#  Non-minimal non-Abelian quantum vector fields in curved spacetime

L. L. Salcedo<EMAIL_ADDRESS>Departamento de Física Atómica, Molecular y
Nuclear and
Instituto Carlos I de Física Teórica y Computacional,
Universidad de Granada, E-18071 Granada, Spain.

###### Abstract

The quantum effective action of non-minimal vector fields with Abelian or non-
Abelian gauge degrees of freedom in curved spacetime is studied. The Proca or
Yang-Mills fields are coupled to a local mass-like term acting in both
coordinate and gauge spaces. Pathologies due to gauge invariance in the
ultraviolet are avoided through the introduction of a non-Abelian version of
the Stueckelberg field. It is found that the breaking of gauge invariance
induced by the mass term affects only the tree level part of the effective
action. The ultraviolet divergent part of the effective action to one loop is
obtained using the method of covariant symbols and dimensional regularization.
Formulas are given valid for any spacetime dimension and explicit results are
shown for the two-dimensional case. As already happened for a single vector
field, the ultraviolet divergences are local but not of polynomial type.

###### Contents

  1. I Introduction
  2. II Formulation of the problem
  3. III The effective action
    1. III.1 The background gauge field
    2. III.2 One-loop effective action
  4. IV The Stueckelberg field
    1. IV.1 The non-Abelian Stueckelberg field
    2. IV.2 The one-loop effective action revisited
    3. IV.3  Contributions to the effective action
  5. V  Some notational conventions
    1. V.1  Covariant derivatives
    2. V.2  Operators $Z_{\mu_{1}\cdots\mu_{n}}$
    3. V.3  Integrals and traces
  6. VI  Results for $\Gamma^{\mathrm{div}}_{1}$
    1. VI.1  Results for $\Gamma_{\mathrm{gh}}$ and $\Gamma_{L,1}$
    2. VI.2  Results for $\Gamma_{S}$
      1. VI.2.1 Contributions to $\Gamma_{S}$
      2. VI.2.2 $\Gamma_{S}^{\mathrm{div}}$ in $d=2$
    3. VI.3  Results for $\Gamma_{L,0}$
      1. VI.3.1  Preliminaries
      2. VI.3.2 Special case of separable mass term
      3. VI.3.3 Method of contour integration
      4. VI.3.4 $\Gamma_{L,0}$ in $d=2$
      5. VI.3.5 Verification of metric-related symmetries in $\Gamma_{L,0}$
  7. VII  Computation of the diagonal matrix elements
    1. VII.1  Method of covariant symbols
    2. VII.2  Calculation of $\Gamma^{\mathrm{div}}_{S}$ in $d=2$
    3. VII.3  Calculation of $\Gamma^{\mathrm{div}}_{L,0}$ in $d=2$
    4. VII.4 Extraction of the UV divergent component
  8. VIII Summary and conclusions
  9. A Derivation of some formulas
    1. A.1 Derivation of Eq. (15)
    2. A.2 Derivation of Eq. (24)
    3. A.3 Derivation of Eq. (26)
  10. B The operators $Z_{\mu_{1}\cdots\mu_{n}}$
    1. B.1 Definition and properties of $Z^{R}_{\mu_{1}\cdots\mu_{n}}$
    2. B.2 Relation with the operator $\hat{{\mathcal{R}}}_{\mu\nu}$ of Barvinsky:1985an
    3. B.3 Derivation of Eqs. (55)
  11. C Canonical form of $\Gamma_{S}$
  12. D Perturbative expansions
  13. E Method of non-covariant symbols
    1. E.1 Covariant vs non-covariant method of symbols
    2. E.2 Method of non-covariant symbols in curved spacetime
  14. F  Sample calculation using the method of covariant symbols

##  I Introduction

Vector fields play a prominent role in the Standard Model of particles, as
mediators of gauge interactions. In turn there is currently a growing interest
on the role that various types of vector fields could play in relativistic
gravity and cosmology Dimopoulos:2006ms ; Dimopoulos:2008yv ; Koivisto:2008xf
; Yokoyama:2008xw ; Esposito-Farese:2009wbc ; Graham:2015rva ;
DeFelice:2016yws ; Emami:2016ldl ; Heisenberg:2018vsk . As noted in
Heisenberg:2018vsk  “Imposing the conditions of Lorentz symmetry, unitarity,
locality and a (pseudo-)Riemannian space–time, any attempt of modifying
gravity inevitably introduces new dynamical degrees of freedom. They could be
additional scalar, vector or tensor fields.” The subject has received a
further boost with the discovery of ghost free consistent non-linear theories
of Proca interactions Heisenberg:2014rta ; Allys:2015sht ; Allys:2016kbq ;
ErrastiDiez:2019ttn ; deRham:2020yet ; GallegoCadavid:2020dho . The crucial
issue of the quantum stability of these theories has been analyzed in
Heisenberg:2020jtr ; deRham:2021yhr .

In this work we consider a set of $N$ vector fields in curved spacetime
endowed with Abelian (Proca) and/or non-Abelian (Yang-Mills) internal degrees
of freedom. No self-interactions are included beyond those implied by the
Yang-Mills structure, but the vector fields are coupled to an external mass-
like $x$-dependent tensor field which is allowed to arbitrarily mix them (see
Eq. (1)). Our focus is on the proper quantization of such a theory and on the
structure of the quantum fluctuations.

Early work studying the subject of quantum fluctuations for vectors fields was
carried out in Barvinsky:1985an ; Furlani:1997fh ; Gorbar:2003yt ;
Buchbinder:2007xq  (see Felipe:2018pdw ; Franchino-Vinas:2019upg ;
Gomez:2019tbj ; Sanongkhun:2019ntn ; ErrastiDiez:2020dux ; Marques:2021jfy ;
Martini:2021slj ; Hell:2021oea  for recent related work). Particular non-
minimal couplings (the minimal case being a standard mass term) were
considered in Novello:1979ik  at the classical level and in Davies:1984vm  at
the quantum level. The quantized theory for general non-minimal couplings was
first studied in Toms:2015fja  for particular spacetime backgrounds. There it
was found that pathologies arise in the quantization of the theory since the
mass term couples effectively as a metric field. Technically the problem is
that the mass-like field breaks gauge invariance but does not suppress the
fluctuations in the longitudinal polarization at large wavenumbers. In other
words, the principal symbol of the fluctuation operator is singular. General
backgrounds were considered in Buchbinder:2017zaa  solving the above-mentioned
pathology by means of a Stueckelberg field. In this way a proper gauge
symmetry is present in the theory and one can proceed through a standard gauge
fixing procedure. However, approximations were introduced in the analysis of
Buchbinder:2017zaa  giving rise to a non-local result. A full solution to the
problem of computing the ultraviolet (UV) divergent part of the effective
action, within dimensional regularization, was obtained in Ruf:2018vzq  using
the Schwinger-DeWitt technique and later in Garcia-Recio:2019iia  using the
method of covariant symbols, finding perfect concordance in both calculations.
The pathologies identified in Toms:2015fja  translate to the fact that the UV
divergences are local but not polynomial in the mass-like external field.

The results just noted refer to a single vector field. Here we address the
case of several vector fields. This allows to consider the non-Abelian
scenario. In fact, we consider sets of vector fields organized in Abelian and
non Abelian multiplets. We treat the Abelian and non-Abelian versions
simultaneously since the formalism is identical in both cases. In the absence
of a mass term there would be a gauge symmetry present in the Lagrangian. This
symmetry is explicitly broken by the mass term. Nevertheless we obtain the
remarkable result that the breaking only affects to the effective action at
tree level, while the contributions from one or more loops are fully gauge
invariant. This results in an important simplification of the calculation.
Another insight comes from the introduction of the Stueckelberg field in our
present non-Abelian setting (see Ruegg:2003ps  for a review on this subject).
In the Abelian case the Stueckelberg field appears through
$A_{\mu}=B_{\mu}+\partial_{\mu}\phi$. In this way a $U(1)$ gauge symmetry
arises from $B_{\mu}\to B_{\mu}+\partial_{\mu}\Lambda$, $\phi\to\phi-\Lambda$.
In the non-Abelian case a literal translation of this prescription would take
the form $A^{a}_{\mu}=(B^{\Omega})^{a}_{\mu}$ where $\Omega$ refers to a non-
Abelian gauge transformation with parameters $\phi^{a}$ (in Lie algebra of the
gauge group). The resulting theory enjoys a non-Abelian gauge invariance and
one can then proceed to fix the gauge through the Fadeev-Popov method. Such
approach is in principle correct but exceedingly complicated as the dependence
on $\phi^{a}$ is non-linear. In particular this would imply a reorganization
of the loop expansion from the original theory (field $A_{\mu}^{a}$) to that
with $B_{\mu}^{a}$ and $\phi^{a}$. We develop a completely different approach
where the Stueckelberg field is introduced linearly also in the non-Abelian
setting.

Once the Stueckelberg field is introduced the (UV regulated) quantum theory is
no longer pathological and it is possible to proceed to a systematic
computation of its effective action. Our focus is on the UV part of the
effective action to one-loop order. As already known from the study of the
Abelian case Ruf:2018vzq ; Garcia-Recio:2019iia , the mass term acts as an
effective second metric tensor. In the present case this is in fact a non-
Abelian effective metric. Presumably the generalized Schwinger-DeWitt
technique Barvinsky:1985an  can be adapted to this situation, but such an
approach is not presently available. Instead here we apply the method of
covariant symbols Salcedo:2006pv . This method is simple to use and allows to
formulate the loop momentum integration while preserving manifest covariance
under diffeomorphism and gauge transformations. Details of the method are
provided below.

The problem studied here is an extension of that already solved for $N=1$
(just one vector field), so some specific features of that case are inherited
in the more general setting analyzed in this work. In particular, the loop
momentum integrals cannot be written in closed form. For $N>1$ this problem is
even worse as the propagator are now matrices with respect to the gauge
indices. Also for this reason some contributions to the effective action (see
$\Gamma_{L,0}$ below) cannot be expressed in a standard form involving just
integration over $x$, $p$ and traces in internal space, and it is necessary to
resort to a parametric form, with integration over one more parameter.
Unfortunately, while the problem is well-posed and the method fully
appropriate to solve it, we have found an unexpected impediment, namely, the
number of terms obtained for the physically relevant case of four spacetime
dimensions is prohibitively large (at least hundreds of terms are generated).
In view of this we develop the formulas for the general case but only present
detailed results for two spacetime dimensions (note that there are no UV
divergences for odd dimensions within dimensional regularization).

In Sec. II we expose the theory to be analyzed. In Sec. III the background
field approach is introduced for the effective action. It is shown that its
quantum part admits a gauge-covariant treatment. The effective action to one
loop is constructed, showing its limitations in the UV sector. Those obstacles
are overcome in Sec. IV by introducing the Stueckelberg field. The non-
pathological one-loop effective action is constructed, and then decomposed
into various contributions to be computed subsequently. Sec. V introduces some
notational conventions. Sec. VI introduces general considerations to undertake
the calculation and presents explicit results for $d=2$. Some non-trivial
symmetries related to metric deformations are also verified. The actual
calculations are worked out in Sec. VII. To this end, the method of covariant
symbols is reviewed first, and its application to the various contributions is
discussed, including the extraction of the coefficients of the UV divergence.
The conclusions are summarized in Sec. VIII. The proof of some formulas is
provided in App. A. Properties of the operator $Z_{\mu\nu}$ and its relation
to $\hat{{\mathcal{R}}}_{\mu\nu}$ of Barvinsky:1985an  are discussed in App.
B. The canonical form of $\Gamma_{S}$ using a basis of standard operators is
displayed in App. C. Explicit results for perturbative mass expansions are
presented in App. D. Details of the method of non-covariant symbols, used as a
check of the calculations, are given in App. E. Finally, the method of
covariant symbols is illustrated through a sample computation in App. F.

##  II Formulation of the problem

We consider $N$ real vector fields $\hat{{\mathcal{A}}}^{a}_{\mu}(x)$,
$a=1,\ldots,N$ in an Euclidean $d$ dimensional spacetime with metric
$g_{\mu\nu}(x)$ and action111The ugly notation $\hat{{\mathcal{A}}}^{a}_{\mu}$
and $\hat{{\mathcal{F}}}_{a}^{\mu\nu}$ will soon be traded by $A^{a}_{\mu}$
and $F^{\mu\nu}$.

$S[\hat{{\mathcal{A}}};{\mathcal{M}},g]=\int
d^{d}x\sqrt{g}\left(\frac{1}{4}\hat{{\mathcal{F}}}_{a}^{\mu\nu}\hat{{\mathcal{F}}}^{a}_{\mu\nu}+\frac{1}{2}{\mathcal{M}}_{ab}^{\mu\nu}\hat{{\mathcal{A}}}^{a}_{\mu}\hat{{\mathcal{A}}}^{b}_{\nu}\right),$
(1)

where ${\mathcal{M}}_{ab}^{\mu\nu}(x)$ is a positive definite222As a matrix
with indices $(\mu a)$ and $(\nu b)$. local mass term fulfilling the symmetry
condition

${\mathcal{M}}_{ab}^{\mu\nu}(x)={\mathcal{M}}_{ba}^{\nu\mu}(x).$ (2)

Coordinate indices are raised, lowered and contracted with the metric
$g_{\mu\nu}$.

The $N$ fields are organized in $n$ gauge sectors. Each sector has gauge
symmetry of the type either $\mathrm{SU}(n_{i})$ or $\mathrm{U}(1)$ and the
full gauge group is the direct product of these.333More generally, one could
take a Lie subgroup of $\mathrm{SO}(N)$ and the results and formulas derived
in this work hold equally well in that case. The fields fall in the Lie
algebra of the group, i.e., the adjoint representation, with gauge coupling
$g_{i}$ in the gauge sector $i$. Without loss of generality one can choose
$g_{i}=1$ for the Abelian factors. Hence $N=\sum_{i=1}^{n}N_{i}$ where
$N_{i}=n_{i}^{2}-1$ for an $\mathrm{SU}(n_{i})$ sector and $N_{i}=1$ for a
$\mathrm{U}(1)$ one. With a standard normalization of the fields, the field
strength tensor is

$\hat{{\mathcal{F}}}^{a}_{\mu\nu}=\partial_{\mu}\hat{{\mathcal{A}}}^{a}_{\nu}-\partial_{\nu}\hat{{\mathcal{A}}}^{a}_{\mu}+g_{a}f_{abc}\hat{{\mathcal{A}}}^{b}_{\mu}\hat{{\mathcal{A}}}^{c}_{\nu}$
(3)

where $f_{abc}$ are the structure constants of the gauge group. Since the
gauge group is a direct product, the structure constants are block diagonal,
one block for each gauge sector, and $g_{a}=g_{i}$ is the coupling of the
$i$-th sector (the $g_{a}$ take a common value within each block). Of course,
in a $\mathrm{U}(1)$ sector the structure constant vanishes. The field
strength tensor $\hat{{\mathcal{F}}}^{a}_{\mu\nu}(x)$ is covariant under gauge
and coordinate transformations. The kinetic part of the action is block
diagonal while the mass term may mix different gauge sectors.

When all the gauge sectors are of the $\mathrm{U}(1)$ type the theory is
Abelian and reduces to a generalized Proca field with $N$ flavors.
Nevertheless, since all the cases can be treated within the same scheme, we
will refer to the internal space as gauge space.

Regarding the symmetries, the kinetic term is fully local gauge invariant but
such symmetry is reduced to a global one by the mass term: the action is
invariant under
${\mathcal{M}}\to{\mathcal{M}}^{\Omega}=\Omega{\mathcal{M}}\Omega^{-1}$ for a
global transformation $\Omega$ in the gauge group. If some gauge sectors are
equivalent, namely, with equal gauge group $\mathrm{SU}(n_{i})$ o
$\mathrm{U}(1)$ and same $g_{i}$, there is an additional global symmetry under
rotations among those equivalent sectors, with a corresponding rotation of
${\mathcal{M}}$. A further symmetry, special for the case of $d=4$ spacetime
dimensions, is that of local Weyl-like transformations, namely, the action is
unchanged under the simultaneous replacements
$g_{\mu\nu}(x)\to\xi(x)g_{\mu\nu}(x)$ and
${\mathcal{M}}^{\mu\nu}_{ab}(x)\to\xi^{-2}(x){\mathcal{M}}^{\mu\nu}_{ab}(x)$.

##  III The effective action

###  III.1 The background gauge field

Within the background field approach Abbott:1980hw  the field is split as a
background plus a fluctuation,
$\hat{{\mathcal{A}}}^{a}_{\mu}(x)=A^{a}_{\mu}(x)+{\mathcal{A}}^{a}_{\mu}(x)$.

In this approach the effective action $\Gamma[A;{\mathcal{M}},g]$ follows from

$Z=e^{-\Gamma[A;{\mathcal{M}},g]}=\int{\mathcal{D}}{\mathcal{A}}e^{-S[{\mathcal{A}}+A]+\int
d^{d}x\sqrt{g}\,J{\mathcal{A}}}$ (4)

where $A_{\mu}^{a}(x)$ is the background field and the current
$J^{\mu}_{a}(x)$ is adjusted so that
$\langle{\mathcal{A}}_{\mu}^{a}(x)\rangle=0$. As usual

$J^{\mu}_{a}(x)=\frac{1}{\sqrt{g(x)}}\frac{\delta\Gamma[A;{\mathcal{M}},g]}{\delta
A_{\mu}^{a}(x)}.$ (5)

In the background gauge field approach, the field ${\mathcal{A}}_{\mu}^{a}(x)$
transforms homogeneously under local gauge transformations, the inhomogeneity
being saturated by the transformation of $A^{a}_{\mu}(x)$. Correspondingly,
the gauge covariant derivative relies on $A^{a}_{\mu}(x)$ as gauge connection.

We will use a single covariant derivative $\nabla_{\mu}$ containing coordinate
and gauge connections Vassilevich:2003xt . The coordinate connection is that
of Levi-Civita for the metric and the gauge connection is that of the
background field $A_{\mu}^{a}$. So for instance, for coordinate-scalar and
coordinate-vector fields $\phi^{a}$ and ${\mathcal{B}}^{a}_{\mu}$,
respectively, both in the adjoint gauge representation,

$\begin{split}\nabla_{\mu}\phi^{a}&=\partial_{\mu}\phi^{a}+g_{a}f_{abc}A_{\mu}^{b}\phi^{c},\\\
\nabla_{\mu}{\mathcal{B}}_{\nu}^{a}&=\partial_{\mu}{\mathcal{B}}_{\nu}^{a}-\Gamma_{\mu\nu}^{\lambda}{\mathcal{B}}_{\lambda}^{a}+g_{a}f_{abc}A_{\mu}^{b}{\mathcal{B}}_{\nu}^{c}.\end{split}$
(6)

Throughout, coordinate indices are contracted with the metric $g_{\mu\nu}$ and
gauge-vector indices with $\delta_{ab}$.

The effective action can be split into the classical or tree-level component
$S[A]$ and the quantum correction $\Gamma_{Q}[A]$ which contains graphs with
one or more loops,

$\Gamma[A]=S[A]+\Gamma_{Q}[A].$ (7)

Here we find a fundamental result given by the following

Theorem: $\Gamma_{Q}[A]$ is invariant under local gauge transformations and
all the gauge breaking in the effective action is saturated by the mass term
at tree level. That is,

$\Gamma_{Q}[A;{\mathcal{M}},g]=\Gamma_{Q}[A^{\Omega};{\mathcal{M}}^{\Omega},g],$
(8)

where $\Omega(x)$ is any local gauge transformation, and $A^{\Omega}$,
${\mathcal{M}}^{\Omega}$ are the gauge-transformed fields.

Proof: The reason is fairly simple. The semiclassical expansion follows from a
Taylor expansion of the action $S[{\mathcal{A}}+A]-\int
d^{d}x\sqrt{g}\,J{\mathcal{A}}$ in powers of the fluctuation ${\mathcal{A}}$.
The zeroth order gives the classical action, and the first order in
${\mathcal{A}}$ cancels due to the equations of motion, i.e. the choice of
$J^{\mu}$. The quantum component $\Gamma_{Q}$ depends only on terms which are
quadratic or higher order in ${\mathcal{A}}$. The breaking of gauge invariance
would come solely from the mass term
$\frac{1}{2}{\mathcal{M}}_{ab}^{\mu\nu}{\mathcal{A}}^{a}_{\mu}{\mathcal{A}}^{b}_{\nu}$
but this is covariant since the field ${\mathcal{A}}_{\mu}^{a}$ transforms
homogeneously under gauge transformations.

The property Eq. (8) is important because it allows to use a gauge-covariant
formalism for $\Gamma_{Q}$.

###  III.2 One-loop effective action

The one-loop effective action follows from the quadratic part of the action

$\Gamma_{1}[A]=\frac{1}{2}\log\operatorname{Det}_{1}\left(\frac{\delta^{2}S[{\mathcal{A}}+A]}{\delta{\mathcal{A}}{}^{2}}\right)\Bigg{|}_{{\mathcal{A}}=0}$
(9)

The subindex $1$ in $\operatorname{Det}_{1}$ indicates that the determinant is
to be evaluated in the space ${\mathcal{A}}_{\mu}(x)$, i.e., of coordinate
vectors. Also, for the gauge degrees of freedom, the determinant is taken in
the adjoint gauge representation space. This is not explicitly indicated but
will be implicit in all formulas as no other gauge representations will be
present.

