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\section{\label{sec:intro}Introduction}

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%%$\slash{a} \ \slash{V}\  \slash{hola}\quad \spr{m} \spr{e}$

Research on the theory of massive vector fields started with   \cite{Pro36},
and reached a major milestone with the standard electroweak  theory  which  is
unitary and renormalizable, and successful.

In this paper, we would like to review a contribution of Ernst C.G.
Stueckelberg in 1938, namely his introduction of the scalar
``$B$--field'' of
positive--definite metric, accompanying a massive Abelian vector field
\cite{Stu38I,Stu38II,Stu38III}. We shall see that this idea had many different
applications which went far beyond the original motivations of its author.
Stueckelberg also invented in that year the general formulation of baryon
number conservation \cite[p. 317]{Stu38III}.

We will not review other important contributions by Stueckelberg. It is,
nevertheless, noteworthy that among his discoveries one can include the
picture of  antiparticles  as particles moving backwards in time, implying pair
creation and annihilation \cite{Stu41,Stu42}, the
propagator \cite{Stu46}, \cite{Riv48,Riv49,Stu50} and the renormalization group
\cite{Stu51,Stu53}.

We shall mainly discuss three important topics, (1) hidden symmetry,  (2)
renormalizability, and (3) infrared divergences.

It was believed by many that only massless vector theories were
gauge--invariant. Then  \cite{Pau41} showed
 that Stueckelberg's formalism for a massive vector field satisfied a
restricted $U(1)$ gauge invariance, similar to the one encountered
in quantum electrodynamics, but with the gauge function $\Lambda(x)$ restricted
by the massive
Klein--Gordon equation.

Much later, Delbourgo,  Twisk and  Thompson found that Stueckelberg's
lagrangian for real vector fields, complemented with ghost terms, is actually
BRST invariant \cite{Del88}. The BRST symmetry   \cite{Bec74,Bec75,Tyu75}
allows for a systematic and
convenient exploration of gauge symmetries, and in fact   the
$S$--matrix elements of a
BRST--invariant theory are independent of the
gauge--fixing terms\cite{Lee76}. The BRST
symmetry facilitates considerably the effort to prove the perturbative
renormalizability of a theory, as well as its unitarity
\cite{Alv83,Bec81}, see also \cite{Zin75}, \cite{Kra98} and the textbook
\cite{Col84}.

Charged vector theories (for example
non-Abelian gauge theories) are trickier.
The electroweak theory, with spontaneously broken $SU(2)_L\times U(1)_Y$
symmetry, is renormalizable \cite{Hoo71b}. As is well known, this theory
comprises two charged and
two neutral vector bosons.  A technical problem remains:  the infrared
divergences of massless (vector) field theories.

One important aim of the present paper, therefore, is to construct a BRST
invariant $SU(2)_L\times U(1)_Y$ theory with a massive photon, which calls for
a Stueckelberg field, along with the appropriate ghost terms. This development
was suggested in \cite{Sto00}. A complication is
that the
BRST--invariant infrared regulator not only gives a mass to the
photon, but also changes its couplings to the fermions, the Weinberg angle, and
more. But  the new terms in the lagrangian (or their modifications) are
proportional at least to the photon's mass squared, and thus very small.
Indeed, from a Cavendish experiment and the known value of the galactic
magnetic field one finds the stringent upper
limit $m_\gamma<10^{-16}$~eV \cite[p. 249]{PDBook}.

 This model has the important advantage of providing an infrared cut-off for
 photon interactions, while preserving BRST invariance. Thus the infrared
 catastrophe is avoided without spoiling ultraviolet renormalizability and
 unitarity.

 This paper is organized as follows.

 In section \ref{sec:quamasvecfi}, we review the original formalism and
 motivations of Proca and Stueckelberg.

 In section \ref{sec:brs}, we review the BRST  invariance properties of the
 massive neutral vector field together with the Stueckelberg
 $B$--field.

In section \ref{sec:uone} we consider in some detail the case of a $U(1)$ gauge
field coupled to matter, and study the Stueckelberg mass mechanism in the
absence and in the presence of spontaneous symmetry breaking and the usual
Higgs mechanism. This section is a warm-up for the core of the paper, in
section
\ref{sec:sm}, where we write out the $SU(2)_L\times U(1)_Y$ lagrangian with a
Stueckelberg field and a mass term for the hypercharge vector boson, as well as
with the usual spontaneous symmetry breaking. We write out the full ghost
sector and check the BRST invariance of the full lagrangian. We then turn to
some of the phenomenological consequences of the model. In particular, we study
mass matrices, mixings and currents, and we scratch the surface of some
conundra  related to anomalies and the electric charge. Curiously, one should
now distinguish the quantum field $A_\mu$ responsible for photon scattering
from
the external $A_\mu^{\rm e.m.}$ which enters in the calculation of, \sl e.g.
\rm the   $(g-2)$ value of the electron: for a massless photon the two fields
coincide.

Section \ref{sec:influ} is devoted to a historical review of the influence of
Stueckelberg's three 1938 papers. In the forties and fifties, the
renormalizability of the massive Abelian Stueckelberg theory was painfully
established. There was a long debate in the sixties and seventies about the
non-Abelian case, which waned when massive
Yang--Mills with spontaneous symmetry breakdown and the Higgs mechanism was
shown to be unitary and renormalizable \cite{Hoo71b}. The problem is still open
in 2002: no renormalizable and unitary
non-Abelian Stueckelberg model has been found, but it has not been completely
proven that it is impossible to do so. Renormalizable models of massive
Yang--Mills without Higgs mechanism nor Stueckelberg field have
been exhibited, but they are not unitary.

 On the other hand, Stueckelberg fields were introduced very early in string
 theory by Pierre Ramond and collaborators, both  for the formulation of the
antisymmetric partner to the graviton
\cite{Kal74} and in covariant string field
theory \cite{Mar75,Ram86}.
Stueckelberg  fields   turned out  to be crucial in the covariant quantization
of the spacetime
supersymmetric  string \cite{Ber90b}. They have also proven useful in the study
of dualities in field and string theory, see  section \ref{sec:pubel}.

 
 
 
\section{\label{sec:quamasvecfi}Quantization of the massive vector field}

The electromagnetic potential is described by a neutral vector field $A_\mu$
obeying Maxwell's equations. Its quantization gives rise to a massless
particle, the photon, which has only two physical degrees of freedom, its two
transverse polarizations or, equivalently, its two helicities (+1 and -1). The
vector field $A_\mu$ has, however, four components.
This is an example of how physicists   introduce
apparently unphysical entities in order to simplify the theory. Indeed, with
the four--vector $A_\mu$ one can construct a manifestly
Lorentz--invariant theory of the potential
$A_\mu$ interacting with a current $j_\mu$.

How can one reduce the four components of $A_\mu$ to the two physical degrees
of freedom of the photon? First, four becomes three in a
Lorentz--invariant way
by imposing the Lorentz subsidiary condition $\partial^\mu A_\mu=0$ (for
quantum subtleties, see below). With gauge invariance and the mass--shell
condition, only two components survive.

Contrariwise, if one adds to the wave equation of $A_\mu$ a mass term, the
gauge
invariance is lost, because the field $A_\mu$ transforms inhomogeneously  and
thus the mass term in the lagrangian is not invariant. The three components of
$A_\mu$ left by the Lorentz condition are then interpreted as belonging to a
massive vector field, that is a massive particle of spin one. This
spin--one
object has now a longitudinal polarization, in addition to the two transverse
ones.

Stueckelberg's wonderful trick consists in introducing an extra physical scalar
field $B$, in addition to the four components $A_\mu$, for a total of five
fields, to describe covariantly the three polarizations of a massive vector
field. With the Stueckelberg mechanism, which we shall exhibit in more detail
below, not only is Lorentz covariance manifest, but also, and most
interestingly, gauge invariance is also manifest. One could say that the
Stueckelberg field restores the gauge symmetry which had been broken by the
mass term.

 %Ramond
 
 

\subsection{Proca} The original  aim of \cite{Pro36} was to describe the four
states of electrons and positrons by a Lorentz
four--vector. The motivation was to
imitate the procedure of \cite{Pau34}, who had quantized
the scalar field obeying the
Klein--Gordon equation and had
interpreted the conserved current as carrying electric charge rather than
probability. This, they thought, eliminated negative probabilities. Of course,
Proca's choice is inadequate for describing spin--$\frac 1 2$ particles, and it
didn't make much sense either a decade after Dirac's equation.
 Nevertheless, Proca's  mathematical formalism describes well
a  massive real or complex  vector field.

 Proca's equation of motion for a free complex vector field $V_\mu$ reads as
follows \cite{Pro36}:
 \be \partial^\mu F^V_{\mu\nu}(x) + m^2 V_\nu(x)  =0 \label{22} \ee
 where the field strength is
 \be F^V_{\mu\nu}=\partial_\mu
V_\nu -\partial_\nu V_\mu \label{21} \ee
with $\partial_\mu =\partial/\partial  x^\mu$ and the sign in \eq{22}
depends on the metric, which is $(+ - - -)$
throughout this paper. Differentiating the equations of motion \eq{22} with
respect to $x^\nu$ yields immediately the Lorentz condition \be \partial^\mu
V_\mu=0 \label{23} \ee Notice that a
non-zero mass is crucial for \eq{23} to
follow from the equations of motion.

Hence, $V_\mu(x)$ describes a
spin--one particle of
non-zero mass $m$.
Following the lucid presentation in \cite{Wen43,Wen48}, equation \eq{22} can be
derived from a lagrangian density for the complex $V_\mu$: \be {\cal L}= -
\frac12 F_{\mu\nu}^{V\dagger} F^{V\mu\nu} +m^2 V_\mu^\dagger V^\mu \label{24}
\ee where $\,^\dagger$ means hermitian conjugation. The hamiltonian density
 following from the above lagrangian density has three positive terms
involving the spatial components $V_i$ ($i=1,2,3$), and one negative term
depending on $V_0$. This last term can be eliminated using the Lorentz
condition \eq{23} and the resulting hamiltonian is thus positive definite.

It is then possible to use the canonical formalism to find, first, the
commutation relations for the spatial components of the vector field, and then,
using again the Lorentz condition \eq{23}, the commutations relations for its
temporal component. The result is \cite{Wen43} \be \left[ V_\mu(x) , V_\nu(y)
\right] =\left[ V_\mu^\dagger (x) , V^\dagger _\nu(y) \right] = 0 \label{25}
\ee
\be \left[ V_\mu(x) , V^\dagger _\nu(y) \right] = - i \left( g_{\mu\nu}
+\frac{1}{m^2} \partial_\mu\partial_\nu \right) \Delta_m (x-y) \label{26} \ee
Here, the massive generalization $\Delta_m(x)$ of the
Jordan--Pauli function
obeys \be \left(\partial^2 +m^2\right) \Delta_m(x)=0 \label{27}\ee where
$\partial^2=\partial_\mu\partial^\mu$.

The commutator \eq{26} differs from the corresponding expression in QED by the
second term in the
right--hand side, proportional to $1/m^2$, which is either
absent
(Stueckelberg--Feynman gauge) or with coefficient $-1/\partial^2$ instead of
$1/m^2$
(Landau gauge). After 1945 it became clear that the term $m^{-2}
\partial_\mu\partial_\nu$ gives rise to (quadratic) divergences at high
energies which cannot be eliminated by the renormalization procedure



\subsection{Stueckelberg}

Stueckelberg's formalism for the vector field differs from Proca's. His
motivations made sense at the time, so let us sketch briefly the historical
framework. Recall that   Yukawa, in order to
explain the nuclear forces, postulated the existence of a massive particle
which
would mediate them, just as the photon mediates the Coulomb force between
charged particles. The first attempt \cite{Yuk35}   called for  the exchanged
particle to be a component of a Lorentz
four--vector (all computations were carried in the static approximation). Then,
\cite{Yuk37}   proposed a scalar particle instead.
  \cite{Stu38I} showed  that choosing a scalar would lead to a
repulsive instead of attractive nuclear interaction\footnote{To find
phenomenological agreement, one needs to consider an isospin triplet of
pseudoscalar intermediate particles \cite{Kem38a}. \cite{Stu37} and
\cite{Bha38} first noticed that the
Yukawa particle could decay into electrons. See also the criticisms of Yukawa's
scalar proposal
in \cite{Kem38b,Kem38c,Ser38} and the comments in  \cite[p. 434]{Pai86}.} and
then  turned  to reconsider the exchange of a massive charged vector particle.
The guiding
principle of this
research  was to develop  a formalism as close as possible to QED.

Instead of Proca's equation of motion \eq{22}, \cite{Stu38II}  wrote simply
\be \left(\partial^2 +m^2\right) A_\mu (x) =0 \label{28} \ee
which follows from the covariant lagrangian density \be {\cal L}= -\partial_\mu
A_\nu^\dagger \partial^\mu A^\nu +m^2 A_\mu^\dagger A^\mu \label{29} \ee The
difference between this lagrangian and Proca's \eq{24} (other than the change
in notation between $V_\mu$ and $A_\mu$) is a term $ \partial_\mu A_\nu^\dagger
\partial^\nu A^\mu $ which, up to total derivatives, is $( \partial^\nu
A_\nu^\dagger )(\partial^\mu A_\mu )$. This term, present in Proca's lagrangian
but not in Stueckelberg's, is responsible for being able to derive the Lorentz
condition \eq{23} from Proca's lagrangian.

So following the QED track leads, not surprisingly, to the fact that the gauge
condition must be imposed as a supplementary ingredient besides the covariant
lagrangian: a disadvantage of Stueckelberg's procedure is thus that the Lorentz
subsidiary condition does not follow from the equation of motion, as was the
case with Proca. This feature has terrible consequences on the positivity of
the hamiltonian, which is now
\be {\cal H} = - \sum_{\lambda=0,1,2,3} \left (
\partial_\lambda A_\mu^\dagger \right)\left(\partial_\lambda A^\mu \right) -m^2
A_\mu^\dagger A^\mu \label{210}\ee
The explicit sum over $\lambda$ gives no trouble, but
the implicit sum over $\mu$ does, since $A_\mu=g_{\mu\nu}A^\nu$ and thus the
contribution from $A_0$ to the energy density is negative, whereby it is not
possible to conclude that \eq{210} is positive definite. In Proca's formalism,
the negative term with $A_0$ can be eliminated with the subsidiary condition
which follows from the field equations,
but now we do not have it automatically. Where does the subsidiary condition
come from, then?

