is linked to sigma models (gauged and
ungauged) in different respects. Sect. II B 1 contains the
generalisation to a position dependent mass, which cor-
responds to introducing the Higgs degree of freedom. In
Sect. II B 2 non-diagonal mass terms are admitted. This
is necessary to accommodate the Weinberg–Salammodel,
which is studied as particular case.
Sect. III presents a description of the massive case,
with constant and varying mass, in terms of geometric
variables. In this step the remaining gauge degrees of
freedom are eliminated. The emergent description is in
terms of local colour singlet variables. Finally, Sect. III A
is concerned with the geometric representation of the
Weinberg–Salam model.
The Appendix treats the Abelian case. It allows to
better interpret and understand several of the findings in
the non-Abelian settings. Of course, in the Abelian case
already the Bµν field is gauge invariant. Among other
things, we demonstrate that them→ 0 limit of the gauge
propagator for the Bµν fields is well-defined as opposed
to the ill-defined limit for the Aµ field propagator.
Sect. IV summarises the paper.
II. ANTISYMMETRIC TENSOR FIELDS
A. Massless
Before we investigate massive gauge field theories let
us recall some details about the massless case. The parti-
tion function of a massless non-Abelian gauge field theory
without fermions is given by
[dA] exp{i
xL}, (1)
with the Lagrangian density
L = L0 := − 14g2F
aµν (2)
and the field tensor
µν := ∂µA
ν − ∂νAaµ + fabcAbµAcν . (3)
Aaµ stands for the gauge field, f
abc for the antisymmetric
structure constant, and g for the coupling constant. [28]
Variation of the classical action with respect to the gauge
field gives the classical Yang–Mills equations
Dabµ (A)F
bµν = 0, (4)
http://arxiv.org/abs/0704.1828v1
where the covariant derivative is defined as Dabµ (A) :=
δab∂µ + f
acbAcµ. The partition function in the first-order
formalism can be obtained after multiplying Eq. (1) with
a prefactor in form of a Gaussian integral over an anti-
symmetric tensor field Baµν ,
[dA][dB] exp{i
d4x[L0 − g
BaµνB
aµν ]}. (5)
(”∼=” indicates that in the last step the normalisation of
the partition function has been changed.) Subsequently,
the field Baµν is shifted by
F̃ aµν , where the dual field
tensor is defined as F̃ aµν :=
ǫµνκλF
[dA][dB] ×
× exp{i
x[− 1
aµν − g
aµν ]}. (6)
In this form the partition function is formulated in terms
of the Yang–Mills connection Aaµ and the antisymmetric
tensor field Baµν as independent variables. Variation of
the classical action with respect to these variables leads
to the classical equations of motion
µν = −F̃ aµν and Dabµ (A)B̃bµν = 0, (7)
where B̃aµν :=
ǫκλµνB
aκλ. By eliminating Baµν the origi-
nal Yang–Mills equation (4) is reproduced. Every term in
the classical action in the partition function (6) contains
at most one derivative as opposed to two in Eq. (1). This
explains the name ”first-order” formalism. The classical
action in Eq. (6) is invariant under simultaneous gauge
transformations of the independent variables according
AaµT a =: Aµ → AµU := U [A
µ − iU †(∂µU)]U † (8)
BaµνT a =: Bµν → Bµν
:= UBµνU †, (9)
or infinitesimally,
δAaµ = ∂µθ
a + fabcAbµθ
δBaµν = f
abcBbµνθ
c. (10)
The T a stand for the generators of the gauge group. From
the Bianchi identity Dabµ (A)F̃
bµν = 0 follows a second
symmetry of the BF term alone: Infinitesimally, for un-
changed Aaµ,
µν = ∂µϑ
ν − ∂νϑaµ + fabc(Abµϑcν −Abνϑcµ). (11)
A particular combination of the transformations (10) and
(11), θa = nµAaµ and ϑ
ν = n
µBaµν , corresponds to the
transformation of a tensor under an infinitesimal local
coordinate transformation xµ → xµ − nµ(x),
δBµν = Bλν∂µn
λ +Bµλ∂νn
λ + nλ∂λBµν , (12)
that is a diffeomorphism. Hence, the BF term is dif-
feomorphism invariant, which explains why this theory
is also known as BF gravity. The BB term is not dif-
feomorphism invariant and, hence, imposes a constraint.
The combination of the two terms amounts to an action
of Plebanski type which are studied in the context of
quantum gravity [3, 4].
We now would like to eliminate the Yang–Mills connec-
tion by integrating it out. For fixed Baµν the integrand
of the path integral is not gauge invariant with respect
to gauge transformations of the gauge field Aaµ alone; the
field tensor F aµν transforms homogeneously and the corre-
sponding gauge transformations are not absorbed if Baµν
is held fixed. Therefore, the integral over the gauge group
is in general not cyclic which otherwise would render the
path integral ill-defined. The term in the exponent linear
in the gauge field Aaµ, A
ν∂µB̃
aµν , is obtained by carry-
ing out a partial integration in which surface terms are
ignored. Afterwards it is absorbed by shifting Aaµ by
(B−1)abµν(∂λB̃
bλν), where Babµν := f
abcB̃aµν . In general its
inverse (B−1)abµν , defined by (B
−1)abµνB
νκ = δacgµλ
exists in three or more space-time dimensions [5]. We
are left with a Gaussian integral in Aaµ giving the inverse
square-root of the determinant of Babµν ,
[da] exp{− i
bν}. (13)
In the last expression Babµν appears in the place of an in-
verse gluon propagator, that is sandwiched between two
gauge fields. This analogy carries even further: Interpret-
ing ∂µB̃
aµν as a current, (B−1)abµν(∂λB̃
bλν), the current
together with the ”propagator” (B−1)abµν , is exactly the
abovementioned term to be absorbed in the gauge field
Aaµ. Finally, we obtain,
[dB]Det−
B exp{i
d4x[− g
BaµνB
aµν −
(∂κB̃
aκµ)(B−1)abµν(∂λB̃
bλν)]}. (14)
This result is known from [5, 6, 7]. The exponent in
the previous expression corresponds to the value of the
[dA] integral at the saddle-point value Ăaµ of the gauge
field. It obeys the classical field equation (7). Using
Ăaµ(B) = (B
−1)abµν(∂λB̃
bλν) the second term in the above
exponent can be rewritten as − i
d4xB̃aµνF
aµν [Ă(B)],
which involves an integration by parts and makes its
gauge invariance manifest. The fluctuations aaµ around
the saddle point Ăaµ, contributing to the partition func-
tion (6), are Gaussian because the action in the first-order
formalism is only of second order in the gauge field Aaµ.
They give rise to the determinant (13). What happens
if a zero of the determinant is encountered can be un-
derstood by looking at the Abelian case discussed in Ap-
pendix A. There the BF term does not fix a gauge for the
integration over the gauge field Aµ because the Abelian
field tensor Fµν is gauge invariant. If it is performed
nevertheless one encounters a functional δ distribution
which enforces the vanishing of the current ∂µB̃
µν . In
this sense the zeros of the determinant in the non-Abelian
case arise if B̃aµν is such that the BF term does not totally
fix a gauge for the [dA] integration, but leaves behind a
residual gauge invariance. It in turn corresponds to van-
ishing components of the current ∂µB̃
aµν . (Technically,
there then is at least one flat direction in the otherwise
Gaussian integrand. The flat directions are along those
eigenvectors of B possessing zero eigenvalues.)
When incorporated with the exponent, which requires
a regularisation [8], the determinant contributes a term
proportional to 1
ln detB to the action. This term to-
gether with the BB term constitutes the effective poten-
tial, which is obtained from the exponent in the partition
function after dropping all terms containing derivatives of
fields. The effective potential becomes singular for field
configurations for which detB = 0. It is gauge invari-
ant because all contributing addends are gauge invariant
separately.
The classical equations of motion obtained by varying
the action in Eq. (14) with respect to the dual antisym-
metric tensor field B̃aµν are given by
g2B̃aµν = (g
µ − gρµgσν )∂ρ(B−1)abσκ(∂λB̃bλκ)−
−(∂ρB̃dρκ)(B−1)dbκµfabc(B−1)ceνλ(∂σB̃eσλ),
which coincides with the first of Eqs. (7) with the
field tensor evaluated at the saddle point of the action,
F aµν [Ă(B)]. Taking into account additionally the effect
due to fluctuations of Aaµ contributes a term proportional
to δDetB
δB̃aµν
det−1B to the previous equation.
B. Massive
In the massive case the prototypical Lagrangian is of
the form L = L0 +Lm, where Lm := m
aµ. (Due to
our conventions the physical mass is given by mphys :=
mg.) This contribution to the Lagrangian is of course
not gauge invariant. Putting it, regardlessly, into the
partition function, gives
[dA][dU ] exp{i
x[L0 + m
aµ]}, (16)
which can be interpreted as the unitary gauge represen-
tation of an extended theory. In order to see this let us
split the functional integral over Aaµ into an integral over
the gauge group [dU ] and gauge inequivalent field config-
urations [dA]′. Usually this separation is carried out by
fixing a gauge according to
[dA]′ :=
[dA]δ[fa(A) − Ca]∆f (A). (17)
fa(A) = Ca is the gauge condition and ∆f (A) stands
for the Faddeev–Popov determinant defined through
[dA]δ[fa(A) − Ca]∆f (A) [29]. Introducing this
reparametrisation into the partition function (16) yields,
[dA]′[dU ] exp(i
d4x{− 1
F aµνF
aµν +
[Aµ − iU †(∂µU)]a[Aµ − iU †(∂µU)]a}).
L0 is gauge invariant in any case and remains thus un-
affected. In the mass term the gauge transformations
appear explicitly [9]. We now replace all of these gauge
transformations with an auxiliary (gauge group valued)
scalar field Φ, U † → Φ, obeying the constraint
!≡ 1. (19)
The field Φ can be expressed as Φ =: e−iθ, where
θ =: θaT a is the gauge algebra valued non-Abelian gen-
eralisation of the Stückelberg field [10]. For a massive
gauge theory they are a manifestation of the longitudi-
nal degrees of freedom of the gauge bosons. In the con-
text of symmetry breaking they arise as Goldstone modes
(”pions”). In the context of the Thirring model these ob-
servations have been made in [11]. There it was noted
as well that the θ is also the field used in the canonical
Hamiltonian Batalin–Fradkin–Vilkovisky formalism [12].
We can extract the manifestly gauge invariant classical
Lagrangian
Lcl := − 14g2F
aµν +m2tr[(DµΦ)
†(DµΦ)], (20)
where the scalars have been rearranged making use of the
product rule of differentiation and the cyclic property of
the trace and whereDµΦ := ∂µΦ−iAµΦ. Eq. (20) resem-
bles the Lagrangian density of a non-linear gauged sigma
model. In the Abelian case the fields θ decouple from
the dynamics. For non-Abelian gauge groups they do
not and one would have to deal with the non-polynomial
coupling to them.
In the following we show that these spurious degrees of
freedom can be absorbed when making the transition to a
formulation based on the antisymmetric tensor field Baµν .
Introducing the antisymmetric tensor field into the corre-
sponding partition function, like in the previous section,
results in,
[dA][dΦ][dB] exp(i
×{− g
aµν − 1
aµν +
[Aµ − iΦ(∂µΦ†)]a[Aµ − iΦ(∂µΦ†)]a}).