Therefore we need to isolate the terms quadratic in ${\mathcal{A}}$ from the
action $S[{\mathcal{A}}+A]$. After the shift
$\hat{{\mathcal{A}}}={\mathcal{A}}+A$, the field strength tensor in Eq. (3)
becomes

$\hat{{\mathcal{F}}}^{a}_{\mu\nu}={\mathcal{F}}^{a}_{\mu\nu}+\nabla_{\mu}{\mathcal{A}}^{a}_{\nu}-\nabla_{\nu}{\mathcal{A}}^{a}_{\mu}+g_{a}f_{abc}{\mathcal{A}}^{b}_{\mu}{\mathcal{A}}^{c}_{\nu}$
(10)

with

${\mathcal{F}}^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+g_{a}f_{abc}A^{b}_{\mu}A^{c}_{\nu}.$
(11)

In the shifted variables, the quadratic part of the kinetic term of the action
takes the form444We will occasionally place all the coordinate indices as
lower indices when no ambiguity arises. Repeated coordinate indices are always
contracted with the metric $g_{\mu\nu}$.

$S^{(2)}_{\rm kin}[{\mathcal{A}}]=\int
d^{d}x\sqrt{g}\left(\frac{1}{4}(\nabla_{\mu}{\mathcal{A}}_{\nu}^{a}-\nabla_{\nu}{\mathcal{A}}_{\mu}^{a})^{2}-\frac{1}{2}F_{\mu\nu}^{ab}{\mathcal{A}}_{\mu}^{a}{\mathcal{A}}^{b}_{\nu}\right)$
(12)

where we have the introduced field strength tensor $F^{ab}_{\mu\nu}$ as an
antisymmetric matrix in gauge space (as well as in coordinate space)

$F_{\mu\nu}^{ab}\equiv g_{c}f_{acb}{\mathcal{F}}^{c}_{\mu\nu}.$ (13)

Note that $F_{\mu\nu}^{ab}$ vanishes in the Abelian sectors.

In what follows we adopt the convention that covariant derivatives are
indicated by adding new indices to the left, hence
$\phi_{\mu}^{a}\equiv\nabla_{\mu}\phi^{a}$,
${\mathcal{B}}_{\mu\nu}^{a}\equiv\nabla_{\mu}{\mathcal{B}}^{a}_{\nu}$, etc.
The only exception to this rule are the operators $Z_{\mu_{1}\mu_{2}\ldots}$
and $Z^{R}_{\mu_{1}\mu_{2}\ldots}$.555Some conventions used in this work are
summarized in Sec. V. With this convention

$S^{(2)}_{\rm kin}[{\mathcal{A}}]=\int
d^{d}x\sqrt{g}\left(\frac{1}{4}({\mathcal{A}}_{\mu\nu}^{a}-{\mathcal{A}}_{\nu\mu}^{a})^{2}-\frac{1}{2}F_{\mu\nu}^{ab}{\mathcal{A}}_{\mu}^{a}{\mathcal{A}}^{b}_{\nu}\right)$
(14)

Using integration by parts and Bianchi identities, the quadratic part of the
kinetic term can be written as (see App. A)

$\begin{split}S^{(2)}_{\rm kin}[{\mathcal{A}}]&=\int
d^{d}x\sqrt{g}\Bigg{(}\frac{1}{2}({\mathcal{A}}_{\mu\nu}^{a})^{2}-\frac{1}{2}({\mathcal{A}}_{\mu\mu}^{a})^{2}\\\
&\quad-
F_{\mu\nu}^{ab}{\mathcal{A}}_{\mu}^{a}{\mathcal{A}}^{b}_{\nu}+\frac{1}{2}{\mathcal{R}}_{\mu\nu}{\mathcal{A}}^{a}_{\mu}{\mathcal{A}}^{a}_{\nu}\Bigg{)}\end{split}$
(15)

where ${\mathcal{R}}_{\mu\nu}$ is the Ricci tensor. Thus, adding the mass
term,

$S^{(2)}_{\rm mass}[{\mathcal{A}}]=\int
d^{d}x\sqrt{g}\left(\frac{1}{2}{\mathcal{M}}_{ab}^{\mu\nu}{\mathcal{A}}^{a}_{\mu}{\mathcal{A}}^{b}_{\nu}\right),$
(16)

the full quadratic Lagrangian controlling the one-loop fluctuations is

${\mathcal{L}}^{(2)}(x)=\frac{1}{2}{\mathcal{A}}_{\mu}K^{\mu\nu}_{0}{\mathcal{A}}_{\nu}$
(17)

where

$K_{0}^{\mu\nu}=-g^{\mu\nu}\nabla^{2}+\nabla^{\mu}\nabla^{\nu}-2F^{\mu\nu}+{\mathcal{R}}^{\mu\nu}+{\mathcal{M}}^{\mu\nu}.$
(18)

Here, and also in what follows, we use a matric notation for the gauge
indices, which will be implicit. As advertised, the Lagrangian
${\mathcal{L}}^{(2)}(x)$ is manifestly gauge invariant.

The term $+\nabla^{\mu}\nabla^{\nu}$ in $K_{0}^{\mu\nu}$ is a direct
consequence of gauge invariance of the kinetic energy part of the action (1)
and is needed to retain just three polarizations in the Proca field. While the
differential operator $K_{0}$ needs not be singular in presence of a positive
definite mass term ${\mathcal{M}}^{\mu\nu}$, its principal symbol, i.e., the
$O(\nabla^{2})$ leading UV divergent component is singular, since the
longitudinal polarizations are not penalized at large wavenumbers. The fact
that the principal symbol is singular introduces pathologies in the effective
action which prevent to carry out an extraction of the UV divergent terms.

In the special case of a standard Proca field, with a constant scalar mass,
the UV divergent part of the effective action is a polynomial in the mass
Barvinsky:1985an , but this is no longer so for a non-constant mass term even
in the Abelian case Ruf:2018vzq . This confirms that $K_{0}$ cannot be
directly used as the fluctuation operator.

##  IV The Stueckelberg field

###  IV.1 The non-Abelian Stueckelberg field

In order to bypass the above mentioned pathologies in the UV we will adapt the
Stueckelberg approach introduced in Buchbinder:2017zaa  (and also applied in
Ruf:2018vzq ; Garcia-Recio:2019iia ) for the non-minimal Proca field to the
non-Abelian case. To this end, we rewrite the partition function as

$Z=\int{\mathcal{D}}{\mathcal{A}}\,e^{-S[{\mathcal{A}}+A]+\int
d^{d}x\sqrt{g}\,J{\mathcal{A}}}\int{\mathcal{D}}\chi\,e^{-S_{\rm gf}[\chi]}$
(19)

where $S_{\rm gf}[\chi]$ can be any action. The partition function is
unchanged as the new factor is just a constant.666It does not depend on
${\mathcal{A}}_{a}^{\mu}(x)$, $J^{\mu}_{a}(x)$ nor
${\mathcal{M}}^{\mu\nu}_{ab}(x)$. It is also independent of the metric if no
derivatives are involved. We take a standard choice

$S_{\rm gf}[\chi]=\int d^{d}x\sqrt{g}\frac{1}{2}\chi^{a}\chi_{a},$ (20)

where $\chi^{a}(x)$ is a coordinate scalar and a gauge vector (i.e., in the
adjoint representation).

Subsequently a change of variables
$({\mathcal{A}},\chi)\to({\mathcal{B}},\phi)$ is applied in (19), where
${\mathcal{B}}_{\mu}^{a}(x)$ is a real coordinate-vector and gauge-vector
field and $\phi^{a}(x)$ is real coordinate-scalar and gauge-vector field:

$\begin{split}&Z=\int{\mathcal{D}}{\mathcal{B}}\,{\mathcal{D}}\phi\,J[{\mathcal{B}},\phi]\,e^{-S[{\mathcal{A}}+A]-S_{\rm
gf}[\chi]+\int d^{d}x\sqrt{g}\,J{\mathcal{A}}},\\\
&J[{\mathcal{B}},\phi]=\operatorname{Det}\left(\frac{\partial({\mathcal{A}},\chi)}{\partial({\mathcal{B}},\phi)}\right).\end{split}$
(21)

By construction the effective action does not depend on the detailed choice of
gauge-fixing function(al) $\chi[{\mathcal{B}},\phi]$, moreover, the
expectation value of any functional written in the form
$F[{\mathcal{A}},\chi]$ is independent of this choice (unless the very
functional $F$ depends on it). This property provides identities for the
gauge-fixing dependence of the expectation values Collins:1984xc .

We choose a linear change of variables. Besides simplicity, the virtue of such
a choice is that the loop expansion in the new variables coincides with that
in the old ones. Specifically we take

${\mathcal{A}}_{\mu}^{a}={\mathcal{B}}_{\mu}^{a}+\nabla_{\mu}\phi^{a},\qquad\chi^{a}=\nabla_{\mu}{\mathcal{B}}^{a}_{\mu},$
(22)

or, using our notational convention for the covariant derivatives,

${\mathcal{A}}_{\mu}^{a}={\mathcal{B}}_{\mu}^{a}+\phi_{\mu}^{a},\qquad\chi^{a}={\mathcal{B}}^{a}_{\mu\mu}.$
(23)

The corresponding Fadeev-Popov determinant is easily obtained as (see App. A)

$J[{\mathcal{B}},\phi]=\operatorname{Det}_{0}\left(\delta_{ab}\,\nabla^{\alpha}\nabla_{\alpha}\right).$
(24)

The subindex $0$ in $\operatorname{Det}_{0}$ indicates that the determinant is
to be evaluated in the $\phi^{a}$ space, i.e. the coordinate-scalar space. As
already noted, the reference to the adjoint gauge representation is not
explicitly displayed as its presence is ubiquitous and no other gauge
representation will be needed. With our choice of a linear change of
variables, the determinant $J$ does not depend on the quantum fields
${\mathcal{B}},\phi$. It depends on $A_{\mu}^{a}$ and $g_{\mu\nu}$. As usual
the determinant can be implemented through a complex ghost field with
quadratic action.

It is worth noticing that one could have introduced the Stueckelberg field in
a different manner, to wit, through the change of variable
${\mathcal{A}}={\mathcal{B}}^{\Omega}$ in Eq. (4), where $\Omega(x)$ is an
arbitrary gauge transformation, and $\phi^{a}(x)$ enters through
$\Omega=e^{i\phi}$. In addition the measure ${\mathcal{D}}{\mathcal{A}}$ is
replaced by ${\mathcal{D}}{\mathcal{B}}{\mathcal{D}}\Omega$.777Or just
${\mathcal{D}}{\mathcal{B}}{\mathcal{D}}\phi$. The two measures
${\mathcal{D}}\Omega$ and ${\mathcal{D}}\phi$ are equivalent. As is well-known
the Jacobian of a ultra-local change of variables such as
$\partial\Omega/\partial\phi$ has no effect in dimensional regularization. In
this way the full theory $S[{\mathcal{B}}^{\Omega}]$ becomes gauge invariant
even in presence of the mass term. Then one fixes the gauge as usual with the
Fadeev-Popov method. A more involved question is how to introduce the
background gauge machinery. In such alternative approach the change of
variables from $({\mathcal{A}},\chi)$ to $({\mathcal{B}},\phi)$ is not linear
and so it should be considerably more complicated than the method adopted
above. In the Abelian case the two approaches are equivalent.

###  IV.2 The one-loop effective action revisited

The introduction of the gauge-fixing action $S_{\rm gf}[\chi]$ adds an
irrelevant constant to the effective action. Hence in variables
$({\mathcal{B}},\phi)$ the effective action is just

$\begin{split}\Gamma_{1}[A]&=\frac{1}{2}\log\operatorname{Det}_{1+0}\left(\frac{\delta^{2}(S[{\mathcal{A}}+A]+S_{\rm
gf}[\chi])}{\delta({\mathcal{B}},\phi){}^{2}}\right)\Bigg{|}_{\begin{subarray}{c}{\mathcal{B}}=0\\\
\phi=0\end{subarray}}\\\
&\quad-\log\operatorname{Det}_{0}(\delta_{ab}\,\nabla^{2}),\end{split}$ (25)

where the last term comes from the Fadeev-Popov determinant. The subindex
$1+0$ indicates the direct sum of coordinate-vector and -scalar spaces.

The kinetic energy term Eq. (15) in variables $({\mathcal{B}},\phi)$ becomes
(see App. A)

$\begin{split}S^{(2)}_{\rm kin}[{\mathcal{A}}]&=\int
d^{d}x\sqrt{g}\Bigg{(}\frac{1}{2}({\mathcal{B}}_{\mu\nu}^{a})^{2}-\frac{1}{2}({\mathcal{B}}_{\mu\mu}^{a})^{2}-F_{\mu\nu}^{ab}{\mathcal{B}}_{\mu}^{a}{\mathcal{B}}^{b}_{\nu}\\\
&\quad+\frac{1}{2}{\mathcal{R}}_{\mu\nu}{\mathcal{B}}^{a}_{\mu}{\mathcal{B}}^{a}_{\nu}+F^{ab}_{\mu\mu\nu}\phi^{a}{\mathcal{B}}^{b}_{\nu}+\frac{1}{2}F^{ab}_{\mu\mu\nu}\phi^{a}\phi^{b}_{\nu}\Bigg{)},\end{split}$
(26)

while $S_{\rm gf}[\chi]$ is already quadratic, namely,

$\mathrm{S}_{\rm gf}[\chi]=\int
d^{d}x\sqrt{g}\frac{1}{2}({\mathcal{B}}_{\mu\mu}^{a})^{2}.$ (27)

This contribution removes the problematic longitudinal term in the kinetic
energy. The price to pay is the introduction of a kinetic term for $\phi$
which has a metric-like coupling to the mass tensor, namely, the last term in

$\begin{split}\mathrm{S}^{(2)}_{\rm mass}[{\mathcal{A}}]&=\int
d^{d}x\sqrt{g}\Bigg{(}\frac{1}{2}{\mathcal{M}}^{\mu\nu}_{ab}{\mathcal{B}}_{\mu}^{a}{\mathcal{B}}_{\nu}^{b}+{\mathcal{M}}^{\mu\nu}_{ab}\phi_{\mu}^{a}{\mathcal{B}}_{\nu}^{b}\\\
&\quad+\frac{1}{2}{\mathcal{M}}^{\mu\nu}_{ab}\phi_{\mu}^{a}\phi_{\nu}^{b}\Bigg{)}.\end{split}$
(28)

In summary, the new full quadratic Lagrangian controlling the one-loop
fluctuations is

${\mathcal{L}}^{(2)}=\frac{1}{2}({\mathcal{B}},\phi)K({\mathcal{B}},\phi)^{T}+\omega^{*}_{a}\nabla^{2}\omega^{a}.$
(29)

The ghost field $\omega_{a}$ is a complex fermionic coordinate-scalar and
gauge-vector field. On the other hand $K$ is a second order differential
operator acting on the space $({\mathcal{B}},\phi)$,

$\begin{split}&K=\\\
&\begin{pmatrix}-\nabla^{2}g^{\mu\nu}-2F^{\mu\nu}+{\mathcal{R}}^{\mu\nu}+{\mathcal{M}}^{\mu\nu}&-F_{\alpha\alpha}{}^{\mu}+{\mathcal{M}}^{\mu\alpha}\nabla_{\alpha}\\\
F_{\alpha\alpha}{}^{\nu}-\nabla_{\alpha}{\mathcal{M}}^{\alpha\nu}&\frac{1}{2}\\{F_{\alpha\alpha}{}^{\beta},\nabla_{\beta}\\}-\nabla_{\alpha}{\mathcal{M}}^{\alpha\beta}\nabla_{\beta}\end{pmatrix}.\end{split}$
(30)

The matric gauge indices are implicit. $\\{\;,\;\\}$ denotes anticommutator.

Note that several differential operators can be read from the last term in Eq.
(26), namely, $F_{\alpha\alpha}{}^{\beta}\nabla_{\beta}$,
$\nabla_{\beta}F_{\alpha\alpha}{}^{\beta}$ or
$\frac{1}{2}\\{F_{\alpha\alpha}{}^{\beta},\nabla_{\beta}\\}$. All of them are
equivalent in the $\phi$-$\phi$ sector, since the matrix $F_{\mu\nu}$ is
antisymmetric and the Hilbert space spanned by $\phi^{a}$ is real. However the
functional integral over ${\mathcal{B}}$ and $\phi$ is only related to the
determinant of the symmetric version of $K$, the one presented in Eq. (30).

As expected, after the introduction of the Stueckelberg field, the principal
symbol of the operator $K$ is no longer singular. Nevertheless, even though
the technical problems have been sorted, the pathologies still will reflect on
the effective action, in particular one finds that the UV divergences do not
depend polynomially on ${\mathcal{M}}$, as already happened in the case $N=1$
studied in Ruf:2018vzq ; Garcia-Recio:2019iia .

The leading UV divergent terms, with two derivatives, are at the diagonal of
the matrix $K$. Particularly problematic will be the term
$-\nabla_{\alpha}{\mathcal{M}}^{\alpha\beta}\nabla_{\beta}$ in the
$\phi$-$\phi$ sector. Because the leading divergence in the covariant
derivatives is Abelian, only the symmetric component of
${\mathcal{M}}^{\mu\nu}$ is truly of second order. Hence we will introduce the
separation of the mass tensor into symmetric and antisymmetric components:

$\begin{split}{\mathcal{M}}^{\mu\nu}_{ab}&=M^{\mu\nu}_{ab}+Q^{\mu\nu}_{ab},\qquad
M^{\mu\nu}_{ab}=M^{\nu\mu}_{ab}=M^{\mu\nu}_{ba},\qquad\\\
Q^{\mu\nu}_{ab}&=-Q^{\nu\mu}_{ab}=-Q^{\mu\nu}_{ba}.\end{split}$ (31)

It can be noted that $M^{\mu\nu}$ must be positive definite and should
dominate $Q^{\mu\nu}$, which is not. The mass term from $Q$ is subdivergent
since it is of first order in the derivatives. Indeed, after integration by
parts,

$\begin{split}&\int
d^{d}x\sqrt{g}\frac{1}{2}Q^{\mu\nu}_{ab}\phi_{\mu}^{a}\phi_{\nu}^{b}=\\\
&\quad\int
d^{d}x\sqrt{g}\left(-\frac{1}{2}Q^{\mu\mu\nu}_{ab}\phi^{a}\phi_{\nu}^{b}-\frac{1}{4}Q^{\mu\nu}_{ac}F^{cb}_{\mu\nu}\phi^{a}\phi^{b}\right).\end{split}$
(32)

Then the (symmetric) fluctuation operator $K$ takes the final form

$K=\begin{pmatrix}-\nabla^{2}g^{\mu\nu}+Y^{\mu\nu}&-\Phi^{\mu}+{\mathcal{M}}^{\mu\alpha}\nabla_{\alpha}\\\
\Phi^{\nu}-\nabla_{\alpha}{\mathcal{M}}^{\alpha\nu}&-\nabla_{M}^{2}+\frac{1}{2}\\{P^{\beta},\nabla_{\beta}\\}+W\end{pmatrix},$
(33)

where we have introduced the following shorthand notation

$\begin{split}Y_{\mu\nu}&={\mathcal{M}}_{\mu\nu}-2F_{\mu\nu}+{\mathcal{R}}_{\mu\nu},\qquad\Phi_{\mu}=F_{\alpha\alpha\mu},\qquad\\\
P_{\mu}&=F_{\alpha\alpha\mu}-Q_{\alpha\alpha\mu},\qquad
W=-\frac{1}{4}\\{Q_{\mu\nu},F_{\mu\nu}\\},\qquad\end{split}$ (34)

as well as

$\nabla_{M}^{2}\equiv\nabla_{\alpha}M^{\alpha\beta}\nabla_{\beta}.$ (35)

As already noted, the field $M^{\mu\nu}(x)$, which was seemingly UV-
subdominant in the original action, is in fact UV-dominant in the sector of
the field $\phi$ in $K$ and acts effectively as a second (inverse) metric. In
the Abelian case ($N=1$ ) such metric is an ordinary one, and even so it
introduced a considerable amount of complication in the calculation of the
effective action in Ruf:2018vzq ; Garcia-Recio:2019iia . In the setting
discussed in this work the “effective metric” $M^{\mu\nu}(x)$ is a non-Abelian
one in gauge space, so we can certainly expect a higher degree of difficulty
in the resources needed to attack this problem.

