# A generalized Stueckelberg-Higgs gauge theory

C. A. Bonin1 , G. B. de Gracia2, A. A. Nogueira3 and B. M. Pimentel2
1Departamento de Matemática e Estatística, Universidade Estadual de Ponta
Grossa (DEMAT/UEPG),
Avenida Carlos Cavalcanti, CEP 4748, 84030-900, Ponta Grossa, PR, Brazil
2 Instituto de Física Teórica, Universidade Estadual Paulista (IFT/Unesp)
Rua Dr. Bento Teobaldo Ferraz 271, Bloco II Barra Funda, CEP 01140-070 São
Paulo, SP, Brazil
3 Instituto de Física, Universidade Federal de Goiás (IF/UFG),
Av. Esperança, CEP 74690-900, Goiânia, Goiás, Brasil
carlosbonin@gmail.comgb9950<EMAIL_ADDRESS>(corresponding
<EMAIL_ADDRESS>
###### Abstract

The aim of this work is to discuss and explore some generalized aspects of
generation of photon mass respecting gauge symmetry. We introduce the
generalized Stueckelberg and Higgs gauge theories and present the classical
and quantum frameworks related to them. We construct the quantum theory by
writing the transition amplitude in the Fadeev-Senjanovic formalism and put it
in a covariant form by the Fadeev-Popov method. We also analyze the
independence of the transition amplitude by gauge parameters via BRST
symmetry. Even in the generalized context, the Stueckelberg structure
influences the quantization process of the Higgs theory in the ’t Hooft’s
fashion, in which an intimate relationship between the Stueckelberg
compensating field and the Goldstone boson arises.

Keywords: Constrained Systems; Gauge Theories

##  1 Introductory Aspects

The current understanding of how to formulate and interpret the laws of nature
describing the fundamental interactions between matter and radiation is in
terms of gauge theories [1, 2, 3]. The Quantum Mechanical view of such
interactions is through the assumption of the existence of mediating
particles. The origin of this idea can be traced back to the (now famous)
Yukawa theory [4]. The core of the Yukawa theory is that the particular
interaction conceived in it is short-ranged due to a non-vanishing mass for
the corresponding mediating particle. That particle turned out to the pion,
which is associated to a scalar field. However, concerning gauge interactions,
extensive studies of hypothetical mass for vector fields were done by Proca
[5] and nowadays there are numerous experimental efforts to search for
(nonzero) mass for photons [6]. As the question of mass in the context of
gauge symmetry attracted more attention, Stueckelberg explored it with its
gauge-symmetry mass generation mechanisms [7, 8]. In the present day,
Stueckelberg’s particle find its application in Cosmology, as an Ultralight
Dark Matter Candidate used to explain the anomalous rotation curves of
galaxies [50]. It is quite natural to think that these ideas influenced the
spontaneous symmetry-breaking mass generation mechanism (of the Higgs type)
[9, 10, 11, 12] in which there is an interplay between the gauge fields and
the Nambu-Goldstone bosons, whose general result is the gauge field acquiring
a nonvanishing mass [13]. The Higgs model led to the formulation of the
Weinberg-Salam electroweak theory, in which massive vector bosons and photons
mediate the interaction [14]. These concepts paved the way to the
understanding of general and important properties related to the mechanisms of
mass generation for gauge fields such as unitarity and renormalization [15].
The concept of mass generation also appears in other contexts, like in the
Nambu-Jona-Lasinio model [16, 17] with the dynamical breaking of chiral
symmetry [18] and in gauged Thirring model where the gauge symmetry does not
prohibit the photon to acquiring mass dynamically, as shown by some of us
[19]. It is also known that in plasma physics there is a shielding phenomenon
where the interaction mediated by the gauge field becomes short-ranged with
the gauge field acquiring a mass - the so-called Debye mass [20] and there is
also the possibility of exploring the concept of restoration of symmetry
(Jackiw-Dolan) above a critical temperature [21].

On the other hand Bopp-Podolsky-Schwed proposed a higher-order derivative
Lagrangian with gauge symmetry in which, by dimensional reasons, an extra free
parameter is introduced [22]. That can be identified as the Podolsky’s mass.
This generalized electrodynamics gives the correct finite expressions for
self-force of charged particles, as shown by Frenkel-Zayats [23], interesting
effects produced by the presence of sources [24], and, as general rule, the
theory offers better ultraviolet behavior in a sense closely related to the
Pauli-Villars-Rayski regularization scheme [25, 26]. Although we have a
characteristic length (the inverse of the Podolsky mass) associated with the
size of the charged particles, seen for example in the electron-positron
scattering [27], the possible source for this mass, derive from the structure
behind the self-interaction of the particles and their sizes, is mysterious.
But it is worth noticing that Podolsky electrodynamics breaks the dual
symmetry [28] and lead us to speculate the existence of some mechanism that
breaks the dual symmetry and generate mass. From the point of view of degrees
of freedom, generalized photons have five degrees (Maxwell plus Proca) [29]
and in the free case (without interactions) generalized electrodynamics is
equivalent to the Lee-Wick theory, wherein the gauge field is complex [30,
31]. These five degrees of freedom also have influence in the study of zero
point energy [32, 33].

The generalized electrodynamics respect some important general properties such
as causality from the point of view of causal perturbation theory [34],
stability from the concept of Lagrangian anchor [35], and renormalizability as
well [36].

In what follows, we will try to understand the mass generation mechanism in
the generalized electrodynamics context, following the ideas in the previous
works [37, 38, 39]. This work is organized as follows. In section 2 we define
what we call the generalized Stueckelberg model and we discuss its classical
aspects and construct the phase space for the model. In section 3, we
construct the transition amplitude by the Fadeev-Senjanovic procedure and
write it in the Fadeev-Popov covariant form. In Sec. 4 we analyse the BRST
symmetry and the invariance of the physics by gauge choices. In Sec. 5 we use
the Stueckelberg structure to quantize the generalized Higgs gauge theory.
Finally section 6 contains our conclusions and perspectives.

##  2 The Generalized Stueckelberg Gauge Theory

In the present section we will introduce the generalized Stueckelberg model
and we will make a brief discussion of its canonical structure, including the
constraint analysis, the canonical Hamiltonian, and its equations of motion.