Removing the gauge scalars Φ from the mass term by a
gauge transformation of the gauge field Aaµ makes them
explicit in the BF term,
[dA][dΦ][dB] exp{i
d4x[− g
BaµνB
aµν −
−tr(ΦF̃µνΦ†Bµν) + m
aµ]}. (22)
In the next step we would like to integrate over the Yang–
Mills connection Aaµ. Already in the previous expression,
however, we can perceive that the final result will only
depend on the combination of fields Φ†BµνΦ. [The Φ
field can also be made explicit in the BB term in form
of the constraint (19).] Therefore, the functional inte-
gral over Φ only covers multiple times the range which
is already covered by the [dB] integration. Hence the
degrees of freedom of the field Φ have become obsolete
in this formulation and the [dΦ] integral can be factored
out. Thus, we could have performed the unitary gauge
calculation right from the start. In either case, the final
result reads,
[dB]Det−
M exp{i
x[− g
aµν −
(∂κB̃
aκµ)(M−1)abµν(∂λB̃
bλν)]}, (23)
where Mabµν := B
µν − m2δabgµν , which coincides with
[13]. Mabµν and hence (M
−1)abµν transform homogeneously
under the adjoint representation. In Eq. (14) the cen-
tral matrix (B−1)abµν in the analogous term transformed
in exactly the same way. There this behaviour ensured
the gauge invariance of this term’s contribution to the
classical action. Consequently, the classical action in the
massive case has the same invariance properties. In par-
ticular, the aforementioned gauge invariant classical ac-
tion describes a massive gauge theory without having to
resort to additional scalar fields. For detB 6= 0, the limit
m → 0 is smooth. For detB = 0 the conserved current
components alluded to above would have to be separated
appropriately in order to recover the corresponding δ dis-
tributions present in these situations in the massless case.
Again the effective action is dominated by the term
proportional to 1
detM. The contribution from the mass
to M shifts the eigenvalues from the values obtained for
B. Hence the singular contributions are typically ob-
tained for eigenvalues of B of the order of m2. The ef-
fective potential is again gauge invariant, for the same
reason as in the massless case.
The classical equations of motion obtained by variation
of the action in Eq. (21) are given by,
µν = −F̃ aµν ,
Dabµ (A)B̃
bµν = −m2[Aν − iΦ(∂νΦ†)]a,
0 = δ
x{[Aaµ − iΦ(∂µΦ†)]a}2. (24)
In these equations a unique solution can be chosen, that
is a gauge be fixed, by selecting the scalar field Φ. Φ ≡ 1
gives the unitary gauge, in which the last of the above
equations drops out. The general non-Abelian case is
difficult to handle already on the classical level, which
is one of the main motivations to look for an alternative
formulation. In the non-Abelian case, the equation of
motion obtained from Eq. (23) resembles strongly the
massless case,
g2B̃aµν = (g
µ − gρµgσν )∂ρ(M−1)abσκ(∂λB̃bλκ)−
−(∂ρB̃dρκ)(M−1)dbκµfabc(M−1)ceνλ(∂σB̃eσλ),
insofar as all occurrences of (B−1)abµν have been replaced
by (M−1)abµν . Incorporation of the effect of the Gaussian
fluctuations of the gauge field Aaµ would give rise to a con-
tribution proportional to δB
δB̃aµν
det−1M in the previous
equation.
Before we go over to more general cases of massive
non-Abelian gauge field theories, let us have a look at
the weak coupling limit: There the BB term in Eq. (21)
is neglected. Subsequently, integrating out the Baµν field
enforces F aµν ≡ 0. [This condition also arises from the
classical equation of motion (24) for g=0.] Hence, for
vanishing coupling exclusively pure gauge configurations
of the gauge field Aaµ contribute. They can be combined
with the Φ fields and one is left with a non-linear reali-
sation of a partition function,
g=0∼=
[dΦ] exp{im2
d4x tr[(∂µΦ
†)(∂µΦ)]}, (26)
of a free massless scalar [13]. Setting g = 0 interchanges
with integrating out the Baµν field from the partition func-
tion (21). Thus, the partition function (23) with g = 0 is
equivalent to (26). That a scalar degree of freedom can
be described by means of an antisymmetric tensor field
has been noticed in [14].
1. Position-dependent mass and the Higgs
One possible generalisation of the above set-up is ob-
tained by softening the constraint (19). This can be seen
as allowing for a position dependent mass. The new
degree of freedom ultimately corresponds to the Higgs.
When introducing the mass m as new degree of freedom
(as ”mass scalar”) we can restrict its variation by in-
troducing a potential term V (m2), which remains to be
specified, and a kinetic term K(m), which we choose in
its canonical form K(m) = 1
(∂µm)(∂
µm). It gives a
penalty for fast variations of m between neighbouring
space-time points. The fixed mass model is obtained in
the limit of an infinitely sharp potential with its mini-
mum located at a non-zero value for the mass. Putting
together the partition function in unitary gauge leads to,
[dA][dm] exp{i
d4x[− 1
F aµνF
aµν +
aµ +K(m) + V (m2)]}, (27)
where we have introduced the normalisation constant
N := dim R, with R standing for the representation of
the scalars. This factor allows us to keep the canonical
normalisation of the mass scalar m. We can now repeat
the same steps as in the previous section in order to iden-
tify the classical Lagrangian,
Lcl := − 14g2F
aµν +N−1tr[(Dµφ)†(Dµφ)] + V (|φ|2),
where now φ := mΦ. In order to reformulate the parti-
tion function in terms of the antisymmetric tensor field
we can once more repeat the steps in the previous sec-
tion. Again the spurious degrees of freedom represented
by the field Φ can be factored out. Finally, this gives
[15],
[dB][dm]Det−
M exp{i
d4x[− g
BaµνB
aµν −
(∂κB̃
aκµ)(M−1)abµν(∂λB̃
bλν) +
+K(m) + V (m2)]}, (28)
whereMabµν = B
µν −m2N−1δbgµν depends on the space-
time dependent mass m. The determinant can as usual
be included with the exponent in form of a term pro-
portional to 1
detM, the pole of which will dominate the
effective potential. As just mentioned, however,M is also
a function of m. Hence, in order to find the minimum,
the effective potential must also be varied with respect
to the mass m.
Carrying the representation in terms of antisymmetric
tensor fields another step further, the partition function
containing the kinetic term K(m) of the mass scalar can
be expressed as Abelian version of Eq. (26),
[db][da] exp{i
d4x[− 1
b̃µνf
µν + 1
µ]} =
[dm] exp{i
d4x[ 1
(∂µm)(∂
µm)]}, (29)
where here the mass scalar m is identified with the
Abelian gauge parameter. Combining the last equation
with the partition function (28) all occurrences of the
mass scalar m can be replaced by the phase integral
dxµaµ. The bf term enforces the curvature f
to vanish which constrains aµ to pure gauges ∂µm and
the aforementioned integral becomes path-independent.
2. Non-diagonal mass term and the Weinberg–Salam model
The mass terms investigated so far had in common that
all the bosonic degrees of freedom they described pos-
sessed the same mass. A more general mass term would
be given by Lm := m
ab. Another similar ap-
proach is based on the Lagrangian Lm := m
tr{AµAµΨ}
where Ψ is group valued and constant. We shall begin our
discussion with this second variant and limit ourselves to
a Ψ with real entries and trΨ = 1, which, in fact, does
not impose additional constraints. Using this expression
in the partition function (27) and making explicit the
gauge scalars yields,
[dA][dm] exp{i
d4x[− 1
F aµνF
aµν +
tr{(Dµφ)†(Dµφ)Ψ} + V (m2)]}. (30)
Expressed in terms of the antisymmetric tensor field
Baµν , the corresponding partition function coincides with
Eq. (28) but with Mabµν replaced by M
µν := B
m2tr{T aT bΨ}gµν.
Let us now consider directly the SU(2)× U(1)
Weinberg–Salam model. Its partition function can be
expressed as,
[dA][dψ] exp{i
x[− 1
aµν +
∂ µ + iAµ)(
∂ µ − iAµ)ψ + V (|ψ|2)]},
where ψ is a complex scalar doublet, Aµ := A
a, with
a ∈ {0; . . . ; 3}, T a here stands for the generators of
SU(2) in fundamental representation, and, accordingly,
T 0 for g0
times the 2 × 2 unit matrix, with the U(1)
coupling constant g0. The partition function can be
reparametrised with ψ = mΦψ̂, where m =
|ψ|2, Φ
is a group valued scalar field as above, and ψ̂ is a con-
stant doublet with |ψ̂|2 = 1. The partition function then
becomes,
[dA][dΦ][dm] exp(i
x{− 1
aµν +
tr[Φ†(
∂ µ + iAµ)(
µ − iAµ)ΦΨ] +
(∂µm)(∂
µm) + V (m2)}), (32)
where
Ψ = ψ̂ ⊗ ψ̂†. (33)
Making the transition to the first order formalism leads
[dA][dB][dΦ][dm] exp(i
d4x{− g
BaµνB
aµν −
F aµνB̃
aµν +K(m) + V (m2) +
tr[Φ†(
∂ µ + iAµ)(
∂ µ − iAµ)ΦΨ]}). (34)
As in the previous case, a gauge transformation of the
gauge field Aaµ can remove the gauge scalar Φ from the
mass term (despite the matrix Ψ). Thereafter Φ only
appears in the combination Φ†BµνΦ and the integral [dΦ]
merely leads to repetitions of the [dB] integral. [The U(1)
part drops out completely right away.] Therefore the [dΦ]
integration can be factored out,
[dA][dB][dm] exp{i
d4x[− g
BaµνB
aµν −
F aµνB̃
aµν + m
tr(AµA
µΨ) +
+K(m) + V (m2)]}. (35)
The subsequent integration over the gauge fields A
µ leads
[dB][dm]Det−
M exp{i
d4x[− g
BaµνB
aµν −
(∂κB̃
aκµ)(M−1)abµν(∂λB̃
bλν) +
+K(m) + V (m2)]}, (36)
where M
µν := B
µν −m2tr(T aT bΨ)gµν .
From hereon we continue our discussion based on the
mass matrix
ab := 1
tr({T a, T b}Ψ), (37)
which had already been mentioned at the beginning of
Sect. II B 2. mab is real and has been chosen to be sym-
metric. (Antisymmetric parts are projected out by the
contraction with the symmetric A
bµ.) Thus it pos-
sesses a complete orthonormal set of eigenvectors µ
j with
the associated real eigenvalues mj , m
= 6Σjmjµ
With the help of these normalised eigenvectors one can
construct projectors π
j := 6Σjµ
j and decompose the
mass matrix, mab = mjπ
j . The projectors are com-
plete, 1ab = Σjπ
j , idempotent 6Σjπ
j = π
j , and
satisfy π
j 6=k
= 0. The matrix B
µν , the antisym-
metric tensor field Baµν , and the gauge field A
µ can
also be decomposed with the help of the eigenvectors:
µν = µ
, where bjkµν := µ
µν = b
where bjµν := B
j ; and A
µ = a
j , where a
µ := A
Using this decomposition in the partition function (36)
leads to,
[db][dm]Det−
m exp{i
d4x[− g
bjµνb
jµν −
(∂κb̃
jκµ)(m−1)jkµν(∂λb̃
kλν) +
+K(m) + V (m2)]}, (38)
where mjkµν := b
µν −m2
jlδklgµν .