From the Lagrangian in eq. (29), the effective action to one-loop is thus

$\Gamma_{1}[A;{\mathcal{M}},g]=\Gamma_{K}[A;{\mathcal{M}},g]+\Gamma_{\mathrm{gh}}[A;g]$
(36)

with

$\begin{split}\Gamma_{K}[A;{\mathcal{M}},g]&=\frac{1}{2}\operatorname{Tr}_{1+0}\log(K),\qquad\\\
\Gamma_{\mathrm{gh}}[A;g]&=-\operatorname{Tr}_{0}\log(\nabla^{2}).\end{split}$
(37)

###  IV.3  Contributions to the effective action

As just said the effective action can be split into

$\Gamma_{1}=\Gamma_{K}+\Gamma_{\mathrm{gh}}.$ (38)

The operator $K$ can be split into UV leading $O(\nabla^{2})$ and subdivergent
$O(\nabla)$ components

$\begin{split}K&=K_{L}+K_{S},\qquad\\\
K_{L}&\equiv\operatorname{diag}(-\nabla^{2}g^{\mu\nu},-\nabla^{2}_{M}).\end{split}$
(39)

Correspondingly we also separate the effective action as

$\Gamma_{K}=\frac{1}{2}\operatorname{Tr}_{1+0}(K_{L}(1+K_{L}^{-1}K_{S}))=\Gamma_{L}+\Gamma_{S}$
(40)

with

$\begin{split}\Gamma_{L}&=\frac{1}{2}\operatorname{Tr}_{1+0}\log(K_{L}),\qquad\\\
\Gamma_{S}&=\frac{1}{2}\operatorname{Tr}_{1+0}\log(1+K_{L}^{-1}K_{S}).\end{split}$
(41)

The relation
$\operatorname{Tr}(\log(AB))=\operatorname{Tr}(\log(A))+\operatorname{Tr}(\log(B))$
is only guaranteed for sufficiently convergent pseudo-differential operators
$A$ and $B$. Nevertheless, it is expected to correctly reproduce the UV
divergent terms within dimensional regularization when $A$ and $B$ are both
coordinate-scalar operators.

$\Gamma_{L}$ can be split further as

$\Gamma_{L}=\Gamma_{L,1}+\Gamma_{L,0}$ (42)

with

$\begin{split}\Gamma_{L,1}&=\frac{1}{2}\operatorname{Tr}_{1}\log(-\nabla^{2}g^{\mu\nu}),\qquad\\\
\Gamma_{L,0}&=\frac{1}{2}\operatorname{Tr}_{0}\log(-\nabla^{2}_{M}).\end{split}$
(43)

For the purpose of obtaining the UV divergences of the effective action the
subdivergent terms can be treated perturbatively

$\Gamma_{S}=\frac{1}{2}\operatorname{Tr}_{1+0}\log(1+K_{L}^{-1}K_{S})=\sum_{n=1}^{\infty}\Gamma_{S,n}$
(44)

where

$\Gamma_{S,n}=-\frac{1}{2n}\operatorname{Tr}_{1+0}\left((-K_{L}^{-1}K_{S})^{n}\right).$
(45)

Since $K_{L}^{-1}K_{S}=O(\nabla^{-1})$, terms with $n>d$ are UV finite in $d$
spacetime dimensions. Thus, collecting the various contributions,

$\Gamma^{\mathrm{div}}_{1}=\Gamma^{\mathrm{div}}_{\mathrm{gh}}+\Gamma^{\mathrm{div}}_{L,0}+\Gamma^{\mathrm{div}}_{L,1}+\sum_{n=1}^{d}\Gamma^{\mathrm{div}}_{S,n}\,.$
(46)

The contributions to $\Gamma^{\mathrm{div}}_{1}[A;{\mathcal{M}},g]$ are
analyzed in the following sections, after introducing some notation.

##  V  Some notational conventions

###  V.1  Covariant derivatives

Let us first recall that the covariant derivative operator $\nabla_{\mu}$
contains all connections (and not only the Christoffel symbols), and also our
convention that covariant derivatives are indicated by adding coordinate
indices to the left. E.g.,

$R_{\rho\mu\nu\alpha\beta}\equiv[\nabla_{\rho},R_{\mu\nu\alpha\beta}],\qquad{\mathcal{R}}_{\lambda\lambda\mu}=\frac{1}{2}{\mathbf{R}}_{\mu}.$
(47)

Here $R_{\mu\nu\alpha\beta}$, ${\mathcal{R}}_{\mu\nu}$, and ${\mathbf{R}}$
denote the Riemann tensor, the Ricci tensor and the scalar curvature,
respectively.

All quantities in the fluctuation operator $K$ are to be regarded as operators
acting on the vector space spanned by the fields ${\mathcal{B}}_{\mu}$ and
$\phi$. Hence $\nabla_{\mu}$ acts on such quantities through the commutator.
In particular $g_{\mu\nu}$, ${\mathcal{M}}^{\mu\nu}$ and $F_{\mu\nu}$ are
purely multiplicative operators, which means that they are ordinary functions
(possibly matrices in gauge space).888Of class
${\mathcal{C}}(\underline{\nabla},\underline{Z})$ in the notation of
Salcedo:2006pv .

###  V.2  Operators $Z_{\mu_{1}\cdots\mu_{n}}$

We will make use of the operator $Z_{\mu\nu}$, which is defined as

$Z_{\mu\nu}\equiv[\nabla_{\mu},\nabla_{\nu}].$ (48)

This operator is multiplicative because its action on a quantity does not
involve derivatives of that quantity; $Z_{\mu\nu}$ is diagonal in $x$-space.
However, it is not purely multiplicative because it acts (is not diagonal) on
coordinate indices. For instance, for a purely multiplicative tensor field
$V_{\mu\nu}$

$[Z_{\mu\nu},V_{\alpha\beta}]=R_{\mu\nu\alpha\lambda}V_{\lambda\beta}+R_{\mu\nu\beta\lambda}V_{\alpha\lambda}+[F_{\mu\nu},V_{\alpha\beta}].$
(49)

The operator $Z_{\mu\nu}$ admits a natural separation between coordinate and
gauge actions

$Z_{\mu\nu}=Z^{R}_{\mu\nu}+F_{\mu\nu}$ (50)

where $Z^{R}_{\mu\nu}$ acts only on coordinate indices. As illustrated in (49)
the operator $Z^{R}_{\mu\nu}$ acts on every coordinate index in turn.

Higher order operators $Z_{\mu_{1}\cdots\mu_{n}}$, with $n$ covariant
derivatives, are defined recursively (see App. B) so that they are also
multiplicative. Letting $I=\mu_{1}\cdots\mu_{n}$ denote a string of coordinate
indices, the operators $Z_{I}$ have again a clean separation between
coordinate and gauge

$Z_{I}=Z^{R}_{I}+F_{I},$ (51)

and also fulfill

$[Z^{R}_{I},V_{\mu_{1}\mu_{2}}]=R_{I\mu_{1}\lambda}V_{\lambda\mu_{2}}+R_{I\mu_{2}\lambda}V_{\mu_{1}\lambda}.$
(52)

In fact these operators are an antihermitian version of the derivatives of the
operator $\hat{{\mathcal{R}}}_{\mu\nu}$ of Barvinsky:1985an . Specifically

$Z^{R}_{\alpha_{1}\cdots\alpha_{n}\mu\nu}=\nabla_{\alpha_{1}}\cdots\nabla_{\alpha_{n}}\hat{{\mathcal{R}}}_{\mu\nu}+C_{\alpha_{1}\cdots\alpha_{n}\mu\nu},$
(53)

where the $C_{I}$ are purely multiplicative operators constructed with the
Riemann tensor:

$\begin{split}C_{\mu\nu}&=0,\qquad\\\
C_{\alpha_{1}\cdots\alpha_{n}\mu\nu}&=\frac{1}{2}R_{\lambda\alpha_{2}\alpha_{3}\cdots\alpha_{n}\mu\nu\alpha_{1}\lambda}+\frac{1}{2}R_{\alpha_{1}\lambda\alpha_{3}\cdots\alpha_{n}\mu\nu\alpha_{2}\lambda}+\cdots\\\
&\quad+\frac{1}{2}R_{\alpha_{1}\cdots\alpha_{n-1}\lambda\mu\nu\alpha_{n}\lambda}.\end{split}$
(54)

Eventually we will need to take traces of operators with a factor $Z^{R}_{I}$
on the left. The formulas for the coordinate-scalar and -vector spaces are,
respectively,

$\begin{split}\operatorname{tr}_{0}(Z^{R}_{I}{\mathcal{O}})&=-\operatorname{tr}_{0}(C_{I}{\mathcal{O}}),\qquad\\\
\operatorname{tr}_{1}(Z^{R}_{I}{\mathcal{O}}^{\mu}{}_{\nu})&=\operatorname{tr}_{1}\left((R_{I}{}^{\mu}{}_{\lambda}-g^{\mu}{}_{\lambda}C_{I}){\mathcal{O}}^{\lambda}{}_{\nu}\right).\end{split}$
(55)

Further details and proofs are given in App. B.

###  V.3  Integrals and traces

We will use the shorthand notation

$\langle X\rangle_{x}\equiv\int d^{d}x\,\sqrt{g}\,X,$ (56)

as well as

$\langle X\rangle_{x,g}\equiv\int
d^{d}x\,\sqrt{g}\,\operatorname{tr}_{g}(X)=\langle\operatorname{tr}_{g}(X)\rangle_{x},$
(57)

where $\operatorname{tr}_{g}(~{})$ refers to trace over gauge space. In
particular $\operatorname{tr}_{g}(1)=N$, the dimension of the gauge space is
the number of dynamical real vector fields in the theory.

In addition, for integrals over a momentum variable in Sec. VII,

$\langle
X\rangle_{p}\equiv\frac{1}{\sqrt{g}}\int\frac{d^{d+2\epsilon}p}{(2\pi)^{d}}\,X,$
(58)

where $d+2\epsilon$ refers to dimensional regularization. We also use
combination such as $\langle X\rangle_{x,p}$ for $\langle\langle
X\rangle_{p}\rangle_{x}$, etc.

##  VI  Results for $\Gamma^{\mathrm{div}}_{1}$

###  VI.1  Results for $\Gamma_{\mathrm{gh}}$ and $\Gamma_{L,1}$

The contributions $\Gamma_{\mathrm{gh}}$ and $\Gamma_{L,1}$ to the one-loop
effective action are given by Eqs. (37) and (43), respectively. The
computation of their UV divergent part is straightforward in dimension
regularization, in $d+2\epsilon$ dimensions, using the identity

$\operatorname{Tr}\,\log(-\nabla^{2})\big{|}_{\mathrm{div}}=\frac{1}{(4\pi)^{d/2}}\frac{1}{\epsilon}\int
d^{d}x\,\sqrt{g}\,\operatorname{tr}(b_{d/2}(x)),$ (59)

where the trace is taken in the corresponding space and $b_{n}$ is the $n$-th
heat-kernel coefficient of the Laplacian Vassilevich:2003xt . For $d=2$ and
$d=4$ the required coefficients are

$\begin{split}b_{1}&=\frac{1}{6}{\mathbf{R}},\qquad\\\
b_{2}&=\frac{1}{12}Z^{2}_{\mu\nu}+\frac{1}{180}R_{\mu\nu\alpha\beta}^{2}-\frac{1}{180}{\mathcal{R}}_{\mu\nu}^{2}+\frac{1}{72}{\mathbf{R}}^{2}\,.\end{split}$
(60)

As they stand, these formulas hold for an arbitrary space since $Z_{\mu\nu}$
takes care of all required curvatures (coordinate, gauge or other in more
general cases). For the space of coordinate tensors of rank $r$ in $d$
dimensions (and adjoint gauge representation) one easily finds

$\begin{split}\operatorname{tr}_{r}(Z^{2}_{\mu\nu})&=\operatorname{tr}_{r}(F^{2}_{\mu\nu})+\operatorname{tr}_{r}((Z^{R}_{\mu\nu})^{2})\\\
&=d^{r}\operatorname{tr}_{g}(F^{2}_{\mu\nu})-rd^{r-1}NR_{\mu\nu\alpha\beta}^{2},\end{split}$
(61)

where as already said $\operatorname{tr}_{g}(~{})$ denotes the trace over
gauge space. Of course this result is fully consistent with eq. (55).

Therefore, for $d=2$,

$\begin{split}\Gamma_{\mathrm{gh}}^{\mathrm{div}}&=-\frac{1}{4\pi\,\epsilon}\left\langle\frac{1}{6}\,{\mathbf{R}}\right\rangle_{x,g}\,,\\\
\Gamma_{L,1}^{\mathrm{div}}&=\frac{1}{4\pi\,\epsilon}\left\langle\frac{1}{6}\,{\mathbf{R}}\right\rangle_{x,g}\,.\end{split}$
(62)

These two contributions cancel each other, as they should in $d=2$.

For $d=4$,

$\begin{split}\Gamma_{\mathrm{gh}}^{\mathrm{div}}&=-\frac{1}{(4\pi)^{2}\,\epsilon}\Bigg{\langle}\frac{1}{12}F_{\mu\nu}^{2}\\\
&\quad+\frac{1}{180}R_{\mu\nu\alpha\beta}^{2}-\frac{1}{180}{\mathcal{R}}_{\mu\nu}^{2}+\frac{1}{72}{\mathbf{R}}^{2}\Bigg{\rangle}_{x,g},\\\
\Gamma_{L,1}^{\mathrm{div}}&=\frac{1}{(4\pi)^{2}\,\epsilon}\Bigg{\langle}\frac{1}{6}F_{\mu\nu}^{2}\\\
&\quad+-\frac{11}{360}R_{\mu\nu\alpha\beta}^{2}-\frac{1}{90}{\mathcal{R}}_{\mu\nu}^{2}+\frac{1}{36}{\mathbf{R}}^{2}\Bigg{\rangle}_{x,g}.\end{split}$
(63)

###  VI.2  Results for $\Gamma_{S}$

####  VI.2.1 Contributions to $\Gamma_{S}$

The UV divergent part of $\Gamma_{S}$ in $d$ dimensions is contained in
$\sum_{n=1}^{d}\Gamma_{S,n}$ where $\Gamma_{S,n}$ is given in Eq. (45). The
quantity $K_{L}$, defined in Eq. (39), is homogeneous in $\nabla$ of degree
$+2$ while $K_{S}\equiv K-K_{L}$ contains terms of degrees $0$ and $1$. We
will expand $\Gamma_{S,n}$ in powers of $\nabla$ keeping terms up to
$O(\nabla^{-4})$ which is sufficient for $\Gamma_{1}^{\mathrm{div}}$ in
spacetime dimensions $d\leq 4$. The trace cyclic property can be used to
collect equivalent terms and also we choose whenever possible to bring the
trace to the scalar-coordinate space.

Introducing the notation

$\Delta\equiv\frac{1}{\nabla^{2}},\qquad\Delta_{M}\equiv\frac{1}{\nabla_{M}^{2}},$
(64)

this procedure yields the following expressions

$\begin{split}\Gamma_{S,1}&=\operatorname{Tr}_{1}\Big{(}-\frac{1}{2}\Delta\,Y_{\mu\nu}\Big{)}+\operatorname{Tr}_{0}\Big{(}-\frac{1}{2}\Delta_{M}W-\frac{1}{4}\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}\Big{)}\\\
\Gamma_{S,2}&=\operatorname{Tr}_{1}\Big{(}-\frac{1}{4}\Delta\,Y_{\mu\alpha}\Delta\,Y_{\alpha\nu}\Big{)}+\operatorname{Tr}_{0}\Big{(}\frac{1}{2}\Delta_{M}\Phi_{\mu}\Delta\,\Phi_{\mu}+\frac{1}{2}\Delta_{M}\nabla_{\mu}{\mathcal{M}}_{\mu\nu}\Delta\,{\mathcal{M}}_{\nu\alpha}\nabla_{\alpha}\\\
&-\frac{1}{2}\Delta_{M}\Phi_{\mu}\Delta\,{\mathcal{M}}_{\mu\nu}\nabla_{\nu}-\frac{1}{2}\Delta_{M}\nabla_{\mu}{\mathcal{M}}_{\mu\nu}\Delta\,\Phi_{\nu}\\\
&-\frac{1}{4}\Delta_{M}W\Delta_{M}W-\frac{1}{16}\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}\Delta_{M}\\{P_{\nu},\nabla_{\nu}\\}-\frac{1}{4}\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}\Delta_{M}W\Big{)}\\\
\Gamma_{S,3}&=\operatorname{Tr}_{0}\Big{(}\frac{1}{2}\Delta_{M}\nabla_{\mu}{\mathcal{M}}_{\mu\nu}\Delta\,Y_{\nu\alpha}\Delta\,{\mathcal{M}}_{\alpha\beta}\nabla_{\beta}+\frac{1}{2}\Delta_{M}W\Delta_{M}\nabla_{\mu}{\mathcal{M}}_{\mu\nu}\Delta\,{\mathcal{M}}_{\nu\alpha}\nabla_{\alpha}+\frac{1}{4}\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}\Delta_{M}\nabla_{\nu}{\mathcal{M}}_{\nu\alpha}\Delta\,{\mathcal{M}}_{\alpha\beta}\nabla_{\beta}\\\
&-\frac{1}{4}\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}\Delta_{M}\nabla_{\nu}{\mathcal{M}}_{\nu\alpha}\Delta\,\Phi_{\alpha}-\frac{1}{4}\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}\Delta_{M}\Phi_{\nu}\Delta\,{\mathcal{M}}_{\nu\alpha}\nabla_{\alpha}\\\
&-\frac{1}{8}\Delta_{M}W\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}\Delta_{M}\\{P_{\nu},\nabla_{\nu}\\}-\frac{1}{48}\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}\Delta_{M}\\{P_{\nu},\nabla_{\nu}\\}\Delta_{M}\\{P_{\alpha},\nabla_{\alpha}\\}+O(\nabla^{-5})\Big{)}\\\
\Gamma_{S,4}&=\operatorname{Tr}_{0}\Big{(}-\frac{1}{4}\Delta_{M}\nabla_{\mu}{\mathcal{M}}_{\mu\nu}\Delta\,{\mathcal{M}}_{\nu\alpha}\nabla_{\alpha}\Delta_{M}\nabla_{\beta}{\mathcal{M}}_{\beta\rho}\Delta\,{\mathcal{M}}_{\rho\sigma}\nabla_{\sigma}+\frac{1}{8}\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}\Delta_{M}\\{P_{\nu},\nabla_{\nu}\\}\Delta_{M}\nabla_{\alpha}{\mathcal{M}}_{\alpha\beta}\Delta\,{\mathcal{M}}_{\beta\rho}\nabla_{\rho}\\\
&-\frac{1}{128}\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}\Delta_{M}\\{P_{\nu},\nabla_{\nu}\\}\Delta_{M}\\{P_{\alpha},\nabla_{\alpha}\\}\Delta_{M}\\{P_{\beta},\nabla_{\beta}\\}+O(\nabla^{-5})\Big{)}\end{split}$
(65)

The functional traces are of the form

$\operatorname{Tr}_{r}(A)=\int d^{d}x\sqrt{g}\,\operatorname{tr}_{r}(\langle
x|A|x\rangle),\qquad r=0,1$ (66)

where $d$ is the spacetime dimension and $\operatorname{tr}_{0}$ or
$\operatorname{tr}_{1}$ refer to the trace over coordinate labels, scalar or
vector, respectively, and also include trace over gauge labels. In detail, the
diagonal (in $x$-space) matrix element $\langle x|A|x\rangle$ is of the form
$\langle x,I^{\prime},a|A|x,I,b\rangle$ where $a,b$ are gauge labels in the
adjoint representation and $I,I^{\prime}$ are coordinate labels. For
$\operatorname{Tr}_{0}(~{})$ these coordinate labels are absent while for
$\operatorname{Tr}_{1}(~{})$ they are of the type $I=\mu$, $I^{\prime}=\nu$.

At this point one could already attempt the computation the functional traces
to obtain $\Gamma^{\mathrm{div}}_{S}$. Nevertheless it is convenient to first
simplify the expressions by bringing the operators to a canonical form. This
is in the same spirit as the universal functional traces of Barvinsky:1985an .
The goal is to put together terms involving powers of $\nabla$ (to wit
$\nabla_{\mu}$, $\Delta$ and $\Delta_{M}$) on one side and the purely
multiplicative terms on another. That is, bring the various operators in (65)
to the form

$A=\sum_{n}{\mathcal{O}}_{n}\,A_{n}\,,$ (67)

where ${\mathcal{O}}_{n}$ form a basis of pseudo-differential operators and
$A_{n}$ are purely multiplicative operators. We have chosen to put the latter
on the right-hand side. Thus, for the diagonal matrix elements

$\langle x,I^{\prime},a|A|x,I,b\rangle=\sum_{n,c}\langle
x,I^{\prime},a|{\mathcal{O}}_{n}|x,I,c\rangle\,(A_{n}(x))_{cb},$ (68)

or just $\langle x|A|x\rangle=\sum_{n}\langle
x|{\mathcal{O}}_{n}|x\rangle\,A_{n}(x)$. The coefficients $A_{n}$ do not
modify the UV degree of divergence of the term, which is controlled by
${\mathcal{O}}_{n}$, so the hardest work is computing $\langle
x|{\mathcal{O}}_{n}|x\rangle$.