We start with the following Lagrangian density

$\displaystyle{\cal
L}_{S}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}a^{2}_{0}\partial_{\nu}F^{\nu\mu}\partial_{\rho}F^{\rho}_{\phantom{\rho}\mu}+\frac{1}{2}M^{2}S_{\mu}{S^{*}}^{\mu};$
(2.1) $\displaystyle
S_{\mu}=A_{\mu}+\frac{1}{m_{s}}\partial_{\mu}B+\frac{i}{m_{p}^{2}}\partial_{\lambda}F^{\lambda}_{\phantom{\lambda}\mu},$
(2.2)

in which $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ are the
components of the field-strength tensor, $B$ the Stueckelberg compensating
field, and $a^{2}_{0}$, $M$, $m_{s}$, and $m_{P}$ are nonvanishing real
parameters. (2.1) is invariant under the following $U\left(1\right)$ local
gauge transformation

$\displaystyle A_{\mu}\left(x\right)\rightarrow
A_{\mu}\left(x\right)+\partial_{\mu}\alpha\left(x\right);$ (2.3)
$\displaystyle B\left(x\right)\rightarrow
B\left(x\right)-m_{s}\alpha\left(x\right),$ (2.4)

where $\alpha$ is a (smooth) Poincaré-scalar field.

The previous Lagrangian (2.1) can be rewritten in a more convenient form as

${\cal
L}_{S}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}a^{2}\partial_{\mu}F^{\mu\beta}\partial^{\nu}F_{\nu\beta}+\frac{1}{2}M^{2}\left(A_{\mu}+\frac{1}{m_{s}}\partial_{\mu}B\right)\left(A^{\mu}+\frac{1}{m_{s}}\partial^{\mu}B\right),$
(2.5)

where we have defined the (squared) changed Podolsky parameter
$a^{2}={1}/{m^{2}}$ according to

$a^{2}\equiv a^{2}_{0}+\left(\frac{M}{m_{p}^{2}}\right)^{2}.$ (2.6)

The Euler-Lagrange equations that follow from this Lagrangian density read

$\displaystyle(1+a^{2}\Box)(\eta_{\mu\nu}\Box-\partial_{\mu}\partial_{\nu})A^{\nu}+M^{2}\left(A_{\mu}+\frac{\partial_{\mu}B}{m_{s}}\right)=$
$\displaystyle\,0;$ (2.7) $\displaystyle\partial_{\mu}A^{\mu}+\frac{\Box
B}{m_{s}}=$ $\displaystyle\,0,$ (2.8)

with explicit gauge symmetry, in view of the transformations (2.3-2.4).
Furthermore, it interesting to notice that (2.8) can also be obtained as the
derivative of (2.7).

Now we will construct the phase space in the Hamilton formalism for the theory
using the Dirac’s methodology. In order to write the Hamiltonian we will seek
firstly the generator of the spacetime translations in the view of the Noether
theorem, whose components are [40, 41]

$\mathcal{T}^{\mu}_{\phantom{\mu}\lambda}=\frac{\partial{\cal
L}}{\partial(\partial_{\mu}A_{\nu})}\partial_{\lambda}A_{\nu}-\partial_{\theta}\left[\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\partial_{\theta}A_{\nu})}\right]\partial_{\lambda}A_{\nu}+\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\partial_{\theta}A_{\nu})}\partial_{\theta}\partial_{\lambda}A_{\nu}+\frac{\partial{\cal
L}}{\partial(\partial_{\mu}B)}\partial_{\lambda}B-\eta^{\mu}_{\lambda}{\cal
L}.$ (2.9)

The canonical Hamiltonian $H_{c}=\int
d^{3}x\,\mathcal{T}^{0}_{\phantom{0}0}\equiv\int d^{3}x\mathcal{H}_{c}$ is
written explicitly as

$H_{c}=\int
d^{3}x\left(\Pi^{\mu}\partial_{0}A_{\mu}+\Phi^{\mu}\partial_{0}\Gamma_{\mu}+\varPi\partial_{0}B-{\cal
L}\right),$ (2.10)

where $\Gamma_{\mu}\equiv\partial_{0}A_{\mu}$ and with the following
Ostrogradsky canonical momenta

$\displaystyle\Pi^{\mu}$ $\displaystyle=$ $\displaystyle\,F^{\mu
0}+\frac{1}{m_{s}^{2}}\left(\eta^{\mu
j}\partial_{j}\partial_{l}F^{l0}-\partial_{0}\partial_{\nu}F^{\nu\mu}\right);$
(2.11) $\displaystyle\Phi^{\mu}$ $\displaystyle=$
$\displaystyle\,\frac{1}{m_{s}^{2}}\left(\partial_{\nu}F^{\nu\mu}-\eta^{\mu
0}\partial_{j}F^{j0}\right);$ (2.12) $\displaystyle\varPi$ $\displaystyle=$
$\displaystyle\,\frac{M^{2}}{m_{s}}\left(A_{0}+\frac{1}{m_{s}}\partial_{0}B\right).$
(2.13)

Some of these momenta furnish the primary constraints of the
theory111Throughout this work the symbol $\approx$ is used to denote weak
equality [42].

$\displaystyle\varphi_{1}\left(x\right)$ $\displaystyle=$
$\displaystyle\,\Phi_{0}\left(x\right)\approx 0;$ (2.14)
$\displaystyle\varphi_{2}\left(x\right)$ $\displaystyle=$
$\displaystyle\,\Pi_{0}\left(x\right)-\overrightarrow{\partial}\cdot\overrightarrow{\Phi}\left(x\right)\approx
0,$ (2.15)

Now, we are able to write the canonical Hamiltonian density (2.10) in its
explicit form as

$\displaystyle\mathcal{H}_{C}=$
$\displaystyle\,-\overrightarrow{\Pi}\cdot\overrightarrow{\Gamma}-\frac{m^{2}}{2}\left|\overrightarrow{\Phi}\right|^{2}+\Phi^{k}\partial_{j}F^{jk}-\frac{1}{2}\left|\overrightarrow{\Gamma}+\overrightarrow{\partial}A^{0}\right|^{2}-\frac{1}{2m^{2}}\left(\overrightarrow{\partial}^{2}A^{0}+\overrightarrow{\partial}\cdot\overrightarrow{\Gamma}\right)^{2}+$
$\displaystyle\,+\frac{1}{4}F^{jk}F_{jk}+\frac{m^{2}_{s}}{2M^{2}}\varPi^{2}-m_{s}A^{0}\varPi+\frac{M^{2}}{2}\left|\overrightarrow{A}-\frac{1}{m_{s}}\overrightarrow{\partial}B\right|^{2}.$
(2.16)

The Ostrogradsky analysis leads us to the following non-vanishing fundamental
Poisson brackets:

$\displaystyle\left\\{A^{\mu}\left(\mathbf{x},t\right),\Pi_{\nu}\left(\mathbf{y},t\right)\right\\}_{P}$
$\displaystyle=$
$\displaystyle\,\delta^{\mu}_{\nu}\delta^{(3)}(\mathbf{x}-\mathbf{y});$ (2.17)
$\displaystyle\left\\{\Gamma^{\mu}\left(\mathbf{x},t\right),\Phi_{\nu}\left(\mathbf{y},t\right)\right\\}_{P}$
$\displaystyle=$
$\displaystyle\,\delta^{\mu}_{\nu}\delta^{(3)}(\mathbf{x}-\mathbf{y});$ (2.18)
$\displaystyle\left\\{B\left(\mathbf{x},t\right),\varPi\left(\mathbf{y},t\right)\right\\}_{P}$
$\displaystyle=$ $\displaystyle\,\delta^{(3)}(\mathbf{x}-\mathbf{y}).$ (2.19)

Now, following the Dirac methodology, we define the primary Hamiltonian
density ${\cal H}_{p}={\cal H}_{c}+C_{1}\varphi^{1}+C_{2}\varphi^{2}$, where
$\varphi^{1}$ and $\varphi^{2}$ are the constraints (2.14) and (2.15) and
$C_{1}$ and $C_{2}$ are the respective Lagrange multipliers. The consistence
conditions for each constraint, namely,

$\partial_{0}\varphi_{j}\left(\mathbf{x},t\right)=\int
d^{3}{y}\\{\varphi_{j}\left(\mathbf{x},t\right),{\cal
H}_{p}\left(\mathbf{y},t\right)\\}_{P}\approx 0,$ (2.20)

gives us only one additional constraint: the Gauss’ law

$\varphi_{3}\left(x\right)\equiv
m_{s}\varPi\left(x\right)-\overrightarrow{\partial}\cdot\overrightarrow{\Pi}\left(x\right)\approx
0,$ (2.21)

whose consistency condition is tautological. As such, we have the full set of
constraints $\left\\{\varphi_{1},\varphi_{2},\varphi_{3}\right\\}$ and the
extended Hamiltonian density ${\cal H}_{E}={\cal H}_{p}+C_{3}\varphi^{3}$ with
a new Lagrange multiplier $C_{3}$. Moreover, this set of constraints is first
class, which means
$\left\\{\varphi_{j}\left(\mathbf{x},t\right),\varphi_{k}\left(\mathbf{y},t\right)\right\\}_{P}=0$
for all $j,k\in\left\\{1,2,3\right\\}$.

Using the extended Hamiltonian as the generator of the time evolution, we find
the Hamilton equations222For any field $\phi\left(\mathbf{x},t\right)$ we have
$\dot{\phi}\left(\mathbf{x},t\right)=\left\\{\phi\left(\mathbf{x},t\right),\int
d^{3}y\mathcal{H}_{e}\left(\mathbf{y},t\right)\right\\}_{P}$.

$\displaystyle\dot{A}^{\mu}\left(x\right)=$
$\displaystyle\,\delta^{\mu}_{k}\left[\Gamma^{k}\left(x\right)-\partial_{k}C_{3}\left(x\right)\right]+\delta^{\mu}_{0}C_{2}\left(x\right);$
(2.22) $\displaystyle\dot{\Gamma}^{\mu}\left(x\right)=$
$\displaystyle\,-\delta^{\mu}_{k}\left[m^{2}\Phi_{k}\left(x\right)+\partial_{j}F^{jk}\left(x\right)-\partial_{k}C_{2}\left(x\right)\right]+\delta^{\mu}_{0}C_{1}\left(x\right);$
(2.23) $\displaystyle\dot{B}\left(x\right)=$
$\displaystyle\,\frac{m_{s}^{2}}{M^{2}}\varPi\left(x\right)-m_{s}A^{0}\left(x\right)+m_{s}C_{3}\left(x\right);$
(2.24) $\displaystyle\dot{\Pi}_{\mu}\left(x\right)=$
$\displaystyle\,-\delta^{k}_{\mu}\overrightarrow{\partial}^{2}\Phi_{k}\left(x\right)-\partial^{j}_{\mu}\overrightarrow{\partial}\cdot\overrightarrow{\Phi}\left(x\right)-\delta^{0}_{\mu}\left\\{\overrightarrow{\partial}\cdot\overrightarrow{\Gamma}\left(x\right)+\overrightarrow{\partial}^{2}A_{0}\left(x\right)\right.$
$\displaystyle\,\left.-\frac{1}{m^{2}}\overrightarrow{\partial}^{2}\left[\overrightarrow{\partial}^{2}A^{0}\left(x\right)+\overrightarrow{\partial}\cdot\overrightarrow{\Gamma}\left(x\right)\right]\right\\}-\frac{1}{2}\left(\delta^{j}_{\mu}\partial_{k}-\delta^{k}_{\mu}\partial_{j}\right)F^{kj}\left(x\right)$
$\displaystyle\,+\delta^{0}_{\mu}m_{s}\varPi\left(x\right)-M^{2}\delta^{k}_{\mu}\left[A^{k}-\frac{1}{m_{s}}\partial_{k}B\left(x\right)\right];$
(2.25) $\displaystyle\dot{\Phi}_{\mu}\left(x\right)=$
$\displaystyle\,\delta^{k}_{\mu}\left\\{\Gamma^{k}\left(x\right)+\partial_{k}A^{0}\left(x\right)-\Pi_{k}\left(x\right)-\frac{1}{m^{2}}\partial_{k}\left[\overrightarrow{\partial}^{2}A^{0}\left(x\right)+\overrightarrow{\partial}\cdot\overrightarrow{\Gamma}\left(x\right)\right]\right\\};$
(2.26) $\displaystyle\dot{\varPi}\left(x\right)=$
$\displaystyle\,-\frac{M^{2}}{m_{s}}\left[\overrightarrow{\partial}\cdot\overrightarrow{A}\left(x\right)-\frac{1}{m_{s}}\overrightarrow{\partial}^{2}B\left(x\right)\right].$
(2.27)

As we can see, there are some arbitrariness in the Hamilton equations due to
the Lagrange multiplier dependence. Dirac conjectured that the first-class
constraints are the generators of the gauge transformations [43]. In order to
set this arbitrariness aside, we must impose (three) extra constraints on the
system in such a way that the new set of constraints
$\Omega=\left\\{\omega_{1},\omega_{2},\omega_{3},\omega_{4},\omega_{5},\omega_{6}\right\\}$
is second class. This means that for any $j\in\left\\{1,2,3,4,5,6\right\\}$
there is at least one $k\in\left\\{1,2,3,4,5,6\right\\}$ for which
$\left\\{\omega_{j}\left(\mathbf{x},t\right),\omega_{k}\left(\mathbf{y},t\right)\right\\}_{P}\neq
0$. As a result, the Hamilton equations are dynamically consistent with the
Euler-Lagrange equations. In order to do that, we firstly notice that, thanks
to (2.27), the time derivative of equation (2.24) leads to

$\partial_{\mu}A^{\mu}\left(x\right)+\frac{\Box}{m_{s}}B\left(x\right)=\frac{\partial
C_{3}\left(x\right)}{\partial t}.$ (2.28)

Comparison with (2.8) tells us that the Lagrange multiplier $C_{3}$ must be
time independent. However, the left-hand-side of (2.28) is a Lorentz scalar,
which means its right-hand-side must be invariant under Lorentz as well. The
only way a Lorentz scalar is time-independent is if it is event-independent.
Therefore, the Lagrange multiplier $C_{3}$ is a constant which, without loss
of generality, may be taken to be zero.