In QED, the same problem arises: $\partial^\mu A_\mu=0$ does not follow from
the
equations of motion.  \cite{Fer32} proposed   to impose instead $\partial^\mu
A_\mu \left|{\bf
phys}\right>=0$, with $\left|{\bf phys}\right>$ an admissible physical state of
the system; see also the discussion in
\cite[pp. 354--355]{Pai86}. Even this condition is too strong, however, because
it restricts
the
space of physical states to nothing. But its spirit is correct. In fact, we
only
need $\left<{\bf phys}'\right|\partial^\mu A_\mu \left|{\bf phys}\right>=0$. We
impose then \cite{Gup50,Ble50} the weaker, and sufficient, condition
\be\partial^\mu A_\mu ^{(-)}\left|{\bf phys}\right>=0 \label{212}\ee where
$A_\mu ^{(-)}$ involves only
free--field annihilation operators ($A_\mu = A_\mu
^{(-)} +A_\mu ^{(+)}$, with $A_\mu ^{(+)}$ involving only creation operators).
The Hilbert space still has indefinite metric, but the space of physical states
is of positive definite norm.

Could the same trick be used in the massive vector field theory? Suprisingly,
because of the
non-zero mass, one cannot impose the operator condition
\eq{212}, since it comes into conflict with the canonical commutation
relations. The commutation relations of Stueckelberg's vector field are the
same as those of the QED's photon in what later was to be called the Feynman
gauge: \ba &&\left[ A_\mu(x) , A_\nu(y) \right] = 0 \nn \\ &&\left[ A_\mu(x) ,
A^\dagger_\nu(y) \right] = -i g_{\mu\nu} \Delta_m(x-y) \label{211} \ea except
that $\Delta_m$ obeys the
Klein--Gordon equation with $m\not=0$. It is easy
to derive from \eq{211} the commutator \be \left[ \partial^\mu A_\mu(x) ,
\partial^\nu A^\dagger_\nu(y) \right] = i\,\partial^2 \Delta_m(x-y) \label{213}
\ee In QED, $\partial^2 \Delta_0=0$, so \eq{212} is consistent indeed. But in
the
massive vector case, since $\partial^2 \Delta_m = -m^2 \Delta_m$, the
subsidiary condition \eq{212} is inconsistent with the commutation
relations.

Stueckelberg brilliantly solved this puzzle by introducing  a new additional
scalar field, which he
simply called $B(x)$, and is now known as the Stueckelberg field
\cite[p. 243]{Stu38I}, \cite[p. 302]{Stu38II}. Note that the Hilbert space
norm of the Stueckelberg field is positive, a simple fact which has caused much
confusion when overlooked.

In the original formulation, Stueckelberg's $B$ field obeys the same equation
\eq{28}
as the vector field $A_\mu$; both fields are complex and with the same mass
$m$: \be \left( \partial^2
  +m^2 \right) B(x) =0 \label{214}\ee
  In close analogy with \eq{211},
  \ba &&\left[ B(x), B(y)\right] =0 \nn \\ &&
\left[ B(x), B^\dagger(y)\right] =i \Delta_m(x-y) \label{215}\ea The subsidiary
condition \eq{212} is replaced (in the
Gupta--Bleuler version), however, by \be
S(x)\left|{\bf phys}\right>\equiv \left( \partial^\mu A_\mu(x) +m\;
B(x)\right)^{(-)} \left|{\bf phys}\right> =0\label{216}\ee and one verifies
easily that $S(x)$ commutes both with $S(y)$ and with $S^\dagger(y)$. A short
explicit calculation shows that, after imposing the subsidiary condition
\eq{216}, the hamiltonian is positive definite.

 Instead of \eq{29}, the Stueckelberg lagrangian density is now
 \cite[p. 313]{Stu38III}
 \be {\cal
 L}_{\rm Stueck} = - \partial _\mu A_\nu^\dagger \partial^\mu A^\nu + m^2
 A^\dagger_\mu A^\mu + \partial _\mu B^\dagger \partial^\mu B -m^2 B^\dagger B
 \label{Lnew} \ee which describes consistently and covariantly a free massive
 charged vector field, accompanied by the Stueckelberg scalar. An enormous
 advantage of Stueckelberg's formalism is the absence of derivatives in the
 commutation relations. These derivatives would make the theory more singular
at  higher energies.

On the other hand, the number of degrees of freedom has now been increased to
five, instead of the required three for a massive vector field. The situation
is
somewhat similar to that encountered in QED. The subsidiary condition \eq{216}
can be used to decrease the number of components to four. In addition,
Stueckelberg's theory satisfies a new gauge invariance \cite{Pau41}, a feature
that explains the lasting success of this formalism in the literature up to our
days. The fact that a supplementary condition has to be imposed on physical
states, like in QED, just means that the theory enjoys a gauge invariance which
must be fixed, like in QED. Pauli's gauge transformations are the following
(see
also the discussion of this gauge invariance in \cite{Gla53}): \ba &A_\mu
(x)\to
A_\mu'(x) = A_\mu(x) +\partial_\mu \Lambda(x) \label{2177}\\ &B(x)\to B'(x) =
B(x) +m \Lambda(x) \label{217}\ea with the complex gauge function $\Lambda$
subject to the same field equation as $B$ and $A_\mu$:
\be
\left(\partial^2 +m^2\right) \Lambda(x) =0 \label{218}\ee

The gauge invariance is
manifest if we rewrite the lagrangian \eq{Lnew} as \ba & {\cal L}_{\rm Stueck}
=& - \frac12 F_{\mu\nu}^\dagger F^{\mu\nu} +m^2 \left( A_\mu ^\dagger - \frac1m
\partial_\mu B^\dagger \right) \left( A^\mu - \frac1m \partial^\mu B \right)
\nn\\ && - \left( \partial^\mu A_\mu ^\dagger + m B^\dagger \right) \left(
\partial_\nu A^\nu + m B \right) \label{fenetre} \ea
where \be F_{\mu\nu}(x)= \partial_\mu A_\nu(x) -\partial_\nu A_\mu(x) \ee
and we have dropped a total derivative.
The first two terms are
invariant for arbitrary $\Lambda(x)$ while the invariance of the last term
requires \eq{218}. This gauge invariance is responsible for lowering the number
of
local degrees of freedom to three, the required number for a massive vector
field. The important difference between the above gauge transformation and the
usual Abelian gauge transformation in QED is that here the gauge parameter
$\Lambda(x)$ is restricted by \eq{218}. This dynamical restriction does not
change the number of degrees of freedom, however. Note, in passing, that the
vector field in QED is real, whereas here we are following the original papers
by Stueckelberg which dealt with complex fields: the restriction to real fields
causes no trouble. The discussion on the number of
fields and physical degrees of freedom does not change if these fields and
degrees of freedom are complex or real.

The second term in equation \eq{fenetre}, which gives rise to the mass term for
the vector and to the kinetic term for the Stueckelberg scalar, clearly
displays another way of thinking of the Stueckelberg mechanism, in terms of
representations of the Lorentz group. Spin one representations can be built
from a vector or, with the help of the momentum operator, from a scalar. The
Stueckelberg trick is to couple the spin one $A_\mu$ with the spin one
$\partial_\mu B$ to have enough degrees of freedom for a gauge-invariant
massive vector field.
 
To conclude the comparison between Proca's and Stueckelberg's formalism, let us
note that the latter can be brought quite close to the former. Indeed, if one
defines \cite{Pau41}
\be V_\mu \equiv A_\mu -\frac1m \partial_\mu B \label{220}\ee one sees
that Stueckelberg's lagrangian \eq{fenetre} is the sum of Proca's \eq{24} plus
an extra term: \be {\cal L}_{\rm Stueck}= {\cal L}_{\rm Proca} - ( \partial^\mu
A_\mu^\dagger + m \, B^\dagger) ( \partial^\nu A_\nu + m \, B) \label{ssss} \ee
Note that $V_\mu(x)$ is
gauge--invariant under \eq{2177} ($V'_\mu=V_\mu$) and
so is $F_{\mu\nu} = F_{\mu\nu}^V$. The supplementary condition \eq{216} is
the same as Proca's \eq{23} on the Stueckelberg field's mass shell.

 On physical states, for which \be\left< {\bf phys} \right| \partial^\nu A_\nu
 + m \, B | {\bf phys'}>=0\label{klop} \ee Stueckelberg's lagrangian
(\ref{ssss}) coincides
 with Proca's.

We shall discuss the physical relevance of Stueckelberg's $B$ field in the next
section, in the context of real vector and Stueckelberg fields, for there is
little point in pursuing the original complex field formalism, which will turn
out to be not renormalizable. As a general clue, keep in mind that for
renormalization purposes it is advantageous to keep $A_\mu$ and $B$ independent
as long as possible: it is not a good policy to eliminate $B$ and recover the
cumbersome Proca lagrangian!
 
 
 
\section{\label{sec:brs}BRST Invariance}

We confine ourselves to real vector fields in this section. Stueckelberg's
theory  of a massive vector field $A_\mu$ accompanied by the Stueckelberg
scalar field $B$ \cite{Stu38I,Stu38II} satisfies a  gauge invariance
despite the presence
of the mass term for $A_\mu$ \cite{Pau41}: \ba \delta A_\mu(x) = \partial_\mu
\Lambda(x) \label{a31}\\ \delta B(x) = m\; \Lambda(x) \label{31}\ea where the
real gauge parameter $\Lambda(x)$ is restricted by \eq{218}.

To see this, we start from the lagrangian \eq{Lnew} specialized to real fields
\be {\cal L}_{\rm Stueck}=-\frac12 \partial^\mu A^\nu \; \partial_\mu A_\nu
+\frac12 m^2 A^\mu A_\mu +\frac12 \partial^\mu B \;\partial_\mu B -\frac12 m^2
\; B^2 \label{32}\ee and rewrite it (up to total derivatives) as \be
{\cal L}_{\rm Stueck}=-\frac14 F_{\mu\nu} ^2 +\frac12 m^2 \left( A^\mu -
\frac1m \partial^\mu B\right) ^2 -\frac12\left( \partial_\mu A^\mu +m
\,B\right)^2 \label{33}\ee

Let us consider now the supplementary condition \eq{216} on physical states,
which
involve both the Stueckelberg and the vector fields. The Stueckelberg field
actually
participates in the definition of
asymptotic states, so there one certainly cannot ignore it \cite{Pic02}!
\cite{Gla53}
 pointed out  that the condition \eq{216} can be viewed
as a practical definition of the Stueckelberg $B$ field. The massive photon's
three physical components fix the value of $B$ through the physical state
condition, consistently. (Note that in the original literature, these massive
photons were called mesons.) See section \ref{sec:hs} below for a
more detailed discussion of these points.

Consider now the expression \be {\cal L}_{gf}=-\frac1{2\alpha} \left(
\partial_\mu A^\mu + \alpha \; m \, B\right)^2 \label{35}\ee which is very much
like the 't~Hooft
gauge--fixing term in the Abelian Higgs model, where $B$
stands for the Goldstone boson. We see that Stueckelberg's lagrangian \eq{33}
corresponds to the Proca lagrangian supplemented by the 't~Hooft
gauge--fixing
term in the Stueckelberg (or Feynman) gauge $\alpha=1$ \cite[p. 37]{Tay76}.
This
is crucial for understanding Stueckelberg's contribution in modern terms. The
point, clarified by 't~Hooft, is that we can allow $\alpha$ to be any real
parameter, and thus Stueckelberg's lagrangian \eq{33} is better written for
arbitrary $\alpha$ as \be {\cal L}_{\rm Stueck}=-\frac14 F_{\mu\nu}  ^2
+\frac12 m^2 \left( A^\mu -\frac1m \partial^\mu B\right) ^2 -
\frac1{2\alpha}\left( \partial_\mu A^\mu +\alpha\,m \,B\right)^2
\label{3333}\ee
 
The restriction \eq{218} on $\Lambda$ is satisfied because of the equation of
motion for $B$, which is simply $(\partial^2 +m^2)B=0$.

The gauge fixing term \eq{35} could spoil renormalizability, but \cite{Del88}
showed that it does not. Indeed, if one adds the appropriate  terms with the
well--known ghosts
\cite{Fad67} to
Stueckelberg's lagrangian \eq{32}, one obtains a  theory invariant under the
BRST symmetry \cite{Bec74,Bec75,Tyu75}, as we
shall show shortly.
 

 For completeness, and to come closer to QED, consider also the lagrangian for
a
fermion minimally coupled to the massive vector field \be {\cal L} _f =
\bar\psi
\left[\gamma^\mu \left( i \partial_\mu +g\, A_\mu \right) - M \right] \psi
\label{36}\ee It is invariant under the special gauge transformation \ba
&& A_\mu'(x) = A_\mu(x)+\frac1m \partial_\mu B(x) \label{t7}\\  && \psi'(x) =
{\rm  e}^{iB(x)/m} \psi(x) \label{3333777} \\ &&{\bar\psi}'(x) = {\rm
e}^{-iB(x)/m}
\bar\psi(x)
\label{37} \ea
 
On the other hand, consider the Proca vector interaction
\be {\cal L}_{f}'= \bar \psi \left[ \gamma^\mu \left( i \partial_\mu + g V_\mu
\right) -M \right] \psi \label{34bis} \ee
The substitution $V_\mu = A_\mu + \frac1m \partial_\mu B$ yields
\be {\cal L}_{f}'= \bar \psi \left[ \gamma^\mu \left( i \partial_\mu + g A_\mu
+ \frac{g}m \partial_\mu B \right) -M \right] \psi \label{34bisbille} \ee
This shows that the bad high--energy behavior of the Proca lagrangian comes
from the term $\partial_\mu B$. However, the transformations  \eq{3333777} and
\eq{37} eliminate this term and leave us with \eq{36} \cite{Sto00}.
 
This  is an instance of a general $U(1)$ gauge
transformation \eq{31} where the function $\Lambda(x)$ is chosen proportional
to $ B(x)$. The elimination of $\partial_\mu B$ from the interaction is a
crucial ingredient of renormalizability.
Nevertheless, the field $B(x)$ must be renormalized in a
gauge--invariant way as well \cite{Gla53}.
 
We now turn to the BRS symmetry for the Stueckelberg theory coupled to a
fermion, ${\cal L}={\cal L}_{\rm Stueck} +{\cal L}_f$, given by eqs. \eq{3333}
and \eq{36} above.
 
Let $\omega(x)$ and $\omega^*(x)$ be independent scalar anticommuting  fields.
First,
read off from the infinitesimal gauge transformations   the following  BRST
transformation ${\bf s}$: \ba &&{\bf s}\,A_\mu
=\partial_\mu \omega \label{38aa}\\ &&{\bf s}\,B=m\,\omega \\ && {\bf s}\,\psi=
 i\,g\;\omega
\psi \\ &&{\bf s}\,\bar\psi= -i\,g\; \omega \bar\psi \\ &&{\bf s}\,\omega =0
\label{38}
\ea Note that  $\omega(x)$ is an anticommuting scalar, $\omega(x)^2=0$, and
thus the BRST transformation $\bf s$ is nilpotent, ${\bf s}^2=0$, even
off--shell.
 