To obtain the expansion in Eq. (67) we apply the following commutation
identities,

$\begin{split}[\nabla_{\mu},X_{I}]&=X_{\mu I},\\\
[\Delta,X_{I}]&=\Delta\big{(}X_{\mu\mu I}-2\nabla_{\mu}X_{\mu
I}\big{)}\Delta,\\\
[\Delta,\nabla_{\mu}]&=\Delta\big{(}2\nabla_{\nu}Z_{\mu\nu}-\nabla_{\nu}{\mathcal{R}}_{\mu\nu}+Z_{\nu\nu\mu}+\frac{1}{4}{\mathbf{R}}_{\mu}\big{)}\Delta,\\\
[\Delta_{M},\nabla_{\mu}]&=\Delta_{M}\big{(}\nabla_{\alpha}\nabla_{\beta}M_{\mu\alpha\beta}-\nabla_{\alpha}M_{\beta\mu\beta\alpha}+\nabla_{\alpha}M_{\alpha\beta}Z_{\mu\beta}+Z_{\mu\alpha}\nabla_{\beta}M_{\alpha\beta}-Z_{\mu\alpha}M_{\beta\beta\alpha}\big{)}\Delta_{M},\\\
[\Delta,Z^{R}_{I}]&=\Delta\big{(}\nabla_{\mu}R_{\nu
I\mu\nu}-2\nabla_{\mu}Z^{R}_{\mu I}+Z^{R}_{\mu\mu I}\big{)}\Delta,\\\
[\Delta_{M},Z^{R}_{I}]&=\Delta_{M}\left(\nabla_{\mu}M_{\mu\nu}\big{(}R_{\alpha
I\nu\alpha}-2Z^{R}_{\nu I}\big{)}+M_{\mu\nu}\big{(}Z^{R}_{\mu\nu
I}-\frac{1}{2}R_{\mu\alpha I\nu\alpha}-\frac{1}{2}R_{\alpha\mu
I\nu\alpha}\big{)}+M_{\mu\mu\nu}\big{(}Z^{R}_{\nu I}-\frac{1}{2}R_{\alpha
I\nu\alpha}\big{)}\right)\Delta_{M}.\end{split}$ (69)

In these formulas $I=\mu_{1}\cdots\mu_{n}$ represents any (possibly empty)
string of coordinate indices (e.g. $\alpha\beta$) and $\mu I$ the new string
adding $\mu$ to the left (e.g. $\mu\alpha\beta$). $X_{I}$ represents a purely
multiplicative operator, that is, any coordinate tensor (in particular may be
a coordinate scalar) and a matrix in gauge space, not involving $\nabla_{\mu}$
nor $Z_{I}^{R}$.

The usefulness of the commutation relations (69) is that the true degree of UV
divergence of the operator on the left-hand side is actually smaller than the
nominal one. So for instance, $[\Delta,X_{I}]$ would be nominally of
$O(\nabla^{-2})$ while this commutator is actually of order $O(\nabla^{-3})$.
Note that $Z_{I}$ and $Z^{R}_{I}$ count as degree $O(\nabla^{0})$ as they do
not add to the UV degree of divergence of a term, as follows from the property
in Eq. (140).

A very conspicuous and relevant absence in the list of commutators is the
combination $[\Delta_{M},X_{I}]$. This is not in the list because that
commutator is still of $O(\nabla^{-2})$ unless $[X_{I},M_{\mu\nu}]=0$. This
absence prevents from putting together all terms involving $\nabla$ and is a
consequence of the non-Abelian character of the theory.

For $\Gamma_{S}$, the operators required in the basis are

$\displaystyle({\mathcal{O}}_{1})_{\mu}$
$\displaystyle=\Delta_{M}\nabla_{\mu}$ $\displaystyle{\mathcal{O}}_{2}$
$\displaystyle=\Delta_{M}$ (70) $\displaystyle({\mathcal{O}}_{3})_{\mu\nu}$
$\displaystyle=\Delta_{M}\Delta\nabla_{\mu}\nabla_{\nu}$
$\displaystyle({\mathcal{O}}_{4}[X])_{\mu\nu}$
$\displaystyle=\Delta_{M}X\Delta_{M}\nabla_{\mu}\nabla_{\nu}$
$\displaystyle({\mathcal{O}}_{5})_{\mu}$
$\displaystyle=\Delta_{M}\Delta\nabla_{\mu}$
$\displaystyle({\mathcal{O}}_{6})_{\mu\nu\alpha}$
$\displaystyle=\Delta_{M}\Delta^{2}\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}$
$\displaystyle({\mathcal{O}}_{7}[X])_{\mu}$
$\displaystyle=\Delta_{M}X\Delta_{M}\nabla_{\mu}$
$\displaystyle({\mathcal{O}}_{8}[X])_{\mu\nu\alpha}$
$\displaystyle=\Delta_{M}X\Delta_{M}\Delta\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}$
$\displaystyle({\mathcal{O}}_{9}[X,X^{\prime}])_{\mu\nu\alpha}$
$\displaystyle=\Delta_{M}X\Delta_{M}X^{\prime}\Delta_{M}\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}$
$\displaystyle{\mathcal{O}}_{10}$ $\displaystyle=\Delta_{M}\Delta$
$\displaystyle({\mathcal{O}}_{11})_{\mu\nu}$
$\displaystyle=\Delta_{M}\Delta^{2}\nabla_{\mu}\nabla_{\nu}$
$\displaystyle({\mathcal{O}}_{12})_{\mu\nu\alpha\beta}$
$\displaystyle=\Delta_{M}\Delta^{3}\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}\nabla_{\beta}$
$\displaystyle{\mathcal{O}}_{13}[X]$ $\displaystyle=\Delta_{M}X\Delta_{M}$
$\displaystyle({\mathcal{O}}_{14}[X])_{\mu\nu}$
$\displaystyle=\Delta_{M}X\Delta_{M}\Delta\nabla_{\mu}\nabla_{\nu}$
$\displaystyle({\mathcal{O}}_{15}[X])_{\mu\nu\alpha\beta}$
$\displaystyle=\Delta_{M}X\Delta_{M}\Delta^{2}\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}\nabla_{\beta}$
$\displaystyle({\mathcal{O}}_{16}[X,X^{\prime}])_{\mu\nu}$
$\displaystyle=\Delta_{M}X\Delta_{M}X^{\prime}\Delta_{M}\nabla_{\mu}\nabla_{\nu}$
$\displaystyle({\mathcal{O}}_{17}[X,X^{\prime}])_{\mu\nu\alpha\beta}$
$\displaystyle=\Delta_{M}X\Delta_{M}X^{\prime}\Delta_{M}\Delta\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}\nabla_{\beta}$
$\displaystyle({\mathcal{O}}_{18}[X,X^{\prime},X^{\prime\prime}])_{\mu\nu\alpha\beta}$
$\displaystyle=\Delta_{M}X\Delta_{M}X^{\prime}\Delta_{M}X^{\prime\prime}\Delta_{M}\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}\nabla_{\beta}$
$\displaystyle{\mathcal{O}}_{19}$ $\displaystyle=\Delta$
$\displaystyle{\mathcal{O}}_{20}$ $\displaystyle=\Delta^{2}$

Here $X,X^{\prime},X^{\prime\prime}$ are arbitrary purely multiplicative
operators, possibly with coordinate indices. The presence of operators in the
basis with such insertions of purely multiplicative operators is a direct
consequence of $[\Delta_{M},X]$ being of $O(\nabla^{-2})$ in the non-Abelian
case. Nevertheless, the computation of the diagonal matrix elements of the
operators ${\mathcal{O}}_{n}$ can be done for generic
$X,X^{\prime},X^{\prime\prime}$.

The operators ${\mathcal{O}}_{1}$ through ${\mathcal{O}}_{18}$ are required
for the terms $\operatorname{Tr}_{0}(~{})$ of $\Gamma_{S}$ in $d=4$
dimensions, ${\mathcal{O}}_{19}$ and ${\mathcal{O}}_{20}$ appear in the terms
$\operatorname{Tr}_{1}(~{})$. The explicit expansion of the operators in
$\Gamma_{S}$, Eq. (65), in the basis (70) is presented in App. C.

The traces indicated in Eq. (65) can then be obtained from

$\operatorname{Tr}_{r}(A)=\left\langle\sum_{n}\operatorname{tr}_{r}(\langle
x|{\mathcal{O}}_{n}|x\rangle\,A_{n}(x))\right\rangle_{x},\qquad r=0,1\,.$ (71)

The diagonal matrix elements of the operators ${\mathcal{O}}_{n}$ can be
regarded as a generalization of the well-known universal functional traces
introduced in Barvinsky:1985an  and computed there using a Schwinger-DeWitt
technique. The same technique was adapted in Ruf:2018vzq  for the Abelian
case, where $M^{\mu\nu}$ acts effectively as a second metric. Our problem
involves considerably more complicated operators since now such effective
metric is gauge non-Abelian. We will use the method of covariant symbols,
already applied in Garcia-Recio:2019iia  in the Abelian setting. Of course, it
would be interesting to adapt the Schwinger-DeWitt approach to the present
non-Abelian case.

####  VI.2.2 $\Gamma_{S}^{\mathrm{div}}$ in $d=2$

Operator | degree | $-2$ | $-4$ | $-6$
---|---|---|---|---
${\mathcal{O}}_{1}$ | $-1$ | 2 | 265 |
${\mathcal{O}}_{2}$ | $-2$ | 1 | 11 | 3303
${\mathcal{O}}_{3}$ | $-2$ | 1 | 44 |
${\mathcal{O}}_{4}$ | $-2$ | 1 | 123 |
${\mathcal{O}}_{5,6}$ | $-3$ |  | 2 |
${\mathcal{O}}_{7,8}$ | $-3$ |  | 6 |
${\mathcal{O}}_{9}$ | $-3$ |  | 12 |
${\mathcal{O}}_{10\,\text{-}18}$ | $-4$ |  | 1 |
${\mathcal{O}}_{19}$ | $-2$ | 1 | 1 |
${\mathcal{O}}_{20}$ | $-4$ |  | 1 |
Table 1: For the various operators in Eq. (70), the second column displays the
UV divergence degree, while the columns labeled with $-2,-4,-6$, show the
number of terms of each degree in the expansion of the diagonal matrix element
of the operator.

Details of the calculation of the divergent part of the diagonal matrix
elements will be given in Sec. VII. However, from the data displayed in Table
1, we already anticipate that the number of terms contributing to
$\Gamma_{S}^{\mathrm{div}}$ is very large for $d=4$ (namely, $265$ terms from
${\mathcal{O}}_{1}$, $123$ from ${\mathcal{O}}_{4}$, etc). In view of this we
will only display explicit results for $d=2$.

In $d=2$, Eq. (65) reduces to

$\begin{split}\Gamma_{S,1}&=\operatorname{Tr}_{1}\Big{(}-\frac{1}{2}\Delta\,Y_{\mu\nu}\Big{)}+\operatorname{Tr}_{0}\Big{(}-\frac{1}{2}\Delta_{M}W-\frac{1}{4}\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}\Big{)}\\\
\Gamma_{S,2}&=\operatorname{Tr}_{0}\Big{(}\frac{1}{2}\Delta_{M}\nabla_{\mu}{\mathcal{M}}_{\mu\nu}\Delta\,{\mathcal{M}}_{\nu\alpha}\nabla_{\alpha}\\\
&\quad-\frac{1}{16}\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}\Delta_{M}\\{P_{\nu},\nabla_{\nu}\\}+O(\nabla^{-3})\Big{)}\end{split}$
(72)

This can be rewritten using the basis of operators ${\mathcal{O}}_{n}$ in Eq.
(70) (see App. C)

$\begin{split}\Gamma_{S,1}&=\operatorname{Tr}_{1}\Big{(}-\frac{1}{2}{\mathcal{O}}_{19}\,Y_{\mu\nu}\Big{)}+\operatorname{Tr}_{0}\Big{(}-\frac{1}{2}{\mathcal{O}}_{2}W-\frac{1}{2}({\mathcal{O}}_{1})_{\mu}P_{\mu}\\\
&\quad+\frac{1}{4}{\mathcal{O}}_{2}\,P_{\mu\mu}+O(\nabla^{-3})\Big{)}\\\
\Gamma_{S,2}&=\operatorname{Tr}_{0}\Big{(}\frac{1}{2}({\mathcal{O}}_{3})_{\mu\nu}{\mathcal{M}}_{\mu\alpha}{\mathcal{M}}_{\alpha\nu}-\frac{1}{4}({\mathcal{O}}_{4}[P_{\mu}])_{\mu\nu}\,P_{\nu}+O(\nabla^{-3})\Big{)}\end{split}$
(73)

Details of the calculation are given in Sec. VII.2. The result is999Note that
$Y_{\mu\mu}=M_{\mu\mu}+{\mathbf{R}},\quad
P_{\mu\mu}=\frac{1}{2}[F_{\mu\nu},Q_{\mu\nu}]$.

$\begin{split}\Gamma_{S,1}^{\mathrm{div}}&=\frac{1}{4\pi\epsilon}\Big{\langle}-\frac{1}{2}Y_{\mu\mu}-\frac{1}{2}\hat{N}_{M}W+\frac{1}{4}\hat{N}_{M}P_{\mu\mu}\\\
&\quad-\frac{1}{2}\hat{N}_{M}M_{\mu\nu}\hat{N}_{M}M_{\mu\alpha\beta}\hat{N}_{M}P_{\lambda}\hat{k}_{\nu}\hat{k}_{\alpha}\hat{k}_{\beta}\hat{k}_{\lambda}\\\
&\quad+\frac{1}{2}\hat{N}_{M}M_{\mu\alpha\beta}\hat{N}_{M}M_{\mu\nu}\hat{N}_{M}P_{\lambda}\hat{k}_{\nu}\hat{k}_{\alpha}\hat{k}_{\beta}\hat{k}_{\lambda}\Big{\rangle}_{x,\,g,\,\mathrm{ang},},\\\
\Gamma_{S,2}^{\mathrm{div}}&=\frac{1}{4\pi\epsilon}\Big{\langle}\frac{1}{2}\hat{N}_{M}{\mathcal{M}}_{\mu\alpha}{\mathcal{M}}_{\alpha\nu}\,\hat{k}_{\mu}\hat{k}_{\nu}-\frac{1}{4}\hat{N}_{M}P_{\mu}\hat{N}_{M}P_{\nu}\hat{k}_{\mu}\hat{k}_{\nu}\Big{\rangle}_{x,\,g,\,\mathrm{ang}}\,.\end{split}$
(74)

Here $\hat{k}_{\mu}$ is a normalized momentum variable,
$g^{\mu\nu}\hat{k}_{\mu}\hat{k}_{\nu}=1$, whereas

$\hat{N}_{M}\equiv(\hat{k}_{\mu}\hat{k}_{\nu}M^{\mu\nu})^{-1}\,.$ (75)

The symbol $\langle~{}\rangle_{\mathrm{ang}}$ denotes angular average over
$\hat{k}_{\mu}$,

$\langle X\rangle_{\mathrm{ang}}\equiv\frac{\Gamma(d/2)}{2\pi^{d/2}}\int
d^{d-1}\Omega_{\hat{k}}\,X\,.$ (76)

This average is to be applied together with $\langle~{}\rangle_{x,g}$, already
introduced in Sec. V.3.

Eq. (74) is our final result for $\Gamma^{\mathrm{div}}_{S}$ in $d=2$. The
expression for $d=4$ is qualitatively similar, but considerably longer.
Further perturbative results are given in App. D.

While the angular averages in Eq. (74) cannot be evaluated in closed form in
general, they are perfectly convergent and well-defined. In any case, these
integrals introduce a wild local but non polynomial dependence on the field
$M^{\mu\nu}_{ab}(x)$ in the divergent part of the effective action, implying
that the UV divergences cannot be removed by polynomial counterterms,
rendering the theory not renormalizable in a standard sense. This was true
already when $N=1$, the Abelian case studied in Toms:2015fja ;
Buchbinder:2017zaa ; Ruf:2018vzq ; Garcia-Recio:2019iia .

###  VI.3  Results for $\Gamma_{L,0}$

####  VI.3.1  Preliminaries

The remaining contribution to the effective action is $\Gamma_{L,0}$,

$\Gamma_{L,0}=\frac{1}{2}\operatorname{Tr}_{0}\log(-\nabla_{M}^{2})=\frac{1}{2}\operatorname{Tr}_{0}\log(-\nabla_{\alpha}M^{\alpha\beta}\nabla_{\beta}).$
(77)

This is the effective action of a scalar field with action

$S_{L,0}=\int
d^{d}x\,\sqrt{g}\,\frac{1}{2}\phi_{\mu}^{a}\,M^{\mu\nu}_{ab}\phi^{b}_{\nu}.$
(78)

In the special case of $N=1$, that is, when there is no gauge sector (the case
considered in Toms:2015fja ; Buchbinder:2017zaa ; Ruf:2018vzq ; Garcia-
Recio:2019iia ) one can combine $g_{\mu\nu}$ and $M^{\alpha\beta}$ into a new
metric $\tilde{g}_{\mu\nu}$, namely,
$\sqrt{g}\,M^{\mu\nu}=\sqrt{\tilde{g}}\,\tilde{g}^{\mu\nu}$ deRham:2014wfa ;
Ruf:2018vzq , in such a way that

$S_{L,0}=\int
d^{d}x\,\sqrt{\tilde{g}}\,\frac{1}{2}\phi_{\mu}\,\tilde{g}^{\mu\nu}\phi_{\nu}\qquad(N=1).$
(79)

Hence $\Gamma_{L,0}=\frac{1}{2}\operatorname{Tr}_{0}\log(-\tilde{\nabla}^{2})$
and the heat-kernel result (59) immediately applies. Note that this method
works in any spacetime dimension except $d=2$, since in that case
$\det(\sqrt{\tilde{g}}\tilde{g}_{\mu\nu})\equiv 1$.

Unfortunately no such simplification takes place for $N>1$. In fact the
situation is even worse, namely, $\Gamma^{\mathrm{div}}_{L,0}$ does not admit
a standard form, like that displayed in Eq. (112). By definition, we say that
the terms in an expansion of a diagonal matrix element $\langle
x|{\mathcal{O}}|x\rangle$ adopt a standard form when all the pieces are
covariant with no “free” $\nabla_{\mu}$ operators (all $\nabla_{\mu}$ are in
the form $[\nabla_{\mu},~{}]$) and only remains to carry out a momentum
integration. Also labeled operators Salcedo:2004yh  are not permitted in a
standard form. The momentum integration can be traded by another parameter,
e.g. a proper time as in a Schwinger parametrization.

While a standard form can always be achieved for matrix elements of the type
$\langle x|f(\nabla,X)|x\rangle$ ($X$ being purely multiplicative operators)
when the dependence on $\nabla_{\mu}$ is of rational type, this is not
guaranteed for more general functions $f$. However, it is often the case that
a standard form exist for a pseudo-differential operator of the type
$\log(\hat{f})$ with $\hat{f}$ a differential operator. A well-known instance
of standard form for a $\operatorname{Tr}\log$ is that found by Chan (for flat
spacetime to fourth order in a derivative expansion) Chan:1986jq . To second
order

$\begin{split}\operatorname{Tr}\log(-\nabla_{\mu}^{2}+X)&=\\\
&\left\langle\int\frac{d^{d}k}{(2\pi)^{d}}\left(-\log(N_{C})+\frac{k^{2}}{d}[\nabla_{\mu},N_{C}]^{2}+O(\nabla^{4})\right)\right\rangle_{x,g},\qquad\\\
N_{C}&\equiv(k^{2}+X)^{-1}.\end{split}$ (80)

Extensions of this formula exist to sixth order Caro:1993fs , also for curved
spacetime Salcedo:2007bt  and for finite temperature Moral-Gamez:2011wcb . Yet
the techniques applied in those cases cannot be translated to evaluating
$\Gamma_{L,0}$ because in the present case what plays the role a metric,
$M^{\mu\nu}$, is actually a non-trivial matrix in gauge space and appears
already in the leading term $\log(k_{\mu}k_{\nu}M^{\mu\nu})$. The obstruction
to a standard form does not depend on the method used to compute the effective
action, it is intrinsic to $\Gamma_{L,0}$ in the non-Abelian case.

####  VI.3.2 Special case of separable mass term

Here we make an apart to mention the special case of a mass term separable in
coordinate and gauge spaces, that is,

$M^{\mu\nu}_{ab}(x)=T_{ab}(x)G^{\mu\nu}(x).$ (81)

The fluctuation operator can then be written as

$\begin{split}\nabla_{\alpha}M^{\alpha\beta}\nabla_{\beta}&=\nabla_{\alpha}TG^{\alpha\beta}\nabla_{\beta}=T\nabla_{\alpha}G^{\alpha\beta}\nabla_{\beta}+T_{\alpha}G^{\alpha\beta}\nabla_{\beta}\\\
&=T(\nabla_{G}^{2}+L^{\alpha}\,\nabla_{\alpha})\end{split}$ (82)

where we have defined

$\nabla_{G}^{2}\equiv\nabla_{\alpha}G^{\alpha\beta}\nabla_{\beta},\qquad
L^{\mu}\equiv T^{-1}T_{\nu}\,G^{\nu\mu}.$ (83)

Now, the factor $T(x)$ can be dropped from
$\operatorname{Tr}(\log(-\nabla_{M}^{2}))$, being ultralocal (carries no
derivative operators) hence

$\begin{split}\Gamma_{L,0}&=\frac{1}{2}\operatorname{Tr}_{0}\log(-\nabla_{G}^{2}-L^{\alpha}\nabla_{\alpha})\\\
&=\frac{1}{2}\operatorname{Tr}_{0}\log(-\nabla_{G}^{2})+\frac{1}{2}\operatorname{Tr}_{0}\log(1+\Delta_{G}\,L^{\alpha}\nabla_{\alpha})\end{split}$
(84)

with $\Delta_{G}=1/\nabla_{G}^{2}$. The first term is just like $\Gamma_{L,0}$
of the Abelian case $N=1$ (with $G^{\mu\nu}$ instead of $M^{\mu\nu}$). The
second term can be expanded in powers of
$\Delta_{G}\,L^{\alpha}\nabla_{\alpha}$, similarly as done for $\Gamma_{S}$ in
(44). So this contribution does admit a standard form.