Making use of the concrete form of mab given in
Eq. (37), inserting Ψ from Eq. (33), and subsequent diag-
onalisation leads to the eigenvalues 0, 1
and 1
These correspond to the photon, the two W bosons and
the heavier Z boson, respectively. The thus obtained tree-
level Z to W mass ratio squared consistently reproduces
the cosine of the Weinberg angle in terms of the coupling
constants, cos2 ϑw =
g2+g2
. Due to the masslessness of
the photon one addend in the sum over l in the expression
µν above does not contribute. Still, the totalm
µν does
not vanish like in the case of a single massless Abelian
gauge boson (see Appendix A). Physically this corre-
sponds to the coupling of the photon to the W and Z
bosons.
III. GEOMETRIC REPRESENTATION
The fact that the antisymmetric tensor field Baµν trans-
forms homogeneously represents already an advantage
over the formulation in terms of the inhomogeneously
transforming gauge fields Aaµ. Still, B
µν contains de-
grees of freedom linked to the gauge transformations (9).
These can be eliminated by making the transition to
a formulation in terms of geometric variables. In this
section we provide a classically equivalent description of
the massive gauge field theories in terms of geometric
variables in Euclidean space for two colours by adapt-
ing Ref. [16] to include mass. The first-order action is
quadratic in the gauge-field Aaµ.[30] Thus the evaluation
of the classical action at the saddle point yields the ex-
pression equivalent to the different exponents obtained
after integrating out the gauge field Aaµ in the various
partition functions in the previous section. In Euclidean
space the classical massive Yang–Mills action in the first
order formalism reads
d4x(LBB + LBF + LAA), (39)
where
LBB = − g
BaµνB
µν , (40)
LBF = + i4ǫ
µνκλBaµνF
κλ, (41)
LAA = −m
µ. (42)
At first we will investigate the situation for the unitary
gauge mass term LAA and study the role played by the
scalars Φ afterwards.
As starting point it is important to note that a metric
can be constructed that makes the tensor Baµν self-dual
[7]. In order to exploit this fact, it is convenient to define
the antisymmetric tensor (j ∈ {1; 2; 3})
µν := η
ν , (43)
with the self-dual ’t Hooft symbol η
[17] [31] and the
tetrad eAµ . From there we construct a metric gµν in terms
of the tensor T jµν
gµν ≡ eAµ eAν = 16ǫ
jklT jµκT
kκλT lλν , (44)
where
jµν := 1
g)3 := 1
(ǫjklT
T kµ2ν2T
×(ǫj′k′l′T j
×ǫµ1ν1κ1λ1ǫµ2ν2κ2λ2ǫµ3ν3κ3λ3 (46)
Subsequently, we introduce a triad daj such that
Baµν =: d
µν . (47)
This permits us to reexpress the BB term of the classical
Lagrangian,
LBB = − g
µνhjkT
µν , (48)
where hjk := d
k. Putting Eqs. (47) and (45) into the
saddle point condition
ǫκλµνDabµ (Ă)B
κλ = +im
2Ăaν (49)
gives
µ (Ă)(
gdbjT
jµν) = +im2Ăaν . (50)
In the following we define the connection coefficients γµ|kj
as expansion parameters of the covariant derivative of the
triads at the saddle point in terms of the triads,
Dabµ (Ă)d
j =: γµ|kj dak. (51)
This would not be directly possible for more than two
colours, as then the set of triads is not complete. The
connection coefficients allow us to define covariant deriva-
tives according to
∇µ|kj := ∂µδkj + γµ|kj . (52)
These, in turn, permit us to rewrite the saddle point
condition (49) as
dak∇µ|kj (
gT jµν) = im2Ăaν , (53)
and the mass term in the classical Lagrangian becomes
LAA = 12m2 [∇µ|
gT iµν)]hkl[∇κ|lj(
gT jκν)]. (54)
In the limit m→ 0 this term enforces the covariant con-
servation condition ∇µ|ki (
gT iµν) ≡ 0, known for the
massless case. It results also directly from the saddle
point condition (53). Here dak∇µ|ki (
gT iµν) are the di-
rect analogues of the Abelian currents ǫµνκλ∂µBκλ, which
are conserved in the massless case [see Eq. (A6)] and dis-
tributed following a Gaussian distribution in the massive
case [see Eq. (A10)].
The commutator of the above covariant derivatives
yields a Riemann-like tensor Rkjµν
Rkjµν := [∇µ,∇ν ]kj . (55)
By evaluating, in adjoint representation (marked by
)̊, the following difference of double commutators
[D̊µ(Ă), [D̊ν(Ă), d̊j ]]− (µ↔ ν) in two different ways, one
can show that
i[d̊j , F̊µν(Ă)] = d̊kR
jµν , (56)
or in components,
F aµν(Ă) =
ǫabcdbjdckR
jµν , (57)
where dajdak := δ
defines the inverse triad, daj = hjkdak.
Hence, we are now in the position to rewrite the remain-
ing BF term of the Lagrangian density. Introducing
Eqs. (47) and (57) into Eq. (41) results in
LBF = i4
gT jµνRklµνǫjmkh
lm. (58)
Let us now repeat the previous steps with a mass term
in which the gauge scalars Φ are explicit,
LΦAA := −
[Aµ − iΦ(∂µΦ†)]a[Aµ − iΦ(∂µΦ†)]a. (59)
In that case the saddle point condition (49) is given by,
µ (Ă)B̃
κλ = im
2[Ăν − iΦ(∂νΦ†)]a, (60)
or in the form of Eq. (53), that is with the left-hand side
replaced,
dak∇µ|kj (
gT jµν) = im2[Ăν − iΦ(∂νΦ†)]a. (61)
Reexpressing LΦAA with the help of the previous equation
reproduces exactly the unitary gauge result (54) for the
mass term.
Finally, the tensor B appearing in the determinant
(13), which accounts for the Gaussian fluctuations of the
gauge field Aaµ, formulated in the new variables reads
gfabcdai T
iµν . Now all ingredients are known
which are needed to express the equivalent of the parti-
tion function (16) in terms of the new variables. For a
position-dependent mass the discussion does not change
materially. The potential and kinematic term for the
mass scalar m have to be added to the action.
Contrary to the massless case the Aaµ dependent part
of the Euclidean action is genuinely complex. Without
mass only the T-odd and hence purely imaginary BF
term was Aaµ dependent. With mass there contributes
the additional T-even and thus real mass term. There-
fore the saddle point value Ăaµ for the gauge field becomes
complex. This is a known phenomenon and in this con-
text it is essential to deform the integration contour of
the path integral in the partition function to run through
the saddle point [18]. For the Gaussian integrals which
are under consideration here, in doing so, we do not re-
ceive additional contributions. The imaginary part IĂaµ
of the saddle point value of the gauge field transforms
homogeneously under gauge transformations. The com-
plex valued saddle point of the gauge field which is in-
tegrated out does not affect the real-valuedness of the
remaining fields, here Baµν . In this sense the field B
represents a parameter for the integration over Aaµ. The
tensor T jµν is real-valued by definition and therefore the
same holds also for the triad daj [see Eq. (47)]. hkl is
composed of the triads and, consequently, real-valued as
well. The imaginary part of the saddle point value of the
gauge field, IĂaµ, enters the connection coefficients (51).
Through them it affects the covariant derivative (52) and
the Riemann-like tensor (55). More concretely the con-
nection coefficients γµ|kj can be decomposed according to
Dabµ (RĂ)dbj = (Rγµ|kj )dak, (62)
abc(IĂcµ)dbj = (Iγµ|kj )dak, (63)
with the obvious consequences for the covariant deriva-
tive,
∇µ|kj = R∇µ|kj + iI∇µ|kj , (64)
R∇µ|kj = ∂µδkj +Rγµ|kj , (65)
I∇µ|kj = Iγµ|kj . (66)
This composition reflects in the mass term,
RLAA = 12m2 {[R∇µ|
gT iµν)]hkl[R∇κ|lj(
gT jκν)]−
−[Iγµ|kj (
gT iµν)]hkl[Iγκ|lj(
gT jκν)]}
ILAA = 22m2 [R∇µ|
gT iµν)]hkl[I∇κ|lj(
jκν)]
on one hand, and in the Riemann-like tensor,
RRkjµν = [R∇µ,R∇ν ]kj − [I∇µ, I∇ν ]kj (67)
IRkjµν = [R∇µ, I∇ν ]kj + [I∇µ,R∇ν ]kj . (68)
on the other. The connection to the imaginary part of
Ăaµ is more direct in Eq. (57) which yields,
RF aµν (Ă) = 12ǫ
abcdbjdckRRkjµν , (69)
IF aµν (Ă) = 12ǫ
kIRkjµν , (70)
Finally, the BF term becomes,
RLBF = − 14
gT jµνǫjmkh
lmIRklµν , (71)
ILBF = + 14
gT jµνǫjmkh
lmRRklµν . (72)
Summing up, at the complex saddle point of the [dA] in-
tegration the emerging Euclidean LAA and LBF are both
complex, whereas before they were real and purely imag-
inary, respectively. Both terms together determine the
saddle point value Ăaµ. Therefore, they become coupled
and cannot be considered separately anymore. This was
already to be expected from the analysis in Minkowski
space in Sect. II, where the matrixMabµν combines T-odd
and T-even contributions, which originate from LAA and
LBF , respectively. There the different contributions be-
come entangled when the inverse (M−1)abµν is calculated.
A. Weinberg–Salam model
Finally, let us reformulate the Weinberg–Salam model
in geometric variables. We omit here the kinematic term
K(m) and the potential term V (m2) for the sake of
brevity because they do not interfere with the calcula-
tions and can be reinstated at every time. The remaining
terms of the classical action are
x(LAbelBB + LAbelBF + LBB + LBF + LAA),
LAA := −m
µ, (73)
LAbelBB := − g
µν , (74)
LAbelBF := + i4ǫ
µνκλB0µνF
κλ, (75)
and LBB as well as LBF have been defined in Eqs. (40)
and (41), respectively.
The saddle point conditions for the [dA] integration
with this action are given by
ǫκλµνDabµ (Ă)B
κλ = +im
abAbν , (76)
κλ = +im
ν . (77)
For the following it is convenient to use linear combina-
tions of these equations, which are obtained by contrac-
tion with the eigenvectors µ
of the matrix mab—defined
between Eqs. (37) and (38)—,
ǫκλµν [µalD
µ (Ă)B
κλ + µ
l ∂µB
κλ] = im
abAbν .(78)
The non-Abelian term on the left-hand side can be
rewritten using the results from the first part of Sect. III.
The right-hand side may be expressed in terms of eigen-
values of the matrix mab. We find (no summation over
Xaν = im2mla
ν , (79)
where
aν := daj∇µ|
gT kµν) + 1
l ∂µB
κλ. (80)
The mass term can be decomposed in the eigenbasis of
ab as well and, subsequently, be formulated in terms of
the geometric variables,
LAA = −m
(m̄−1)abXaνXbν , (81)
where
(m̄−1)ab :=
∑∀ml 6=0
. (82)
With the help of these relations and the results from
the beginning of Sect. III we are now in the posi-
tion to express the classical action in geometric vari-
ables: The mass term is given in the previous expres-
sion. It describes a Gaussian distribution of a com-
posite current. The components of the current are su-
perpositions of Abelian and non-Abelian contributions.
This mixture is caused by the symmetry breaking pat-
tern SU(2)L × U(1)Y → U(1)em which leaves unbroken
U(1)em and not the U(1)Y which is a symmetry in the
unbroken phase. The Abelian antisymmetric fields B0µν
in LAbelBB are gauge invariant and we leave LAbelBB as de-
fined in Eq. (74). In geometric variables LBB is given
by Eq. (48) and LBF by Eq. (58). At the end the ki-
netic term K(m) and the potential term V (m2) should
be reinstated.