The crucial property of the BRST transformation $ {\bf s}$ is that it is
nilpotent, $ {\bf s} ^2=0$, even
off--shell, \sl i.e. \rm without using the field equations. The important point
is that the gauge parameter $\Lambda$ was
constrained by the
Klein--Gordon equation, whereas $\omega$ is free.

The fermionic lagrangian ${\cal L}_f$ and the first two terms of \eq{3333},
which we can denote
simply as ${\cal L}_{g}$, are invariant under $\bf s$. Letting
\be {\cal L}_{\rm Stueck} = {\cal  L}_{g}+{\cal L}_{\rm gf} \ee
we can consider instead of the particular
gauge--fixing term
\eq{35}, which is Stueckelberg's original one for $\alpha=1$,  a more general
one \cite{Del88} \be {\cal L}_{\rm gf} =
{\bf s}\left[ \omega^* \left( {\cal G}(A_\mu,B,\psi,\bar\psi) + \frac\alpha2 \,
\,b \right)
\right] \label{39}\ee which is invariant under the nilpotent $\bf s$ with \ba
 &&{\bf  s}\,\omega^* = b\label{310xx}\\ &&{\bf s}\,b=0 \label{310}\ea The
auxiliary field
$b$ is just
a
Lagrange multiplier. The local functional ${\cal G}(A_\mu,B)$ is arbitrary and
will be chosen in a specific form only for calculational convenience. The
global parameter $\alpha$ does not
transform under $\bf s$ and labels a family of different but equivalent gauge
slices.

To check the invariance, observe that $\bf s$ anticommutes with both $\omega$
and $\omega^*$. Using \eq{310}, the
gauge--fixing lagrangian \eq{39} can be
rewritten as \be {\cal L}_{\rm gf} = -\omega ^*({\bf s}\,{\cal G})+ b\,{\cal G}
+\frac\alpha2 \;b^2 \label{311}\ee Note that, crucially, \be {\bf s}\,{\cal
L}_{\rm gf} = \omega ^*({\bf s}\,^2 {\cal G}) -b\, ({\bf s}\,{\cal G}) + b\,
({\bf s}\,{\cal G}) =0 \ee Explicitly, \be {\cal L}_{\rm gf} =\frac12 \left(
\sqrt\alpha b + \frac1{\sqrt\alpha} {\cal G} \right) ^2 -\frac1{2\alpha} {\cal
G}^2 -\omega^* ( {\bf s}\, {\cal G}) \label{lgfexp}\ee
The auxiliary scalar $b$ field
\cite{Nak66,Lau67} has indefinite metric, in sharp contrast to Stueckelberg's
$B$ field, which
propagates and whose metric is positive. It can be eliminated using its own
algebraic
equations of motion, which is equivalent to the gaussian redefinition in the
functional formalism: \be \frac{\delta {\cal L}_{\rm gf} }{\delta b}=0
\Rightarrow b= -
\frac1\alpha \;{\cal G} \label{312}\ee so that \be {\cal L}_{\rm gf} = -
\omega^*({\bf s}\,{\cal G} ) -\frac1{2\alpha} {\cal G}^2 \label{313} \ee

 %In this $b$--free version, ${\bf s}^2=0$ is not true ($s^2\omwga^*\not=0$)
% but ${\bf s}{\cal L}=0$ remains true if
%we define \be {\bf s}\,\omega^*=
%-\frac1\alpha {\cal G} \label{snou}\ee
 

There are infinitely many gauge choices for ${\cal G}$, all providing
the same
$S$--matrix. Popular choices are the covariant gauge ${\cal
G}=\partial^\mu A_\mu$ or the
't~Hooft--like gauge ${\cal G}=(\partial^\mu
A_\mu + \alpha m B) $, which gives \eq{35}. In these cases, the high energy
behavior of the vector field propagator goes like $g_{\mu\nu} /k^2$, so these
theories are
power--counting renormalizable (being also unitary). For ${\cal
G}=0$, one recovers the Proca theory, which superficially seemed
non-renormalizable. So it turns out that Proca's theory is actually
renormalizable and unitary, after all.

 
 
We choose \be {\cal G}=\partial^\mu A_\mu + \alpha\, m\; B\label{314}\ee so
that \be {\bf s}\,{\cal G}=(\partial^\mu \partial_\mu \omega +\alpha \,m^2
\;\omega) \label{315}\ee and thus \be {\cal L}_{\rm gf} = -
\omega^*\left(\partial^2+ \alpha \, m^2 \right) \omega -\frac1{2\alpha}
(\partial^\mu A_\mu + \alpha\, m\; B) ^2 \label{316}\ee
The ghost term
 decouples as in QED.
 In the Stueckelberg or
Feynman gauge $\alpha=1$, we recover  the lagrangian ${\cal L}_{\rm Stueck}$
in eq.~\eq{33}, whose BRST invariance is thus established. It is now a
canonical exercise to show the renormalizability and unitarity of the
Stueckelberg model \cite{Pic02}.
 
 
 

 \section{\label{sec:uone}Massive $U(1)$ gauge field}

Let us analyze the full $U(1) $ massive gauge field theory, including
spontaneous symmetry breakdown, before turning to the standard model in
the
next section. This will allow us to highlight the differences and similarities
between the Higgs and Stueckelberg mechanisms, which can be implemented
simultaneously. The starting lagrangian has three
components
\be {\cal L}_\circ = {\cal L}_g +{\cal L}_s+{\cal L}_f \label{lag0}\ee
where the gauge, scalar and fermion pieces are as follows: \be {\cal L}_g =-
\frac14 (\partial_\mu A_\nu -\partial_\nu A_\mu)^2 + \frac12 \left(
\partial_\mu
B-m \;A_\mu\right)^2 \label{lagg}\ee \be {\cal L}_s = \left| \partial _\mu \Phi
- i\, e\, A_\mu \Phi \right|^2 -\lambda \left( \Phi^\dagger \Phi -\frac{f^2}2
 \right)^2 \label{lags}\ee \be {\cal L}_f = \bar\psi \left(
% i \gamma^\mu   \partial_\mu +g\gamma^\mu A_\mu
 i \slash   \partial +g\slash A
 -M\right) \psi \label{lagf}\ee In this
 lagrangian,  $e$, $g$, $\lambda$, $m$, $f$ and $M$ are parameters (the first
 three massless, the last three of mass dimension one). They are all customary
 except for the photon mass $m$, accompanied by the Stueckelberg field $B(x)$,
 which is a scalar commuting field with positive metric.
 It is useful to introduce the covariant derivatives
 \be
 D_\mu \Phi = \partial_\mu \Phi - i e A_\mu \Phi \ee
 and
 \be D_\mu \psi = \partial_\mu \psi -i g A_\mu \psi \ee
 The vacuum expectation
 value of the complex scalar field, $\left<\Phi\right> = f/\sqrt2$ is taken to
 be a real modulus. We will distinguish between the cases with $f$ zero or
non-zero, since in the latter case the $U(1)$ gauge symmetry is spontaneously
broken
 \cite{Eng64,Hig64,Gur64}. Since $m\not=0$, in both cases the photon is
massive.

Each of the three pieces of the above lagrangian is invariant under the BRST
transformation $\bf s$ \be {\bf s}\, {\cal L}_g ={\bf s}\,{\cal L}_s= {\bf
s}\,{\cal L}_f =0\label{sl}\ee defined by
(\ref{38aa}--\ref{38}) and \ba  &&{\bf  s}\,
\Phi  =  i\,e\;\omega \Phi \\ &&  {\bf s}\,
\Phi^\dagger  =  -i\,e\, \omega \Phi ^\dagger \label{sfie}\ea
Note also that $ {\bf s}\,
F_{\mu\nu}=0$, $ {\bf s}\, D_\mu\Phi = i\,e\,\omega \,D_\mu\Phi$, $ {\bf s}\,
D_\mu\psi = i\,g\,\omega\, D_\mu\psi$, and $f$, just like all other parameters,
is inert, $ {\bf s}\, f=0$.

 

In order to quantize ${\cal L}_\circ$, we must fix the gauge, as discussed in
section \ref{sec:brs}. To do so, we add to the lagrangian a
gauge--fixing piece
${\cal L}_{gf}$ given by \eq{39}. After eliminating the auxiliary $b$, the
result is a
gauge--fixed lagrangian
\be
{\cal L}= {\cal L}_\circ - \frac1{2\alpha} {\cal G}^2 -\omega^* ( {\bf s}\,
{\cal G}) \label{laggg}\ee and we can choose ${\cal G}$ as we wish. It is
convenient to include a covariant gauge condition in ${\cal G}$, as well as a
term that cancels, up to total derivatives, the quadratic mixing terms in
${\cal   L}_\circ$ involving one derivative.
 
 
Define the local current $j^\mu$ as
\be j_\mu (x)= \frac{\delta {\cal L}_\circ}{\delta A^\mu(x)}\ee
By explicit computation, this current is \be j_\mu= m(mA_\mu -\partial_\mu B)
+ie \left( \Phi^\dagger D_\mu \Phi - \Phi D_\mu \Phi^\dagger \right) +g
\bar\psi \gamma_\mu \psi \ee
This current is BRST invariant, ${\bf s}j_\mu =0$,
 and  conserved, $\partial^\mu j_\mu=0$ (from  the field equations for
$A_\mu$).
Since physical states 1) contain no ghosts nor antighosts and 2) are
annihilated by the gauge condition $\cal G$,   the field equations for $A_\mu$
from the full gauge-fixed lagrangian \eq{laggg} imply that the expectation of
the divergence of the current vanishes between physical states:
\be \left< {\bf phys} \left | \partial^\mu j_\mu \right| {\bf phys}' \right> =0
\ee
So the current is indeed conserved in the quantum theory.
 
 

 
Before ending  this section, let us note that the
Stueckelberg model  can be
viewed as a free Abelian Higgs model,
\be {\cal L_g} =-\frac14 F_{\mu\nu}^2 +\left| (\partial_\mu -i e A_\mu) \Phi
\right|^2 \ee
where the module of the complex scalar field is fixed, and its phase is the
Stueckelberg field,
 \be \Phi = \frac1{\sqrt2} \frac{m}{e}\,{\rm e}^{ieB(x)/m} \ee
 This cute formulation is due to \cite{Kib65}. We shall not exploit it in what
follows.
 
 

 


\subsection{Massive electrodynamics}

Let us first work out the simple case with
$f=0$, so that the $U(1)$ symmetry is unbroken in perturbation theory -- and
still the photon is massive. We choose \be {\cal G}= \partial_\mu A^\mu
+\alpha\, m\, B \label{gfffuu}\ee to cancel the cross-term between $A_\mu$ and
$B$ in ${\cal L}_\circ$, whereby \ba &{\cal L}=& -\frac14 F_{\mu\nu}^2
+\frac{m^2}2 A_\mu^2 -\frac1{2\alpha} (\partial\cdot A)^2\nn\\ & &+\frac12
(\partial_\mu B)^2 -\frac{\alpha m^2}2 B^2 \nn\\ &&-\omega^* (\partial^2
+\alpha\, m^2) \omega\nn\\ &&+ {\cal L}_s+{\cal L}_f -m\,\partial_\mu ( B\,
A^\mu) \label{lnneew}\ea with ${\cal L}_s$ given by \eq{lags} with $f=0$, and
${\cal L}_f $ given by  \eq{lagf}. Since there is no spontaneous symmetry
breakdown, the gauge--fixing is conveniently chosen independent of the matter
fields, the complex scalar and the Dirac fermion. The gauge
sector contains one massive vector (the photon) with mass $m$, a commuting
scalar Stueckelberg
$B$--field with mass $\sqrt\alpha\, m$, and a pair of
anticommuting ghost--antighost scalars, also with mass $\sqrt\alpha\, m$. We
can  integrate out the two conjugate
Faddeev--Popov ghosts $\omega$ and $\omega^*$, since they do not couple to
other fields and they never appear in external asymptotic states. We cannot,
however, integrate out the Stueckelberg $B$--field: it is a free field
 but, as discussed above,
it plays a role in the definition of physical states and it undergoes a
non-trivial renormalization.



\subsection{Spontaneously broken $U(1)$}

What happens if $f\not=0$, that is if
the photon acquires a mass both through the Stueckelberg trick and through the
Higgs mechanism? It is now convenient to choose a different
gauge--fixing
functional ${\cal G}$, similar to 't~Hooft's, to cancel not only the mixing
between the photon and the Stueckelberg field, but also that between the photon
and the Goldstone. The price we pay is a quadratic mixing term between the
Stueckelberg and Goldstone fields, which then have to be redefined through a
global rotation.
We can parametrize $\Phi$ in cartesian or polar forms.



\subsubsection{Cartesian parametrization}

Although awkward in the Abelian case, this parametrization is the one we will
use in the standard model. Recalling that the vacuum expectation value
$f/\sqrt2$ of $\Phi$ is real, we write \be \Phi= \frac1{\sqrt2} \left( \phi_1+i
\,\phi_2 +f \right) \label{k1}\ee
It is worth noting explicitly that \ba && {\bf s} \,\phi_1 = -e \omega \phi_2
\\
&& {\bf s}\,\phi_2 = e \omega (\phi_1 + f ) \ea

We choose the
gauge--fixing function \be {\cal G}= \partial^\mu A_\mu + \alpha
( mB + e f \phi_2) \ee
 Up to a
total
derivative (proportional to $ \partial_\mu \left[ A^\mu S\right]$), this
gauge--fixing function eliminates the mixing terms between the vector field and
the gradients $\partial_\mu B$ and $\partial_\mu \phi_2$ of the scalars.

We find the tremendous lagrangian \be {\cal L}= {\cal L} _2 + {\cal L} _3 +
{\cal L} _4 + {\cal L} _{gh} +{\cal L}_f \ee
 where  the quadratic piece
\ba &{\cal L}_2=&-\frac14 F_{\mu\nu}^2 -
\frac1{2\alpha} (\partial\cdot A)^2 +\frac12 m_\gamma^2 \; A_\mu^2+\frac12
(\partial_\mu G)^2\nn\\ &&+\frac12 (\partial_\mu S)^2 -\frac\alpha2
\,m_\gamma^2\;S^2 +\frac12 (\partial_\mu \phi_1)^2 -\frac12 m_H^2\,
\phi_1^2\ea
has been diagonalized through
 \be \pmatrix{S\cr G\cr }
= \pmatrix{ \cos\beta & \sin\beta\cr -\sin\beta & \cos\beta \cr} \pmatrix{B\cr
\phi_2\cr} \label{krot}\ee with the angle $\beta$ given by
 \be \tan \beta =ef/m \label{lrrr} \ee
and the short-hands for the Higgs and photon masses
\be m_H^2 =2 \lambda f^2 \label{masssahigs}\ee
\be m_\gamma = \sqrt{ m^2 + e^2 f^2} \label{masafo}\ee
Note that $\sin\beta=ef/m_\gamma$ and $\cos \beta = m/m_\gamma$. The
gauge--fixing function is just
\be {\cal G}= \partial^\mu A_\mu +\alpha m_\gamma S\ee
whereas \ba &&{\bf s} \, S = \left( m_\gamma  + \frac {e^2 f}{m_\gamma} \phi_1
\right) \omega \\
&&{\bf s} \, G =  \frac {e m}{m_\gamma} \phi_1  \omega \ea
Note that the two contributions in quadrature to the photon mass are always
positive, so we cannot envisage a cancellation. The phase conventions are such
that $e$, $m$ and $f$ are always positive.
 