It is noteworthy that there is an ambiguity in the separation (81) of Weyl-
transformation type, namely,

$G^{\mu\nu}(x)\to\lambda(x)G^{\mu\nu}(x),\qquad
T_{ab}(x)\to\lambda^{-1}(x)T_{ab}(x),$ (85)

where $\lambda(x)$ is local but both scalar coordinate and gauge singlet. Such
ambiguity can be used to fix some gauge condition on $G^{\mu\nu}(x)$ or
$T_{ab}(x)$, or as a check of the calculation, since the sum of the two terms
in (84) should be $\lambda$-independent.

We do not pursue the subject of the separable mass case any further in this
work.

####  VI.3.3 Method of contour integration

Coming back to the case of a general mass term $M^{\mu\nu}_{ab}(x)$, to deal
with the logarithm we follow here the approach of introducing a parametric
integral. This is based on the observation that a standard form would easily
follow for a rational function instead of the $\log$.

The contour integration method is based on the identity

$\Gamma_{L,0}=\int_{\gamma}\frac{dz}{2\pi
i}\,\log(z)\,\frac{1}{2}\operatorname{Tr}_{0}\left(\frac{1}{z+\nabla_{M}^{2}}\right)$
(86)

where the path $\gamma$ is meant to enclose counterclockwise the spectrum of
$-\nabla_{M}^{2}$. More precisely, the path $\gamma$ on the $z$ complex plane
starts at $-\infty$ towards the origin just above the negative real semiaxis,
encircles $z=0$ clockwise and goes back to $-\infty$ just below the negative
real semiaxis.101010The $\zeta$-function result would follows from
understanding $\log(z)$ as $-\frac{dz^{-s}}{ds}$ at $s=0$ in the sense of
analytical continuation in the $s$ variable, taking the derivative with
respect to $s$ only after the $\operatorname{Tr}(~{})$ has been computed
Seeley:1967ea . Hence, introducing a convenient notation,

$\langle X\rangle_{z}\equiv\int_{\gamma}\frac{dz}{2\pi
i}\,\log(z)\,X,\qquad{\mathcal{O}}_{2}^{\prime}\equiv\frac{1}{z+\nabla_{M}^{2}},$
(87)

we can express Eq. (86) as

$\Gamma_{L,0}=\left\langle\frac{1}{2}\operatorname{Tr}_{0}({\mathcal{O}}_{2}^{\prime})\right\rangle_{z}.$
(88)

The calculation of the diagonal matrix element $\langle
x|{\mathcal{O}}_{2}^{\prime}|x\rangle$ is formally identical to that of
$\langle x|{\mathcal{O}}_{2}|x\rangle$, unfortunately the presence of the
$z$-integration implies that relevant quantity
$\langle{\mathcal{O}}_{2}^{\prime}\rangle_{z}=\log(-\nabla_{M}^{2})$ is
actually of degree $O(\nabla^{0})$ instead of $O(\nabla^{-2})$. Hence, the
amount of work required to evaluate $\langle
x|\langle{\mathcal{O}}_{2}^{\prime}\rangle_{z}|x\rangle$, is similar to that
of $\langle x|{\mathcal{O}}_{2}|x\rangle$ for $d+2$ dimensions to achieve the
same degree of UV divergence (see Sec. VII.3).

The convergence problem would not improve if instead of a $z$-integration the
sought-for rational dependence on $\nabla_{\mu}$ were obtained through the
variation of $\Gamma_{L,0}$ under a deformation of $M^{\mu\nu}$,

$\delta\Gamma_{L,0}=\frac{1}{2}\operatorname{Tr}_{0}\left(\Delta_{M}\nabla_{\mu}\delta
M^{\mu\nu}\nabla_{\nu}\right),$ (89)

as this requires to evaluate $\langle
x|\Delta_{M}\nabla_{\mu}\nabla_{\nu}|x\rangle$ which is still of
$O(\nabla^{0})$.

####  VI.3.4 $\Gamma_{L,0}$ in $d=2$

Once again the number of terms is prohibitively large in $d=4$ (it requires
${\mathcal{O}}_{2}$ to $O(p^{-6})$ in Table 1) and we present results for
$d=2$. The calculation gives

$\begin{split}&\Gamma_{L,0}^{\mathrm{div}}=\frac{1}{4\pi\epsilon}\Big{\langle}\\\
&\quad-\hat{N}_{M}\,M_{\mu\nu}\,\hat{N}_{M}\,M_{\mu\alpha\beta}\,\hat{N}_{M}\,M_{\rho\sigma}\,\hat{N}_{M}\,M_{\rho\lambda\eta}\,\hat{N}_{M}\,\hat{k}_{\nu}\,\hat{k}_{\alpha}\,\hat{k}_{\beta}\,\hat{k}_{\sigma}\,\hat{k}_{\lambda}\,\hat{k}_{\eta}\\\
&\quad-\hat{N}_{M}\,M_{\mu\nu\alpha}\,\hat{N}_{M}\,M_{\mu\beta}\,\hat{N}_{M}\,M_{\rho\sigma\lambda}\,\hat{N}_{M}\,M_{\rho\eta}\,\hat{N}_{M}\,\hat{k}_{\nu}\,\hat{k}_{\alpha}\,\hat{k}_{\beta}\,\hat{k}_{\sigma}\,\hat{k}_{\lambda}\,\hat{k}_{\eta}\\\
&\quad+\frac{1}{2}\hat{N}_{M}\,M_{\mu\nu}\,\hat{N}_{M}\,M_{\mu\alpha\beta}\,\hat{N}_{M}\,M_{\alpha\rho\sigma}\,\hat{N}_{M}\,\hat{k}_{\nu}\,\hat{k}_{\beta}\,\hat{k}_{\rho}\,\hat{k}_{\sigma}\\\
&\quad-\frac{1}{2}\hat{N}_{M}\,M_{\mu\nu}\,\hat{N}_{M}\,M_{\alpha\alpha\beta}\,\hat{N}_{M}\,M_{\mu\rho\sigma}\,\hat{N}_{M}\,\hat{k}_{\nu}\,\hat{k}_{\beta}\,\hat{k}_{\rho}\,\hat{k}_{\sigma}\\\
&\quad+\frac{1}{2}\hat{N}_{M}\,M_{\mu\nu\alpha}\,\hat{N}_{M}\,M_{\beta\mu\rho}\,\hat{N}_{M}\,M_{\beta\sigma}\,\hat{N}_{M}\,\hat{k}_{\nu}\,\hat{k}_{\alpha}\,\hat{k}_{\rho}\,\hat{k}_{\sigma}\\\
&\quad-\frac{1}{2}\hat{N}_{M}\,M_{\mu\nu\alpha}\,\hat{N}_{M}\,M_{\beta\beta\rho}\,\hat{N}_{M}\,M_{\mu\sigma}\,\hat{N}_{M}\,\hat{k}_{\nu}\,\hat{k}_{\alpha}\,\hat{k}_{\rho}\,\hat{k}_{\sigma}\\\
&\quad-\frac{1}{2}\hat{N}_{M}\,M_{\mu\nu}\,\hat{N}_{M}\,M_{\alpha\beta}\,\hat{N}_{M}\,M_{\mu\rho}\,\hat{N}_{M}\,{\mathbf{R}}\,\hat{k}_{\nu}\,\hat{k}_{\alpha}\,\hat{k}_{\beta}\,\hat{k}_{\rho}\\\
&\quad+\frac{1}{2}\hat{N}_{M}\,M_{\mu\mu\nu}\,\hat{N}_{M}\,M_{\alpha\alpha\beta}\,\hat{N}_{M}\,\hat{k}_{\nu}\,\hat{k}_{\beta}\\\
&\quad-\frac{1}{2}\hat{N}_{M}\,M_{\mu\nu}\,F_{\mu\alpha}\,\hat{N}_{M}\,M_{\alpha\beta}\,\hat{N}_{M}\,\hat{k}_{\nu}\,\hat{k}_{\beta}\\\
&\quad-\frac{1}{2}\hat{N}_{M}\,M_{\mu\nu}\,\hat{N}_{M}\,F_{\mu\alpha}\,M_{\alpha\beta}\,\hat{N}_{M}\,\hat{k}_{\nu}\,\hat{k}_{\beta}\\\
&\quad+\frac{1}{3}\hat{N}_{M}\,M_{\mu\nu}\,\hat{N}_{M}\,M_{\mu\alpha}\,\hat{N}_{M}\,{\mathbf{R}}\,\hat{k}_{\nu}\,\hat{k}_{\alpha}\\\
&\quad+\frac{1}{24}\hat{N}_{M}\,M_{\mu\mu}\,\hat{N}_{M}\,{\mathbf{R}}\Big{\rangle}_{x,g,\mathrm{ang},z}\end{split}$
(90)

In this formula
$\hat{N}_{M}\equiv(-z+\hat{k}_{\mu}\hat{k}_{\nu}M^{\mu\nu})^{-1}$. Besides
integration over $x$, trace over gauge space and angular average, an integral
over $z$, as defined in Eq. (87), is applied.

Integration by parts with respect to $x$ has been used to have at most one
covariant derivative on $M_{\mu\nu}$. Also the following two-dimensional
identities have been used:

$\begin{split}R_{\mu\nu\alpha\beta}&=\frac{1}{2}(g_{\mu\alpha}\,g_{\nu\beta}-g_{\mu\beta}\,g_{\nu\alpha}){\mathbf{R}},\qquad\\\
{\mathcal{R}}_{\mu\nu}&=\frac{1}{2}g_{\mu\nu}{\mathbf{R}}\,.\end{split}$ (91)

In addition integration by parts identities in momentum space have been
applied to bring the expression to a manifestly Hermitian form.

Perturbative expansions are presented in App. D.

####  VI.3.5 Verification of metric-related symmetries in $\Gamma_{L,0}$

The Weyl-like transformation noted at the end of Sec. II,
$g_{\mu\nu}(x)\to\xi(x)g_{\mu\nu}(x)$,
${\mathcal{M}}^{\mu\nu}_{ab}(x)\to\xi^{-2}(x){\mathcal{M}}^{\mu\nu}_{ab}(x)$,
is a symmetry of the full action only in $d=4$. However, in any spacetime
dimension, $\Gamma_{L,0}$ has a large symmetry as this term only depends on
the pair $(g_{\mu\nu},M^{\mu\nu})$ through the combination
$\sqrt{g}M^{\mu\nu}_{ab}$. This follows from (78) since $\phi^{a}$ are
coordinate scalars and so $\phi_{\mu}^{a}$ does not depend on the metric
tensor.

It is convenient to distinguish two types of symmetry transformations leaving
invariant $\Gamma_{L,0}$, which will be called transverse and longitudinal,
respectively:

  * $i)$

A transverse transformation corresponds to leaving $M^{\mu\nu}$ intact, while
$g_{\mu\nu}$ changes arbitrarily but keeping $\sqrt{g}$ invariant, thus
preserving the volume element.

  * $ii)$

A longitudinal transformations correspond to the simultaneous change

$\begin{split}g_{\mu\nu}(x)&\to\xi(x)g_{\mu\nu}(x),\\\
M^{\mu\nu}_{ab}(x)&\to\xi^{-d/2}(x)M^{\mu\nu}_{ab}(x),\end{split}$ (92)

in $d$ spacetime dimensions.

We have checked that both symmetries are fulfilled by the expression in Eq.
(90). It is sufficient to consider the infinitesimal case.111111Here we need
to refer to results to be established in Sec. VII.3. For transverse
transformations the first order variations are

$\begin{split}\delta
g_{\mu\nu}&=\omega_{\mu\nu}\quad\text{with}\quad\omega^{\lambda}{}_{\lambda}=0,\\\
\delta M^{\mu\nu}&=\delta p_{\mu}=\delta F_{\mu\nu}=\delta N_{M}=0,\\\ \delta
M_{\alpha}{}^{\mu\nu}&=\frac{1}{2}(\omega_{\alpha}{}^{\mu}{}_{\lambda}+\omega_{\lambda}{}^{\mu}{}_{\alpha}-\omega^{\mu}{}_{\alpha\lambda})M^{\lambda\nu}\\\
&\quad+\frac{1}{2}(\omega_{\alpha}{}^{\nu}{}_{\lambda}+\omega_{\lambda}{}^{\nu}{}_{\alpha}-\omega^{\nu}{}_{\alpha\lambda})M^{\mu\lambda},\\\
\delta(g_{\mu\nu}{\mathbf{R}})&=\omega^{\lambda}{}_{\mu\lambda\nu}+\omega^{\lambda}{}_{\nu\lambda\mu}-\omega^{\lambda}{}_{\lambda\mu\nu}\quad(d=2).\end{split}$
(93)

It should be noted that the metric does not appear explicitly anywhere in Eq.
(115), and the metric is only contained in $R_{\mu\nu}{}^{\alpha}{}_{\beta}$,
${\mathcal{R}}_{\mu\nu}$ and the covariant derivatives. This is because the
same is true for the covariant symbols of $\nabla_{\mu}$ and the
multiplicative operators, and no explicit metric appears in the original
expression of ${\mathcal{O}}_{2}^{\prime}$. After using the two-dimensional
identities (91) the metric does enter in Eq. (116) in the combination
$g_{\mu\nu}{\mathbf{R}}$. In $d=2$ the variation of this quantity depends only
on derivatives of $\omega_{\mu\nu}$. Also note that the integral
$\langle~{}\rangle_{x,p}$ does not contain a net $\sqrt{g}$. When the above
first order variations are applied to (116), and after integration by parts in
$x$-space to remove terms with $\omega_{\mu\nu\alpha\beta}$ (two derivatives),
one obtains an expression which vanishes upon integration by parts in
$p$-space. We have checked this to third order in an expansion of the type
$M^{\mu\nu}=m^{2}g^{\mu\nu}+H^{\mu\nu}$ in powers of $H^{\mu\nu}$. The
integration over $z$ plays no role in the cancellation of terms.

For first order longitudinal variations one has instead (in $d=2$)

$\begin{split}\delta g_{\mu\nu}&=wg_{\mu\nu},\quad\delta p_{\mu}=\delta
F_{\mu\nu}=0,\quad\delta M^{\mu\nu}=-wM^{\mu\nu},\\\ \delta
M_{\alpha}{}^{\mu\nu}&=-wM_{\alpha}{}^{\mu\nu}+\frac{1}{2}w_{\lambda}g^{\mu}{}_{\alpha}M^{\lambda\nu}+\frac{1}{2}w_{\lambda}g^{\nu}{}_{\alpha}M^{\mu\lambda}\\\
&\quad-\frac{1}{2}w^{\mu}M_{\alpha}{}^{\nu}-\frac{1}{2}w^{\nu}M^{\mu}{}_{\alpha},\\\
\delta(g_{\mu\nu}{\mathbf{R}})&=-w^{\lambda}{}_{\lambda}.\end{split}$ (94)

The terms with $w$ without a covariant derivative correspond to global
transformations. It is easy to verify that (116) remains invariant under
global longitudinal transformations by applying the simultaneous rescaling
$z\to\xi^{-1}z$ (which in turn implies $N_{M}\to\xi N_{M}$ or $\delta
N_{M}=wN_{M}$). Therefore it is only necessary to keep terms with $w_{\mu}$
and $w_{\mu\nu}$ in the variation of (116). After variation and integration by
parts in $x$ to remove terms with $w_{\mu\nu}$ (two derivatives), one finds
that the result cancels upon integration over $p$. Once again we have checked
this through third order in an expansion in powers of $H^{\mu\nu}$.

The cancellation of transverse and longitudinal variations of the effective
action is a highly non trivial check of Eq. (90).

##  VII  Computation of the diagonal matrix elements

###  VII.1  Method of covariant symbols

The diagonal matrix elements can be computed using the method of covariant
symbols. This method was introduced in Pletnev:1998yu  for flat spacetime and
extended to curved spacetime in Salcedo:2006pv  where it is described in great
detail. It has also been extended to finite temperature in Moral-Gamez:2011wcb
; Brauner:2011vb . A summary can be found in Garcia-Recio:2019iia  (Sec. 3.4
and App. B). Nevertheless, as the method is not widely known, and to have a
more self-contained work, we give some details here.

For an operator $\hat{\mathcal{O}}$, constructed with $\nabla_{\mu}$ plus some
purely multiplicative fields, such those in the basis (70), its covariant
symbol will be denoted $\overline{{\mathcal{O}}}$. This quantity is obtained
by applying two successive similarity transformations121212This formula is
schematic, see full construction in Salcedo:2006pv .

$\overline{{\mathcal{O}}}:=e^{-\frac{1}{2}\\{\nabla_{\mu},\partial^{\mu}\\}}e^{-\xi^{\alpha}p_{\alpha}}\,\hat{{\mathcal{O}}}\,e^{\xi^{\beta}p_{\beta}}e^{\frac{1}{2}\\{\nabla_{\nu},\partial^{\nu}\\}}\big{|}_{\xi^{\mu}=0}\,.$
(95)

Here $\\{,\\}$ denotes anticommutator, $\xi^{\mu}$ are the Riemann coordinates
corresponding to the affine connection in $\nabla_{\mu}$ located at the point
$x$ where the diagonal matrix element $\langle x|\hat{{\mathcal{O}}}|x\rangle$
will be evaluated. $p_{\mu}$ is a momentum variable and
$\partial^{\mu}\equiv\partial/\partial p_{\mu}$. For convenience, we use a
purely imaginary variable, $p_{\mu}=ik_{\mu}$, $k\in\mathbb{R}^{d}$ but
$d^{d}p\equiv d^{d}k$. This definition of covariant symbol holds for
$\nabla_{\mu}$ having any affine connections (e.g., with torsion); here we
assume the Levi-Civita connection for the coordinate indices, plus the gauge
connection.

The original operator $\hat{\mathcal{O}}$ acts only in $x$-space (and possibly
in some internal space) while its covariant symbol $\overline{{\mathcal{O}}}$
acts both on $x$\- and $p$-spaces. The first remarkable property of
$\overline{{\mathcal{O}}}$ is that this operator is multiplicative with
respect to $x$ (although not with respect to $p$). This means that it commutes
with functions of $x$ which are coordinate-scalar and gauge-singlet.

The second important property of the map
$\hat{{\mathcal{O}}}\mapsto\overline{{\mathcal{O}}}$ is that, being a
similarity transformation, it is a faithful algebra homomorphism (which also
preserves hermiticity). Therefore, the covariant symbol of a pseudo-
differential operator of the type $f(\nabla_{\mu},X)$ is simply
$f(\overline{\nabla}_{\mu},\overline{X})$. This implies that it is sufficient
to obtain (once and for all) the covariant symbol of a few basic blocks and
once this is done there is no need to go back to the original definition in
Eq. (95). For instance,

$\begin{split}\overline{\nabla}_{\mu}&=p_{\mu}-\frac{1}{4}\\{Z_{\alpha\mu},\partial^{\alpha}\\}+\frac{1}{12}\\{[Z_{\alpha\mu},p_{\beta}],\partial^{\alpha}\partial^{\beta}\\}+O(p^{-2}),\\\
\overline{X}&=X-X_{\alpha}\partial^{\alpha}+\frac{1}{2!}X_{\alpha\beta}\partial^{\alpha}\partial^{\beta}+O(p^{-3}),\\\
\overline{Z}_{\mu\nu}&=Z_{\mu\nu}-\frac{1}{2}\\{Z_{\alpha\mu\nu},\partial^{\alpha}\\}+O(p^{-2}).\end{split}$
(96)

Extensive formulas can be found in Salcedo:2006pv ; Garcia-Recio:2019iia . As
advertised, explicit $\nabla_{\mu}$ are no longer present and the covariant
symbol is a multiplicative operator. The expressions take the form of an
expansion in powers of $\nabla/p$, i.e., with terms ordered by the number of
covariant derivatives or equivalently by powers of $p_{\mu}$ (counting
$\partial^{\mu}$ as $O(p^{-1})$). Here $X$ is any purely multiplicative
operator (i.e., not containing $\nabla_{\mu}$ nor $Z^{R}_{I}$), so in
particular

$\overline{M}_{\mu\nu}=M_{\mu\nu}-M_{\alpha\mu\nu}\partial^{\alpha}+\frac{1}{2!}M_{\alpha\beta\mu\nu}\partial^{\alpha}\partial^{\beta}+O(p^{-3}).$
(97)

And of course,
$\overline{Z}_{\mu\nu}=[\overline{\nabla}_{\mu},\overline{\nabla}_{\nu}]$ is
verified.