Before we go into details as how to find the other unknown functions, whose
revelations are more involved, let us notice that the zero-th components of
equations (2.22) and (2.23) read

$\displaystyle C_{1}\left(x\right)=$
$\displaystyle\,\dot{\Gamma}^{0}\left(x\right);$ (2.29) $\displaystyle
C_{2}\left(x\right)=$ $\displaystyle\,\dot{A}^{0}\left(x\right).$ (2.30)

The main goal of the present procedure is obtaining Hamilton equations which
are equivalent to the Euler-Lagrange equations. We already used one of them to
discover the Lagrange multiplier $C_{3}$. In order to find the other
functions, we shall use the other Euler-Lagrange equations. To be more
precise, let us look into the zero-th component of (2.7):

$\displaystyle\left[M^{2}-\left(1+\frac{\Box}{m^{2}}\right)\overrightarrow{\partial}^{2}\right]A^{0}\left(x\right)=\left(1+\frac{\Box}{m^{2}}\right)\frac{\partial\left(\overrightarrow{\partial}\cdot\mathbf{A}\left(x\right)\right)}{\partial
t}-\frac{M^{2}}{m_{s}}\frac{\partial B\left(x\right)}{\partial t}.$ (2.31)

This equation can be solved formally as

$\displaystyle
A^{0}\left(x\right)=\left[M^{2}-\left(1+\frac{\Box}{m^{2}}\right)\overrightarrow{\partial}^{2}\right]^{-1}\frac{\partial}{\partial
t}\left[\left(1+\frac{\Box}{m^{2}}\right)\overrightarrow{\partial}\cdot\mathbf{A}\left(x\right)-\frac{M^{2}}{m_{s}}B\left(x\right)\right].$
(2.32)

In complete analogy with [52], we choose the modified generalized Coulomb
condition

$\left(1+\frac{\Box}{m^{2}}\right)\overrightarrow{\partial}\cdot\mathbf{A}\left(x\right)\approx\frac{M^{2}}{m_{s}}B\left(x\right).$
(2.33)

This condition and its consistency - that is, the fact that it must hold for
all time - imply $A^{0}\left(x\right)\approx 0$ and
$\dot{A}^{0}\left(x\right)=\Gamma^{0}\left(x\right)\approx 0$. These
equalitlies, through relations (2.29) and (2.30), define the last two unknown
Lagrange multiplier to be both identically equal to zero as well.

So, we are lead to choose the following gauge conditions:

$\displaystyle\Sigma_{1}\left(x\right)\equiv$
$\displaystyle\,\Gamma^{0}\left(x\right)\approx 0;$ (2.34)
$\displaystyle\Sigma_{2}\left(x\right)\equiv$
$\displaystyle\,A^{0}\left(x\right)\approx 0;$ (2.35)
$\displaystyle\Sigma_{3}\left(x\right)\equiv$
$\displaystyle\,\left(1+\frac{\Box}{m^{2}}\right)\overrightarrow{\partial}\cdot\mathbf{A}\left(x\right)-\frac{M^{2}}{m_{s}}B\left(x\right)\approx
0.$ (2.36)

Taking into account the first-class constraints (2.14), (2.15), and 2.21), as
well as the gauge conditions (2.34), (2.35), and (2.36), we define:

$\displaystyle\omega_{j}\left(x\right)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\varphi_{j}\left(x\right),&\mbox{for
}j\in\left\\{1,2,3\right\\};\\\ \Sigma_{j-3}\left(x\right)&\mbox{for
}j\in\left\\{4,5,6\right\\}.\end{array}\right.$ (2.39)

Therefore, we achieved our goal that was to find a set of constraints that is
second class:
$\Omega=\left\\{\omega_{1},\omega_{2},\omega_{3},\omega_{4},\omega_{5},\omega_{6}\right\\}$,
as evidenced by the nonvanishing determinant of the matrix
$\left[\Omega_{jk}\right]$, whose $jk$-th component is
$\Omega_{jk}\equiv\left\\{\omega_{j}\left(\mathbf{x},t\right),\omega_{k}\left(\mathbf{y},t\right)\right\\}_{P}$:

$\displaystyle\left[\Omega_{jk}\right]$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccccc}0&0&0&-1&0&0\\\ 0&0&0&0&-1&0\\\
0&0&0&0&0&\left(1+\frac{\Box}{m^{2}}\right)\overrightarrow{\partial}^{2}+M^{2}\\\
1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&-\left(1+\frac{\Box}{m^{2}}\right)\overrightarrow{\partial}^{2}-M^{2}&0&0&0\end{array}\right]\delta^{\left(3\right)}\left(\mathbf{x}-\mathbf{y}\right).$
(2.46)

Before proceeding to the quantization of the theory, let us make a quick
counting of the degrees of freedom. We have a set of $18$ variables
$\left\\{A^{\mu}\left(x\right),\Gamma^{\mu}\left(x\right),B,\Pi_{\nu}\left(x\right),\Phi_{\mu}\left(x\right),\varPi\left(x\right)\right\\}$
and $6$ constraints - the $\Omega$ set - to describe the phase space. So, we
ended up with $12$ independent fields in the phase space or, alternatively,
$6$ degrees of freedom in the configuration space. This is the same number of
degrees of freedom a theory comprising the Podolsky field with a real scalar
field has $\left(5+1\right)$ which, in its turn, it the same one as Maxwell,
Proca, and Scalar $(2+3+1)$ [19]. This result is interesting and show how the
gauge symmetry shield the longitudinal degree of freedom.

##  3 The (quantum) transition amplitude

Since we have constructed the phase space of the model following Dirac
methodology, in this section we will write down the transition amplitude of
the quantum theory in a covariant fashion [44].