The cubic, quartic and ghost  lagrangians are
 \ba {\cal L}_3 &=& -e A^\mu (\phi_1 \partial_\mu \phi_2 -\phi_2
\partial_\mu \phi_1)
+e^2 f \phi_1 A_\mu^2 -\lambda f \phi_1 \left( \phi_1^2 +\phi_2^2 \right)
\nn\\ &=& -e\,A^\mu \left[ \cos\beta (\phi_1 \partial_\mu
G-G \partial_\mu \phi_1) +\sin\beta (\phi_1 \partial_\mu S-
 S \partial_\mu \phi_1) \right]\nn \\ &&+
e^2 f \phi_1 A_\mu^2
-\lambda f \phi_1 \left[ \phi_1^2 +(\cos\beta\, G +\sin\beta\,
 S)^2 \right]
\ea
\ba
{\cal L}_4&=& \frac{e^2}2 A_\mu^2
\left( \phi_1^2 + \phi_2^2\right) -\frac{\lambda}{4} \left(\phi_1^2
+ \phi_2^2\right) ^2 \nn\\ &=&
\frac{e^2}2
A_\mu^2 \left[ \phi_1^2 + (\cos\beta\, G +\sin\beta\,
 S)^2\right] -\frac{\lambda}{4}\left[\phi_1^2 + (\cos\beta \, G
+\sin\beta\, S)^2\right] ^2
\ea
\ba
{\cal L}_{gh}=-\omega^*\left[ \partial^2
+\alpha\,m_\gamma^2 +\alpha  e^2 f\, \phi_1 \right]\omega \label{kcartt}\ea
The couplings of the
two scalars $G$ and $S$  (identical except for a
$\sin\beta$ or $\cos\beta$ weight) are both derivative and non-derivative.
Furthermore, the Faddeev--Popov ghosts couple to the Higgs field $\phi_1$.

In the limit $m\to0$, $\beta\to\pi/2$ and the massless $G$ decouples,
whereas the surviving $S$ coincides with the original $\phi_2$. In this
limit, of course, the photon mass is due only to the Higgs mechanism.
Contrariwise, when $f\to0$, $\beta\to0$, the field $S\to B$ decouples, and
and only $G$ remains coupled. Curiously, in this limit, the surviving
field is again $\phi_2$. So in both extreme limits, $f\to0$ and $m\to0$, the
original Stueckelberg field decouples. Also, the lagrangian in both limits is
identical, except, of course, that the photon mass \eq{masafo} is either $ef$
or $m$. In general, for $f\not=0$ and $m\not=0$, there are altogether three
propagating scalar fields with different masses, the Higgs $\phi_1$, the
Goldstone $G$, and the Stueckelberg $S$.



\subsubsection{Polar parametrization}

Letting \be \Phi(x)=\frac1{\sqrt2} {\rm
e}^{i\theta(x)/f} (H(x)+f) \label{polfi}\ee the scalar part of the lagrangian
is
\be {\cal L}_s= \frac12 (\partial_\mu H)^2 +\frac12 \left( \partial_\mu \theta
-
e \,f\, A_\mu \right)^2 \left(1 + \frac{H}{f} \right)^2 - \lambda \, H^2
\left(f + \frac{H}{2}\right)^2 \label{lsss}\ee Note that the Goldstone field
$\theta$ is massless and couples only through its derivatives. Due to
spontaneous symmetry breaking, the photon $A_\mu$ has acquired a mass $ef$,
which adds in quadrature to the Stueckelberg mass $m$.

It would be tempting  to carry out the  gauge transformation
 \ba &&\Phi \to
{\rm e} ^{ -i\theta/f} \Phi = \frac1{\sqrt2}
(f+H) \\
&&A_\mu \to A_\mu -\frac1{e\,f} \partial_\mu \theta \\
&& B\to B-\frac{m}{ef}
\theta \\ && \psi \to {\rm e} ^{ -i\, g\theta/(e\, f)} \psi \label{llu} \ea
whereby the Goldstone $\theta$ would disappear completely from the classical
lagrangian: \ba {\cal L} & =& -\frac14 F_{\mu \nu}^2
+\frac12 (m A_\mu-\partial_\mu B) ^2 +\frac12 (\partial_\mu H)^2
\nn\\ &&
+\frac{e^2}2 A_\mu ^2 (H+f)^2 -2 \frac\lambda4 \left( H^2 +
2f H \right)^2 +{\cal L}_f \ea
 Note, however, that the choice $\Lambda=-\theta/(ef)$ in \eq{llu} means
that we must require $(\partial^2 +m^2 )\theta=0$, which is not consistent with
the field equations  for $\theta$ from the original lagrangian \eq{lag0}.
Thus, this ``unitary'' gauge is not allowed due to the presence of the
Stueckelberg field. Scalars are, indeed, trickier than
fermions.

Therefore,  we choose \be {\cal G}= \partial_\mu A^\mu +\alpha\,
m\, B + \alpha\, e\, f\, \theta \label{ffff}\ee whereby the quadratic piece of
the gauge-fixed lagrangian is
\ba &{\cal L}_2=& -\frac14 F_{\mu\nu}^2
+\frac{m_\gamma^2}2 A_\mu^2 -\frac1{2\alpha} (\partial_\mu A^\mu)^2
=\bar\psi(i\slash\partial -M)\psi \nn\\
&&+\frac12 (\partial _\mu B)^2 +\frac12 (\partial _\mu \theta)^2 -\frac\alpha2
\left(m \,B +e \,f\, \theta\right)^2\nn\\ &&+\frac12 (\partial _\mu H)^2-
\frac{m_H^2}2 H^2\nn\\ &&+\omega^* \left(\partial ^2 +\alpha\;
m_\gamma^2\right)
\omega -\partial _\mu \left[ A_\mu (e \,f\, \theta + m
\,B)\right]\label{lqqua}\ea where   the photon and Higgs masses  are given by
 \eq{masafo} and \eq{masssahigs}.
To derive this expression, it is useful
to keep in mind that in the polar parametrization, $ {\bf s}\, H=0$ and $ {\bf
s}\,
\theta=e \,f\, \omega$ (and, as usual, ${\bf s}\, f=0$). Note that the last
line of \eq{lqqua} can be ignored in the computation of
$S$--matrix elements.

Observe that the photon mass \eq{masafo} squared is the sum of two
contributions, one from the Stueckelberg mechanism and the other from the Higgs
mechanism. Note also that the $B$ and $\theta$ fields mix, so we must rotate
them into mass eigenstates. One of these is massless, and the other has its
mass
squared equal to the gauge parameter times the photon mass squared. We will
call
them $G'$ and $S'$ (after Goldstone and Stueckelberg, respectively): \be
\pmatrix{S'\cr G'\cr } = \pmatrix{ \cos\beta & \sin\beta\cr -\sin\beta &
\cos\beta
\cr} \pmatrix{B\cr \theta\cr} \label{lrot}\ee with the angle $\beta$ defined by
eq. \eq{lrrr}
 
Dropping the total derivative $-m_\gamma\;\partial_\mu(A^\mu\, S')$ and the
non-interacting ghost-antighost system, we end up with the following
gauge-fixed
quantum lagrangian, appropriate for perturbative evaluations of
$S$--matrix
elements: \ba &{\cal L}=& -\frac14 F_{\mu\nu}^2 +\frac{m_\gamma^2}2 A_\mu^2 -
\frac1{2\alpha} (\partial_\mu A^\mu)^2 +{\cal L}_f \nn\\ &&+\frac12 (\partial
_\mu S')^2 -
\frac\alpha2 m_\gamma^2\; S^{'2} +\frac12 (\partial _\mu G')^2 \nn\\ &&+\frac12
(\partial _\mu H)^2-\frac{m_H^2}2 H^2-\frac{\lambda}{4}  H^4  -\lambda f
\,H^3
\nn\\ &&+\left( \frac{H}f +\frac{H^2}{2f^2} \right) \left[
\frac1{m_\gamma} (e \,f\,\partial_\mu S' +m\, \partial_\mu G') -e\, f\, A_\mu
\right]^2 \label{liifi}\ea
The most salient feature of this lagrangian is that
there are two independent fields with derivative couplings to the Higgs.
The
two scalars $G'$ and $S'$ have the same masses as $G$ and $S$ in the
polar parametrization, but now their couplings (identical except for a
$\sin\beta$ or $\cos\beta$ weight) are only derivative.
Furthermore, the Faddeev--Popov ghosts decouple.

Note
that in the limit $m\to0$, that is when the photon mass is due solely to the
spontaneous symmetry breakdown, the massless field $G'$ decouples and the
massive field $S'$ coincides with the original Goldstone $\theta$. This
reproduces, of course, the usual Higgs mechanism.
The limit $f\to0$ is rather singular in this polar parametrization \eq{polfi},
but in this limit the massive $S'$ coincides with $B$ and decouples whereas the
massless $G'$ is just $\theta$ and remains coupled.







\section{\label{sec:sm}Electroweak theory with a massive photon}

  We now come to one of the main points of this paper which is to allow a
consistent regularization of the infrared divergences due to the photon by
giving it a finite mass. To achieve this, we use the Stueckelberg mechanism
described earlier of introducing an auxiliary scalar field $B$ of positive
metric while preserving the BRST invariance of the standard electroweak theory
  \cite{Sal68,Wei67,Gla61a} .
  This allows a separate treatment of infrared and ultraviolet divergences in
the perturbative expansion. It does not suffice to add an explicit mass term
for the photon, however, even with the Stueckelberg trick, because it would
spoil the $SU(2)\times U(1)$ symmetry. Instead,  following \cite{Sto00} and
\cite{Gra01}, we give a Stueckelberg mass to the vector field $V_\mu$
corresponding to the hypercharge Abelian factor $U(1)_Y$of the gauge group.
  After the symmetry breakdown $SU(2)_L \times U(1)_Y \to U(1)_{\rm em}$, the
photon field $A_\mu$ inherits a mass proportional to the original Stueckelberg
mass for the hypercharge vector boson.  Empirically, this mass is strictly
bound \cite{Luo03,PDBook}.
  As we shall show, it is necessary to modify many of the
parameters in the electroweak theory, albeit by very small amounts.
The spontaneously broken electroweak
theory is still BRST invariant, as pointed out by \cite{Sto00}
and checked by \cite{Gra01}, and in addition it is free of
infrared divergences. Of course, infrared divergences in QCD remain.

It is perhaps worth stressing at this early point that the
Stueckelberg mechanism is implemented in the Abelian factor of the
standard model gauge group to give mass to the Abelian gauge boson without any
symmetry breaking.   We do not know of any mechanism for generating
a Stueckelberg mass if none is present to start with. Thus, if
there is a grand unification of the standard model into a gauge
group without Abelian factors, then there is no reason to expect
a Stueckelberg mechanism at low energies. On the other hand, if at high
energies the
gauge group contains a $U(1)$ factor, its gauge boson can acquire a
Stueckelberg mass without symmetry breakdown and this mass can then tumble
down.
It would be
surprising that in the framework of string theory such an Abelian mass
generation was forbidden. If it were the case, it would be nice to understand
why.
 
Our approach is rather that since gauge invariance allows for an Abelian vector
mass term, it should be considered.
Accordingly, the Stueckelberg mass $m$ for
the Abelian $U(1)_Y$ factor is a free parameter of the standard model, just
like
$\theta_{QCD}$.
We are aware, however, that the introduction of this new mass scale entails
unavoidably
a new hierarchy problem,  notably with respect to the electroweak breaking
scale $f$ ($m << f$). Of course, dimensional transmutation yields an intrinsic
mass scale for strong interactions, $\Lambda_{QCD}$, so it is somewhat
symmetric, perhaps, that the Stueckelberg mechanism produces a mass scale for
the Abelian factor of the standard model gauge group. These three unrelated
masses, associated with each of the three gauge group factors of the standard
model,  accompany generically  the three distinct phases of a gauge theory
(confinement, spontaneous symmetry breaking, or Abelian).
This could bring up an additional problem, since this third phase is usually
called Coulomb precisely because the static potential is of infinite range. But
a massive photon implies that the elecrostatic potential deviates from Coulomb,
so it is not of infinite range. In fact, we are not quite clear about what
properties the external ``classical'' electromagnetic fields inherit from the
modified massive quantum photon. The issue does not arise in Stueckelberg QED,
but it does in the Stueckelberg standard model: the Noether current leaving
invariant the vacuum does not couple to the asymptotic (and massive) photon. We
shall return to these tricky issues below.

%Charge radius of neutrino.

%Furry

%2 $\gamma$ Landau--Yang C +

%$G$-parity ?

%$\nu \bar\nu $ bound state

%Cavendish

%Astrophysical field (Earth, Jupiter, Sun?, Galaxy)



 
 
Other authors have investigated in different directions.
\cite{Cve91} exploited the Stueckelberg formalism, but they constructed a
standard
model with the gauge symmetry realized non-linearly, which is equivalent to
taking the Higgs mass to infinity. In a similar vein,
\cite{Gro93a}  integrated out the
Higgs field from the standard model  and obtained an effective lagrangian
close
to Proca's \cite{Gro93b,Gro94,Gro95,Dit95a,Dit95b,Dit96}. Our approach is
orthogonal to
these viewpoints, since we keep the physical Higgs field and our photon is not
massless. Also, we ensure exact
quantum unitarity and renormalizability.


The hypercharge is normalized such that $Q=T_3 +Y/2$. We follow
the careful notation of \cite{Tay76}, and leave to the Appendix some details of
our conventions, as well as the long formulas.
 

 For starters, we concentrate in section \ref{sec:gausec} on the gauge sector,
which consists of the
 vectors, scalars and ghosts. We leave the lagrangian and   BRST
transformations of the
 fermion matter fields  to
 section \ref{sec:matter}. For simplicity, we do not include any right--handed
gauge singlets, so neutrinos are massless.
 Including them, with the phenomenologically required large Majorana mass, does
not change the analysis in any substantial way: it would be tantalizing to
speculate that neutrino masses are related to the Stueckelberg mechanism. We
present here the Stueckelberg modification of the minimal standard theory,
without any neutrino masses.