It might seem that $Z_{\mu\nu}$ or $Z_{\alpha\mu\nu}$ would commute with
$p_{\lambda}$ or $\partial^{\lambda}$ in Eq. (96), but in fact this is not so.
Since $p_{\mu}$ and $\partial^{\mu}$ carry coordinate indices, Eq. (140)
applies and one has instead

$[Z_{\mu\nu},p_{\alpha}]=R_{\mu\nu\alpha\beta}p_{\beta},\qquad[Z_{\mu\nu},\partial^{\alpha}]=R_{\mu\nu\alpha\beta}\partial^{\beta}\,.$
(98)

The Eqs. in (96) are written so that the hermiticity properties are manifest.
Note that the metric does not appear in the covariant symbols of
$\nabla_{\mu}$ or $X$, in fact those formulas hold for a completely arbitrary
torsionless connection.131313In this sense $\overline{\nabla}_{\mu}$ or
$\overline{X}$ are truly universal. The concrete connection will show up
through the action of $Z_{\mu\nu}$, as in Eq. (55). In our case we have also
$\overline{g}_{\mu\nu}=g_{\mu\nu}$ since $\nabla_{\mu}$ is metric preserving.

The last important property of the covariant symbols to be noted is their
relation with diagonal matrix elements,

$\langle
x|\hat{\mathcal{O}}|x\rangle=\frac{1}{\sqrt{g}}\int\frac{d^{d}p}{(2\pi)^{d}}\overline{{\mathcal{O}}}\,\mathbf{1}.$
(99)

Here the object $\mathbf{1}$ (which is usually not written explicitly) is the
function identically equal to 1 in momentum space. In practice this means that
the operator $\partial^{\mu}$ in $\overline{{\mathcal{O}}}$ vanishes when it
is placed on the right. On the other hand $\partial^{\mu}$ is also zero when
placed on left due to the integral over $p_{\mu}$.

The quantity $\overline{{\mathcal{O}}}\mathbf{1}$ (with all the
$\partial^{\mu}$ already canceled by moving them to the right) is just a
function of $x$ and $p$, and is closer to the standard definition of the
symbol of a pseudo-differential operator. Hence the function
$\overline{{\mathcal{O}}}\mathbf{1}$ (multiplicative with respect to
$p_{\mu}$) is what is needed to obtain the diagonal matrix elements but the
full covariant symbol $\overline{{\mathcal{O}}}$ is the object carrying a
faithful algebra representation.

If instead of Eq. (95) only the first similarity transformation is applied,
$\hat{{\mathcal{O}}}\mapsto
e^{-\xi^{\alpha}p_{\alpha}}\,\hat{{\mathcal{O}}}\,e^{\xi^{\beta}p_{\beta}}$,
one obtains $f(\nabla,X)\mapsto f(\nabla+p,X)$, and the result is the method
of non-covariant symbols,

$\langle
x|f(\nabla,X)|x\rangle=\frac{1}{\sqrt{g}}\int\frac{d^{d}p}{(2\pi)^{d}}f(\nabla+p,X).$
(100)

The integrand is not a multiplicative operator (nor manifestly covariant) but
it becomes so after integration over $p_{\mu}$. What the additional similarity
transformation achieves in Eq. (95) is precisely to have an integrand which is
manifestly covariant by systematically applying integration by parts in
$p$-space. Further details on the method of non-covariant symbols are given in
App. E.

The manifest covariance of $\overline{{\mathcal{O}}}$ (and hence
$\overline{{\mathcal{O}}}\mathbf{1}$) follows from using Riemann coordinates
at $x$ (rather than the Synge function as in the Schwinger-DeWitt approach),
so the method of covariant symbols is suited to computing diagonal matrix
elements (within a derivative expansion approach), but not for non-diagonal
matrix elements. On the other hand the method makes no assumptions on
$\hat{\mathcal{O}}$ so it works equally well even for a pseudo-Laplacian like
$\nabla^{2}_{M}$ with a “metric” which is non-Abelian.

We will illustrate the use of the method of covariant symbols below, but its
application is straightforward. $\overline{{\mathcal{O}}}$ is obtained from
$f(\nabla,X)\mapsto f(\overline{\nabla},\overline{X})$, and
$\overline{\nabla}_{\mu}$, $\overline{X}$ are taken from the already compiled
tables to the required order. The quantity $\overline{{\mathcal{O}}}$ so
obtained contains only purely multiplicative operators (covariant derivatives
of $X$), plus $Z_{\mu_{1}\cdots\mu_{n}}$, $p_{\mu}$ and $\partial^{\mu}$.

The natural next step is to remove all $\partial^{\mu}$ by moving them to the
right where they vanish, acting on all dependence on $p_{\mu}$. In doing so
the commutator between $Z_{\mu_{1}\cdots\mu_{n}}$ and $\partial^{\mu}$ has to
be applied. This generates some Riemann tensors. One can also choose to move
some of the $\partial^{\mu}$ to the left (where they also vanish) if this
produces a smaller number of terms. The two choices are related through
integration by parts in momentum space.

Since $p_{\mu}$ does not commute with $Z_{\mu_{1}\cdots\mu_{n}}$, in order to
carry out the momentum integration it will be convenient to move all the
$Z_{\mu_{1}\cdots\mu_{n}}$ together, to the right (or to the left) using their
commutation relations. This produces more coordinate curvatures and also gauge
curvatures $F_{\mu\nu}$. All these manipulations produce a diagonal matrix
element which does not assume a particular vector space for the action of the
operator $\hat{{\mathcal{O}}}$. The concrete space is used when the operators
$Z_{\mu_{1}\cdots\mu_{n}}$ are removed after its action is evaluated, as in
Eq. (55).

Once the $\partial^{\mu}$ have been eliminated and all the
$Z_{\mu_{1}\cdots\mu_{n}}$ are together (or eliminated) it only remains to
carry out the momentum integration, if possible. When the non-polynomial
dependence on $p_{\mu}$ comes from a single (and Abelian) metric the integrals
can often be obtained explicitly. If there are two metrics $g_{\mu\nu}$ and
$M^{\mu\nu}$ (are even more so when the latter is non-Abelian) the integrals
cannot be evaluated in closed form in general. In this case one should be
aware of ambiguities in the final expression due to integration by parts in
momentum space. Complicated expressions can occasionally reach a simpler form
through integration by parts in $p_{\mu}$. For this reason the numbers quoted
in Table 1 are upper bounds.

An explicit application of the method of the covariant symbols to illustrate
the procedure just described is displayed in App. F, by computing one of the
universal functional traces of Barvinsky:1985an .

In what follows we proceed to give details of the calculation of
$\Gamma^{\mathrm{div}}_{S}$ and $\Gamma^{\mathrm{div}}_{L,0}$.

###  VII.2  Calculation of $\Gamma^{\mathrm{div}}_{S}$ in $d=2$

The starting point is Eq. (73). Applying the covariant symbols formula Eq.
(99),

$\langle
x|\hat{\mathcal{O}}|x\rangle=\langle\overline{{\mathcal{O}}}\rangle_{p}$ (101)

(with $\langle~{}\rangle_{p}$ defined in Eq. (58) and the $\mathbf{1}$ is
implicit) one has141414For a multiplicative $X$, $\langle
x|{\mathcal{O}}X|x\rangle=\langle x|{\mathcal{O}}|x\rangle X(x)$, consistently
$\langle\overline{{\mathcal{O}}}\,\overline{X}\rangle_{p}=\langle\overline{{\mathcal{O}}}\rangle_{p}X(x)$
since $\overline{X}-X$ contains $\partial^{\mu}$ but not $p_{\mu}$ (see Eq.
(96)), and such terms vanish inside $\langle~{}\rangle_{p}$.

$\begin{split}\Gamma_{S,1}&=\Bigg{\langle}\operatorname{tr}_{1}\Big{(}-\frac{1}{2}\overline{{\mathcal{O}}}_{19}\,Y_{\mu\nu}\Big{)}\\\
&\quad+\operatorname{tr}_{0}\Big{(}-\frac{1}{2}\overline{{\mathcal{O}}}_{2}W-\frac{1}{2}(\overline{{\mathcal{O}}}_{1})_{\mu}P_{\mu}+\frac{1}{4}\overline{{\mathcal{O}}}_{2}\,P_{\mu\mu}+O(p^{-3})\Big{)}\Bigg{\rangle}_{x,p}\\\
\Gamma_{S,2}&=\\\
&\Bigg{\langle}\operatorname{tr}_{0}\Big{(}\frac{1}{2}(\overline{{\mathcal{O}}}_{3})_{\mu\nu}{\mathcal{M}}_{\mu\alpha}{\mathcal{M}}_{\alpha\nu}-\frac{1}{4}(\overline{{\mathcal{O}}_{4}[P_{\mu}]})_{\mu\nu}\,P_{\nu}+O(p^{-3})\Big{)}\Bigg{\rangle}_{x,p}\end{split}$
(102)

Since $\overline{\nabla}_{\mu}=O(p_{\mu})$, while for multiplicative operators
$\overline{X}=O(1)$ for large $p_{\mu}$, the terms $O(\nabla^{-n})$ in
$\hat{{\mathcal{O}}}$ become $O(p^{-n})$ in $\overline{{\mathcal{O}}}$.

Within dimensional regularization $\langle p^{-n}\rangle_{p}$ vanishes for all
$n$ with the exception $n=d$. Hence we need to isolate terms $1/p^{2}$ in
$d=2$, and in particular more UV convergent terms can be neglected.

Of the basic operators present in the formula, ${\mathcal{O}}_{2,3,4,19}$, are
of $O(\nabla^{-2})$. Therefore the covariant symbols of the latter only
require the leading terms of the building blocks:

$\begin{split}\overline{\nabla}_{\mu}&=p_{\mu}+O(p^{-1}),\qquad\overline{\Delta}=-N_{g}+O(p^{-3}),\qquad\\\
\overline{\Delta}_{M}&=-N_{M}+O(p^{-3}),\qquad\overline{X}=X+O(p^{-1}),\end{split}$
(103)

where $X$ is any purely multiplicative operator. Here we have introduced the
definitions

$N_{g}\equiv(-g^{\mu\nu}p_{\mu}p_{\nu})^{-1},\quad
N_{M}\equiv(-M^{\mu\nu}p_{\mu}p_{\nu})^{-1}.$ (104)

Note that $N_{M}$ is a matrix in gauge space. This produces

$\begin{split}\overline{{\mathcal{O}}}_{2}&=-N_{M}+O(p^{-3}),\qquad\\\
\overline{{\mathcal{O}}}_{3}&=N_{M}N_{g}p_{\mu}p_{\nu}+O(p^{-3}),\qquad\\\
\overline{{\mathcal{O}}_{4}[X]}_{\mu\nu}&=N_{M}XN_{M}p_{\mu}p_{\nu}+O(p^{-3}),\qquad\\\
\overline{{\mathcal{O}}}_{19}&=-N_{g}+O(p^{-3}).\end{split}$ (105)

The terms shown explicitly are homogeneous of degree $p^{-2}$.

The remaining operator ${\mathcal{O}}_{1}$ is $O(\nabla^{-1})$ and its
covariant symbol $O(p^{-1})$. To isolate the $1/p^{2}$ term we have to take
one more term in the expansion. The expansion in (96) is effectively in powers
of $\nabla/p$, so terms with one more covariant derivative are needed. Note
that in this counting $Z_{\mu_{1},\cdots,\mu_{n}}$ counts as $O(\nabla^{n})$.
From its definition
$({\mathcal{O}}_{1})_{\lambda}\equiv\Delta_{M}\nabla_{\lambda}$, one obtains

$(\overline{{\mathcal{O}}}_{1})_{\lambda}=\overline{\Delta}_{M}\overline{\nabla}_{\lambda}$
(106)

with

$\overline{\Delta}_{M}=(\overline{\nabla}_{\mu}\overline{M}_{\mu\nu}\overline{\nabla}_{\nu})^{-1}.$
(107)

The expansion of $\overline{\nabla}_{\lambda}$ in Eq. (103) is already
sufficient, but $\overline{\Delta}_{M}$ needs to be expanded to order
$1/p^{3}$ which in turn requires $\overline{M}_{\mu\nu}$ to order $1/p$

$\begin{split}\overline{\Delta}_{M}&=\left(p_{\mu}(M_{\mu\nu}-M_{\alpha\mu\nu}\partial^{\alpha})p_{\nu}+O(1)\right)^{-1}\\\
&=-N_{M}+N_{M}p_{\mu}M_{\nu\mu\alpha}\partial^{\nu}p_{\alpha}N_{M}+O(p^{-4}).\end{split}$
(108)

Therefore,

$\begin{split}(\overline{{\mathcal{O}}}_{1})_{\lambda}&=-N_{M}p_{\lambda}+N_{M}p_{\mu}M_{\nu\mu\alpha}\partial^{\nu}p_{\alpha}N_{M}p_{\lambda}+O(p^{-3}).\end{split}$
(109)

To the order needed, the operators $\overline{{\mathcal{O}}}_{2,3,4,19}$ do
not have any $\partial^{\mu}$ hence they already coincide with
$\overline{{\mathcal{O}}}_{n}\mathbf{1}$. For $\overline{{\mathcal{O}}}_{1}$
the momentum derivatives can be moved to the right using

$[\partial^{\mu},p_{\nu}]=\delta^{\mu}_{\nu},\qquad[\partial^{\mu},N_{M}]=2p_{\nu}N_{M}M^{\mu\nu}N_{M}$
(110)

In this way151515Actually for convenience here $\partial^{\mu}$ has been moved
to the left (and then removed) so the RHS differs from
$\overline{{\mathcal{O}}}\mathbf{1}$ by terms which vanish upon momentum
integration.

$\begin{split}&(\overline{{\mathcal{O}}}_{1})_{\lambda}\mathbf{1}=-N_{M}p_{\lambda}-N_{M}M_{\mu\mu\nu}N_{M}p_{\lambda}p_{\nu}\\\
&\quad-2N_{M}M_{\mu\nu}N_{M}M_{\mu\alpha\beta}N_{M}p_{\lambda}p_{\nu}p_{\alpha}p_{\beta}\\\
&\quad+O(p^{-3})\end{split}$ (111)

The first term is homogeneous of degree $1/p$ and the two other explicit terms
are homogeneous of degree $1/p^{2}$. The first term vanishes within momentum
integration due to parity. For the same reason all odd order terms have been
omitted in Table 1. In any case only the contributions $1/p^{d}$ are relevant
in $d$ dimensions for $\Gamma^{\mathrm{div}}_{1}$.

No operators $Z_{I}$ appear in the expansions of
$\overline{{\mathcal{O}}}_{1,2,3,4,19}$ to the order required in Eq. (102).
The most UV divergent operator is ${\mathcal{O}}_{1}=O(\nabla^{-1})$, so its
leading term is $1/p$ and the term relevant in $d=2$, $1/p^{2}$, comes from
contributions of the type $\nabla/p^{2}$, while $Z_{I}$ needs at least two
covariant derivatives.

Operators $Z_{I}$ do appear in $d=4$. Just as $\partial^{\mu}$, the operators
$Z^{R}_{I}$ have to be resolved before having a useful expression for $\langle
x|{\mathcal{O}}_{n}|x\rangle$. To this end, the operators $Z^{R}_{I}$ can be
moved to the left or to the right, using their commutation relations noted in
(140), including (98). In accordance with the choice in Eq. (67), where the
multiplicative operators $A_{n}$ have been placed at the right, $Z^{R}_{I}$
should be moved to the left. This allows to apply the rules in Eq. (55) (see
App. B).

It remains to insert in Eq. (102) the various expressions for
$\overline{{\mathcal{O}}}_{n}\mathbf{1}$ just obtained . In the resulting
expression, the terms involving $P_{\mu}$ in $\Gamma_{S,1}$ are not manifestly
Hermitian. This can be fixed by applying integration by parts in $p$ and $x$
spaces161616As shown in App. C of Garcia-Recio:2019iia , $p_{\mu}$ can be
treated as a constant when integrating by parts in $x$-space, when
$\nabla_{\mu}$ and $Z^{R}_{I}$ are no longer present. as well as the trace
cyclic property. This gives,

$\begin{split}\Gamma_{S,1}^{\mathrm{div}}&=\Big{\langle}\frac{1}{2}N_{g}Y_{\mu\mu}+\frac{1}{2}N_{M}W-\frac{1}{4}N_{M}P_{\mu\mu}\\\
&\qquad+\frac{1}{2}N_{M}M_{\mu\nu}N_{M}M_{\mu\alpha\beta}N_{M}P_{\lambda}p_{\nu}p_{\alpha}p_{\beta}p_{\lambda}\\\
&\qquad-\frac{1}{2}N_{M}M_{\mu\alpha\beta}N_{M}M_{\mu\nu}N_{M}P_{\lambda}p_{\nu}p_{\alpha}p_{\beta}p_{\lambda}\Big{\rangle}_{x,\,p,\,g},\\\
\Gamma_{S,2}^{\mathrm{div}}&=\left\langle\frac{1}{2}N_{g}N_{M}{\mathcal{M}}_{\mu\alpha}{\mathcal{M}}_{\alpha\nu}\,p_{\mu}p_{\nu}-\frac{1}{4}N_{M}P_{\mu}N_{M}P_{\nu}p_{\mu}p_{\nu}\right\rangle_{x,\,p,\,g}\,.\end{split}$
(112)

The integrand is a homogeneous function of $p$ of degree $-2$. It only remains
to extract the $1/\epsilon$ coefficient to isolate the UV divergent
contributions. The details are given below, in Sec. VII.4. An application of
the rules provided there immediately produces the result quoted in Eq. (74).

In Eq. (111) there is one term of degree $p^{-1}$ and two terms of degree
$p^{-2}$. Table 1 shows the number of terms of each degree for the diagonal
matrix elements of the operators ${\mathcal{O}}_{n}$. Only even orders are
displayed since odd orders vanish upon integration over $p_{\mu}$ in any
parity preserving regularization, such as dimensional regularization. For a
given operator, the number of terms increases rapidly with (minus) the degree.
Nevertheless the number of terms displayed in the table is an upper bound,
this number is subject to variations due to various identities which allow to
write a given expression in different forms. Such identities include
integration by parts in momentum space and reordering of the covariant
derivatives acting on a tensor due to the Jacobi identity,

$[\nabla_{\mu},[\nabla_{\nu},X]]=[\nabla_{\nu},[\nabla_{\mu},X]]+[Z_{\mu\nu},X].$
(113)

Furthermore, integration by parts in $x$-space and trace cyclic property is
allowed within the functional trace operations $\operatorname{Tr}_{0,1}$ in
Eq. (71). We have not attempted a systematic minimization of the number of
terms as there is no practical procedure to do this, and in any case we do not
expect a significant reduction in the length of the expressions. An exception
is the operator ${\mathcal{O}}_{2}$ at $p^{-4}$ which is used below, Eq.
(115), in the computation of $\Gamma_{L,0}$, in Sec. VI.3.4.

Nevertheless, it should be noted that recently important progress has been
achieved in the counting and classification of allowed independent terms in
effective field theories, through the construction of Hilbert series of the
operator basis Henning:2017fpj . In this technique the basic blocks (fields or
composite operators) plus their symmetrized derivatives are identified with
representations of the $d$-dimensional conformal group.171717Alternatively,
cohomological techniques can be applied Henning:2017fpj . Computation of the
Clebsch-Gordan series then allows to obtain generating functions for basis
operators and count them. The key point is that both, equation of motion as
well as integration by part identities, are automatically accounted for, in
addition to spacetime and internal group symmetries. The method has been
successfully applied to pure Einstein relativity and also to general
relativity combined with the Standard Model of particle physics
Ruhdorfer:2019qmk . The adaptation of the Hilbert series technique to obtain
basis of operators in diagonal matrix elements and the effective action
contributions as those displayed in Eq. (112) or Eq. (116) would be extremely
interesting, and more so in $d=4$ where the number of terms becomes huge.
Serious complications arise due to the presence of an additional momentum
variable, with its own integration by parts identities, and also the existence
of constraints relating some of the building blocks, such as $M^{\mu\nu}$ and
$N_{M}$. No attempt will be made here to adapt the promising technique of
Hilbert series to working with covariant symbols, we defer such a study to
future work.

###  VII.3  Calculation of $\Gamma^{\mathrm{div}}_{L,0}$ in $d=2$

Due to the similarity between the operators ${\mathcal{O}}^{\prime}_{2}$ and
${\mathcal{O}}_{2}$, the expression of their diagonal matrix elements are
identical when written in terms of $N_{M}$, with the only proviso of using the
new definition

$N_{M}=(-z-M^{\mu\nu}p_{\mu}p_{\nu})^{-1}$ (114)

for ${\mathcal{O}}^{\prime}_{2}$, instead of that in (104).

The UV divergent terms in $\Gamma_{L,0}$ must have exactly $d$ covariant
derivatives. The operator ${\mathcal{O}}_{2}^{\prime}$ is of UV degree
$O(\nabla^{-2})$, so the leading term in
$\overline{{\mathcal{O}}^{\prime}}_{2}$ is $O(p^{-2})$ and no derivatives.
Since one still needs to expand its covariant symbol to $d$ covariant
derivatives, the order $p^{-d-2}\nabla^{d}$ is needed. The proper divergence
$O(p^{-d})$ is only recovered after integration over $z$, a parameter of
dimension squared mass.