With the full set of constraints, the next step is to obtain the transition
amplitude for the quantum theory. It is originally written as

$\displaystyle
Z=\int\left[\prod_{\mu=0}^{3}\mathcal{D}A^{\mu}\mathcal{D}\Pi_{\mu}\mathcal{D}\Gamma^{\mu}\mathcal{D}\Phi_{\mu}\right]\mathcal{D}B\mathcal{D}\varPi\sqrt{\det{\left[\Omega_{jk}\right]}}\left[\prod_{l=1}^{6}\delta\left(\omega_{l}\right)\right]e^{i\int
d^{4}x\mathcal{L}_{C}}$ (3.1)

where $\mathcal{L}_{C}$ is the Lagrangian density in the canonical form,
defined in terms of (2.16) as

$\displaystyle\mathcal{L}_{C}=\Pi_{\mu}\dot{A}^{\mu}+\Phi_{\mu}\dot{\Gamma}^{\mu}+\varPi\dot{B}-\mathcal{H}_{C}.$
(3.2)

After integrating over some variables we are left with the following
transition amplitude:

$\displaystyle
Z=\int\left[\prod_{\mu=0}^{3}\mathcal{D}A^{\mu}\right]\mathcal{D}B\det\left(\left(1+\frac{\Box}{m^{2}}\right)\overrightarrow{\partial}^{2}+M^{2}\right)\delta\left(\left(1+\frac{\Box}{m^{2}}\right)\overrightarrow{\partial}\cdot\mathbf{A}-\frac{M^{2}}{m_{s}}B\right)e^{i\int
d^{4}x\mathcal{L}_{S}},$ (3.3)

where $\mathcal{L}_{S}$ is the Lagrangian density (2.5).

Although the transition amplitude above is correct, it is not Lorentz
covariant. This is due to the fact that it is written in the modified
generalized Coulomb gauge (2.33). Such a gauge is valid in a particular
inertial frame. In order to write $Z$ is a covariant form, that is, a form
that remains the same in every inertial frame, we define the modified
generalized Lorenz gauge as

$\displaystyle\left(1+\frac{\Box}{m^{2}}\right)\partial_{\mu}A^{\mu}\left(x\right)=g\frac{M^{2}}{m_{s}}B\left(x\right)+f\left(x\right)$
(3.4)

where $g$ is a dimensionless real number and $f$ an arbitrary Lorentz scalar
function.

Using the so-called Fadeev-Popov trick, we can rewrite (3.3) as

$\displaystyle
Z=\int\left[\prod_{\mu=0}^{3}\mathcal{D}A^{\mu}\right]\mathcal{D}B\det\left(\left(1+\frac{\Box}{m^{2}}\right)\Box+gM^{2}\right)\delta\left(\left(1+\frac{\Box}{m^{2}}\right)\partial_{\nu}A^{\nu}-g\frac{M^{2}}{m_{s}}B-f\right)e^{i\int
d^{4}x\mathcal{L}_{S}}.$ (3.5)

Now, we define a measure $\mathcal{M}$ which is a functional of the scalar
function $f$:

$\displaystyle\int d\mathcal{M}\left[f\right]=1.$ (3.6)

As shown in [49], an adequate choice for such a measure is
$d\mathcal{M}\left[f\right]=\mathcal{D}f$
$\exp\left\\{-i\int
d^{4}x\left[f\left(x\right)\right]^{2}/\left[2\left(\xi+0^{+}\right)\right]+\ln\left(-2i\pi\xi\right)/2\right\\}$.
With this, up to a (possibly ill-defined) normmalization constant, the
transition amplitude assumes the simple, covariant form:

$\displaystyle
Z=\int\left[\prod_{\mu=0}^{3}\mathcal{D}A^{\mu}\right]\mathcal{D}B\mathcal{D}\bar{c}\mathcal{D}c\,e^{i\int
d^{4}x\mathcal{L}_{{eff}}},$ (3.7)

where the effective Lagrangian density $\mathcal{L}_{{eff}}$ is defined as

$\displaystyle\mathcal{L}_{{eff}}\equiv\mathcal{L}-\frac{1}{2\xi}\left[\left(1+\frac{\Box}{m^{2}}\right)\partial_{\mu}A^{\mu}-g\frac{M^{2}}{m_{s}}B\right]^{2}+\bar{c}\left[\left(1+\frac{\Box}{m^{2}}\right)\Box+gM^{2}\right]c,$
(3.8)

where $\bar{c}$ and $c$ are Grassmannian scalar fields.

Next, we shall study the BRST symmetry of this theory.

##  4 BRST symmetry and fictitious parameters

As we know, to write the classical Maxwell equations in a covariant form
$A_{\mu}=(A_{0},\vec{A})$ we need 4 degrees of freedom and we have a gauge
symmetry ($A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\alpha$). However, the
physics is described by 2 degrees of freedom, defined by the behavior of
electric and magnetic fields. So, to fix this extra degrees of freedom in
describing electromagnetic fields, we have the Coulomb gauge condition
[$A_{0}=0;\vec{\nabla}\vec{A}=0$]. For other hand, if we want to keep the
covariance explicitly and be less restrictive, we can choose the Lorenz gauge
condition [$\partial_{\mu}A^{\mu}=0$]. Now if we want to quantize the Maxwell
equations in the functional integration formalism, first we make the Dirac
constraint analysis and by the Fadeev-Senjanovic formalism, write the
transition amplitude naturally in the physical Coulomb gauge condition
($Z_{\text{Coulomb}}$). Nevertheless, if we want covariance explicitly, we use
the Fadeev-Popov identity and write the covariant transition amplitude
($Z_{\text{covariant}}$). As we are aware, in the covariant transition
amplitude the Lorenz gauge condition are generalized to the $R_{\xi}$ gauges
[$\frac{1}{2\xi}(\partial_{\mu}A^{\mu})^{2}$] and more than that, we have the
’t Hooft-Veltman gauges
[$\frac{1}{2\xi}(\partial_{\mu}A^{\mu}+gA_{\mu}A^{\mu})^{2}$] [46], in which
we have a fictitious interaction between the ghosts and the gauge field,
characterized by the $g$ parameter. Surprisingly the transition amplitude does
not depend on this $g$ parameter due to the BRST symmetry [37, 38], so this is
a dummy parameter and we can take $g=0$. The previous statement is linked to
the fact that physics (S-matrix) does not depend on $g$ [47], when we include
the fermions and study the effective action. The same line of reasoning can be
applied in generalized electrodynamics (Podolsky), wherein we have generalize
covariant gauge choices and a dummy parameter.

Now the question that arises is whether the same methodology of the fictitious
parameters applies to the ’t Hooft gauge
[$\frac{1}{2\xi}(\partial_{\mu}A^{\mu}+gqv\zeta)^{2}$] [48] when we study the
Higgs model and the spontaneous symmetry breaking. Here $gqv$ represent a
fictitious mass to the Goldstone boson $\zeta$. As the transition amplitude
for the generalized Stueckelberg theory in eq. (3.8) has the same ’t Hooft
gauge structure in the limit of $m\rightarrow\infty$, with the Stueckelberg
field $B$ playing the role of the Goldstone boson, let us take a look on the
parameter $g$ and see if it is fictitious.