Additional contributions from fluctuations give rise to
an addend (on the level of the Lagrangian) proportional
ln detm, wherem can be expressed in the new vari-
ables, mjkµν = f
abcdal µ
gT lµν −m2
ljδklgµν .
Repeating the entire calculation not in unitary gauge,
but with explicit gauge scalars Φ, yields exactly the same
result because the mass term and the saddle point con-
dition change in unison, such that Eq. (79) is obtained
again. This has already been demonstrated explicitly for
a massive Yang–Mills theory just before Sect. III A.
IV. SUMMARY
We have discussed the formulation of massive gauge
field theories in terms of antisymmetric tensor fields
(Sect. II) and of geometric variables (Sect. III). The
description in terms of an antisymmetric tensor field
Baµν has the advantage that it transforms homogeneously
under gauge transformations, whereas the usual gauge
field Aaµ transforms inhomogeneously, which complicates
a gauge-independent treatment of massive gauge field
theories. In fact, the (Stückelberg-like) degrees of free-
dom needed for a gauge-invariant formulation in terms
of a Yang–Mills connections are directly absorbed in
the antisymmetric tensor fields. No scalar field is re-
quired in order to construct a gauge invariant massive
theory in terms of the new variables. After recapitu-
lating the massless case in Sect. IIA, we have treated
the massive setting in Sect. IIB. After the fixed mass
case, at the beginning of Sect. IIB, this section encom-
passes also a position dependent mass (Sect. IIB1), that
is the Higgs degree of freedom, and a non-diagonal mass
term (Sect. IIB2). This is required for describing the
Weinberg–Salam model. In this context, we have identi-
fied the degrees of freedom which represent the different
electroweak gauge bosons in the Baµν representation by a
gauge-invariant eigenvector decomposition.
The Abelian section (App. A) serves as basis for an
easier understanding of some issues arising in the non-
Abelian case, like for example vanishing conserved cur-
rents. In that section we also address the massless limits
of propagators in the Aµ and Bµν representations, respec-
tively. We notice that while the limit is ill-defined for the
Aµ fields it is well-defined for the Bµν fields. That is due
to the consistent treatment of gauge degrees of freedom
in the latter case.
In Sect. III we continue with a description of massive
gauge field theories in terms of geometric variables in
four space-time dimensions and for two colours. Thereby
we can eliminate the remaining degrees of freedom which
are still encoded in the Baµν fields. After deriving the
expressions for a fixed mass and in the presence of the
Higgs degree of freedom, respectively, we also investigate
the Weinberg–Salam model (Sect. III A).
Acknowledgments
DDD would like to thank Gerald Dunne and Stefan
Hofmann for helpful, informative and inspiring discus-
sions. Thanks are again due to Stefan Hofmann for read-
ing the manuscript.
APPENDIX A: ABELIAN
1. Massless
The partition function of an Abelian gauge field theory
without fermions is given by
[dA] exp{i
xL} (A1)
with the Lagrangian density
L = L0 := − 14g2FµνF
µν (A2)
and the field tensor
Fµν := ∂µAν − ∂νAµ. (A3)
g stands for the coupling constant. The transition to the
first-order formalism can be performed just like in the
non-Abelian case, which is treated in the main body of
the paper. We find the partition function,
[dA][dB] ×
× exp{i
d4x[− 1
F̃µνB
µν − g
µν ]}. (A4)
Here the antisymmetric tensor field Bµν , like the field
tensor Fµν , is gauge invariant. The classical equations of
motion are given by
µν = 0 and g2Bµν = −F̃µν , (A5)
which after elimination of Bµν reproduce the Maxwell
equations one would obtain from Eq. (A2). Now we can
formally integrate out the gauge field Aµ. As no gauge
is fixed by the BF term because the Abelian field ten-
sor Fµν is gauge invariant this gives rise to a functional
δ distribution. This constrains the allowed field configu-
rations to those for which the conserved current ∂µB̃
vanishes,
[dB]δ(∂µB̃
µν) exp{i
d4x[− g
µν ]}.
2. Massive
In the massive case the Lagrangian density becomes
L = L0 + Lm, where Lm := m
µ. First, we here
repeat some steps carried out above in the non-Abelian
case: We can directly write down the partition function
in unitary gauge. Regauging like in Eq. (18) leads to
[dA]′[dU ] exp(i
d4x{− 1
[Aµ − iU †(∂µU)][Aµ − iU †(∂µU)]}).(A7)
The corresponding gauge-invariant Lagrangian then
reads,
Lcl := − 14g2FµνF
µν + m
(DµΦ)
†(DµΦ), (A8)
with the constraint Φ†Φ
= 1. Constructing a partition
function in the first-order formalism from the previous
Lagrangian yields,
[dA][dΦ][dB] ×
× exp(i
d4x{− 1
Bµν F̃
µν − g
[Aµ − iΦ(∂µΦ†)][Aµ − iΦ(∂µΦ†)]}). (A9)
The Φ fields can be absorbed entirely in a gauge-
transformation of the gauge field Aµ. The integration
over Φ decouples. This can also be seen by putting the
parametrisation Φ = e−iθ into the previous equation and
carrying out the [dA] integration,
[dB][dθ] exp{i
d4x[− g
(∂κB̃
κµ)gµν(∂λB̃
λν)− (∂µθ)(∂κBκµ)]}.
(A10)
The only θ dependent term in the exponent is a total
derivative and drops out, leading to a factorisation of
the θ integral.
A third way which yields the same final result, starts
by integrating out the θ field first. This gives a transverse
mass term∼ Aµ(gµν− ∂µ∂ν� )A
ν . Integration overAµ then
leads to the same result as before.
Instead of a vanishing current ∂µB̃
µν like in the mass-
less case, in the massive case the current has a Gaussian
distribution. The distribution’s width is proportional to
the mass of the gauge boson.
m → 0 limit
In the gauge-field representation the massless limit for
the classical actions discussed above are smooth. In
terms of the Bµν field the mass m ends up in the denom-
inator of the corresponding term in the action. Together
with the m dependent normalisation factors arising form
the integrations over the gauge-field in the course of the
derivation of the Bµν representation, however, the limit
m → 0 still yields the m = 0 result for the partition
function (A6).
Still, it is known that the perturbative propagator for
a massive photon is ill-defined if the mass goes to zero:
Variation of the exponent of the Abelian massive parti-
tion function in unitary gauge with respect to Aκ and Aλ
gives the inverse propagator for the gauge fields,
(G−1)κλ = [(p2 −m2phys)gκλ − pκpλ], (A11)
which here is already transformed to momentum space.
The corresponding equation of motion,
(G−1)κλGλµ
= gκµ, (A12)
is solved by
Gλµ =
p2 −m2phys
m2phys
p2 −m2phys
, (A13)
with boundary conditions (an ǫ prescription) to be spec-
ified and mphys := mg. This propagator diverges in the
limit m→ 0.
In the representation based on the antisymmetric ten-
sor fields, variation of the exponent of the partition func-
tion (A10) with respect to the fields B̃µν and B̃κλ yields
the inverse propagator
(G−1)µν|κλ = gµκgνλ − gνκgµλ +
+m−2phys(∂
µ∂κgνλ − ∂ν∂κgµλ −
− ∂µ∂λgνκ + ∂ν∂λgµκ), (A14)
already expressed in momentum space. Variation with
respect to B̃µν instead of Bµν corresponds only to a
reshuffling of the Lorentz indices and gives an equiva-
lent description. The antisymmetric structure of the in-
verse propagator is due to the antisymmetry of B̃µν . The
equation of motion is then given by
(G−1)µν|κλGκλ|ρσ
= gµρ g
σ − gµσgνρ (A15)
and solved by
2Gκλ|ρσ = (gκρgλσ − gκσgλρ)−
p2 −m2phys
×(pκpρgλσ − pκpσgλρ −
pλpρgκσ + pλpσgκρ). (A16)
Here we observe that the limit m→ 0 is well-defined,
2Gκλ|ρσ
m→0−−−→ gκρgλσ − gκσgλρ −
(pκpρgλσ − pκpσgλρ −
− pλpρgκσ + pλpσgκρ). (A17)
This is due to the consistent treatment of the gauge de-
grees of freedom in the second approach.
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Karabali and Nair [20] also together with Kim [21] have
carried out a gauge invariant analysis of non-Abelian
gauge field theories. As pointed out by Freidel et al. [22]
this approach appears to be extendable to 3+1 dimen-
sions making use of the so-called ”corner variables” [23].
[28] The functional integral over a gauge field is ill-defined as
long as no gauge is fixed. We have to keep this fact in
mind at all times and will discuss it in detail when a field
is really integrated out.
[29] Due to the possible occurrence of what is known as Gri-
bov copies [24] this method might not achieve a unique
splitting of the two parts of the integration. For our il-
lustrative purposes, however, this is not important.
[30] For three colours and four space-time dimensions there
exists a treatment of the massless setting due to Lunev
[25], different from the one which here is extended to the
massive case.
[31] One possible representation is ηi
= ǫamn, η
4n = −δ
m4 = −δ
m, and η
= 0, where m,n ∈ {1; 2; 3}.
http://arxiv.org/abs/hep-th/9807069
http://arxiv.org/abs/hep-th/0609166
http://arxiv.org/abs/gr-qc/9905087
http://arxiv.org/abs/gr-qc/0210094
http://arxiv.org/abs/hep-th/9507036
http://arxiv.org/abs/hep-th/9603151
http://arxiv.org/abs/hep-th/0108097
http://arxiv.org/abs/hep-th/0609146
http://arxiv.org/abs/hep-th/9705087
http://arxiv.org/abs/hep-th/9804132
http://arxiv.org/abs/hep-th/0007188
http://arxiv.org/abs/hep-th/0604185
http://arxiv.org/abs/hep-th/0604184
http://arxiv.org/abs/hep-th/9503133

  We discuss gravity-like formulations of massive Abelian and non-Abelian gauge
field theories in four space-time dimensions with particular emphasis on the
issue of gauge invariance. Alternative descriptions in terms of antisymmetric
tensor fields and geometric variables, respectively, are analysed. In both
approaches St\"uckelberg degrees of freedom factor out. We also demonstrate, in
the Abelian case, that the massless limit for the gauge propagator, which does
not exist in the vector potential formulation, is well-defined for the
antisymmetric tensor fields.

Gauge invariance in gravity-like descriptions of massive gauge field theories
Dennis D. Dietrich
Institut für Theoretische Physik, Universität Heidelberg, Heidelberg, Germany
(Dated: January 13, 2019)
We discuss gravity-like formulations of massive Abelian and non-Abelian gauge field theories in
four space-time dimensions with particular emphasis on the issue of gauge invariance. Alternative
descriptions in terms of antisymmetric tensor fields and geometric variables, respectively, are anal-
ysed. In both approaches Stückelberg degrees of freedom factor out. We also demonstrate, in the
Abelian case, that the massless limit for the gauge propagator, which does not exist in the vector
potential formulation, is well-defined for the antisymmetric tensor fields.