\subsection{\label{sec:gausec}The gauge sector}

The starting gauge-invariant  lagrangian is a sum of pieces associated with the
gauge, the scalar, and the fermion fields: \be {\cal L}_\circ={\cal L}_g+{\cal
L}_s  +{\cal L}_f\label{m0}\ee

The gauge lagrangian contains the usual kinetic terms for  the vector fields
$\vec W^\mu$
and $V^\mu$, as well as the Stueckelberg mass for $V^\mu$, along with the
kinetic term for the Stueckelberg field $B$ (of zero hypercharge and weak
isospin): \be{\cal L}_g =-\frac14 \vec{F}_{\mu\nu} ^2 -\frac14
{F}_{\mu\nu}^2 +\frac12 \left(m\; V_\mu -\partial_\mu B\right)^2 \label{mg}\ee
where the gauge potential field strengths are \ba &{F}_{\mu\nu}= \partial_\mu
V_\nu -\partial_\nu V_\mu \\ &\vec{F}_{\mu\nu}= \partial_\mu {\vec W} _\nu -
\partial_\nu {\vec W}_\mu +g {\vec W}_\mu \times {\vec W}_\nu \label{mmg}\ea


The scalar lagrangian, including Higgs and Goldstones,  is \be {\cal
L}_s=\left| D_\mu \Phi \right|^2-\lambda
\left( \Phi^\dagger \Phi - \frac{f^2}2 \right)^2 \label{ms}\ee where the scalar
weak isodoublet is \be \Phi = \frac1{\sqrt2} \left( H+f +i\,\vec\tau \cdot \vec
\phi\right) \pmatrix{ 0 \cr 1 \cr} =  \pmatrix{ i\,
\phi_- \cr \frac1{\sqrt2}\left( H-i\,\phi_3 +f\right) \cr} \label{mfi}\ee
and its covariant derivative is
\be
D_\mu \Phi = \left( \partial_\mu -i\frac{g}2 \vec\tau \cdot \vec W_\mu -i
\frac{g'}2 V_\mu \right) \Phi \label{mcvd}\ee The minima of the potential are
located at $|\Phi^\dagger\Phi|=f^2/2$, and we choose the vacuum to be given by
$<\Phi>=f/\sqrt2$, with
$f$ real. The scalar lagrangian is spelled out in section \ref{a:scal} of the
Appendix.

%xxxxxxxx Explicitly, \ba &{\cal L}_s =& \frac12 (\partial_\mu
% \vec\phi)^2 +\frac12 (\partial_\mu H)^2 -\lambda f^2 H^2  +\frac{f^2}2
% \left( g^2 \vec W_\mu^2 + \frac {g'^2}4 V_\mu^2 -g g' V_\mu W_3^\mu
% \right) \nn \\ && -\lambda f\; H (H^2 + \vec \phi^2 ) +2f g g' V_\mu (\phi_2
% W_1^\mu -\phi_1 W_2^\mu) +f H (g^2 \vec W_\mu^2 +\frac {g^{\prime2}}4 V_\mu^2
% -g
% g' W_3^\mu V_\mu ) \nn \\ && -\frac\lambda4 (\vec \phi^2 +H^2)^2  +2gg'
% H V_\mu(\phi_2 W_1^\mu -\phi_1 W_2^\mu) -2gg' \phi_3 V_\mu(\phi_1 W_1^\mu
% +\phi_1 W_1^\mu) \nn \\ && +\frac12 (H^2 + \phi_3^2) (g^2 \vec W_\mu^2 +
% \frac{g^{\prime2}}4 V_\mu^2 -g g' W_3^\mu V_\mu)  \nn \\ &&
% +\frac12 (\phi_1^2 + \phi_2^2)
% (g^2 \vec W_\mu^2 + \frac{g^{\prime2}}4 V_\mu^2 +g g' W_3^\mu V_\mu)
% \quad xxxxx\ea

The gauge lagrangian \eq{mg} is invariant under the BRST transformation $\bf s$
defined by \ba && {\bf s}\, V_\mu = \partial_\mu \omega \\ && {\bf s}\, B = m\;
\omega \\ && {\bf s}\, {\vec W}_\mu = \partial_\mu \vec \omega -g \,\vec\omega
\times \vec W _\mu \label{msvbw}\ea where $\omega$ and $\vec\omega$ are
anticommuting scalars. The BRST operator $\bf s$ is nilpotent, ${\bf s}^2=0$,
with \ba & &{\bf s} \,\omega=0 \\ && {\bf s} \,\vec\omega = -\frac{g}2
\vec\omega
\times \vec \omega \label{somega}\ea Note that ${\bf s}\, F_{\mu\nu}=0$ and
${\bf s}\, {\vec F}_{\mu\nu}=-g \, \vec\omega \times {\vec F}_{\mu\nu}$.
 

The scalar lagrangian \eq{ms} is also BRST invariant with the additional
definition \be {\bf s}\, \Phi =\frac{i}2 \left(g\, \vec\tau \cdot \vec \omega
+g'\, \omega \right)\Phi \label{sfi}\ee which implies that ${\bf s}\,
(D_\mu\Phi) =(i/2) (g\, \vec\tau \cdot \vec \omega +g'\, \omega )D_\mu\Phi$.
The BRST transforms of the components of the scalar doublet defined in equation
\eq{mfi} are \ba &&{\bf s}\, \phi_\pm = \frac{g}2 (\pm i \omega_\pm \phi_3 \mp
i \omega_3 \phi_\pm +\omega_\pm  H) \mp i \frac{g'}2 \omega \phi_\pm
+\frac{f\,g}2 \omega_\pm \\
%&&{\bf  s}\, \phi_2 =
%\frac{g}2 (-\omega_3 \phi_1 +\omega_1 \phi_3 +\omega_2 H) -
%\frac{g'}2 \omega \phi_1 +\frac{f\,g}2 \omega_2 \\
 &&{\bf s}\, \phi_3 =
\frac{g}2 ( i\omega_- \phi_+ -i\omega_+ \phi_- +\omega_3 H) -\frac{g'}2 \omega
H
+\frac{f}2 (g\omega_3 -g' w) \label{sficomp}\ea
and \be {\bf s}\, H = -\frac{g}2
\vec\omega\cdot\vec \phi +\frac{g'}2\omega \phi_3 \label{shigs}\ee
 
The fermion lagrangian, discussed in the next section, is also BRST--invariant.
 The nilpotency of the BRST operator and the invariance of the lagrangian under
$\bf s$ ensure the renormalizability of the theory to all orders in
perturbation theory.
 
We add to ${\cal L}_\circ$ a
gauge--fixing lagrangian \be {\cal L}_{gf} = {\bf
s}
\left[ \vec\omega^* \cdot \left(\vec{\cal G}+\frac\alpha2 \vec
b\right)+\omega^* \left({\cal G}+\frac{\alpha'}2 b\right) \right]
\label{slgf}\ee with \ba &{\bf s}\, \vec\omega^* = \vec b \\ &{\bf s}\, \vec b
=0\\ &{\bf s}\, \omega^* = b \\ &{\bf s}\, b =0\label{sghost} \ea

Notice that we keep two different gauge parameters $\alpha$ and $\alpha'$ for
each of the factors in the electroweak gauge group, namely $SU(2)_L$ and
$U(1)_Y$, respectively. Below, we will equate them to simplify some
tree--level
expressions, but in general they must be kept distinct because under
renormalization they behave differently, since there is no symmetry which
favors their equality.

After eliminating the auxiliary
Nakanishi--Lautrup ghosts $\vec b$ and $b$, we find the physical
gauge--fixed lagrangian \be {\cal L}_{\rm ph}= {\cal L}_\circ +
{\cal L}'_{\rm gf} + {\cal L}_{\rm gh} \label{lononop} \ee with \be {\cal
L}'_{\rm gf}= -
\frac1{2\alpha'} {\cal G}^2 -\frac1{2\alpha} {\vec{\cal G}}^2 \label{mlll}\ee
and \be {\cal L}_{\rm gh}= -\omega^* \;{\bf s}\,{\cal G} -\vec\omega^* \cdot
{\bf
 s}\,\vec{\cal G} \label{mlll2}\ee
%For completeness, note that this lagrangian
%is BRST invariant with the modified
%rules
%\ba && {\bf s} \omega^* = -\frac1{\alpha'} {\cal G}\\
%&& {\bf s} \vec\omega^* = -\frac1{\alpha} \vec{\cal G}\ea
% which, however, do not yield a nilpotent $s$.
 

Following 't~Hooft, we choose the following gauge functions \cite{Gra01} \ba
&&{\cal G} = \partial_\mu V^\mu +\alpha'\,m\,B -\alpha'\frac{g'}2 f\, \phi_3 \\
&&\vec{\cal G} = \partial_\mu \vec W^\mu +\alpha\frac{g}2 f\, \vec\phi
\label{kmkm}\ea Notice that the $ SU(2)$ gauge function is just 't~Hooft's,
whereas the $U(1)$ function $\cal G$ contains also a term involving the
Stueckelberg field. These gauge functions have been chosen to give total
derivatives when combined with the terms in the lagrangian with one gauge
boson,
one scalar and one derivative. All total derivatives in the lagrangian we just
drop.

\subsubsection{Mass eigenstates}

Let us collect terms in the lagrangian ${\cal L}_{\rm ph}$ in eq. \eq{lononop}
into three pieces: the fermionic lagrangian ${\cal L}_f$ that we have not even
written out yet, a quadratic lagrangian
${\cal L}_{2}$ with at most two fields, and the rest, which we call the
interaction lagrangian ${\cal L}_{\rm    int}$. They are all spelled out in the
Appendix. We concentrate now on ${\cal L}_2$ and diagonalize it.
 


%The mass terms in the lagrangian \eq{lacuadra}  are
%\ba && -\frac12 (2\lambda f^2) H^2
%+\left( \frac{g f}2 \right)^2  W_+^\mu \vec W_{-\mu}
%+  \frac12 \frac{f^2}4
%\left[ g^2\,  W_3^2 + (g^{'2}+\mu^2)\, V^2 -2g g'\,V W_3 \right] \nn\\
%&&\quad+ \frac{g^2f^2}8 \left[ \alpha\, g^2 \; \vec \phi^2
%   + \alpha' \left( g'\, \phi_3 - \mu\,B\right)^2 \right]\nn\\
%&&\quad +\frac{f^2}4\left[ \alpha \,g^2\,\vec\omega^*\cdot \vec\omega
%   +\alpha'( \,g^{\prime 2}+\mu^2) \omega^*\,\omega
%   -\alpha\,g\,g'\omega_3^* \omega
%   - \alpha'\, g\, g'\omega^* \omega_3 \right]
%\label{scalmas}\ea

 
Just like in the usual standard theory, the charged vector fields $W_\mu^\pm $
have mass
\be M_W = \frac {f\;g}2\label{masw}\ee whereas the charged scalars, $\phi_\pm
$, and the charged ghost-antighost pairs, $\omega^{(*)}_\pm $, have mass
$\sqrt\alpha\, M_W$.

Neutral boson fields mix in pairs. Explicitly, in the bases $(W_3^\mu, V^\mu)$,
$(\phi_3,B)$ and $(\omega_3,\omega)$, the respective square mass matrices for
vectors, scalars and ghosts are \ba &&M_v^2=\frac{f^2}4 \pmatrix{ g^2 & -
g\;g'\cr -g\; g'& g^{\prime2}+\mu^2\cr} \\ &&M_s^2=\frac{f^2}4 \pmatrix{
\alpha\;g^2 + \alpha' g^{\prime2} & -\alpha' \;g'\;\mu \cr -\alpha'\; g'\;\mu&
\alpha'\; \mu^{2}\cr} \\ &&M_g^2=\frac{f^2}4 \pmatrix{ \alpha\;g^2 & -\alpha'
\;g\;g' \cr -\alpha \;g\;g' & \alpha'\; (g^{\prime2}+\mu^{2})\cr} \ea where the
last matrix is understood to be sandwiched between $(\omega^*_3,\omega^*)$ and
$(\omega_3,\omega)$,  and  we have introduced the  rescaled Stueckelberg mass
of the
hypercharge vector boson \be \mu = 2\frac{m}{f} \label{mustu}\ee which behaves
like a coupling constant. These mass matrices reduce to the usual ones of the
standard model when $\mu=0$.

The mass eigenstates are obtained by rotations
\ba &\pmatrix { Z^\mu \cr A^\mu\cr} = \pmatrix
{\cos
\theta_w & -\sin\theta_w \cr \sin \theta_w & \cos\theta_w \cr} \pmatrix{W_3^\mu
\cr V^\mu \cr} \\ &\pmatrix { G\cr S\cr} = \pmatrix {\cos \beta & -\sin\beta
\cr
\sin \beta & \cos\beta \cr} \pmatrix{\phi_3 \cr B \cr} \\ &\pmatrix { \chi_Z
\cr
\chi_A\cr} = \pmatrix {\cos \tilde\theta_w & -\sin \tilde\theta_w \cr \sin
\tilde \theta_w & \cos\tilde \theta_w \cr} \pmatrix{\omega_3 \cr \omega \cr}
\\ &\pmatrix { \chi^*_Z
\cr
\chi^*_A\cr} = \pmatrix {\cos \tilde\theta_w & -\sin \tilde\theta_w \cr \sin
\tilde \theta_w & \cos\tilde \theta_w \cr} \pmatrix{\omega^*_3 \cr \omega^*
\cr}
\label{rotata} \ea
where the mixing angles are defined by
\ba & &\tan 2\theta_w =
\frac{2\,g\,g' }{ g^2 -g^{\prime2} -\mu^2 } \label{rotatapatate}\\ & &\tan
2\beta= \frac{2\,\mu\,g'
}{ \frac{\alpha}{\alpha'}\, g^2 +g^{\prime2} -\mu^2 } \\ & &\tan 2\tilde
\theta_w = \frac{2\,g\,g' }{ g^2 -\frac{\alpha'}\alpha \left( g^{\prime2}
+\mu^2 \right)} \ea The last two expressions simplify if we set
$\alpha'=\alpha$, which we are allowed to do at tree level. In this simpler
case, $\tilde\theta_w =\theta_w$.

Let us point out the main differences with the usual electroweak theory. Since
the hypercharge gauge field $V_\mu$ has a Stueckelberg mass, the Weinberg angle
$\theta_w$ is modified. Just as in the usual theory, the mixing angle between
the associated ghost fields is also the Weinberg angle if the two gauge
parameters $\alpha$ and $\alpha'$ are equal, but not otherwise.  The
new mixing angle is $\beta$, between the longitudinal degrees of freedom of the
$SU(2)_L$ and of the $U(1)$ neutral vector bosons. This is reasonable, since
the latter is the Stueckelberg field, which does not exist in the minimal
electroweak theory. The angle $\beta$ is tiny, proportional to the ratio of the
Stueckelberg mass to the electroweak vacuum expectation value. Again, let us
stress that the charged sector does not change.