Before we proceed, let us define the generating functional333As usual,
$Z_{GF}\left[0,0,0,0\right]=Z$, the transition amplitude.

$\displaystyle
Z_{GF}\left[J,j,\bar{\zeta},\zeta\right]=N\int\left[\prod_{\rho=0}^{3}\mathcal{D}A^{\rho}\right]\mathcal{D}B\mathcal{D}\bar{c}\mathcal{D}ce^{iS\left[A,B,c,\bar{c};J,j,\bar{\zeta},\zeta\right]}$
(4.1)

where $J$, $j$, $\bar{\zeta}$, and $\zeta$ are classical sources (the last two
being also Grassmannian) and

$\displaystyle S\left[A,B,c,\bar{c};J,j,\bar{\zeta},\zeta\right]\equiv\int
d^{4}x\mathcal{L}_{eff}+\int
d^{4}x\left[J_{\mu}\left(x\right)A^{\mu}\left(x\right)+j\left(x\right)B\left(x\right)+\bar{\zeta}\left(x\right)c\left(x\right)+\zeta\left(x\right)\bar{c}\left(x\right)\right].$
(4.2)

We also define the BRS-T transformation as

$\displaystyle A^{\mu}\left(x\right)\rightarrow$
$\displaystyle\,\tilde{A}^{\mu}\left(x\right)\equiv
A^{\mu}\left(x\right)+\delta A^{\mu}\left(x\right);$ (4.3) $\displaystyle
B\left(x\right)\rightarrow$ $\displaystyle\,\tilde{B}\left(x\right)\equiv
B\left(x\right)+\delta B\left(x\right);$ (4.4) $\displaystyle
c\left(x\right)\rightarrow$ $\displaystyle\,\tilde{c}\left(x\right)\equiv
c\left(x\right)+\delta c\left(x\right);$ (4.5)
$\displaystyle\bar{c}\left(x\right)\rightarrow$
$\displaystyle\,\tilde{\bar{c}}\left(x\right)\equiv\bar{c}\left(x\right)+\delta\bar{c}\left(x\right),$
(4.6)

with

$\displaystyle\delta A^{\mu}\left(x\right)\equiv$
$\displaystyle\,\lambda\partial^{\mu}c\left(x\right);$ (4.7)
$\displaystyle\delta B\left(x\right)\equiv$ $\displaystyle\,\lambda
m_{s}c\left(x\right);$ (4.8) $\displaystyle\delta c\left(x\right)\equiv$
$\displaystyle\,0;$ (4.9) $\displaystyle\delta\bar{c}\left(x\right)\equiv$
$\displaystyle\,\frac{\lambda}{\xi}\left[\left(1+\frac{\Box}{m^{2}}\right)\partial_{\mu}A^{\mu}\left(x\right)-g\frac{M^{2}}{m_{s}}B\left(x\right)\right].$
(4.10)

Here, $\lambda$ is a nonvanishing Grassmannian constant.

From now on, we shall write

$\displaystyle\delta_{BRST}Q\equiv\tilde{Q}-Q$ (4.11)

for any quantity $Q$. The symbol $\tilde{Q}$ signifies the quantity $Q$
transformed under (4.3-4.10).

######  Lemma 4.1.

The generating functional (4.1) is invariant under the transformation
(4.3-4.6).

###### Proof.

As a matter of fact, the transformed generating functional is

$\displaystyle\tilde{Z}_{GF}\left[J,j,\bar{\zeta},\zeta\right]=$
$\displaystyle\,N\int\left[\prod_{\rho=0}^{3}\mathcal{D}\tilde{A}^{\rho}\right]\mathcal{D}\tilde{B}\mathcal{D}\tilde{\bar{c}}\mathcal{D}\tilde{c}e^{iS\left[\tilde{A},\tilde{B},\tilde{c},\tilde{\bar{c}};J,j,\bar{\zeta},\zeta\right]}$
(4.12) $\displaystyle=$
$\displaystyle\,N\int\left[\prod_{\rho=0}^{3}\mathcal{D}A^{\rho}\right]\mathcal{D}B\mathcal{D}\bar{c}\mathcal{D}ce^{iS\left[A,B,c,\bar{c};J,j,\bar{\zeta},\zeta\right]}$
(4.13) $\displaystyle=$
$\displaystyle\,Z_{GF}\left[J,j,\bar{\zeta},\zeta\right],$ (4.14)

where we have used the fact that the integration “variables” are dummies. ∎

######  Lemma 4.2.

The integration measure
$\left[\prod_{\rho=0}^{3}\mathcal{D}A^{\rho}\right]\mathcal{D}B\mathcal{D}\bar{c}\mathcal{D}c$
is invariant under the BRS-T transformation (4.3-4.10).

###### Proof.

The integration measure
$\left[\prod_{\rho=0}^{3}\mathcal{D}A^{\rho}\right]\mathcal{D}B\mathcal{D}\bar{c}\mathcal{D}c$
transforms under (4.3-4.10) as

$\displaystyle\left[\prod_{\rho=0}^{3}\mathcal{D}\tilde{A}^{\rho}\right]\mathcal{D}\tilde{B}\mathcal{D}\tilde{\bar{c}}\mathcal{D}\tilde{c}=J\left[\prod_{\rho=0}^{3}\mathcal{D}A^{\rho}\right]\mathcal{D}B\mathcal{D}\bar{c}\mathcal{D}c$
(4.15)

where $J$ is the Jacobian of the transformation which, in simplified notation,
reads

$\displaystyle J$ $\displaystyle=$
$\displaystyle\,\left|\det\left[\begin{array}[]{cccc}\delta^{\mu}_{\nu}&0&\frac{\lambda}{\xi}\left(1+\frac{\Box}{m^{2}}\right)\partial_{\nu}&0\\\
0&1&-\frac{\lambda}{\xi}g\frac{M^{2}}{m_{s}}&0\\\ 0&0&1&0\\\
-\lambda\partial^{\mu}&-\lambda
m_{s}&0&1\end{array}\right]\delta^{\left(4\right)}\left(x-y\right)\right|$
(4.20) $\displaystyle=$
$\displaystyle\delta^{\mu}_{\nu}\delta^{\left(4\right)}\left(x-y\right).$
(4.21)

Therefore,

$\displaystyle\delta_{BRST}\left(\left[\prod_{\rho=0}^{3}\mathcal{D}A^{\rho}\right]\mathcal{D}B\mathcal{D}\bar{c}\mathcal{D}c\right)=0.$
(4.22)

∎

We shall show the parameter $g$ is fictitious through the following

######  Theorem 4.3.

The transition amplitude $Z$ (3.7) is independent of the parameter $g$.