PACS numbers: 11.15.-q, 11.15.Ex, 11.30.Qc, 12.15.-y
I. INTRODUCTION
Massive gauge bosons belong to the fundamental con-
cepts we use for picturing nature. Prominent exam-
ples are found in the physics of electroweak interac-
tions, superconductivity, and confinement. Even more
than in the massless case, gauge invariance is a severe
constraint for the construction of massive gauge field
theories. Usually additional fields beyond the original
gauge field have to be included in order to obtain gauge
invariant expressions.[26] Technically, this is linked to
the fact that the aforementioned gauge field—the Yang–
Mills connection—changes inhomogeneously under gauge
transformations and encodes also spurious degrees of
freedom arising from the construction principle of gauge
invariance. This complicates the extraction of physical
quantities. A variety of approaches has been developed
in order to deal with this situation. Wilson loops [1]
represent gauge invariant but non-local variables.[27] Al-
ternatively, there exist decomposition techniques like the
one due to Cho, Faddeev, and Niemi [2]. Here we first
pursue a reformulation of massive Yang–Mills theories in
terms of antisymmetric gauge algebra valued tensor fields
Baµν (Sect. II) and subsequently continue with a represen-
tation in terms of geometric variables (Sect. III).
In Sect. II A we review the massless case. It is re-
lated to gravity [3] formulated as BF gravity [4] and
thus linked to quantum gravity. The antisymmetric ten-
sor field can be seen as dual field strength and transforms
homogeneously under gauge transformations. This fact
already makes it simpler to keep track of gauge invari-
ance. In Sect. II B the generalisation to the massive case
is presented. In the Baµν field representation the (non-
Abelian) Stückelberg fields, which are commonly present
in massive gauge field theories and needed there in or-
der to keep track of gauge invariance, factor out com-
pletely. In other words, no scalar fields are needed for a
gauge invariant formulation of massive gauge field theo-
ries in terms of antisymmetric tensor fields. The case of
a constant mass is linked to sigma models (gauged and
ungauged) in different respects. Sect. II B 1 contains the
generalisation to a position dependent mass, which cor-
responds to introducing the Higgs degree of freedom. In
Sect. II B 2 non-diagonal mass terms are admitted. This
is necessary to accommodate the Weinberg–Salammodel,
which is studied as particular case.
Sect. III presents a description of the massive case,
with constant and varying mass, in terms of geometric
variables. In this step the remaining gauge degrees of
freedom are eliminated. The emergent description is in
terms of local colour singlet variables. Finally, Sect. III A
is concerned with the geometric representation of the
Weinberg–Salam model.
The Appendix treats the Abelian case. It allows to
better interpret and understand several of the findings in
the non-Abelian settings. Of course, in the Abelian case
already the Bµν field is gauge invariant. Among other
things, we demonstrate that them→ 0 limit of the gauge
propagator for the Bµν fields is well-defined as opposed
to the ill-defined limit for the Aµ field propagator.
Sect. IV summarises the paper.
II. ANTISYMMETRIC TENSOR FIELDS
A. Massless
Before we investigate massive gauge field theories let
us recall some details about the massless case. The parti-
tion function of a massless non-Abelian gauge field theory
without fermions is given by
[dA] exp{i
xL}, (1)
with the Lagrangian density
L = L0 := − 14g2F
aµν (2)
and the field tensor
µν := ∂µA
ν − ∂νAaµ + fabcAbµAcν . (3)
Aaµ stands for the gauge field, f
abc for the antisymmetric
structure constant, and g for the coupling constant. [28]
Variation of the classical action with respect to the gauge
field gives the classical Yang–Mills equations
Dabµ (A)F
bµν = 0, (4)
http://arxiv.org/abs/0704.1828v1
where the covariant derivative is defined as Dabµ (A) :=
δab∂µ + f
acbAcµ. The partition function in the first-order
formalism can be obtained after multiplying Eq. (1) with
a prefactor in form of a Gaussian integral over an anti-
symmetric tensor field Baµν ,
[dA][dB] exp{i
d4x[L0 − g
BaµνB
aµν ]}. (5)
(”∼=” indicates that in the last step the normalisation of
the partition function has been changed.) Subsequently,
the field Baµν is shifted by
F̃ aµν , where the dual field
tensor is defined as F̃ aµν :=
ǫµνκλF
[dA][dB] ×
× exp{i
x[− 1
aµν − g
aµν ]}. (6)
In this form the partition function is formulated in terms
of the Yang–Mills connection Aaµ and the antisymmetric
tensor field Baµν as independent variables. Variation of
the classical action with respect to these variables leads
to the classical equations of motion
µν = −F̃ aµν and Dabµ (A)B̃bµν = 0, (7)
where B̃aµν :=
ǫκλµνB
aκλ. By eliminating Baµν the origi-
nal Yang–Mills equation (4) is reproduced. Every term in
the classical action in the partition function (6) contains
at most one derivative as opposed to two in Eq. (1). This
explains the name ”first-order” formalism. The classical
action in Eq. (6) is invariant under simultaneous gauge
transformations of the independent variables according
AaµT a =: Aµ → AµU := U [A
µ − iU †(∂µU)]U † (8)
BaµνT a =: Bµν → Bµν
:= UBµνU †, (9)
or infinitesimally,
δAaµ = ∂µθ
a + fabcAbµθ
δBaµν = f
abcBbµνθ
c. (10)
The T a stand for the generators of the gauge group. From
the Bianchi identity Dabµ (A)F̃
bµν = 0 follows a second
symmetry of the BF term alone: Infinitesimally, for un-
changed Aaµ,
µν = ∂µϑ
ν − ∂νϑaµ + fabc(Abµϑcν −Abνϑcµ). (11)
A particular combination of the transformations (10) and
(11), θa = nµAaµ and ϑ
ν = n
µBaµν , corresponds to the
transformation of a tensor under an infinitesimal local
coordinate transformation xµ → xµ − nµ(x),
δBµν = Bλν∂µn
λ +Bµλ∂νn
λ + nλ∂λBµν , (12)
that is a diffeomorphism. Hence, the BF term is dif-
feomorphism invariant, which explains why this theory
is also known as BF gravity. The BB term is not dif-
feomorphism invariant and, hence, imposes a constraint.
The combination of the two terms amounts to an action
of Plebanski type which are studied in the context of
quantum gravity [3, 4].
We now would like to eliminate the Yang–Mills connec-
tion by integrating it out. For fixed Baµν the integrand
of the path integral is not gauge invariant with respect
to gauge transformations of the gauge field Aaµ alone; the
field tensor F aµν transforms homogeneously and the corre-
sponding gauge transformations are not absorbed if Baµν
is held fixed. Therefore, the integral over the gauge group
is in general not cyclic which otherwise would render the
path integral ill-defined. The term in the exponent linear
in the gauge field Aaµ, A
ν∂µB̃
aµν , is obtained by carry-
ing out a partial integration in which surface terms are
ignored. Afterwards it is absorbed by shifting Aaµ by
(B−1)abµν(∂λB̃
bλν), where Babµν := f
abcB̃aµν . In general its
inverse (B−1)abµν , defined by (B
−1)abµνB
νκ = δacgµλ
exists in three or more space-time dimensions [5]. We
are left with a Gaussian integral in Aaµ giving the inverse
square-root of the determinant of Babµν ,
[da] exp{− i
bν}. (13)
In the last expression Babµν appears in the place of an in-
verse gluon propagator, that is sandwiched between two
gauge fields. This analogy carries even further: Interpret-
ing ∂µB̃
aµν as a current, (B−1)abµν(∂λB̃
bλν), the current
together with the ”propagator” (B−1)abµν , is exactly the
abovementioned term to be absorbed in the gauge field
Aaµ. Finally, we obtain,
[dB]Det−
B exp{i
d4x[− g
BaµνB
aµν −
(∂κB̃
aκµ)(B−1)abµν(∂λB̃
bλν)]}. (14)
This result is known from [5, 6, 7]. The exponent in
the previous expression corresponds to the value of the
[dA] integral at the saddle-point value Ăaµ of the gauge
field. It obeys the classical field equation (7). Using
Ăaµ(B) = (B
−1)abµν(∂λB̃
bλν) the second term in the above
exponent can be rewritten as − i
d4xB̃aµνF
aµν [Ă(B)],
which involves an integration by parts and makes its
gauge invariance manifest. The fluctuations aaµ around
the saddle point Ăaµ, contributing to the partition func-
tion (6), are Gaussian because the action in the first-order
formalism is only of second order in the gauge field Aaµ.
They give rise to the determinant (13). What happens
if a zero of the determinant is encountered can be un-
derstood by looking at the Abelian case discussed in Ap-
pendix A. There the BF term does not fix a gauge for the
integration over the gauge field Aµ because the Abelian
field tensor Fµν is gauge invariant. If it is performed
nevertheless one encounters a functional δ distribution
which enforces the vanishing of the current ∂µB̃
µν . In
this sense the zeros of the determinant in the non-Abelian
case arise if B̃aµν is such that the BF term does not totally
fix a gauge for the [dA] integration, but leaves behind a
residual gauge invariance. It in turn corresponds to van-
ishing components of the current ∂µB̃
aµν . (Technically,
there then is at least one flat direction in the otherwise
Gaussian integrand. The flat directions are along those
eigenvectors of B possessing zero eigenvalues.)
When incorporated with the exponent, which requires
a regularisation [8], the determinant contributes a term
proportional to 1
ln detB to the action. This term to-
gether with the BB term constitutes the effective poten-
tial, which is obtained from the exponent in the partition
function after dropping all terms containing derivatives of
fields. The effective potential becomes singular for field
configurations for which detB = 0. It is gauge invari-
ant because all contributing addends are gauge invariant
separately.
The classical equations of motion obtained by varying
the action in Eq. (14) with respect to the dual antisym-
metric tensor field B̃aµν are given by
g2B̃aµν = (g
µ − gρµgσν )∂ρ(B−1)abσκ(∂λB̃bλκ)−
−(∂ρB̃dρκ)(B−1)dbκµfabc(B−1)ceνλ(∂σB̃eσλ),
which coincides with the first of Eqs. (7) with the
field tensor evaluated at the saddle point of the action,
F aµν [Ă(B)]. Taking into account additionally the effect
due to fluctuations of Aaµ contributes a term proportional
to δDetB
δB̃aµν
det−1B to the previous equation.
B. Massive
In the massive case the prototypical Lagrangian is of
the form L = L0 +Lm, where Lm := m
aµ. (Due to
our conventions the physical mass is given by mphys :=
mg.) This contribution to the Lagrangian is of course
not gauge invariant. Putting it, regardlessly, into the
partition function, gives
[dA][dU ] exp{i
x[L0 + m
aµ]}, (16)
which can be interpreted as the unitary gauge represen-
tation of an extended theory. In order to see this let us
split the functional integral over Aaµ into an integral over
the gauge group [dU ] and gauge inequivalent field config-
urations [dA]′. Usually this separation is carried out by
fixing a gauge according to
[dA]′ :=
[dA]δ[fa(A) − Ca]∆f (A). (17)
fa(A) = Ca is the gauge condition and ∆f (A) stands
for the Faddeev–Popov determinant defined through
[dA]δ[fa(A) − Ca]∆f (A) [29]. Introducing this
reparametrisation into the partition function (16) yields,
[dA]′[dU ] exp(i
d4x{− 1
F aµνF
aµν +
[Aµ − iU †(∂µU)]a[Aµ − iU †(∂µU)]a}).
L0 is gauge invariant in any case and remains thus un-
affected. In the mass term the gauge transformations
appear explicitly [9]. We now replace all of these gauge
transformations with an auxiliary (gauge group valued)
scalar field Φ, U † → Φ, obeying the constraint
!≡ 1. (19)
The field Φ can be expressed as Φ =: e−iθ, where
θ =: θaT a is the gauge algebra valued non-Abelian gen-
eralisation of the Stückelberg field [10]. For a massive
gauge theory they are a manifestation of the longitudi-
nal degrees of freedom of the gauge bosons. In the con-
text of symmetry breaking they arise as Goldstone modes
(”pions”). In the context of the Thirring model these ob-
servations have been made in [11]. There it was noted
as well that the θ is also the field used in the canonical
Hamiltonian Batalin–Fradkin–Vilkovisky formalism [12].