We present results only for the case $d=2$. For the terms of order exactly
$p^{-4}$, the calculation gives the following result

$\begin{split}&\langle x|{\mathcal{O}}^{\prime}_{2}|x\rangle^{(-4)}=\\\
&\quad\Big{\langle}4{N_{M}}\,M_{\mu\nu}\,{N_{M}}\,M_{\mu\alpha\beta}\,{N_{M}}\,M_{\rho\sigma\lambda}\,{N_{M}}\,M_{\rho\eta}\,{N_{M}}\,p_{\nu}\,p_{\alpha}\,p_{\beta}\,p_{\sigma}\,p_{\lambda}\,p_{\eta}\\\
&\quad+2{N_{M}}\,M_{\mu\nu}\,{N_{M}}\,M_{\mu\alpha\beta}\,{N_{M}}\,M_{\rho\rho\sigma}\,{N_{M}}\,p_{\nu}\,p_{\alpha}\,p_{\beta}\,p_{\sigma}\\\
&\quad+2{N_{M}}\,M_{\mu\mu\nu}\,{N_{M}}\,M_{\alpha\beta\rho}\,{N_{M}}\,M_{\alpha\sigma}\,{N_{M}}\,p_{\nu}\,p_{\beta}\,p_{\rho}\,p_{\sigma}\\\
&\quad+{N_{M}}\,M_{\mu\nu}\,{N_{M}}\,M_{\mu\alpha\beta\rho}\,{N_{M}}\,M_{\alpha\sigma}\,{N_{M}}\,p_{\nu}\,p_{\beta}\,p_{\rho}\,p_{\sigma}\\\
&\quad+{N_{M}}\,M_{\mu\nu}\,{N_{M}}\,M_{\alpha\mu\beta\rho}\,{N_{M}}\,M_{\alpha\sigma}\,{N_{M}}\,p_{\nu}\,p_{\beta}\,p_{\rho}\,p_{\sigma}\\\
&\quad+\frac{2}{3}{N_{M}}\,M_{\mu\nu}\,{N_{M}}\,M_{\alpha\beta}\,{N_{M}}\,M_{\rho\sigma}\,{N_{M}}\,{R}_{\mu\alpha\rho\lambda}\,p_{\nu}\,p_{\beta}\,p_{\sigma}\,p_{\lambda}\\\
&\quad-\frac{2}{3}{N_{M}}\,M_{\mu\nu}\,{N_{M}}\,M_{\alpha\beta}\,{N_{M}}\,M_{\rho\sigma}\,{N_{M}}\,{R}_{\mu\lambda\alpha\rho}\,p_{\nu}\,p_{\beta}\,p_{\sigma}\,p_{\lambda}\\\
&\quad+{N_{M}}\,M_{\mu\mu\nu}\,{N_{M}}\,M_{\alpha\alpha\beta}\,{N_{M}}\,p_{\nu}\,p_{\beta}\\\
&\quad-{N_{M}}\,M_{\mu\nu}\,F_{\mu\alpha}\,{N_{M}}\,M_{\alpha\beta}\,{N_{M}}\,p_{\nu}\,p_{\beta}\\\
&\quad-{N_{M}}\,M_{\mu\nu}\,{N_{M}}\,F_{\mu\alpha}\,M_{\alpha\beta}\,{N_{M}}\,p_{\nu}\,p_{\beta}\\\
&\quad-\frac{1}{6}{N_{M}}\,M_{\mu\nu}\,{N_{M}}\,{{\mathcal{R}}}_{\mu\nu}\Big{\rangle}_{p}.\end{split}$
(115)

The expression obtained after applying the method of covariant symbols has
been simplified by using integration by parts in momentum space, also
achieving a manifest Hermitian form (since ${\mathcal{O}}_{2}$ is Hermitian).

It can be noted that the expression in (115) holds for arbitrary $d$. For
$d=2$ one can use the identities in (91). Also we integrate by parts with
respect to $x$ to have at most one covariant derivative on $M_{\mu\nu}$. This
gives for the effective action

$\begin{split}&\Gamma_{L,0}^{\mathrm{div}}=\frac{1}{2}\Big{\langle}\\\
&\quad-2N_{M}\,M_{\mu\nu}\,N_{M}\,M_{\mu\alpha\beta}\,N_{M}\,M_{\rho\sigma}\,N_{M}\,M_{\rho\lambda\eta}\,N_{M}\,p_{\nu}\,p_{\alpha}\,p_{\beta}\,p_{\sigma}\,p_{\lambda}\,p_{\eta}\\\
&\quad-2N_{M}\,M_{\mu\nu\alpha}\,N_{M}\,M_{\mu\beta}\,N_{M}\,M_{\rho\sigma\lambda}\,N_{M}\,M_{\rho\eta}\,N_{M}\,p_{\nu}\,p_{\alpha}\,p_{\beta}\,p_{\sigma}\,p_{\lambda}\,p_{\eta}\\\
&\quad-
N_{M}\,M_{\mu\nu}\,N_{M}\,M_{\mu\alpha\beta}\,N_{M}\,M_{\alpha\rho\sigma}\,N_{M}\,p_{\nu}\,p_{\beta}\,p_{\rho}\,p_{\sigma}\\\
&\quad+N_{M}\,M_{\mu\nu}\,N_{M}\,M_{\alpha\alpha\beta}\,N_{M}\,M_{\mu\rho\sigma}\,N_{M}\,p_{\nu}\,p_{\beta}\,p_{\rho}\,p_{\sigma}\\\
&\quad-
N_{M}\,M_{\mu\nu\alpha}\,N_{M}\,M_{\beta\mu\rho}\,N_{M}\,M_{\beta\sigma}\,N_{M}\,p_{\nu}\,p_{\alpha}\,p_{\rho}\,p_{\sigma}\\\
&\quad+N_{M}\,M_{\mu\nu\alpha}\,N_{M}\,M_{\beta\beta\rho}\,N_{M}\,M_{\mu\sigma}\,N_{M}\,p_{\nu}\,p_{\alpha}\,p_{\rho}\,p_{\sigma}\\\
&\quad+N_{M}\,M_{\mu\nu}\,N_{M}\,M_{\alpha\beta}\,N_{M}\,M_{\mu\rho}\,N_{M}\,{\mathbf{R}}\,p_{\nu}\,p_{\alpha}\,p_{\beta}\,p_{\rho}\\\
&\quad+N_{M}\,M_{\mu\mu\nu}\,N_{M}\,M_{\alpha\alpha\beta}\,N_{M}\,p_{\nu}\,p_{\beta}\\\
&\quad-
N_{M}\,M_{\mu\nu}\,F_{\mu\alpha}\,N_{M}\,M_{\alpha\beta}\,N_{M}\,p_{\nu}\,p_{\beta}\\\
&\quad-
N_{M}\,M_{\mu\nu}\,N_{M}\,F_{\mu\alpha}\,M_{\alpha\beta}\,N_{M}\,p_{\nu}\,p_{\beta}\\\
&\quad+\frac{2}{3}N_{M}\,M_{\mu\nu}\,N_{M}\,M_{\mu\alpha}\,N_{M}\,{\mathbf{R}}\,p_{\nu}\,p_{\alpha}\\\
&\quad-\frac{1}{12}N_{M}\,M_{\mu\mu}\,N_{M}\,{\mathbf{R}}\Big{\rangle}_{x,p,g,z}\end{split}$
(116)

One could apply further the trace cyclic property, and also the identity
$N_{M}M_{\mu\nu}N_{M}p_{\mu}p_{\nu}=-N_{M}-zN_{M}^{2}$ in one of the terms,
but no simplification would be achieved. The final step is to extract the
$1/\epsilon$ coefficient from the radial part of the momentum integral, as
described in next subsection. This procedure yields the expression quoted in
Eq. (90).

###  VII.4 Extraction of the UV divergent component

Let us consider first the case when the parameter $z$ is not present, as in
Eq. (112). Once the operators $\partial_{\mu}$ and $Z^{R}_{I}$ have been
removed, the structure of a general term $\overline{{\mathcal{O}}}_{n}$ to be
integrated over $p_{\mu}$ is a sum of products with factors $N_{M}$, $N_{g}$,
$p_{\mu}$, as well as $p_{\mu}$-independent multiplicative operators
$M_{I},R_{I},F_{I}$, etc. Hence the momentum integral affects only terms of
the form

$N_{g}^{n}(N_{M})_{a_{1}b_{1}}\cdots(N_{M})_{a_{m}b_{m}}p_{\mu_{1}}\cdots
p_{\mu_{2j}}\equiv N_{g}^{n}N_{M}^{\otimes m}p^{\otimes 2j}.$ (117)

This monomial is homogeneous in $p_{\mu}$ with degree $2(j-n-m)$.

We are only interested in the UV divergent part and to extract it we will use
dimensional regularization, namely, in $d+2\epsilon$ dimensions with
$\epsilon\to 0^{-}$. For the UV divergent part, $d$ can be used instead of
$d+2\epsilon$ in the UV-finite contributions. Within dimensional
regularization, the integral

$I_{d}^{m,j}=\frac{1}{\sqrt{g}}\int\frac{d^{d+2\epsilon}p}{(2\pi)^{d}}N_{g}^{n}N_{M}^{\otimes
m}p^{\otimes 2j},$ (118)

vanishes unless $d=2(n+m-j)$, which is assumed in what follows.

Recalling that $p_{\mu}=ik_{\mu}$ and $d^{d}p_{\mu}\equiv d^{d}k_{\mu}$,

$I_{d}^{m,j}=(-1)^{j}\frac{1}{\sqrt{g}}\int\frac{d^{d+2\epsilon}k_{\mu}}{(2\pi)^{d}}N_{g}^{n}N_{M}^{\otimes
m}k^{\otimes 2j}.$ (119)

Let us introduce a standard tetrad field $e_{\mu}^{A}(x)$,

$g_{\mu\nu}=e^{A}_{\mu}e^{B}_{\nu}\delta_{AB},\qquad
e^{A}_{\mu}e^{\mu}_{B}=\delta^{A}_{B}$ (120)

in such a way that

$\begin{split}k_{\mu}&=e^{A}_{\mu}k_{A},\qquad
d^{d}k_{\mu}=\sqrt{g}d^{d}k_{A},\qquad\\\ N_{g}&=\frac{1}{k_{A}^{2}},\qquad
N_{M}=\frac{1}{k_{A}k_{B}M^{AB}}\end{split}$ (121)

thus

$\begin{split}(I_{d}^{m,j})_{\mu_{1}\cdots\mu_{2j}}&=(I_{d}^{m,j})_{A_{1}\cdots
A_{2j}}e^{A_{1}}_{\mu_{1}}\cdots e^{A_{2j}}_{\mu_{2j}},\qquad\\\
(I_{d}^{m,j})_{A_{1}\cdots
A_{2j}}&=(-1)^{j}\int\frac{d^{d+2\epsilon}k_{A}}{(2\pi)^{d}}\,N_{g}^{n}\,N_{M}^{\otimes
m}\,k_{A}^{\otimes 2j}.\end{split}$ (122)

The UV divergence comes from the radial part of the integral, hence we
introduce spherical coordinates,

$k_{A}=k\hat{k}_{A},\qquad
k=\sqrt{k_{A}^{2}}=N_{g}^{-1/2},\qquad\hat{k}_{A}^{2}=1.$ (123)

This allows to separate the integral into radial and angular-average factors,

$I_{d}^{m,j}=\frac{(-1)^{j}}{(4\pi)^{d/2}}\frac{2}{\Gamma(d/2)}I_{\epsilon}\hat{I}_{d}^{\,m,j},$
(124)

where the angular average is

$\begin{split}\hat{I}_{d}^{\,m,j}&=\frac{\Gamma(d/2)}{2\pi^{d/2}}\int
d^{d-1}\Omega_{\hat{k}}\,\hat{N}_{M}^{\otimes m}\,\hat{k}_{A}^{\otimes
2j}\equiv\langle\hat{N}_{M}^{\otimes m}\,\hat{k}_{A}^{\otimes
2j}\rangle_{\mathrm{ang}},\qquad\\\
\hat{N}_{M}&\equiv\frac{1}{\hat{k}_{A}\hat{k}_{B}M^{AB}}=N_{M}/N_{g},\end{split}$
(125)

and $I_{\epsilon}$ is the radial part (introducing a cutoff mass $m_{0}$ to
avoid a trivial infrared divergence for negative $\epsilon$), and using the
condition $2(n+m-j)=d$,

$I_{\epsilon}=\int_{m_{0}}^{\infty}dkk^{2\epsilon-1}=-\frac{m_{0}^{2\epsilon}}{2\epsilon}=-\frac{1}{2\epsilon}+O(1).$
(126)

In summary, for the UV divergent part, one obtains

$I_{d}^{m,j,\mathrm{div}}=\frac{1}{(4\pi)^{d/2}\Gamma(d/2)}\frac{1}{\epsilon}(-1)^{j+1}\,\hat{I}_{d}^{\,m,j}.$
(127)

As noted, the angular averages $\hat{I}_{d}^{\,m,j}$ are perfectly UV
convergent and well-defined, but they cannot be written in closed form in
general.

The analysis is similar for $\Gamma_{L,0}$, which involves an additional
integration over $z$. As mentioned, in this case and for $d=2$, the relevant
terms are of order $p^{-4}$, and to extract the UV divergent part one must
integrate over $z$ and $p_{\mu}$. The point can be elucidated following the
steps shown above (for $d=2$), noting that now $N_{M}$ contains $z$:

$\begin{split}&\left\langle p^{\otimes
2j}N_{M}^{\otimes(2+j)}\right\rangle_{z,p}=\frac{1}{\sqrt{g}}\int\frac{d^{2+2\epsilon}p}{(2\pi)^{2}}\int_{\gamma}\frac{dz}{2\pi
i}\,\\\ &\quad\times\log(z)\,p^{\otimes
2j}\left(\frac{1}{-z-M^{\mu\nu}p_{\mu}p_{\nu}}\right)^{\otimes(2+j)}\\\
&=(-1)^{j}\int_{m_{0}}^{\infty}dk\,k^{2j+1+2\epsilon}\int\frac{d\Omega}{(2\pi)^{2}}\int_{\gamma}\frac{dz}{2\pi
i}\,\\\ &\quad\times\log(z)\,\hat{k}^{\otimes
2j}\left(\frac{1}{-z+k^{2}M^{\mu\nu}\hat{k}_{\mu}\hat{k}_{\nu}}\right)^{\otimes(2+j)}\end{split}$
(128)

Applying the rescaling $z\to zk^{2}$, and noting that the induced term with
$\log(k^{2})$ vanishes since no singularities are enclosed by the path
$\gamma$ in the $z$ complex plane,

$\begin{split}&\left\langle p^{\otimes
2j}N_{M}^{\otimes(2+j)}\right\rangle_{z,p}=(-1)^{j+1}\frac{m_{0}^{2\epsilon}}{2\epsilon}\int\frac{d\Omega}{(2\pi)^{2}}\int_{\gamma}\frac{dz}{2\pi
i}\,\\\ &\quad\times\log(z)\,\hat{k}^{\otimes
2j}\left(\frac{1}{-z+M^{\mu\nu}\hat{k}_{\mu}\hat{k}_{\nu}}\right)^{\otimes(2+j)}\\\
&\quad=\frac{(-1)^{j+1}}{4\pi\epsilon}\langle\hat{k}^{\otimes
2j}\,\hat{N}_{M}^{\otimes(2+j)}\rangle_{\mathrm{ang},z}+O(1),\end{split}$
(129)

where $\hat{N}_{M}=(-z+M^{\mu\nu}\hat{k}_{\mu}\hat{k}_{\nu})^{-1}$. Applying
this angular average in Eq. (116) yields Eq. (90).

##  VIII Summary and conclusions

In this work we have addressed the problem of quantizing a system of, in
general, non-Abelian vector fields with a completely general local non-minimal
mass term coupling all of them. The case of $N$ Abelian fields is a particular
instance in our formulation. We make use of a background field approach. A
remarkable result is that, although the mass term breaks gauge invariance, the
effective action is fully gauge (as well as coordinate) invariant beyond tree-
level, Eq. (8). The technical problems present in the original theory (namely,
the UV region is blind to the mass term and so requires some type of gauge-
fixing) are satisfactorily removed by introducing a non-Abelian Stueckelberg
field, Eqs. (19) and (22). This is done after background and fluctuation
fields have been separated and the Stueckelberg field only affects the latter.
This auxiliary field is introduced linearly so that the loop structure of the
original theory is preserved. As a consequence the computation of the UV
divergent part of the effective action to one-loop can be carried out
systematically preserving coordinate and gauge symmetries during the
calculation. To this end we apply dimensional regularization and the method of
covariant symbols. This produces terms which are local, i.e., they contain a
finite number derivatives of the external fields (no more that $d$ in $d$
spacetime dimensions), however they are not polynomial with respect to the
mass term, a fact already observed in the simpler case of a single vector
field. The one-loop effective action is expressed in Eq. (46) as a sum of four
terms, $\Gamma_{\mathrm{gh}}$ defined in (37), $\Gamma_{L,1}$ and
$\Gamma_{L,0}$ in (43), and $\Gamma_{S}$ in (44) (and expanded in Eq. (65)).
The formalism is developed for arbitrary spacetime dimensions and we present
explicit results for the two-dimensional case. Regrettably in the four-
dimensional case too many terms are produced (even selecting particular
settings, such as purely Abelian, or flat spacetime, or perturbative
expansions) so their explicit expression would be of little practical use. The
explicit two-dimensional results for the UV divergent component are displayed
in (62) for $\Gamma_{\mathrm{gh}}$ and $\Gamma_{L,1}$, in (74) for
$\Gamma_{S}$, and in (90) for $\Gamma_{L,0}$. Checks have been applied to the
results of the calculation. Particularly stringent are the tests related to
invariance with respect to metric transformations in Sec. VI.3.5. Perturbative
results are also presented in App. D. Although not explicitly discussed in the
text, we have repeated most of the calculations of diagonal matrix elements
using the method of non-covariant symbols Salcedo:2006pv ; Moral-Gamez:2011wcb
to find an identical result, modulo integration by parts in momentum space. In
some cases the latter method has produced shorter expressions. Details of the
non-covariant method are discussed in App. E.

As already pointed out, the non-minimal mass-like coupling discussed in this
work (or also its Abelian version), respects locality but introduces terms in
the effective action which are non-polynomial in the field $M^{\mu\nu}(x)$.
All current efforts for an effective field theory description of general
relativity or the standard model of particles, or both (e.g. Ruhdorfer:2019qmk
) naturally assume an expansion in local and polynomial operators over some
power of the cutoff (a new physics scale),
${\mathcal{O}}_{\alpha}(x)/\Lambda^{n}$, consistently with the renormalization
group analysis of Wilson Wilson:1974mb . In this light, the analyses presented
in Ruf:2018vzq ; Garcia-Recio:2019iia  and in this work should indicate that a
non-minimal coupling of the type Eq. (1) can be ruled out in vector field
theories, also in the non-Abelian setting. If the presence of such non-minimal
coupling could not be prevented through some mechanism (such as the
requirement of strict gauge invariance) one would be impelled to assume a much
larger class of effective field theories, including local but non-polynomial
operators. On the other hand, even in that case re-expansions as that in App.
D would bring the expression again to the standard form, requiring only local
and polynomial composite operators, provided that the scale $m$ can be
interpreted as a proper cutoff of the theory and a separation of the type
$M^{\mu\nu}=m^{2}g^{\mu\nu}+H^{\mu\nu}$ is somehow natural.

The fluctuation operator in Eq. (33) is a rather involved one, due to the
presence of a non-Abelian field $M^{\mu\nu}$ coupling like a metric in the
$\phi$-sector. Chan’s method or even the Schwinger-DeWitt technique are not
readily available to deal with such term. Yet the formalism of covariant
symbols could be applied also to $\Gamma_{L,0}$, upon introduction of a
parametric form to remove the logarithm. In fact the method of covariant
symbols is a practical and easy-to-use tool to obtain diagonal matrix elements
of local operators $f(\nabla,X)$ provided the dependence on $\nabla$ is of
rational type. This latter requirement follows from the fact that, in
practice, the covariant symbols are only obtained as an expansion in powers of
$\nabla/p$. It can also be noted that the covariant symbols depend only on the
connection, the presence of a metric is not required, as the Riemann
coordinates can be defined directly from the connection Gusynin:1990bu . The
method can be used to obtain not only the counterterms but also covariant
derivative expansions of the effective action itself Salcedo:2000hx ;
Salcedo:2007bt ; Garcia-Recio:2009jso . In particular, the treatment of
fermionic modes in curved spacetime poses no special problems once the spin
connection is included in the covariant derivative Motilla . The extension of
the method for finite temperature also exists Moral-Gamez:2011wcb ;
Brauner:2011vb  (but not yet for temperature and curvature at the same time).
The method of covariant symbols should apply whenever the generalized
Schwinger-DeWitt technique applies, so it can be used as an alternative
approach in the analysis quantum field theories in curved spacetime, or
effective field theories involving gravity, or in the study of the newly
developed Proca theories noted in the Introduction.

###### Acknowledgements.

I thank C. Garcia-Recio for suggestions on the manuscript and A.O. Barvinsky
for critical remarks. This work has been partially supported by
MCIN/AEI/10.13039/501100011033 under grant PID2020-114767 GB-I00, by the Junta
de Andalucía (grant No. FQM-225), by the FEDER/Junta de Andalucía-Consejería
de Economía y Conocimiento 2014-2020 Operational Program under grant
A-FQM-178-UGR18, and by the Consejería de Conocimiento, Investigación y
Universidad, Junta de Andalucía and European Regional Development Fund (ERDF),
ref. SOMM17/6105/UGR.