###### Proof.

The derivative of the transition amplitude (3.7) with respect to $g$
reads444We used the widespread notation $\left\langle
O\right\rangle=\int\left[\prod_{\mu=0}^{3}\mathcal{D}A^{\mu}\right]\mathcal{D}B\mathcal{D}\bar{c}\mathcal{D}c\,Oe^{i\int
d^{4}x\mathcal{L}_{{eff}}}$ and $\left\langle
O\right\rangle_{s}=\int\left[\prod_{\mu=0}^{3}\mathcal{D}A^{\mu}\right]\mathcal{D}B\mathcal{D}\bar{c}\mathcal{D}c\,Oe^{iS\left[A,B,c,\bar{c};J,j,\bar{\zeta},\zeta\right]}$
for any operator $O$.

$\displaystyle\frac{\partial Z}{\partial g}=i\int
d^{4}x\left\langle\frac{1}{\xi}\left[\left(1+\frac{\Box}{m^{2}}\right)\partial_{\mu}A^{\mu}\left(x\right)-g\frac{M^{2}}{m_{s}}B\left(x\right)\right]\frac{M^{2}}{m_{s}}B\left(x\right)+M^{2}\bar{c}\left(x\right)c\left(x\right)\right\rangle.$
(4.23)

Computing $\delta_{BRST}Z_{GF}\left[J,j,\bar{\zeta},\zeta\right]$ and taking
into account the lemmas (4.1) and (4.2) furnishes

$\displaystyle i\lambda\int d^{4}z\left\langle
J_{\mu}\left(z\right)\partial^{\mu}c\left(z\right)+j\left(z\right)m_{s}c\left(z\right)+\frac{\zeta\left(z\right)}{\xi}\left[\left(1+\frac{\Box}{m^{2}}\right)\partial_{\mu}A^{\mu}\left(z\right)-g\frac{M^{2}}{m_{s}}B\left(z\right)\right]\right\rangle_{s}=0.$
(4.24)

Applying the functional differential operator ${\delta^{2}}/{\delta
j\left(x\right)\delta\zeta\left(x\right)}$ to both sides of this identity and
subsequently making the classical sources vanish yields

$\displaystyle\lambda\left\langle\frac{1}{\xi}\left[\left(1+\frac{\Box}{m^{2}}\right)\partial_{\mu}A^{\mu}\left(y\right)-g\frac{M^{2}}{m_{s}}B\left(y\right)\right]B\left(x\right)+m_{s}\bar{c}\left(y\right)c\left(x\right)\right\rangle=0.$
(4.25)

Taking into account that $\lambda$ is not zero, choosing $y=x$, and inserting
the result into (4.23) shows us that

$\displaystyle\frac{\partial Z}{\partial g}=0,$ (4.26)

or, in other words, the transition amplitude $Z$ (3.7) is independent of the
parameter $g$. ∎

We usually choose $g=\xi$ when we deal with t’Hooft gauges [49, 48].

##  5 Generalized Higgs gauge theory

What we call the generalized Higgs model is defined by the Lagrangian density

$\displaystyle\mathcal{L}_{H}=$
$\displaystyle\,-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}a_{0}^{2}\partial_{\mu}F^{\mu\nu}\partial_{\rho}F^{\rho}_{\phantom{\rho}\nu}+\nabla_{\mu}\phi\left(\nabla^{\mu}\phi\right)^{*}+\mu_{s}\left|\phi\right|^{2}-\lambda_{s}\left|\phi\right|^{4},$
(5.1)

where the first two terms are like those of (2.1), $\phi$ is a complex scalar
field, $\mu_{s}$ and $\lambda_{s}$ are real parameters with $\lambda_{s}>0$,
and

$\displaystyle\nabla\equiv$
$\displaystyle\,\partial_{\mu}-iq\left(A_{\mu}+iZ_{\mu}\right)$ (5.2)

with $Z_{\mu}\equiv\partial^{\nu}F_{\nu\mu}/m_{P}^{2}$ is an extension of the
minimal coupling.

First of all, we notice that this model is invariant under a
$U\left(1\right)$-gauge symmetry:

$\displaystyle\phi\left(x\right)\rightarrow$
$\displaystyle\,e^{i\alpha\left(x\right)}\phi\left(x\right);$ (5.3)
$\displaystyle\phi^{*}\left(x\right)\rightarrow$
$\displaystyle\,e^{-i\alpha\left(x\right)}\phi^{*}\left(x\right);$ (5.4)
$\displaystyle A_{\mu}\left(x\right)\rightarrow$
$\displaystyle\,A_{\mu}\left(x\right)+\frac{1}{q}\partial_{\mu}\alpha\left(x\right),$
(5.5)

$\alpha$ being a (sufficiently smooth) scalar field.

Up to a total-derivative term, (5.1) can be brought to the form

$\displaystyle\mathcal{L}_{H}=$
$\displaystyle\,-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\partial^{\mu}\phi^{*}\partial_{\mu}\phi+iqA^{\mu}\left(\phi^{*}\partial_{\mu}\phi-\phi\partial_{\mu}\phi^{*}\right)+q^{2}\left|\phi\right|^{2}A^{\mu}A_{\mu}+\mu_{s}\left|\phi\right|^{2}-\lambda_{s}\left|\phi\right|^{4}$
$\displaystyle\,+\frac{1}{2}\left(\frac{2q^{2}}{m_{p}^{4}}\left|\phi\right|^{2}+a_{0}^{2}\right)\partial_{\mu}F^{\mu\nu}\partial_{\rho}F^{\rho}_{\phantom{\rho}\nu},$
(5.6)

where the first line is the one expected from the usual Maxwellian Higgs
model.

In order to proceed, we restrict our analysis to the parameter region in which
$\mu_{s}>0$ and we write the complex scalar field in the polar representation:

$\displaystyle\phi\left(x\right)=$
$\displaystyle\,e^{-i\frac{\theta\left(x\right)}{v}}\left[\frac{\chi\left(x\right)+v}{\sqrt{2}}\right],$
(5.7)

where $\theta$ and $\chi$ are real scalar fields and $v$ is the constant

$v=\sqrt{\frac{\mu_{s}}{\lambda_{s}}}.$ (5.8)

In terms of these newly defined scalar fields, the Lagrangian density (5.6)
takes the form

$\displaystyle\mathcal{L}_{H}=$
$\displaystyle\,-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}\left(a_{0}^{2}+\frac{q^{2}v^{2}}{m_{P}^{4}}\right)\partial_{\mu}F^{\mu\nu}\partial_{\rho}F^{\rho}_{\phantom{\rho}\nu}+\frac{q^{2}v^{2}}{2}\left(A^{\mu}+\frac{1}{qv}\partial^{\mu}\theta\right)\left(A_{\mu}+\frac{1}{qv}\partial_{\mu}\theta\right)$
$\displaystyle\,+\frac{1}{2}\partial^{\mu}\chi\partial_{\mu}\chi-\mu_{s}\chi^{2}-\mathcal{V}_{I}\left[A,\theta,\chi\right],$
(5.9)