We can extract the manifestly gauge invariant classical
Lagrangian
Lcl := − 14g2F
aµν +m2tr[(DµΦ)
†(DµΦ)], (20)
where the scalars have been rearranged making use of the
product rule of differentiation and the cyclic property of
the trace and whereDµΦ := ∂µΦ−iAµΦ. Eq. (20) resem-
bles the Lagrangian density of a non-linear gauged sigma
model. In the Abelian case the fields θ decouple from
the dynamics. For non-Abelian gauge groups they do
not and one would have to deal with the non-polynomial
coupling to them.
In the following we show that these spurious degrees of
freedom can be absorbed when making the transition to a
formulation based on the antisymmetric tensor field Baµν .
Introducing the antisymmetric tensor field into the corre-
sponding partition function, like in the previous section,
results in,
[dA][dΦ][dB] exp(i
×{− g
aµν − 1
aµν +
[Aµ − iΦ(∂µΦ†)]a[Aµ − iΦ(∂µΦ†)]a}).
Removing the gauge scalars Φ from the mass term by a
gauge transformation of the gauge field Aaµ makes them
explicit in the BF term,
[dA][dΦ][dB] exp{i
d4x[− g
BaµνB
aµν −
−tr(ΦF̃µνΦ†Bµν) + m
aµ]}. (22)
In the next step we would like to integrate over the Yang–
Mills connection Aaµ. Already in the previous expression,
however, we can perceive that the final result will only
depend on the combination of fields Φ†BµνΦ. [The Φ
field can also be made explicit in the BB term in form
of the constraint (19).] Therefore, the functional inte-
gral over Φ only covers multiple times the range which
is already covered by the [dB] integration. Hence the
degrees of freedom of the field Φ have become obsolete
in this formulation and the [dΦ] integral can be factored
out. Thus, we could have performed the unitary gauge
calculation right from the start. In either case, the final
result reads,
[dB]Det−
M exp{i
x[− g
aµν −
(∂κB̃
aκµ)(M−1)abµν(∂λB̃
bλν)]}, (23)
where Mabµν := B
µν − m2δabgµν , which coincides with
[13]. Mabµν and hence (M
−1)abµν transform homogeneously
under the adjoint representation. In Eq. (14) the cen-
tral matrix (B−1)abµν in the analogous term transformed
in exactly the same way. There this behaviour ensured
the gauge invariance of this term’s contribution to the
classical action. Consequently, the classical action in the
massive case has the same invariance properties. In par-
ticular, the aforementioned gauge invariant classical ac-
tion describes a massive gauge theory without having to
resort to additional scalar fields. For detB 6= 0, the limit
m → 0 is smooth. For detB = 0 the conserved current
components alluded to above would have to be separated
appropriately in order to recover the corresponding δ dis-
tributions present in these situations in the massless case.
Again the effective action is dominated by the term
proportional to 1
detM. The contribution from the mass
to M shifts the eigenvalues from the values obtained for
B. Hence the singular contributions are typically ob-
tained for eigenvalues of B of the order of m2. The ef-
fective potential is again gauge invariant, for the same
reason as in the massless case.
The classical equations of motion obtained by variation
of the action in Eq. (21) are given by,
µν = −F̃ aµν ,
Dabµ (A)B̃
bµν = −m2[Aν − iΦ(∂νΦ†)]a,
0 = δ
x{[Aaµ − iΦ(∂µΦ†)]a}2. (24)
In these equations a unique solution can be chosen, that
is a gauge be fixed, by selecting the scalar field Φ. Φ ≡ 1
gives the unitary gauge, in which the last of the above
equations drops out. The general non-Abelian case is
difficult to handle already on the classical level, which
is one of the main motivations to look for an alternative
formulation. In the non-Abelian case, the equation of
motion obtained from Eq. (23) resembles strongly the
massless case,
g2B̃aµν = (g
µ − gρµgσν )∂ρ(M−1)abσκ(∂λB̃bλκ)−
−(∂ρB̃dρκ)(M−1)dbκµfabc(M−1)ceνλ(∂σB̃eσλ),
insofar as all occurrences of (B−1)abµν have been replaced
by (M−1)abµν . Incorporation of the effect of the Gaussian
fluctuations of the gauge field Aaµ would give rise to a con-
tribution proportional to δB
δB̃aµν
det−1M in the previous
equation.
Before we go over to more general cases of massive
non-Abelian gauge field theories, let us have a look at
the weak coupling limit: There the BB term in Eq. (21)
is neglected. Subsequently, integrating out the Baµν field
enforces F aµν ≡ 0. [This condition also arises from the
classical equation of motion (24) for g=0.] Hence, for
vanishing coupling exclusively pure gauge configurations
of the gauge field Aaµ contribute. They can be combined
with the Φ fields and one is left with a non-linear reali-
sation of a partition function,
g=0∼=
[dΦ] exp{im2
d4x tr[(∂µΦ
†)(∂µΦ)]}, (26)
of a free massless scalar [13]. Setting g = 0 interchanges
with integrating out the Baµν field from the partition func-
tion (21). Thus, the partition function (23) with g = 0 is
equivalent to (26). That a scalar degree of freedom can
be described by means of an antisymmetric tensor field
has been noticed in [14].
1. Position-dependent mass and the Higgs
One possible generalisation of the above set-up is ob-
tained by softening the constraint (19). This can be seen
as allowing for a position dependent mass. The new
degree of freedom ultimately corresponds to the Higgs.
When introducing the mass m as new degree of freedom
(as ”mass scalar”) we can restrict its variation by in-
troducing a potential term V (m2), which remains to be
specified, and a kinetic term K(m), which we choose in
its canonical form K(m) = 1
(∂µm)(∂
µm). It gives a
penalty for fast variations of m between neighbouring
space-time points. The fixed mass model is obtained in
the limit of an infinitely sharp potential with its mini-
mum located at a non-zero value for the mass. Putting
together the partition function in unitary gauge leads to,
[dA][dm] exp{i
d4x[− 1
F aµνF
aµν +
aµ +K(m) + V (m2)]}, (27)
where we have introduced the normalisation constant
N := dim R, with R standing for the representation of
the scalars. This factor allows us to keep the canonical
normalisation of the mass scalar m. We can now repeat
the same steps as in the previous section in order to iden-
tify the classical Lagrangian,
Lcl := − 14g2F
aµν +N−1tr[(Dµφ)†(Dµφ)] + V (|φ|2),
where now φ := mΦ. In order to reformulate the parti-
tion function in terms of the antisymmetric tensor field
we can once more repeat the steps in the previous sec-
tion. Again the spurious degrees of freedom represented
by the field Φ can be factored out. Finally, this gives
[15],
[dB][dm]Det−
M exp{i
d4x[− g
BaµνB
aµν −
(∂κB̃
aκµ)(M−1)abµν(∂λB̃
bλν) +
+K(m) + V (m2)]}, (28)
whereMabµν = B
µν −m2N−1δbgµν depends on the space-
time dependent mass m. The determinant can as usual
be included with the exponent in form of a term pro-
portional to 1
detM, the pole of which will dominate the
effective potential. As just mentioned, however,M is also
a function of m. Hence, in order to find the minimum,
the effective potential must also be varied with respect
to the mass m.
Carrying the representation in terms of antisymmetric
tensor fields another step further, the partition function
containing the kinetic term K(m) of the mass scalar can
be expressed as Abelian version of Eq. (26),
[db][da] exp{i
d4x[− 1
b̃µνf
µν + 1
µ]} =
[dm] exp{i
d4x[ 1
(∂µm)(∂
µm)]}, (29)
where here the mass scalar m is identified with the
Abelian gauge parameter. Combining the last equation
with the partition function (28) all occurrences of the
mass scalar m can be replaced by the phase integral
dxµaµ. The bf term enforces the curvature f
to vanish which constrains aµ to pure gauges ∂µm and
the aforementioned integral becomes path-independent.
2. Non-diagonal mass term and the Weinberg–Salam model
The mass terms investigated so far had in common that
all the bosonic degrees of freedom they described pos-
sessed the same mass. A more general mass term would
be given by Lm := m
ab. Another similar ap-
proach is based on the Lagrangian Lm := m
tr{AµAµΨ}
where Ψ is group valued and constant. We shall begin our
discussion with this second variant and limit ourselves to
a Ψ with real entries and trΨ = 1, which, in fact, does
not impose additional constraints. Using this expression
in the partition function (27) and making explicit the
gauge scalars yields,
[dA][dm] exp{i
d4x[− 1
F aµνF
aµν +
tr{(Dµφ)†(Dµφ)Ψ} + V (m2)]}. (30)
Expressed in terms of the antisymmetric tensor field
Baµν , the corresponding partition function coincides with
Eq. (28) but with Mabµν replaced by M
µν := B
m2tr{T aT bΨ}gµν.
Let us now consider directly the SU(2)× U(1)
Weinberg–Salam model. Its partition function can be
expressed as,
[dA][dψ] exp{i
x[− 1
aµν +
∂ µ + iAµ)(
∂ µ − iAµ)ψ + V (|ψ|2)]},
where ψ is a complex scalar doublet, Aµ := A
a, with
a ∈ {0; . . . ; 3}, T a here stands for the generators of
SU(2) in fundamental representation, and, accordingly,
T 0 for g0
times the 2 × 2 unit matrix, with the U(1)
coupling constant g0. The partition function can be
reparametrised with ψ = mΦψ̂, where m =
|ψ|2, Φ
is a group valued scalar field as above, and ψ̂ is a con-
stant doublet with |ψ̂|2 = 1. The partition function then
becomes,
[dA][dΦ][dm] exp(i
x{− 1
aµν +
tr[Φ†(
∂ µ + iAµ)(
µ − iAµ)ΦΨ] +
(∂µm)(∂
µm) + V (m2)}), (32)
where
Ψ = ψ̂ ⊗ ψ̂†. (33)
Making the transition to the first order formalism leads
[dA][dB][dΦ][dm] exp(i
d4x{− g
BaµνB
aµν −
F aµνB̃
aµν +K(m) + V (m2) +
tr[Φ†(
∂ µ + iAµ)(
∂ µ − iAµ)ΦΨ]}). (34)
As in the previous case, a gauge transformation of the
gauge field Aaµ can remove the gauge scalar Φ from the
mass term (despite the matrix Ψ). Thereafter Φ only
appears in the combination Φ†BµνΦ and the integral [dΦ]
merely leads to repetitions of the [dB] integral. [The U(1)
part drops out completely right away.] Therefore the [dΦ]
integration can be factored out,
[dA][dB][dm] exp{i
d4x[− g
BaµνB
aµν −
F aµνB̃
aµν + m
tr(AµA
µΨ) +
+K(m) + V (m2)]}. (35)
The subsequent integration over the gauge fields A
µ leads
[dB][dm]Det−
M exp{i
d4x[− g
BaµνB
aµν −
(∂κB̃
aκµ)(M−1)abµν(∂λB̃
bλν) +
+K(m) + V (m2)]}, (36)
where M
µν := B
µν −m2tr(T aT bΨ)gµν .