To expand in powers of the Stueckelberg mass $m$ or, better, in terms of
the rescaled Stueckelberg mass $\mu=2m/f$,  it is useful to introduce the
convenient  parameter
\be
\epsilon=\frac{\mu^2}{g^2+g^{\prime2}} = 4 \frac{m^2}{f^2(g^2+g^{\prime2})}\ee
Note that $\mu$ has mass dimension zero but behaves as a coupling constant,
whereas $\epsilon$ is truly dimensionless.
 The following trigonometric functions of the  modified Weinberg angle are
handy:
\be = \frac{g'}g \left( 1+\epsilon  \right) +{\cal O}(\epsilon^2) \ee
\be {\rm s}_w= \sin \theta_w =
\frac{g'}{\sqrt{g^2+g^{\prime2}}}\left( 1+ \epsilon
\,\frac{g^2}{g^2+g^{\prime2}}  \right) + {\cal  O}(\epsilon^2)  \ee
\be {\rm c}_w= \cos \theta_w = \frac{g}{\sqrt{g^2+g^{\prime2}}} \left(
1- \epsilon \, \frac{g^{\prime2}}{g^2+g^{\prime2}} \right)+{\cal
 O}(\epsilon^2)  \ee
 
Finally, the small mixing angle $\beta$ between the Goldstone $\phi_3$ and the
Stueckelberg  $B$ is approximately
\be \beta = g' \frac{\sqrt{g^2 +g^{\prime2} }}
{\frac\alpha{\alpha'} g^2 +g^{\prime2} } \sqrt\epsilon
+ {\cal O} \left(\epsilon ^{3/2}\right)
\simeq \frac{g'}{\sqrt{g^2 +g^{\prime2} }}
 \sqrt\epsilon
\simeq \sin\theta_w \sqrt\epsilon
\ee
where the last expressions hold at tree level if we choose $\alpha=\alpha'$.

As was to be expected, the photon has a
non-vanishing mass $M_A$ proportional,
to first order, to the original Stueckelberg mass $m= \mu f/2$ of the
hypercharge vector boson: \be M_A  = m \cos
\theta_w+{\cal O}(m^3 )  = M_W  \sqrt\epsilon +{\cal O}(\epsilon^{3/2}) \ee
The $Z^\mu$ vector boson mass $M_Z$ differs
slightly from the usual one because the Weinberg angle is slightly different.
Now it becomes \ba & M_Z &= \frac{f}2 \sqrt{g^2 +g^{\prime2}}
\left(1+\frac\epsilon2 \frac{g^{\prime2} }{g^2 +g^{\prime2}} \right)
+{\cal O}(\epsilon^2) \nn\\ &&
= \frac{M_W}{\cos\theta_w} \left( 1-\frac\epsilon2 \sin^2 \theta_w \right) \ea
The exact mass eigenvalues are given in the Appendix, eq. \eq{mma}.
 
Very nicely, after   rotating by $\beta$  the Goldstone
$\phi_3$ and Stueckelberg $B$ fields, the mass eigenstates $G$ and $S$ have
exactly the same masses as the anticommuting ghosts $\chi_Z$ and $\chi_A$,
obtained by rotating through $\tilde \theta_w$ the ghosts $\omega_3$ and
$\omega$.
 
 If we set $\alpha'=\alpha$ (valid at tree level), then we
find
the simple formulas
\ba M_S=M_{\chi_A} = \sqrt\alpha \; M_A \\ M_G= M_{\chi_Z} = \sqrt\alpha \; M_Z
\ea
The full expressions are given in section \ref{a:mass} of the Appendix.
 
 
 


\subsection{\label{sec:matter}Matter}
 The fermion lagrangian is the sum of a lepton and a quark lagrangians.  Added
to the gauge and scalar lagrangians discussed above, it yields the full
classical lagrangian.
 
The lepton lagrangian is
\ba &{\cal L}_\ell=& \bar R \left( i\, \slash \partial - g' \slash
V \right) R \nn\\ &&+ \bar L \left( i\, \slash\partial -
\frac{g'}2
\slash  V + \frac{g}2 \vec\tau \cdot {\vec {\slash W}} \right) L \nn\\
&&-\left(
y_e
\bar R (\Phi^\dagger L) + {\rm \; h.c.} \right) \label{ml}\ea where $y_e$ is a
Yukawa matrix, the hypercharges of $R$ and $L$ are $-2$ and $-1$, we have
suppressed family indices,  and \be R = e_R =
\frac{1-\gamma_5}2 e \ee \be L = \pmatrix{\nu_L \cr e_L \cr} = \frac{1 +
\gamma_5}2 \pmatrix{\nu \cr e \cr} \label{mlr} \ee

The BRST transformations of the lepton fields are
\ba && {\bf s}\, R = -i g' \omega R \\
&& {\bf s}\, L = \frac{i}2 \left( g  \vec \tau \cdot \vec \omega - g' \omega
\right) L \ea

The quark lagrangian is
\ba &{\cal L}_q=
& i\,\bar Q  \left(  \slash \partial -i \frac{g'}6 \slash V
-i \frac{g}2 \vec\tau \cdot {\vec  {\slash W}} \right) Q \nn\\
&&+ i\,\bar U \left(  \partial_\mu -i \frac{2g'}3  \slash V \right) U \nn\\
  &&+ i\,\bar D  \left(  \slash \partial+i\frac{g'}3  \slash V \right) D \nn\\
  &&  -\left( y_d \bar D (\Phi^\dagger  Q)
  + y_u (\bar Q i \tau^2 \Phi) U + {\rm \;
h.c.} \right) \label{mq}\ea
where \ba
Q=\pmatrix {u _L\cr d _L\cr } \\
U= u_R \\ D= d_R \label{mqud} \ea with
hypercharges $1/3$, $4/3$ and $-2/3$, and we have
suppressed both family and color indices.

The BRST transformations of the quark fields are
\ba && {\bf s}\, U =  i \frac{2 g'}3 \omega U \\
&& {\bf s}\, D =  -i \frac{g'}3 \omega D \\
&& {\bf s}\, Q = \frac{i}2 \left( g  \vec \tau \cdot \vec \omega + \frac{ g'}3
\omega \right) Q \ea

The Yukawa interactions in \eq{ml} and \eq{mq} give the usual mass matrices to
the
fermions, \be M_e=\frac{y_e \; f }{\sqrt2}\ , \quad M_u=\frac{y_u \; f }
{\sqrt2}\   , \quad M_d=\frac{y_d \; f }{\sqrt2} \ee
so that the free fermion lagrangian is
\be {\cal L}_{ff}=i \bar \nu_L  \slash \partial \nu_L
+\sum_{\psi=e,d,u} \left\{ i\bar \psi_L \slash \partial \psi_L +
i\bar \psi_R \slash\partial \psi_R
-M_\psi \bar \psi_R \psi_L -M_\psi^\dagger \bar \psi_L \psi_R
\right\} \ee
They also give rise to the interactions between the scalars and the fermions:
\ba
& {\cal L}_y=&-\frac{1}{\sqrt 2} (H+i\cos \beta G +i \sin \beta S )
\left( y_e \bar e_R e_L +y_d \bar d_R d_L +y_u \bar u_R u_L \right) \nn\\ &&
+i\phi_+ \left( y_e \bar e_R \nu_L +y_d \bar d_R u_L -y_u \bar u_R d_L \right)
\nn\\ && +{\rm h. c.} \ea
where we have already eliminated $\phi_3$  and $B$ in favor of  the mass
eigenstates $G$ and $S$.


The interaction between the fermions and the gauge bosons is cute. From the
covariant kinetic terms for the fermions we find the usual charged current
lagrangian \be {\cal L}_{cc}= \frac{g}{\sqrt 2}
\left( \bar\nu_L \slash W_- e_L + \bar u_L \slash W_- d_L \right)
+ {\rm h.c.}  \label{chargedc}\ee

The neutral currents are rather funny due to the massiveness of
the photon and the modified Weinberg angle. Indeed, the neutral current
lagrangian can be
written as \be {\cal L}_{nc} =\sum_\psi \bar \psi \left(
n_\psi^A \slash A +n_\psi ^Z \slash Z \right) \psi \ee where the
sum runs over all the two-component fermionic fields with non-zero isospin,
$\psi\in\{\nu_L,e_L,e_R,d_L,d_R,u_L,u_R\}$. All the exact couplings
and their expansion to first order in $\epsilon$ are given in section
\ref{a:coup} of the Appendix.
 
To illustrate the novel features, it is instructive to display the approximate
lepton couplings to the photon, with the help of the traditional
 \be e=
\frac{gg'}{\sqrt{g^2+g^{\prime2}}}  \label{cargae}\ee
 which has no particular physical meaning when the Stueckelberg mass does not
vanish:
\ba & n_\nu^A &   \simeq
{\frac\epsilon2} e
\\ & n_{e_L}^A &
\simeq - e
\left(1+\frac\epsilon2 \frac{g^2-g^{\prime2}}{g^2+g^{\prime2}}
\right) \\ & n_{e_R}^A &  \simeq -
e \left(1-\epsilon
\frac{g^{\prime2}}{g^2+g^{\prime2}} \right)      \ea
 
 
These   neutral currents can be rewritten in Dirac spinor notation as follows:
\be
{\cal L}_{nc} = \sum_\psi \left\{
\bar \psi \slash  A (v_\psi^A +a_\psi^A \gamma_5) \psi
+  \bar \psi \slash  Z (v_\psi^Z +a_\psi^Z \gamma_5) \psi
\right\}  \ee
where the sum runs over $\psi\in\{\nu,e,u,d\}$. The leptonic couplings to the
photon are
\ba
&&v_\nu^A= a_\nu^A=-a_e^A
\simeq \frac\epsilon4 e \\
&&v_e^A \simeq  -e \left(1 +
\frac\epsilon4 \frac{ g^2 -3g^{\prime2} } {g^2 +g^{\prime2} } \right)  \ea

Notice the universality of the fermionic axial coupling to the photon,
  only of order $\epsilon$.
  There is also a universality in the fermionic axial couplings to
   the $Z$, which
remain  unchanged at first order from the standard value.
   These universalities extend  to the quarks.

A BRST-consistent mass for the photon in the standard electroweak theory
implies then that it has not only the customary vector couplings (slightly
modified) but also a small non-zero axial coupling!
Indeed, the photon couples differently
to left and right electrons. This means that the vector charge of the electron
is not quite (minus) one or, rather,
and even more curiously, that the left electron's charge is different from the
right electron's. Even more surprisingly, the photon couples also to
the neutrinos.


The expansion to first order in  $\epsilon$ is useful for getting a flavor of
what is going on, and will be commented upon extensively in section
\ref{sec:phenol} below.














\subsection{\label{sec:anomal}Anomalies}

Since the theory we have constructed has exact BRST symmetry, it is no surprise
that there are no anomalies. This is to be expected, since after all we have
not changed the quantum numbers of the matter fields from those in the standard
electroweak theory.  It is quite remarkable, however, that anomalies cancel
directly in the basis of physical vector bosons (the propagating degrees of
freedom, namely the massive photon, $W^\pm$, $Z$, and massless gluon).
The triangle graph with three external photons, for instance, is proportional
to the sum of the cubes of the photonic couplings of the left-handed fermion
fields
minus the sum of the cubes of the  photonic couplings of the right-handed
fields.
Using the two-component form
(\ref{nnnu}-\ref{nnnv}) of the fermion couplings to the neutral vector bosons,
the charged current \eq{chargedc}, and the fact that quarks come in triplets of
$SU(3)$ whereas leptons are singlets thereof, the following exact relationships
between the fermion couplings are verified, where we note on the left the three
gauge bosons at the vertices of the fermion loop,
 including the gluons $G$ and the gravitons $h$:
\ba
& AAA \qquad & (n_\nu^A)^3+ (n_{e_L}^A)^3- (n_{e_R}^A)^3
+3  (n_{u_L}^A)^3+3  (n_{d_L}^A)^3
-3  (n_{u_R}^A)^3-3  (n_{d_R}^A)^3 =0 \\
& AAZ \qquad & (n_\nu^A)^2(n_\nu^Z)+ (n_{e_L}^A)^2 (n_{e_L}^Z)
- (n_{e_R}^A)^2(n_{e_R}^Z)
+3  (n_{u_L}^A)^2(n_{u_L}^Z) \nn\\
&& \qquad\qquad +3  (n_{d_L}^A)^2(n_{d_L}^Z)
-3  (n_{u_R}^A)^2(n_{u_R}^Z)-3  (n_{d_R}^A)^2 (n_{d_R}^Z)=0 \\
& AZZ \qquad & (n_\nu^A)(n_\nu^Z)^2+ (n_{e_L}^A) (n_{e_L}^Z)^2
- (n_{e_R}^A) (n_{e_R}^Z)^2
+3  (n_{u_L}^A) (n_{u_L}^Z)^2 \nn\\
&& \qquad \qquad +3  (n_{d_L}^A)(n_{d_L}^Z) ^2
-3  (n_{u_R}^A) (n_{u_R}^Z)^2-3  (n_{d_R}^A) (n_{d_R}^Z)^2 =0 \\
& ZZZ \qquad & (n_\nu^Z)^3+ (n_{e_L}^Z)^3- (n_{e_R}^Z)^3
+3  (n_{u_L}^Z)^3+3  (n_{d_L}^Z)^3
-3  (n_{u_R}^Z)^3-3  (n_{d_R}^Z)^3 =0\\
& AWW \qquad & (n_\nu^A)+ (n_{e_L}^A)
+3  (n_{u_L}^A)+3  (n_{d_L}^A)  =0\\
& ZWW \qquad & (n_\nu^Z)+ (n_{e_L}^Z)
+3  (n_{u_L}^Z)+3  (n_{d_L}^Z) =0 \\
& AGG \qquad &
   (n_{u_L}^A)+   (n_{d_L}^A)
-   (n_{u_R}^A)-   (n_{d_R}^A) =0\\
& ZGG \qquad &
  (n_{u_L}^Z)+   (n_{d_L}^Z)
-   (n_{u_R}^Z)-   (n_{d_R}^Z) =0\\
& Ahh \qquad &
 (n_{e_R}^A)+   3   (n_{u_R}^A)+3   (n_{d_R}^A) =0\\
& Zhh \qquad &
   (n_{e_R}^Z)+  3   (n_{u_R}^Z)+3   (n_{d_R}^Z) =0\ea


Triangle graphs other than those listed here vanish trivially.  Note that the
actual condition from  the cancellation of the  $Ahh$ and $Zhh$ anomalies \cite{Wit85}
is
really the displayed relation minus the relation from $AWW$ (or $ZWW$).