##  Appendix A Derivation of some formulas

###  A.1 Derivation of Eq. (15)

Applying integration by parts in the first term in (14) (using the notation of
(56))

$\begin{split}&\frac{1}{4}\left\langle({\mathcal{A}}_{\mu\nu}^{a}-{\mathcal{A}}_{\nu\mu}^{a})^{2}\right\rangle_{x}\\\
&=\frac{1}{2}\left\langle({\mathcal{A}}_{\mu\nu}^{a})^{2}-{\mathcal{A}}_{\mu\nu}^{a}{\mathcal{A}}_{\nu\mu}^{a}\right\rangle_{x}=\frac{1}{2}\left\langle({\mathcal{A}}_{\mu\nu}^{a})^{2}+{\mathcal{A}}_{\nu}^{a}{\mathcal{A}}_{\mu\nu\mu}^{a}\right\rangle_{x}\\\
&=\frac{1}{2}\left\langle({\mathcal{A}}_{\mu\nu}^{a})^{2}+{\mathcal{A}}_{\nu}^{a}{\mathcal{A}}_{\nu\mu\mu}^{a}+{\mathcal{A}}_{\nu}^{a}Z_{\mu\nu}^{ab}{\mathcal{A}}_{\mu}^{b}\right\rangle_{x}\\\
&=\frac{1}{2}\left\langle({\mathcal{A}}_{\mu\nu}^{a})^{2}-{\mathcal{A}}_{\nu\nu}^{a}{\mathcal{A}}_{\mu\mu}^{a}+{\mathcal{A}}_{\nu}^{a}(F_{\mu\nu}^{ab}{\mathcal{A}}_{\mu}^{b}+R_{\mu\nu\mu\alpha}{\mathcal{A}}_{\alpha}^{a})\right\rangle_{x}\\\
&=\frac{1}{2}\left\langle({\mathcal{A}}_{\mu\nu}^{a})^{2}-({\mathcal{A}}_{\mu\mu}^{a})^{2}-F_{\mu\nu}^{ab}{\mathcal{A}}_{\mu}^{a}{\mathcal{A}}_{\nu}^{b}+{\mathcal{R}}_{\mu\alpha}{\mathcal{A}}_{\alpha}^{a})\right\rangle_{x}.\end{split}$
(130)

Added to the other term in (14),
$\langle-\frac{1}{2}F_{\mu\nu}^{ab}{\mathcal{A}}_{\mu}^{a}{\mathcal{A}}_{\nu}^{b}\rangle$,
produces (15).

###  A.2 Derivation of Eq. (24)

From the change of variables

${\mathcal{A}}_{\mu}^{a}={\mathcal{B}}_{\mu}^{a}+\phi_{\mu}^{a},\qquad\chi^{a}={\mathcal{B}}^{a}_{\mu\mu}$
(131)

one obtains

$\frac{\partial({\mathcal{A}}^{a}_{\mu},\chi^{a})}{\partial({\mathcal{B}}^{b}_{\nu},\phi^{b})}=\begin{pmatrix}\delta_{ab}g_{\mu}{}^{\nu}&\delta_{ab}\nabla_{\mu}\\\
\delta_{ab}\nabla^{\nu}&0\end{pmatrix}$ (132)

(with rows for the numerator and columns for the denominator). More
conveniently, doing the change of variables in two steps
$({\mathcal{B}},\phi)\to({\mathcal{A}},\phi)\to({\mathcal{A}},\chi)$,

$\frac{\partial({\mathcal{A}}^{a}_{\mu},\chi^{a})}{\partial({\mathcal{B}}^{b}_{\nu},\phi^{b})}=\frac{\partial({\mathcal{A}}^{a}_{\mu},\chi^{a})}{\partial({\mathcal{A}}^{c}_{\lambda},\phi^{c})}\frac{\partial({\mathcal{A}}^{c}_{\lambda},\phi^{c})}{\partial({\mathcal{B}}^{b}_{\nu},\phi^{b})}$
(133)

using $\chi^{a}={\mathcal{A}}_{\mu\mu}^{a}-\phi^{a}_{\mu\mu}$ in the first
factor, produces

$\begin{pmatrix}\delta_{ab}g_{\mu}{}^{\nu}&\delta_{ab}\nabla_{\mu}\\\
\delta_{ab}\nabla^{\nu}&0\end{pmatrix}=\begin{pmatrix}\delta_{ac}g_{\mu}{}^{\lambda}&0\\\
\delta_{ac}\nabla^{\lambda}&-\delta_{ac}\nabla^{2}\end{pmatrix}\begin{pmatrix}\delta_{cb}g_{\lambda}{}^{\nu}&\delta_{cb}\nabla_{\lambda}\\\
0&\delta_{cb}\end{pmatrix}.$ (134)

The second matrix has unit determinant and likewise for the upper-left block
in the first matrix. This produces Eq. (24).

###  A.3 Derivation of Eq. (26)

Starting from (14), and using
${\mathcal{A}}_{\mu}^{a}={\mathcal{B}}_{\mu}^{a}+\phi^{a}_{\mu}$, one obtains
terms of the types ${\mathcal{B}}{\mathcal{B}}$, ${\mathcal{B}}\phi$ and
$\phi\phi$. The terms ${\mathcal{B}}{\mathcal{B}}$ are just those in (15) with
${\mathcal{A}}_{\mu}^{a}\to{\mathcal{B}}_{\mu}^{a}$.

The terms ${\mathcal{B}}\phi$ are given by twice (14) replacing one of the
${\mathcal{A}}$ with ${\mathcal{A}}^{a}_{\mu}\to{\mathcal{B}}^{a}_{\mu}$ and
the other one with ${\mathcal{A}}^{a}_{\mu}\to\phi^{a}_{\mu}$. This produces

$\left(S^{(2)}_{\rm
kin}\right)_{{\mathcal{B}}\phi}=\left\langle\frac{1}{2}(\phi_{\mu\nu}^{a}-\phi_{\nu\mu}^{a})({\mathcal{B}}_{\mu\nu}^{a}-{\mathcal{B}}_{\nu\mu}^{a})-F^{ab}_{\mu\nu}\phi^{a}_{\mu}{\mathcal{B}}^{b}_{\nu}\right\rangle_{x}$
(135)

Using $\phi_{\mu\nu}^{a}-\phi_{\nu\mu}^{a}=F^{ab}_{\mu\nu}\phi^{b}$ and
integration by parts in the other term,

$\begin{split}\left(S^{(2)}_{\rm
kin}\right)_{{\mathcal{B}}\phi}&=\left\langle-\phi^{a}F^{ab}_{\mu\nu}{\mathcal{B}}_{\mu\nu}^{b}+F^{ab}_{\mu\mu\nu}\phi^{a}{\mathcal{B}}^{b}_{\nu}+F^{ab}_{\mu\nu}\phi^{a}{\mathcal{B}}^{b}_{\mu\nu}\right\rangle_{x}\\\
&=\left\langle
F^{ab}_{\mu\mu\nu}\phi^{a}{\mathcal{B}}^{b}_{\nu}\right\rangle_{x}\end{split}$
(136)

Finally, the term $\phi\phi$ is half the previous one after the replacement
${\mathcal{B}}^{a}_{\mu}\to\phi^{a}_{\mu}$. This produces (26).

##  Appendix B The operators $Z_{\mu_{1}\cdots\mu_{n}}$

###  B.1 Definition and properties of $Z^{R}_{\mu_{1}\cdots\mu_{n}}$

The operator $Z_{\mu\nu}$ is defined as

$Z_{\mu\nu}=[\nabla_{\mu},\nabla_{\nu}]=Z^{R}_{\mu\nu}+F_{\mu\nu}$ (137)

$Z^{R}_{\mu\nu}$ acts on coordinate indices and $F_{\mu\nu}$ on gauge indices.
$Z^{R}_{\mu\nu}$ is multiplicative but not “purely multiplicative” (by
definition) as it is not diagonal in the coordinate indices. The higher rank
tensors are defined recursively, namely, (recall that $I=\mu_{1}\cdots\mu_{n}$
stands for a string of coordinate indices)

$Z_{\alpha
I}=[\nabla_{\alpha},Z_{I}]+\frac{1}{2}\\{\nabla_{\lambda},R_{I\alpha\lambda}\\}\,.$
(138)

The extra term ensures that $Z_{I}$ is a multiplicative operator. The same
formula applies to $Z^{R}_{I}$.

The clean separation between coordinate and gauge sectors,

$Z_{I}=Z^{R}_{I}+F_{I},$ (139)

holds too for higher rank tensors. The operators $Z_{I}$, $Z^{R}_{I}$ and
$F_{I}$ are all antihermitian. From its definition $Z^{R}_{I}$ has the
property

$[Z^{R}_{I},V_{\mu_{1}\mu_{2}\cdots}]=R_{I\mu_{1}\lambda}V_{\lambda\mu_{2}\cdots}+R_{I\mu_{2}\lambda}V_{\mu_{1}\lambda\cdots}+\cdots,$
(140)

where $V$ is a coordinate tensor. In particular for a scalar field $\phi(x)$

$[Z^{R}_{I},\phi]=0.$ (141)

Now, let $|0\rangle$ be the scalar function that takes the value $1$ for all
$x$. This is a coordinate scalar. The relation $\nabla_{\mu}|0\rangle=0$
implies

$Z^{R}_{\mu\nu}|0\rangle=0=\langle 0|Z^{R}_{\mu\nu}\,.$ (142)

More generally (using Eq. (138))

$Z^{R}_{I}|0\rangle=C_{I}|0\rangle,\qquad\langle 0|Z^{R}_{I}=-\langle
0|C_{I},$ (143)

with

$\begin{split}C_{\mu\nu}&=0,\\\
C_{\alpha\mu\nu}&=\frac{1}{2}R_{\lambda\mu\nu\alpha\lambda},\\\
C_{\alpha_{1}\cdots\alpha_{n}\mu\nu}&=\frac{1}{2}R_{\lambda\alpha_{2}\alpha_{3}\cdots\alpha_{n}\mu\nu\alpha_{1}\lambda}+\frac{1}{2}R_{\alpha_{1}\lambda\alpha_{3}\cdots\alpha_{n}\mu\nu\alpha_{2}\lambda}+\cdots\\\
&\quad+\frac{1}{2}R_{\alpha_{1}\cdots\alpha_{n-1}\lambda\mu\nu\alpha_{n}\lambda}.\end{split}$
(144)

###  B.2 Relation with the operator $\hat{{\mathcal{R}}}_{\mu\nu}$ of
Barvinsky:1985an

The operator $Z^{R}_{\mu\nu}$ coincides with $\hat{{\mathcal{R}}}_{\mu\nu}$ in
Barvinsky:1985an . The higher rank operators
$(\nabla_{\alpha_{1}}\cdots\nabla_{\alpha_{n}}\hat{{\mathcal{R}}}_{\mu\nu})^{\rho}{}_{\sigma}$
are introduced in Barvinsky:1985an , with the convention that the covariant
derivative connections do not act on the matrix indices $\rho,\sigma$. Using
the notation

$\hat{{\mathcal{R}}}_{\alpha_{1}\cdots\alpha_{n}\mu\nu}\equiv\nabla_{\alpha_{1}}\cdots\nabla_{\alpha_{n}}\hat{{\mathcal{R}}}_{\mu\nu}$
(145)

these operators fulfill the recursion

$\hat{{\mathcal{R}}}_{\alpha
I}=[\nabla_{\alpha},\hat{{\mathcal{R}}}_{I}]+R_{I\alpha\lambda}\nabla_{\lambda},$
(146)

as well as

$\begin{split}[\hat{{\mathcal{R}}}_{I},V_{\mu_{1}\mu_{2}\cdots}]&=[Z^{R}_{I},V_{\mu_{1}\mu_{2}\cdots}]\\\
&=R_{I\mu_{1}\lambda}V_{\lambda\mu_{2}\cdots}+R_{I\mu_{2}\lambda}V_{\mu_{1}\lambda\cdots}+\cdots\end{split}$
(147)

The operators $Z^{R}_{I}$ and $\hat{{\mathcal{R}}}_{I}$ are related through

$Z^{R}_{I}=\hat{{\mathcal{R}}}_{I}+C_{I}.$ (148)

Therefore

$\hat{{\mathcal{R}}}_{I}|0\rangle=0,\qquad\langle
0|\hat{{\mathcal{R}}}_{I}=-2\langle 0|C_{I}.$ (149)

The operators $\hat{{\mathcal{R}}}_{I}$ and $Z^{R}_{I}$ have identical
commutation properties on purely multiplicative fields. The
$\hat{{\mathcal{R}}}_{I}$ are simpler than $Z^{R}_{I}$ when acting on states
on the right (since they vanish on coordinate scalars), while the $Z^{R}_{I}$
have the virtue of being antihermitian.

###  B.3 Derivation of Eqs. (55)

The first relation in (55), the trace in the coordinate-scalar space, follows
from the fact that $Z^{R}_{I}$ coincides with $-C_{I}$ when acting on scalars
on the left,

$\langle\phi|Z^{R}_{I}=\langle 0|\phi Z^{R}_{I}=\langle
0|(Z^{R}_{I}\phi-[Z^{R}_{I},\phi])=-\langle\phi|C_{I}.$ (150)

For the second relation, the trace in the coordinate-vector space, consider an
operator ${\mathcal{O}}^{\mu}{}_{\nu}$ acting on the coordinate-vector space,
$({\mathcal{O}}V)^{\mu}={\mathcal{O}}^{\mu}{}_{\nu}V^{\nu}$. Disregarding the
gauge-space sector for simplicity, the trace can be written as

$\operatorname{tr}_{1}({\mathcal{O}}^{\mu}{}_{\nu})=\sum_{A}u^{A}_{\mu}{\mathcal{O}}^{\mu}{}_{\nu}u^{\nu}_{A}$
(151)

where $u_{A}^{\mu}(x)$ is any local basis of vectors at $x$ and
$u^{A}_{\mu}(x)$ its dual basis, $u^{A}_{\mu}u_{B}^{\mu}=\delta^{A}_{B}$. When
${\mathcal{O}}^{\mu}{}_{\nu}$ is purely multiplicative (i.e., it does not
contain $\nabla_{\mu}$ nor $Z^{R}_{I}$), the trace is simply

$\operatorname{tr}_{1}({\mathcal{O}}^{\mu}{}_{\nu})={\mathcal{O}}^{\mu}{}_{\nu}u^{A}_{\mu}u_{A}^{\nu}={\mathcal{O}}^{\mu}{}_{\nu}g^{\nu}{}_{\mu}={\mathcal{O}}^{\mu}{}_{\mu}.$
(152)

If the operator has a factor $Z^{R}_{I}$ on the left, where $I$ is any string
of coordinate indices, without loss of generality we can assume that it has
the form $Z^{R}_{I}{\mathcal{H}}^{I\mu}{}_{\nu}$. That is, the row-column
indices $\mu\nu$ are not in $Z_{I}^{R}$, and all the indices in $I$ are
different. For instance $Z^{R}_{\alpha}{}^{\alpha\mu}X_{\nu}$ can be rewritten
as $Z^{R}_{\alpha\beta\lambda}g^{\alpha\beta}g^{\mu\lambda}X_{\nu}$. Using now

$\begin{split}\langle u^{A}_{\mu}|Z^{R}_{I}&=\langle
0|u^{A}_{\mu}Z^{R}_{I}=\langle
0|(Z^{R}_{I}u^{A}_{\mu}-[Z^{R}_{I},u^{A}_{\mu}])\\\ &=\langle
0|(-C_{I}u^{A}_{\mu}+R_{I}{}^{\lambda}{}_{\mu}u^{A}_{\lambda})=\langle
u^{A}_{\lambda}|(R_{I}{}^{\lambda}{}_{\mu}-g^{\lambda}{}_{\mu}C_{I}),\end{split}$
(153)

it follows that

$\begin{split}\operatorname{tr}_{1}(Z^{R}_{I}{\mathcal{O}}^{\mu}{}_{\nu})&=\sum_{A}u^{A}_{\mu}Z^{R}_{I}{\mathcal{O}}^{\mu}{}_{\nu}u^{\nu}_{A}=\sum_{A}u^{A}_{\mu}(R_{I}{}^{\mu}{}_{\lambda}-g^{\mu}{}_{\lambda}C_{I}){\mathcal{O}}^{\lambda}{}_{\nu}u^{\nu}_{A}\\\
&=\operatorname{tr}_{1}\left((R_{I}{}^{\mu}{}_{\lambda}-g^{\mu}{}_{\lambda}C_{I}){\mathcal{O}}^{\lambda}{}_{\nu}\right).\end{split}$
(154)

This proves Eqs. (55). If the operator contains more than one factor $Z^{R}$
on the left, the procedure is applied recursively,

$\begin{split}&\operatorname{tr}_{1}(Z^{R}_{I}Z^{R}_{J}{\mathcal{O}}^{\mu}{}_{\nu})\\\
&=\operatorname{tr}_{1}\left((R_{I}{}^{\mu}{}_{\lambda}-g^{\mu}{}_{\lambda}C_{I})Z^{R}_{J}{\mathcal{O}}^{\lambda}{}_{\nu}\right)\\\
&=\operatorname{tr}_{1}\left(Z^{R}_{J}(R_{I}{}^{\mu}{}_{\lambda}-g^{\mu}{}_{\lambda}C_{I}){\mathcal{O}}^{\lambda}{}_{\nu}-[Z^{R}_{J},R_{I}{}^{\mu}{}_{\lambda}-g^{\mu}{}_{\lambda}C_{I}]{\mathcal{O}}^{\lambda}{}_{\nu}\right).\end{split}$
(155)

##  Appendix C Canonical form of $\Gamma_{S}$

The following formulas display the expression of the operators in
$\Gamma_{S,n}$, $n=1,2,3,4$ in Eq. (65) using the basis of operators in (70)
to bring them to the canonical form in (67), by means of the commutation
identities in (69). These expressions hold in any spacetime dimension. Eqs.
(156), (157), (158), and (159) correspond to $\Gamma_{L,1}$, $\Gamma_{L,2}$,
$\Gamma_{L,3}$, and $\Gamma_{L,4}$, respectively.

$\begin{split}\Delta Y_{\mu\nu}&={\mathcal{O}}_{19}Y_{\mu\nu}\\\
\Delta_{M}W&={\mathcal{O}}_{2}\,W\\\
\frac{1}{2}\Delta_{M}\\{P_{\mu},\nabla_{\mu}\\}&=({\mathcal{O}}_{1})_{\mu}P_{\mu}-\frac{1}{2}{\mathcal{O}}_{2}\,P_{\mu\mu}\end{split}$
(156)

$\begin{split}\Delta Y_{\alpha\mu}\Delta
Y_{\mu\beta}&={\mathcal{O}}_{20}\,Y_{\alpha\mu}Y_{\mu\beta}+O(\nabla^{-5})\\\
\Delta_{M}\Phi_{\mu}\Delta\Phi_{\mu}&={\mathcal{O}}_{10}\,\Phi_{\mu}^{2}+O(\nabla^{-5})\\\
\Delta_{M}\nabla_{\mu}{\mathcal{M}}_{\mu\nu}\Delta{\mathcal{M}}_{\nu\alpha}\nabla_{\alpha}&=({\mathcal{O}}_{3})_{\mu\nu}\,{\mathcal{M}}_{\mu\alpha}{\mathcal{M}}_{\alpha\nu}+2({\mathcal{O}}_{6})_{\mu\nu\alpha}\,{\mathcal{M}}_{\nu\mu\beta}{\mathcal{M}}_{\beta\alpha}+4({\mathcal{O}}_{12})_{\mu\nu\alpha\beta}\,{\mathcal{M}}_{\alpha\nu\mu\rho}{\mathcal{M}}_{\rho\beta}\\\
&\quad-({\mathcal{O}}_{11})_{\mu\nu}\,(2{\mathcal{M}}_{\nu\mu\alpha}{\mathcal{M}}_{\beta\alpha\beta}+{\mathcal{M}}_{\alpha\alpha\mu\beta}{\mathcal{M}}_{\beta\nu}+2{\mathcal{M}}_{\alpha\nu\mu\beta}{\mathcal{M}}_{\beta\alpha}+{\mathcal{M}}_{\alpha\beta}{\mathcal{M}}_{\beta\nu}{\mathcal{R}}_{\mu\alpha}\\\
&\quad\quad-2F_{\mu\alpha}{\mathcal{M}}_{\alpha\beta}{\mathcal{M}}_{\beta\nu})-({\mathcal{O}}_{5})_{\mu}\,({\mathcal{M}}_{\mu\nu}{\mathcal{M}}_{\alpha\nu\alpha}+{\mathcal{M}}_{\nu\mu\alpha}{\mathcal{M}}_{\alpha\nu})+O(\nabla^{-5})\\\
\Delta_{M}\Phi_{\mu}\Delta{\mathcal{M}}_{\mu\nu}\nabla_{\nu}&=({\mathcal{O}}_{5})_{\mu}\,\Phi_{\nu}{\mathcal{M}}_{\nu\mu}+{\mathcal{O}}_{10}\,(-\Phi_{\mu}{\mathcal{M}}_{\nu\mu\nu}-\Phi_{\mu\nu}{\mathcal{M}}_{\nu\mu})+2({\mathcal{O}}_{11})_{\mu\nu}\,\Phi_{\mu\alpha}{\mathcal{M}}_{\alpha\nu}+O(\nabla^{-5})\\\