From hereon we continue our discussion based on the
mass matrix
ab := 1
tr({T a, T b}Ψ), (37)
which had already been mentioned at the beginning of
Sect. II B 2. mab is real and has been chosen to be sym-
metric. (Antisymmetric parts are projected out by the
contraction with the symmetric A
bµ.) Thus it pos-
sesses a complete orthonormal set of eigenvectors µ
j with
the associated real eigenvalues mj , m
= 6Σjmjµ
With the help of these normalised eigenvectors one can
construct projectors π
j := 6Σjµ
j and decompose the
mass matrix, mab = mjπ
j . The projectors are com-
plete, 1ab = Σjπ
j , idempotent 6Σjπ
j = π
j , and
satisfy π
j 6=k
= 0. The matrix B
µν , the antisym-
metric tensor field Baµν , and the gauge field A
µ can
also be decomposed with the help of the eigenvectors:
µν = µ
, where bjkµν := µ
µν = b
where bjµν := B
j ; and A
µ = a
j , where a
µ := A
Using this decomposition in the partition function (36)
leads to,
[db][dm]Det−
m exp{i
d4x[− g
bjµνb
jµν −
(∂κb̃
jκµ)(m−1)jkµν(∂λb̃
kλν) +
+K(m) + V (m2)]}, (38)
where mjkµν := b
µν −m2
jlδklgµν .
Making use of the concrete form of mab given in
Eq. (37), inserting Ψ from Eq. (33), and subsequent diag-
onalisation leads to the eigenvalues 0, 1
and 1
These correspond to the photon, the two W bosons and
the heavier Z boson, respectively. The thus obtained tree-
level Z to W mass ratio squared consistently reproduces
the cosine of the Weinberg angle in terms of the coupling
constants, cos2 ϑw =
g2+g2
. Due to the masslessness of
the photon one addend in the sum over l in the expression
µν above does not contribute. Still, the totalm
µν does
not vanish like in the case of a single massless Abelian
gauge boson (see Appendix A). Physically this corre-
sponds to the coupling of the photon to the W and Z
bosons.
III. GEOMETRIC REPRESENTATION
The fact that the antisymmetric tensor field Baµν trans-
forms homogeneously represents already an advantage
over the formulation in terms of the inhomogeneously
transforming gauge fields Aaµ. Still, B
µν contains de-
grees of freedom linked to the gauge transformations (9).
These can be eliminated by making the transition to
a formulation in terms of geometric variables. In this
section we provide a classically equivalent description of
the massive gauge field theories in terms of geometric
variables in Euclidean space for two colours by adapt-
ing Ref. [16] to include mass. The first-order action is
quadratic in the gauge-field Aaµ.[30] Thus the evaluation
of the classical action at the saddle point yields the ex-
pression equivalent to the different exponents obtained
after integrating out the gauge field Aaµ in the various
partition functions in the previous section. In Euclidean
space the classical massive Yang–Mills action in the first
order formalism reads
d4x(LBB + LBF + LAA), (39)
where
LBB = − g
BaµνB
µν , (40)
LBF = + i4ǫ
µνκλBaµνF
κλ, (41)
LAA = −m
µ. (42)
At first we will investigate the situation for the unitary
gauge mass term LAA and study the role played by the
scalars Φ afterwards.
As starting point it is important to note that a metric
can be constructed that makes the tensor Baµν self-dual
[7]. In order to exploit this fact, it is convenient to define
the antisymmetric tensor (j ∈ {1; 2; 3})
µν := η
ν , (43)
with the self-dual ’t Hooft symbol η
[17] [31] and the
tetrad eAµ . From there we construct a metric gµν in terms
of the tensor T jµν
gµν ≡ eAµ eAν = 16ǫ
jklT jµκT
kκλT lλν , (44)
where
jµν := 1
g)3 := 1
(ǫjklT
T kµ2ν2T
×(ǫj′k′l′T j
×ǫµ1ν1κ1λ1ǫµ2ν2κ2λ2ǫµ3ν3κ3λ3 (46)
Subsequently, we introduce a triad daj such that
Baµν =: d
µν . (47)
This permits us to reexpress the BB term of the classical
Lagrangian,
LBB = − g
µνhjkT
µν , (48)
where hjk := d
k. Putting Eqs. (47) and (45) into the
saddle point condition
ǫκλµνDabµ (Ă)B
κλ = +im
2Ăaν (49)
gives
µ (Ă)(
gdbjT
jµν) = +im2Ăaν . (50)
In the following we define the connection coefficients γµ|kj
as expansion parameters of the covariant derivative of the
triads at the saddle point in terms of the triads,
Dabµ (Ă)d
j =: γµ|kj dak. (51)
This would not be directly possible for more than two
colours, as then the set of triads is not complete. The
connection coefficients allow us to define covariant deriva-
tives according to
∇µ|kj := ∂µδkj + γµ|kj . (52)
These, in turn, permit us to rewrite the saddle point
condition (49) as
dak∇µ|kj (
gT jµν) = im2Ăaν , (53)
and the mass term in the classical Lagrangian becomes
LAA = 12m2 [∇µ|
gT iµν)]hkl[∇κ|lj(
gT jκν)]. (54)
In the limit m→ 0 this term enforces the covariant con-
servation condition ∇µ|ki (
gT iµν) ≡ 0, known for the
massless case. It results also directly from the saddle
point condition (53). Here dak∇µ|ki (
gT iµν) are the di-
rect analogues of the Abelian currents ǫµνκλ∂µBκλ, which
are conserved in the massless case [see Eq. (A6)] and dis-
tributed following a Gaussian distribution in the massive
case [see Eq. (A10)].
The commutator of the above covariant derivatives
yields a Riemann-like tensor Rkjµν
Rkjµν := [∇µ,∇ν ]kj . (55)
By evaluating, in adjoint representation (marked by
)̊, the following difference of double commutators
[D̊µ(Ă), [D̊ν(Ă), d̊j ]]− (µ↔ ν) in two different ways, one
can show that
i[d̊j , F̊µν(Ă)] = d̊kR
jµν , (56)
or in components,
F aµν(Ă) =
ǫabcdbjdckR
jµν , (57)
where dajdak := δ
defines the inverse triad, daj = hjkdak.
Hence, we are now in the position to rewrite the remain-
ing BF term of the Lagrangian density. Introducing
Eqs. (47) and (57) into Eq. (41) results in
LBF = i4
gT jµνRklµνǫjmkh
lm. (58)
Let us now repeat the previous steps with a mass term
in which the gauge scalars Φ are explicit,
LΦAA := −
[Aµ − iΦ(∂µΦ†)]a[Aµ − iΦ(∂µΦ†)]a. (59)
In that case the saddle point condition (49) is given by,
µ (Ă)B̃
κλ = im
2[Ăν − iΦ(∂νΦ†)]a, (60)
or in the form of Eq. (53), that is with the left-hand side
replaced,
dak∇µ|kj (
gT jµν) = im2[Ăν − iΦ(∂νΦ†)]a. (61)
Reexpressing LΦAA with the help of the previous equation
reproduces exactly the unitary gauge result (54) for the
mass term.
Finally, the tensor B appearing in the determinant
(13), which accounts for the Gaussian fluctuations of the
gauge field Aaµ, formulated in the new variables reads
gfabcdai T
iµν . Now all ingredients are known
which are needed to express the equivalent of the parti-
tion function (16) in terms of the new variables. For a
position-dependent mass the discussion does not change
materially. The potential and kinematic term for the
mass scalar m have to be added to the action.
Contrary to the massless case the Aaµ dependent part
of the Euclidean action is genuinely complex. Without
mass only the T-odd and hence purely imaginary BF
term was Aaµ dependent. With mass there contributes
the additional T-even and thus real mass term. There-
fore the saddle point value Ăaµ for the gauge field becomes
complex. This is a known phenomenon and in this con-
text it is essential to deform the integration contour of
the path integral in the partition function to run through
the saddle point [18]. For the Gaussian integrals which
are under consideration here, in doing so, we do not re-
ceive additional contributions. The imaginary part IĂaµ
of the saddle point value of the gauge field transforms
homogeneously under gauge transformations. The com-
plex valued saddle point of the gauge field which is in-
tegrated out does not affect the real-valuedness of the
remaining fields, here Baµν . In this sense the field B
represents a parameter for the integration over Aaµ. The
tensor T jµν is real-valued by definition and therefore the
same holds also for the triad daj [see Eq. (47)]. hkl is
composed of the triads and, consequently, real-valued as
well. The imaginary part of the saddle point value of the
gauge field, IĂaµ, enters the connection coefficients (51).
Through them it affects the covariant derivative (52) and
the Riemann-like tensor (55). More concretely the con-
nection coefficients γµ|kj can be decomposed according to
Dabµ (RĂ)dbj = (Rγµ|kj )dak, (62)
abc(IĂcµ)dbj = (Iγµ|kj )dak, (63)
with the obvious consequences for the covariant deriva-
tive,
∇µ|kj = R∇µ|kj + iI∇µ|kj , (64)
R∇µ|kj = ∂µδkj +Rγµ|kj , (65)
I∇µ|kj = Iγµ|kj . (66)
This composition reflects in the mass term,
RLAA = 12m2 {[R∇µ|
gT iµν)]hkl[R∇κ|lj(
gT jκν)]−
−[Iγµ|kj (
gT iµν)]hkl[Iγκ|lj(
gT jκν)]}
ILAA = 22m2 [R∇µ|
gT iµν)]hkl[I∇κ|lj(
jκν)]
on one hand, and in the Riemann-like tensor,
RRkjµν = [R∇µ,R∇ν ]kj − [I∇µ, I∇ν ]kj (67)
IRkjµν = [R∇µ, I∇ν ]kj + [I∇µ,R∇ν ]kj . (68)
on the other. The connection to the imaginary part of
Ăaµ is more direct in Eq. (57) which yields,
RF aµν (Ă) = 12ǫ
abcdbjdckRRkjµν , (69)
IF aµν (Ă) = 12ǫ
kIRkjµν , (70)
Finally, the BF term becomes,
RLBF = − 14
gT jµνǫjmkh
lmIRklµν , (71)
ILBF = + 14
gT jµνǫjmkh
lmRRklµν . (72)
Summing up, at the complex saddle point of the [dA] in-
tegration the emerging Euclidean LAA and LBF are both
complex, whereas before they were real and purely imag-
inary, respectively. Both terms together determine the
saddle point value Ăaµ. Therefore, they become coupled
and cannot be considered separately anymore. This was
already to be expected from the analysis in Minkowski
space in Sect. II, where the matrixMabµν combines T-odd
and T-even contributions, which originate from LAA and
LBF , respectively. There the different contributions be-
come entangled when the inverse (M−1)abµν is calculated.
A. Weinberg–Salam model
Finally, let us reformulate the Weinberg–Salam model
in geometric variables. We omit here the kinematic term
K(m) and the potential term V (m2) for the sake of
brevity because they do not interfere with the calcula-
tions and can be reinstated at every time. The remaining
terms of the classical action are
x(LAbelBB + LAbelBF + LBB + LBF + LAA),
LAA := −m
µ, (73)
LAbelBB := − g
µν , (74)
LAbelBF := + i4ǫ
µνκλB0µνF
κλ, (75)
and LBB as well as LBF have been defined in Eqs. (40)
and (41), respectively.
The saddle point conditions for the [dA] integration
with this action are given by
ǫκλµνDabµ (Ă)B
κλ = +im
abAbν , (76)
κλ = +im
ν . (77)
For the following it is convenient to use linear combina-
tions of these equations, which are obtained by contrac-
tion with the eigenvectors µ
of the matrix mab—defined
between Eqs. (37) and (38)—,
ǫκλµν [µalD
µ (Ă)B
κλ + µ
l ∂µB
κλ] = im
abAbν .(78)
The non-Abelian term on the left-hand side can be
rewritten using the results from the first part of Sect. III.