This exact cancellation takes place independently of the value of the
Stueckelberg mass, and independently of the values of $g$ and $g'$ or, more
accurately, independently of the value of the ``massive'' Weinberg angle
$\theta_w$.

All these anomalies cancel family by family.
The cancellation of anomalies requires, of course, the color factor 3 for
quarks.


The couplings \eq{nnnv} have a very simple interpretation: they
are the charges of the various two-component spinor fields.
Indeed, a single-particle state created with the field $\psi$
has electric charge $n_\psi^A$, and similarly the weak charge of
a (two-component) spinor is $n_\psi^Z$.
 
 


\subsection{\label{sec:courant}Currents}

Recapitulating, the classical lagrangian for the Stueckelberg--modified
standard electroweak theory is
\be {\cal L}_\circ = {\cal L}_ g+{\cal L}_ s+{\cal L}_\ell +{\cal L}_q
\label{llkk}\ee
where the gauge, scalar, lepton and quark lagrangians are given by eqs.
\eq{mg}, \eq{ms},
\eq{ml} and  \eq{mq}, respectively.

\subsubsection{\label{sec:ccurr}Classical currents}

 Let us define the classical currents as
 \ba
 && j_\mu  = \frac{\delta \cal L_\circ} { \delta V^\mu}\\
 && \vec J_\mu = \frac{\delta \cal L_\circ} { \delta \vec W^\mu}\ea
 
The equations of motion for $V_\mu$ and $ \vec W_\mu$ then read as follows:
\ba &&  \partial_\mu F^{\mu\nu} = -j^\nu \\
 &&  \partial_\mu \vec F^{\mu\nu} = -\vec J^\nu \ea
and all four currents are conserved ($ \partial_\mu j^\mu = \partial_\mu \vec J
^\mu =0$).
 
Explicitly,
 \ba & j_\mu =&   m(mV_\mu-\partial_\mu B) \nn\\ &&+ \frac{i g'}2
 \left( \Phi ^\dagger  D_\mu \phi - (D_\mu \Phi)^\dagger \Phi \right)
 \nn\\ &&+\sum_{U,D,R,Q,L} g' Y_f \bar \psi \gamma_\mu \psi \ea
and the customary
 \ba & \vec J_\mu =&  g \; \vec F_{\mu\nu}\times  \vec  W^{\nu}    \nn
 \\ &&+ \frac{i g}2
 \left( \Phi ^\dagger \vec \tau D_\mu \phi - (D_\mu \Phi)^\dagger \vec\tau \Phi
\right)
 \nn\\ &&+\frac{g}2  \sum_{ Q,L}   \bar \psi  \vec \tau \gamma_\mu \psi \ea
% which we can also write as
%   \be \vec J_\mu =   g \; \vec F_{\mu\nu}\times
% \vec  W^{\nu}  + \vec{\cal     J}_\mu \ee
  It turns out that
  $ \partial j = \partial \vec J =0 $
  and $ {\bf s }j  =   0 $,
 whereas % $   {\bf s }   \vec {\cal J} \not=0 $,
   $ {\bf s }   {  J} \not =0 $.
%  and $\partial \vec {\cal J} \not =0$, but
 % $ D \vec {\cal J}=0 $.

 
 
 
   Of course, it is not surprising that
   the conserved  $SU(2)$ current $\vec J$   is not  BRST--invariant   (neither is  the   covariantly
    conserved current):
   it  forms an  $SU(2)$ triplet! Under the restricted BRST transformations
     with $\omega^1 =\omega^2=0$, the third component $J_3$ of the $SU(2)$ current
     is invariant.

   %, and thus a candidate for a local observable, just like $j$.
 
 After spontaneous symmetry breaking,  the current coupled to the physical
massive photon is
  \be J_{\rm A}^\mu =     {\rm c}_w j^\mu + {\rm s}_w J_3^\mu   \ee
 
 
We are interested in linear combinations of $j$ and $J_3$.
 Note that any linear combination of $j_\mu$ and $J^3 _\mu$ is conserved, in
particular the Noether current associated with global $SU(2)\times U(1)$
transformations leaving the vacuum invariant
 \be J_{\rm em}^\mu = \frac1{\sqrt{g^2 +g^{\prime2}}} \left( g j^\mu + g'
J_3^\mu \right) \ee
 
  Somewhat more explicitly,  the fermionic parts of these currents are
   \ba && J_{\rm em}^{(f) \mu} = e
\sum_{U,D,R,Q,L} \bar \psi Q_\psi \gamma^\mu \psi   \\ &&
   J_{\rm A}^{(f) \mu}  =    \sum_{U,D,R,Q,L} \bar \psi
\left( g'{\rm c}_w Y_\psi/2 + g {\rm s}_w  T_\psi^3 \right)  \gamma^\mu \psi
\ea
  with $e$ gievn by eq. \eq{cargae},
 $Q_\psi =Y_\psi/2 +T^3_\psi $,
and $T^3_Q=T^3_L=\tau^3/2$,
$T^3_R=T^3_U=T^3_D=0$. Since $\tan \theta_w = (1+\epsilon)g'/g + {\cal O}
(\epsilon ^2 ) $, the two currents differ for
 a non-zero $\epsilon = 4 m^2 /(f^2 (g^2 +g^{\prime2}
))$.
 
  The gauge and scalar pieces of these two currents are related just like the
above. Indeed, from the expressions
  \ba  &
   J_{\rm A}^{(g) \mu} & =    m  {\rm c}_w \left( mV^\mu -\partial^\mu B
\right)  +i g {\rm s}_w
   \left( F_{\mu\nu} ^+ W^{\nu -} -  F_{\mu\nu} ^- W^{\nu + } \right)  \\ &
   J_{\rm A}^{(s) \mu}  & =
   \frac{ {\rm c}_w g' + {\rm s}_w g }2
   \left( g' V^\mu +g W_3^\mu \right) \phi_+ \phi_-
   -\frac{ {\rm c}_w g g'   }2 \phi_3 \left( W_-^\mu \phi_+ + W_+ ^\mu \phi_-
\right) \nn\\ & & +
   \frac{ {\rm c}_w g' - {\rm s}_w g }2
   \left[ \frac12
    \left( \phi_3^2 +(H+f)^2 \right) \left( g' V^\mu -g W_3^\mu \right)
    + H \partial^\mu \phi_3 - \phi_3 \partial^\mu H + f\partial^\mu \phi_3
   \right]
   \ea
   it suffices to replace the Weinberg angle  by its traditional or
Stueckelberg--free value, ${\rm c}_w \to g/\sqrt{ g^2 +g^{\prime2}} $ and
  ${\rm s}_w \to g'/\sqrt{ g^2 +g^{\prime2}} $ to find the expressions for
$J_{\rm em}^{(g) \mu}$ and $J_{\rm em}^{(s) \mu}$. Note the
enormous simplification in the scalar current, where the second line drops off,
including in particular the term linear in $\partial_\mu \phi_3$.
 
 
  The Noether current which is associated with the global transformation
leaving invariant the vacuum expectation value of the scalar field is just
$J_{\rm em}^\mu $, and thus this vev is invariant under the action of the
electric charge $Q=\int {\rm d}^3 x J_{\rm em}^0 =Y/2 +T_3 $, but not under
that of the
charge associated with $J_A$.
 
 
 
  \subsubsection{\label{sec:qcurr}Quantum currents}
 
  Adding the gauge fixing terms to the lagrangian \eq{llkk}, so that
  \be {\cal L}= {\cal L}_\circ + {\cal L} _{\rm gh} + {\cal L} _{\rm gf} \ee
 where the last two terms are given by eqs. \eq{mlll} and \eq{mlll2},  the
field equations for $V_\mu$ and $ W^3_\mu$ are now
 \ba &&  \partial_\mu F^{\mu\nu} +\frac1{\alpha'} \partial^\nu  {\cal G} =
j^\nu \\
 &&  \partial_\mu   F_3^{\mu\nu}+\frac1{\alpha} \partial^\nu {\cal   G}_3
   -ig \partial^\nu \left( \omega^{*+}
\omega^- - \omega^{*-} \omega^+ \right) =   J_3^\nu +ig
\left( \omega^{*+} \partial^\nu \omega^- - \omega^{*-} \partial^\nu \omega^+
\right)\ea
with the usual          \ba  &&  {\cal G}   =
           \left( \partial
V + \alpha' m B - \frac{\alpha'g'f}2 \phi_3 \right) \\ &&
        {\cal \vec G} =            \left( \partial
W_3 +  \frac{\alpha g f}2 \phi_3 \right)\ea
 
 
 
 Since physical states satisfy the supplementary conditions
  \be \left< {\rm phys}' | {\cal G} | {\rm phys} \right> =0 \ee
 \be \left< {\rm phys}' | {\cal \vec G} | {\rm phys} \right> =0 \ee
and they do not contain any ghosts, it is clear that the divergence of the
currents $j^\mu$ and $\vec J^\mu$ are zero sandwiched between physical states:
   \be \left< {\rm phys}' | \partial j | {\rm phys} \right> =0 \ee
 \be \left< {\rm phys}' | \partial \vec J | {\rm phys} \right> =0 \ee



  \subsubsection{\label{sec:cconc}Conclusions}
 
  From the above analysis, it would seem that the current associated with
external classical fields (the background fields?) is $J_{\rm em}^\mu $,
whereas the current coupled to the quantum asymptotic photon field is $J_{\rm
A}^\mu$.  The latter can be measured in scattering processes like  Bhabha,
Compton or bremsstrahlung. The former's physical relevance stems from the fact
that it is the Noether current of the global $U(1)_{\rm em}$ symmetry leaving
invariant the vacuum.
 
 


\subsection{\label{sec:phenol}Some phenomenology}

Before proceeding to an overview of some of the phenomenological issues associated
with our modification of the standard model, it is perhaps wise to recall the
stringent experimental limits on the mass of the photon. Data analyses are based on
the Maxwell--Proca equations
\ba
&& \partial_\mu F^{\mu\nu}_V = j^\nu + m_\gamma^2 V^\nu \\
&&       \partial_\mu \tilde F^{\mu\nu}_V =      0 \ea
where in addition to the Lorentz gauge following from the field equations one has
imposed the Proca gauge $B=0$. These equations are not gauge-invariant, but they are
the gauge-fixed version of gauge invariant field equations, as we have discussed
in section    \ref{sec:quamasvecfi}. Since gauge-fixing does
not
change the physics, they are perfectly valid starting points for experimentalists.
Interestingly, in an experiment of size $L$, photon mass effects scale like
$(m_\gamma L )^2$, without any resonance effects in the photon frequency \cite{Gol71}.
Measurements of the energy density with a Cavendish torsion experiment
lead to the strong limit \cite{Luo03}
\be m_\gamma <  1.2 \times 10^{-17} \;{ \rm eV} \ee
Direct measurements of the speed of light are five orders of magnitude worse
\cite{Sch99}.
Other limits and methods, as well to references to the early literature,
notably \cite{Sch49}, can
be found in \cite{PDBook,Lak98}.

It is important to keep in mind that the direct limit on the photon's mass is very
strong, so that the modifications to the standard model stemming from the
consistent application of the Stueckelberg mechanism to the hypercharge Abelian
factor, in particular the modified Weinberg angle and fermion couplings to the photon
and the $Z$, are not expected to be competitive.




It is useful  to view the introduction of the BRST--consistent mass for the
photon in the standard model  as a  tiny modification of the latter. Charged
currents do not change, whereas to lowest order in the Stueckelberg mass
parameter, the weak neutral currents remain essentially undisturbed: the photon
acquires a mass and changes its couplings without affecting much the rest of
the theory. This is, of course, fortunate, since the experimental success of
the standard model constitutes the culmination of the quantum understanding of
nature.
 
There are no flavor--changing neutral currents in the theory.

A major problem arises in the computation of charges for bound states. Consider
for example the neutron, or rather the baryon with valence quarks $udd$.
Depending on the chirality of the three quarks, we find different charges or,
more precisely, couplings to the photon.
The results can be summarized in terms of the neutrino's coupling to the
photon,
\be
Q^\epsilon_\nu =n_\nu^A=\frac12 (g{\rm s}_w - g'{\rm c}_w ) \simeq
     \frac{e}2 \epsilon \ee with $e$ defined in eq. \eq{cargae}. In the
following table,  we
denote by $Q({\cal B})$ the coupling of the bound state $\cal B$ (labelled by
its valence
quarks) to the physical asymptotic massive photon.
\ba && Q(u_Ld_Ld_L)=   -Q^\epsilon_\nu
 \\
 && Q(u_Ld_Ld_R)= 0 \label{neu1}\\
 && Q(u_Ld_Rd_R)=  - Q^\epsilon_\nu \\
 && Q(u_Rd_Ld_L)= 2  Q^\epsilon_\nu  \\
 && Q(u_Rd_Ld_R)=  Q^\epsilon_\nu  \label{neu2} \\
 && Q(u_Rd_Rd_R)= 0 \ea
Thus, the Stueckelberg mechanism introduces a breaking of $SU(6)$ and indeed of
$SU(3)$. That the charge of $\Delta ^0$ is not exactly zero is of no particular
experimental relevance, but one should have serious problems accepting the fact
that a neutron's charge depends on its spin. It is comforting that \eq{neu1},
one of the true neutron states, is neutral, but disquieting that the coupling
of
\eq{neu2} to the photon does not vanish.
 
 
 
 The $uud$ bound state (proton) has similarly three different charges depending
on the handedness of the valence quarks, with $u_Lu_Ld_L$ and $u_Lu_Rd_R$
degenerate.
 The fact that left- and right-handed electrons have different charges has the
same origin as the difference in charges for the various bound states of three
valence quarks. This situation  is very problematic not only conceptually, but
also for the stability of
matter. Indeed, it is very hard to escape catastrophic and observable
consequences (for examples, electric fields near grounded metallic conductors)
if matter is not neutral. The total charge of the  hydrogen atoms
can be read off from the following table, showing the total charge (again,
coupling to the photon) of the bound states of a $uud$ baryon  and an electron.
 
 \ba  & e_L & e_R \nn\\
 u_Lu_Ld_L &  0 & -Q^\epsilon_\nu  \\
 u_Lu_Ld_R &  -Q^\epsilon_\nu  &  2 Q^\epsilon_\nu  \\
 u_Lu_Rd_L &   Q^\epsilon_\nu   &  0 \label{22218} \\
 u_Lu_Rd_R &  0 & -Q^\epsilon_\nu  \label{22219} \\
  u_Ru_Rd_L &  2  Q^\epsilon_\nu    &  Q^\epsilon_\nu  \\
  u_Ru_Rd_R &  Q^\epsilon_\nu   &0\ea

 Equations \eq{22218} and \eq{22219} could be interpreted as follows. If one
writes a ``left--handed proton wave--function'' as
 \be p_L = \left( u_L(1) u_R(2) -u_R(1) u_L(2) \right) d_L(3) \ee
 its photon charge $g'{\rm c}_w$ is equal and opposite to the charge of the
right-handed electron $e_R$. Similarly, if
   \be p_R= \left( u_L(1) u_R(2) -u_R(1) u_L(2) \right) d_R(3) \ee
 its charge $\left( g'{\rm c}_w+  g{\rm s}_w \right)/2$ is compensated by the
 $e_L$ charge. For the neutron, only the photonic charge of $n_L$ vanishes.
  This calculation is too naive, however, in the absence of a realistic
three--quark model.
 .