Here, $\mathcal{V}_{I}\left[A,\theta,\chi\right]$ is a term of interaction
among the three fields, whose exact form is unimportant for our purposes.555As
a matter of fact, $\mathcal{V}_{I}\left[A,\theta,\chi\right]$ is the same
interaction potential for the Maxwellian Higgs model added to the following
two terms:
$\left(q^{2}/2m_{P}^{4}\right)\chi^{2}\partial_{\mu}F^{\mu\nu}\partial_{\rho}F^{\rho}_{\phantom{\rho}\nu}$
and
$\left(q^{2}v/m_{P}^{4}\right)\partial_{\mu}F^{\mu\nu}\partial_{\rho}F^{\rho}_{\phantom{\rho}\nu}$.
So, the $\theta$ field remains massless, in accordance to the Golstone
theorem, while the Lagrangian density presents a term of the form
$\left(M^{2}/2\right)A^{\mu}A_{\mu}$. Such term is associated, in first-order
derivative theories, with the nonvanishing mass of the gauge field. More
importantly, we notice that if we define

$\displaystyle q^{2}v^{2}\equiv\,M^{2}\equiv m_{s}^{2}$ (5.10) $\displaystyle
a^{2}\equiv\,a_{0}^{2}+\frac{q^{2}v^{2}}{m_{P}^{4}}\equiv\frac{1}{m^{2}},$
(5.11)

in which we have an infrared (IR) mass $M$ and an ultraviolet mass (UV) $m$ in
term of the Podolsky parameter. Interesting the (IR) and the (UV) regime are
tied by the interaction. In the free case (q=0) we have (5+2) degrees of
freedom (Podolsky+complex scalar; Maxwell+Proca+complex scalar) but when we
turn on the interaction we have (3+3+1) degrees of freedom
(Proca+Proca+Higgs). The first three terms of the previous Lagrangian density
are exactly equal to the Lagrangian density for the generalized Stueckelberg
model (2.5). This allows us to apply the analysis presented in the previous
sections and analogously quantize the generalized Higgs model by writing the
transition amplitude in covariant form:

$\displaystyle
Z=\int\left[\prod_{\mu=0}^{3}\mathcal{D}A^{\mu}\right]\mathcal{D}\theta\mathcal{D}\chi\mathcal{D}\bar{c}\mathcal{D}c\,e^{i\int
d^{4}x\mathcal{L}_{{e}}},$ (5.12)

where the effective Lagrangian density for the generalized Higgs model
$\mathcal{L}_{{e}}$ is defined as

$\displaystyle\mathcal{L}_{{e}}\equiv\mathcal{L}_{H}-\frac{1}{2\xi}\left[\left(1+\frac{\Box}{m^{2}}\right)\partial_{\mu}A^{\mu}-gM\theta\right]^{2}+\bar{c}\left[\left(1+\frac{\Box}{m^{2}}\right)\Box+gM^{2}\right]c,$
(5.13)

where $\bar{c}$ and $c$, like in the Stueckelberg model, are Grassmannian
scalar fields and $g$ is a real parameter. The theorem (4.3) allows us to
choose $g$ as we please.

As we can see, the Fadeev-Popov trick in the Stueckelberg form help to write
the transition amplitude in the t’Hooft shape [49, 48]. The demonstration that
the transition amplitude and reciprocally the physics does not depend on $g$
is the same as Sec. 4.

##  6 Conclusions and final remarks

This paper is devoted to analyse some mechanisms of mass generation in the
context of generalized Stueckelberg-Higgs gauge theories. Our goal was to try
to achieve a better understanding of the ultraviolet and the infra-red regime
of the photon respecting gauge symmetry. The inspiration of this study lies in
understanding the relationship between masses, size of the particles and range
of the interactions.

Firstly we work with the generalized Stueckelberg electrodynamics studying the
classical equation of motion, constructing the dynamics in the phase space,
dealing with the constraints with the Dirac methodology, implementing the
gauge fixing condition, construct the transition amplitude with the Fadeev-
Senjanovic procedure and write it into a covariant form with the Fadeev-Popov
method. It is interesting to see the link between the constraints and the
physical degrees of freedom in the quantization process [29]. As we can see we
have 18 variables and 6 constraints. So we have 12 variables describing the
phase space and 6 degrees of freedom, Maxwell+Proca+Scalar (2+3+1). The
present model can be seen as a toy model to study the quantization process in
the functional integration approach.

Secondly when we write the transition amplitude in a covariant form we can see
the emergence of a t’Hooft parameter $g$, eq. (3.8). We prooved that this
parameter does not contribute to the physics by using BRST symmetry of the
transition amplitude of the theory, following the same line of reasoning of
[37, 38]. So the generalized Stueckelberg gauge theory is one more example
that show the link between the fictitious parameters and BRST symmetry.

Thirdly as a complement of the discussion, following the historical influence
of the Stueckelberg field to the Higgs mechanism and generation of mass [8],
we construct the generalized Higgs model. As can been seen in eq. (5) the
Podolsky parameter depends of the interaction coupling $q$, where we define
the infrared (IR) mass and the ultraviolet mass (UV). So, in this view, the
Podolsky parameter’s actual value is a consequence of the interaction with the
Higgs field. The interesting thing here is that in the language of spontaneous
symmetry breaking the generalized photon eat the Goldstone boson and acquire
an IR and UV masses, therefore this regimes are tied by the interaction. As
consequence, we have the tools to study the influence of the radiative
corrections in the generalized Stueckelberg-Higgs theory, exploring the
proprieties of the effective action in this general context [10, 11, 12].

Finally the next steps of this material would be introduce the coupling with
fermions and to investigate how to implement the Podolsky photon in the
context of electroweak theory $SU(2)\times U(1)$. We think that the relation
between the neutral boson $Z_{0}$ and the photon $A_{\mu}$ are generalized in
the sense of Podolsky with the Stueckelberg and Higgs masses, so we will have
a better UV and IR controlling of divergences. Other context that the
Stueckelberg field could be applied is in $SU(3)\times U(1)$ model for
electroweak interactions [51] where the appearance of more phenomenological
vertices would be interesting. These matters will be further work out and
require elaborations.

## Acknowledgement

G. B. de Gracia thanks CAPES PhD grant (CP), A. A. N. thanks National Post-
Doctoral Program grant (PNPD) for support and B. M. P. thanks CNPq for partial
support.

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