The right-hand side may be expressed in terms of eigen-
values of the matrix mab. We find (no summation over
Xaν = im2mla
ν , (79)
where
aν := daj∇µ|
gT kµν) + 1
l ∂µB
κλ. (80)
The mass term can be decomposed in the eigenbasis of
ab as well and, subsequently, be formulated in terms of
the geometric variables,
LAA = −m
(m̄−1)abXaνXbν , (81)
where
(m̄−1)ab :=
∑∀ml 6=0
. (82)
With the help of these relations and the results from
the beginning of Sect. III we are now in the posi-
tion to express the classical action in geometric vari-
ables: The mass term is given in the previous expres-
sion. It describes a Gaussian distribution of a com-
posite current. The components of the current are su-
perpositions of Abelian and non-Abelian contributions.
This mixture is caused by the symmetry breaking pat-
tern SU(2)L × U(1)Y → U(1)em which leaves unbroken
U(1)em and not the U(1)Y which is a symmetry in the
unbroken phase. The Abelian antisymmetric fields B0µν
in LAbelBB are gauge invariant and we leave LAbelBB as de-
fined in Eq. (74). In geometric variables LBB is given
by Eq. (48) and LBF by Eq. (58). At the end the ki-
netic term K(m) and the potential term V (m2) should
be reinstated.
Additional contributions from fluctuations give rise to
an addend (on the level of the Lagrangian) proportional
ln detm, wherem can be expressed in the new vari-
ables, mjkµν = f
abcdal µ
gT lµν −m2
ljδklgµν .
Repeating the entire calculation not in unitary gauge,
but with explicit gauge scalars Φ, yields exactly the same
result because the mass term and the saddle point con-
dition change in unison, such that Eq. (79) is obtained
again. This has already been demonstrated explicitly for
a massive Yang–Mills theory just before Sect. III A.
IV. SUMMARY
We have discussed the formulation of massive gauge
field theories in terms of antisymmetric tensor fields
(Sect. II) and of geometric variables (Sect. III). The
description in terms of an antisymmetric tensor field
Baµν has the advantage that it transforms homogeneously
under gauge transformations, whereas the usual gauge
field Aaµ transforms inhomogeneously, which complicates
a gauge-independent treatment of massive gauge field
theories. In fact, the (Stückelberg-like) degrees of free-
dom needed for a gauge-invariant formulation in terms
of a Yang–Mills connections are directly absorbed in
the antisymmetric tensor fields. No scalar field is re-
quired in order to construct a gauge invariant massive
theory in terms of the new variables. After recapitu-
lating the massless case in Sect. IIA, we have treated
the massive setting in Sect. IIB. After the fixed mass
case, at the beginning of Sect. IIB, this section encom-
passes also a position dependent mass (Sect. IIB1), that
is the Higgs degree of freedom, and a non-diagonal mass
term (Sect. IIB2). This is required for describing the
Weinberg–Salam model. In this context, we have identi-
fied the degrees of freedom which represent the different
electroweak gauge bosons in the Baµν representation by a
gauge-invariant eigenvector decomposition.
The Abelian section (App. A) serves as basis for an
easier understanding of some issues arising in the non-
Abelian case, like for example vanishing conserved cur-
rents. In that section we also address the massless limits
of propagators in the Aµ and Bµν representations, respec-
tively. We notice that while the limit is ill-defined for the
Aµ fields it is well-defined for the Bµν fields. That is due
to the consistent treatment of gauge degrees of freedom
in the latter case.
In Sect. III we continue with a description of massive
gauge field theories in terms of geometric variables in
four space-time dimensions and for two colours. Thereby
we can eliminate the remaining degrees of freedom which
are still encoded in the Baµν fields. After deriving the
expressions for a fixed mass and in the presence of the
Higgs degree of freedom, respectively, we also investigate
the Weinberg–Salam model (Sect. III A).
Acknowledgments
DDD would like to thank Gerald Dunne and Stefan
Hofmann for helpful, informative and inspiring discus-
sions. Thanks are again due to Stefan Hofmann for read-
ing the manuscript.
APPENDIX A: ABELIAN
1. Massless
The partition function of an Abelian gauge field theory
without fermions is given by
[dA] exp{i
xL} (A1)
with the Lagrangian density
L = L0 := − 14g2FµνF
µν (A2)
and the field tensor
Fµν := ∂µAν − ∂νAµ. (A3)
g stands for the coupling constant. The transition to the
first-order formalism can be performed just like in the
non-Abelian case, which is treated in the main body of
the paper. We find the partition function,
[dA][dB] ×
× exp{i
d4x[− 1
F̃µνB
µν − g
µν ]}. (A4)
Here the antisymmetric tensor field Bµν , like the field
tensor Fµν , is gauge invariant. The classical equations of
motion are given by
µν = 0 and g2Bµν = −F̃µν , (A5)
which after elimination of Bµν reproduce the Maxwell
equations one would obtain from Eq. (A2). Now we can
formally integrate out the gauge field Aµ. As no gauge
is fixed by the BF term because the Abelian field ten-
sor Fµν is gauge invariant this gives rise to a functional
δ distribution. This constrains the allowed field configu-
rations to those for which the conserved current ∂µB̃
vanishes,
[dB]δ(∂µB̃
µν) exp{i
d4x[− g
µν ]}.
2. Massive
In the massive case the Lagrangian density becomes
L = L0 + Lm, where Lm := m
µ. First, we here
repeat some steps carried out above in the non-Abelian
case: We can directly write down the partition function
in unitary gauge. Regauging like in Eq. (18) leads to
[dA]′[dU ] exp(i
d4x{− 1
[Aµ − iU †(∂µU)][Aµ − iU †(∂µU)]}).(A7)
The corresponding gauge-invariant Lagrangian then
reads,
Lcl := − 14g2FµνF
µν + m
(DµΦ)
†(DµΦ), (A8)
with the constraint Φ†Φ
= 1. Constructing a partition
function in the first-order formalism from the previous
Lagrangian yields,
[dA][dΦ][dB] ×
× exp(i
d4x{− 1
Bµν F̃
µν − g
[Aµ − iΦ(∂µΦ†)][Aµ − iΦ(∂µΦ†)]}). (A9)
The Φ fields can be absorbed entirely in a gauge-
transformation of the gauge field Aµ. The integration
over Φ decouples. This can also be seen by putting the
parametrisation Φ = e−iθ into the previous equation and
carrying out the [dA] integration,
[dB][dθ] exp{i
d4x[− g
(∂κB̃
κµ)gµν(∂λB̃
λν)− (∂µθ)(∂κBκµ)]}.
(A10)
The only θ dependent term in the exponent is a total
derivative and drops out, leading to a factorisation of
the θ integral.
A third way which yields the same final result, starts
by integrating out the θ field first. This gives a transverse
mass term∼ Aµ(gµν− ∂µ∂ν� )A
ν . Integration overAµ then
leads to the same result as before.
Instead of a vanishing current ∂µB̃
µν like in the mass-
less case, in the massive case the current has a Gaussian
distribution. The distribution’s width is proportional to
the mass of the gauge boson.
m → 0 limit
In the gauge-field representation the massless limit for
the classical actions discussed above are smooth. In
terms of the Bµν field the mass m ends up in the denom-
inator of the corresponding term in the action. Together
with the m dependent normalisation factors arising form
the integrations over the gauge-field in the course of the
derivation of the Bµν representation, however, the limit
m → 0 still yields the m = 0 result for the partition
function (A6).
Still, it is known that the perturbative propagator for
a massive photon is ill-defined if the mass goes to zero:
Variation of the exponent of the Abelian massive parti-
tion function in unitary gauge with respect to Aκ and Aλ
gives the inverse propagator for the gauge fields,
(G−1)κλ = [(p2 −m2phys)gκλ − pκpλ], (A11)
which here is already transformed to momentum space.
The corresponding equation of motion,
(G−1)κλGλµ
= gκµ, (A12)
is solved by
Gλµ =
p2 −m2phys
m2phys
p2 −m2phys
, (A13)
with boundary conditions (an ǫ prescription) to be spec-
ified and mphys := mg. This propagator diverges in the
limit m→ 0.
In the representation based on the antisymmetric ten-
sor fields, variation of the exponent of the partition func-
tion (A10) with respect to the fields B̃µν and B̃κλ yields
the inverse propagator
(G−1)µν|κλ = gµκgνλ − gνκgµλ +
+m−2phys(∂
µ∂κgνλ − ∂ν∂κgµλ −
− ∂µ∂λgνκ + ∂ν∂λgµκ), (A14)
already expressed in momentum space. Variation with
respect to B̃µν instead of Bµν corresponds only to a
reshuffling of the Lorentz indices and gives an equiva-
lent description. The antisymmetric structure of the in-
verse propagator is due to the antisymmetry of B̃µν . The
equation of motion is then given by
(G−1)µν|κλGκλ|ρσ
= gµρ g
σ − gµσgνρ (A15)
and solved by
2Gκλ|ρσ = (gκρgλσ − gκσgλρ)−
p2 −m2phys
×(pκpρgλσ − pκpσgλρ −
pλpρgκσ + pλpσgκρ). (A16)
Here we observe that the limit m→ 0 is well-defined,
2Gκλ|ρσ
m→0−−−→ gκρgλσ − gκσgλρ −
(pκpρgλσ − pκpσgλρ −
− pλpρgκσ + pλpσgκρ). (A17)
This is due to the consistent treatment of the gauge de-
grees of freedom in the second approach.
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[26] We do not discuss here odd dimensional cases, where
mass can be created topologically by means of Chern–
Simons terms [19].
[27] Again in an odd number of space-time dimensions (2+1)
Karabali and Nair [20] also together with Kim [21] have
carried out a gauge invariant analysis of non-Abelian
gauge field theories. As pointed out by Freidel et al. [22]
this approach appears to be extendable to 3+1 dimen-
sions making use of the so-called ”corner variables” [23].
[28] The functional integral over a gauge field is ill-defined as
long as no gauge is fixed. We have to keep this fact in
mind at all times and will discuss it in detail when a field
is really integrated out.
[29] Due to the possible occurrence of what is known as Gri-
bov copies [24] this method might not achieve a unique
splitting of the two parts of the integration. For our il-
lustrative purposes, however, this is not important.
[30] For three colours and four space-time dimensions there
exists a treatment of the massless setting due to Lunev
[25], different from the one which here is extended to the
massive case.
[31] One possible representation is ηi
= ǫamn, η
4n = −δ
m4 = −δ
m, and η
= 0, where m,n ∈ {1; 2; 3}.
http://arxiv.org/abs/hep-th/9807069
http://arxiv.org/abs/hep-th/0609166
http://arxiv.org/abs/gr-qc/9905087
http://arxiv.org/abs/gr-qc/0210094
http://arxiv.org/abs/hep-th/9507036
http://arxiv.org/abs/hep-th/9603151
http://arxiv.org/abs/hep-th/0108097
http://arxiv.org/abs/hep-th/0609146
http://arxiv.org/abs/hep-th/9705087
http://arxiv.org/abs/hep-th/9804132
http://arxiv.org/abs/hep-th/0007188
http://arxiv.org/abs/hep-th/0604185
http://arxiv.org/abs/hep-th/0604184
http://arxiv.org/abs/hep-th/9503133