 .



\section{\label{sec:influ}The influence of Stueckelberg's 1938 papers}

Stueckelberg's 1938 papers, rather difficult to read then and now, have been
continuously cited from 1941 to the present. The following domains of influence
will be reviewed: renormalization of massive vector field interactions, hidden
symmetry, and  electroweak theory without spontaneous symmetry breaking.

Other topics worthy of attention which have developed from Stueckelberg's 1938
papers include baryon number, as emphasized in \cite[footnote on p.25]{Wig67},
broken chiral symmetry,  and  electromagnetic properties of vector
mesons. We shall not review them here.

We must distinguish two aspects of
Stueckelberg's formalism for massive gauge vector fields (vector mesons in
those days):

1) The decomposition of the massive Proca vector field $V_\mu$, as in equation
\eq{220} above, namely \be V_\mu = A_\mu - \frac1m \: \partial_\mu B
\label{61}\ee has been used in the discussion of the renormalizability problem.
The $\partial_\mu B$ term is responsible for the singular character of Proca's
theory: the commutation relations of the massive vector $A_\mu$ and of the
Stueckelberg field $B$ are local, but the
presence of the derivative of $B$ makes the commutation relations of $V_\mu$
non-local.

2) The replacement of Proca's free lagrangian by Stueckelberg's: \ba & {\cal
L}_{Stueck} (A_\mu, B) &= {\cal L} _{Proca} (A_\mu, B) + {\cal L}_{gf} \nn\\ &&
= -\frac12 (\partial_\mu A_\nu)^2 +\frac12 m^2 A_\mu^2 +\frac12 (\partial_\mu
B)^2 -\frac12 m^2 B^2 \label{62}\ea (We consider the neutral case \eq{32},
which follows from \eq{3333}
with $\alpha=1$;
for charged vector fields the Proca and Stueckelberg lagrangians are given by
\eq{29} and \eq{Lnew}, respectively).
Previously, it was believed that a massive vector theory could not be gauge
invariant, but \cite{Pau41} showed that ${\cal L}_{Stueck}$ was a
counter-example to such belief. We have seen in section \ref{sec:brs} that the
theory
of real massive vector fields is even BRST invariant, which is relevant for its
renormalizability \cite{Del88}.






\subsection{The question of renormalizability}

\subsubsection{Power--counting renormalizability}


It was found in the 1930s that the quantum field theory of electrons and
photons (quantum electrodynamics, QED) was plagued by infinities, already at
low orders in perturbation theory.

In 1949, Dyson showed that, in QED, renormalization of mass and charge of the
electron and renormalization of the wave-functions (or better, the rescaling of
the field operators) removed all the divergences from the $S$--matrix to all
orders in perturbation theory \cite{Dys49a,Dys49b}. This is now known as the
power--counting procedure, because it is based on counting the powers of
four--momenta over which one integrates. Dyson's proof was made more rigorous
by  \cite{Sal51a, Sal51b},  \cite{Wei60}, and later by \cite{Bog76} and others.

After Dyson, it was natural to ask whether massive vector field interactions
were also renormalizable. Vector mesons were first considered, following
Yukawa, as mediators of strong nuclear interactions, with little
phenomenological success.


\cite{Miy48} was the first to use Stueckelberg's lagrangian
\eq{62} extensively in the theory of charged massive vector mesons interacting
with nucleons, mimicking the  treatment of QED in the
super--many--time formalism \cite{Tom46}, which has the advantage of being
manifestly Lorentz invariant. Tomonaga and collaborators applied this formalism
to the
interaction of electrons with photons \cite{Kob47a,Kob47b} and of mesons with
photons \cite{Kan48a,Kan48b}. In the latter case, an additional interaction
term was necessary to satisfy relativistic invariance. Miyamoto showed that,
for the interaction of mesons and nucleons, the additional term is provided
automatically in the Stueckelberg formalism by the scalar $B$--field.

Miyamoto then derived the generalized Schr\"odinger equation, the integrability
conditions, Stueckelberg's auxiliary condition, and the passage to the
Heisenberg picture. He found that Proca's theory is not well adapted to the
many-time formalism, and also that it has problems with the integrability
conditions. Miyamoto's careful and extensive work paved the way for further
research.

In a different vein, using   the many--time formalism of \cite{Dir32},
\cite{Pod48} considered a
non-renormalizable modification of QED with higher
derivatives of the massless $A_\mu$ field. They claimed to get a finite
self-energy for a point source. In the process of trying to quantize the
theory, they introduced the Stueckelberg field $B$ in order to get a consistent
subsidiary condition, similar to Stueckelberg's \eq{klop}.



\subsubsection{1949--1954 : lessons from QED}

The first   definite answer to the question of
renormalizability of vector interactions with the nucleons was provided by
\cite{Mat49a,Mat49b}. This problem depends crucially on the high--energy
behavior of the $S$--matrix, which depends in turn on the power of the
energy--momentum factors.
Four--momenta are the Fourier transform of derivatives, and they may appear in
the commutation relations of the quantized
fields, and hence in the propagators, as well as in the interaction lagrangian.

This point can be illustrated by comparing  Proca's commutation relations for
real massive vector fields,
\be \left[ V_\mu(x) ,V_\nu(y) \right] = -i \left( g_{\mu\nu} + \frac1{m^2}
\partial_\mu \partial_\nu \right) \Delta_m(x-y) \label{63}\ee with
Stueckelberg's: \be\left[ A_\mu(x) ,A_\nu(y) \right] = -i g_{\mu\nu}
\Delta_m(x- y) \label{64}\ee Similarly,  \eq{26} can be compared with \eq{211}
for charged (\sl i.e. \rm non-hermitian) vector fields.
 Proca's theory is clearly more divergent at high energies than
Stueckelberg's. On the other hand, the interaction of Proca's massive vector
field with a charged fermion field $\psi$ is the harmless
\be {\cal L}^I_{Proca} = e\bar\psi \gamma^\mu V_\mu \psi  \label{65}\ee
whereas Stueckelberg's is \be
{\cal L}^I_{Stueck} = e \bar\psi \gamma^\mu \psi \; \left( A_\mu - \frac1m
\partial_\mu B \right) \label{66}\ee The last vertex diverges like $p_\mu$, so
it seems that Stueckelberg's interacting theory is also singular.

According to Matthews, however, the bad terms with $\partial_\mu B$ can be
eliminated from the interaction by a unitary transformation, as follows.
Working
in the Dirac or interaction picture, Matthews (quoting Miyamoto) writes,
instead
of \eq{66}, the interaction term \be {\cal L}^I_{Stueck} = j^\mu\; \left( A_\mu
- \frac1m \partial_\mu B \right) +\frac1{2m^2} (j^\mu n_\mu)^2\label{67}\ee
where
$j^\mu =e \bar\psi \gamma^\mu \psi$ and $n_\mu$ is a normal unit vector to a
general space-like surface. The point is that the last term,
quadratic in $j^\mu$, is absolutely necessary for what they called
``integrability''
 in those days; note that it does not look renormalizable. The
physical states are defined using Stueckelberg's
subsidiary condition \eq{216},
 \be (\partial^\mu A_\mu +mB)^{(-)} |{\bf phys}>  =0 \label{68}\ee
  Now Matthews performs the unitary transformation
\cite{Dys48,Cas49}
\ba && |{\bf phys}> \to |{\bf phys}'> = {\rm e}^{-i G}|{\bf
phys}> \label{69}\\ && G=\frac{1}m \int d\sigma^\mu j_\mu(x) B(x)
\label{610}\ea where the integral is over the
space--like reference surface.
Note the similarity to the gauge transformation
(\ref{t7}--\ref{37}). This redefinition
eliminates the last two terms in \eq{67} and thus  we end up with an
interaction lagrangian exactly like that of the massless photon interacting
with
the electron current in QED, which is  renormalizable \cite{Dys48}. For
charged (non hermitean) vector fields, an additional term spoils the
renormalizability \cite{Cas49}. See also \cite{Bel49a,Bel49b} and \cite{Gup51}.

\cite{Phi54} worked in the same framework as Matthews (Stueckelberg
lagrangian, with the unitary transformation \eq{69}), but criticized the former
for a technicality concerning the integrability conditions. He introduced a
``quasi interaction representation", in which the massive field $A_\mu$ obeys
free field equations, but the ``free'' equations for the $B$ and $\psi$ fields
include the term $j_\mu(x) \partial^\mu B(x)$, which can be eliminated just as
proposed by Matthews.


The review
% \sl The Renormalization of Meson Theories \rm
\cite{Mat51}  established to what meson interactions
Dyson's proof of finiteness of QED could be applied. The result is that the
only interaction of
vectors or pseudovectors with fermions satisfying Dyson's
criteria is the vector interaction of a neutral vector
\cite{Mat49a}. On the other hand, the scalar interactions of scalars and
the pseudoscalar interactions of pseudoscalars require only a finite
number of renormalizations, as in the case of QED.\footnote{In 1951 parity
conservation was still unquestioned.} In addition to the counterterms analogous
to those occurring in QED, one needs a quartic (pseudo)scalar term and, in the
case of scalar mesons, a further cubic term.
See also the textbook presentation of \cite{Ume56} and the discussions of
\cite{Fuj59}.


\subsubsection{1960--1962 : equivalence theorems}




In the late 1950s and early 1960s, the interest in intermediate vector theories
was revived by work  on an isospin $SU(2)$ gauge--invariant
theory of massless vector fields \cite{Yan54}. [For a history of gauge
fields, see \cite{Ora00}.] Furthermore,
 \cite{Fey58}, \cite{Sak58} and \cite{Sud58}  proposed the
universal $V-A$ theory of weak nuclear interactions, which ``can most
beautifully be formulated by assuming an intermediate vector particle,'' as
stated by \cite{Kam60}.  Indeed, \cite{Fey58,Sud58} suggested that the $V-A$
interaction could be mediated by a charged spin--one particle. \cite{Blu58}
proposed to add
a neutral field in the framework of an $SU(2)$ Yang--Mills invariance, and
\cite{Gla61a} added yet another neutral particle to achieve $SU(2)\times U(1)$
invariance. On the other hand, \cite{Fuj59} proposed a massive vector meson to
mediate strong interactions, and \cite{Sak60} identified it with the $\rho$.
\cite{Lee49} had already proposed the idea that the exchange of a boson of
non-specified spin could explain the approximate equality of the
$\beta$--decay and muon interaction couplings (the universality of weak
interactions). \cite{Sch57} had  ``freely invented'' an intermediate vector
boson in strong and weak interactions (with some hints from experiment).
 It was tempting to identify the latter with the
Yang--Mills field. But for empirical reasons, related to
the Fermi theory of weak interactions, this particle ought to be massive. And
yet, the theory of massive charged vector fields seemed not to be
renormalizable. The main reason for this singularity seemed to be the lack of
gauge invariance of such theories.

In this context, \cite{Gla59}  conjectured  that a ``partially conserved"
vector current could lead to a renormalizable theory. This proposal was
refuted independently by \cite{Sal60} and  \cite{Kam60}. Both used the
decomposition \eq{220} $V_\mu =A_\mu -
m^{-1}\partial_\mu B$, with $m$ the mass of the vector meson, and found a
general equivalence theorem for vector meson interactions, from which they
deduced ``a precise criterion for renormalizability in the conventional sense"
 \cite{Kam61}.
They were inspired by
 \cite{Dys48}, who  had already shown that the pseudovector interaction
 \be g \bar\psi \gamma^\mu \gamma_5 \partial_\mu \tilde B \psi \label{D2}\ee of
a
{\sl pseudoscalar } field $\tilde B$ with a nucleon field $\psi$ was equivalent
to an exponential pseudoscalar interaction of $\tilde B$ with $\psi$, using the
unitary transformation \be \psi\to \psi' = {\rm e}^{i g \gamma_5 \tilde B}\psi
\label{D1}\ee which eliminated the pseudovector interaction  and took the mass
term $M\bar\psi\psi$ of the nucleon into \be M \bar\psi'\left( 1-{\rm e}^{-2ig
\gamma_5 \tilde B} \right) \psi' \label{D3}\ee

\cite{Sal60}  applied the same procedure   to a real pseudovector Proca
field $\tilde V_\mu=\tilde A_\mu -\frac1m \partial_\mu \tilde B$. The
pseudovector interaction of $\tilde V_\mu$ with fermions contains the
pseudovector interaction of $\tilde B$. The elimination of the latter yields,
again, a non-renormalizable exponential interaction. The current \be j_\mu = g
\bar \psi \gamma_\mu \gamma_5 \psi \ee is ``partially conserved:" \be
\partial^\mu j_\mu = 2i g M \bar\psi \gamma_5 \psi \ee

 \cite{Kam60} treated the general case  of the interaction of the
neutral vector field $V_\mu$ with an arbitrary complex field of spin 0, 1/2,
or~1, and assumed that the interaction hamiltonian was of the form $H_1+H_2$,
with
$H_1$ gauge invariant and $H_2$ not gauge invariant. Writing $V_\mu = A_\mu -
\frac1m \partial_\mu B$ and applying the analogue of \eq{D1}, the $B$ field is
succesfully eliminated from $H_1$, but reappears in an exponential in $H_2$.

Both Salam's and Kamefuchi's examples contradict Glashow. They had only
considered, nevertheless, the case of Abelian $U(1)$ gauge invariance.


%\subsection{Umezawa and Kamefuchi}

Shortly thereafter, \cite{Ume61} generalized the equivalence theorem of
\cite{Sal60} and \cite{Kam60} to the
non-Abelian isospin $SU(2)$ Yang--Mills
gauge theory, with massive vector mesons. They found that the mass terms spoil
renormalizability.

To prove the equivalence theorems, they used Stueckelberg's lagrangian, which
they introduced in a new and elegant way. Then they extended it to the
Yang--Mills case.

% supprimer ou raccourcir?


Their starting point is the Proca lagrangian for a real vector field $V_\mu$,
with interactions. Introduce in addition to the real scalar Stueckelberg field
$B(x)$
an extra real scalar field $C(x)$, with the wrong energy and the wrong metric
in Hilbert
space: \ba &{\cal L}_{UK} =&-\frac14 F_{\mu\nu}(V)^2 +\frac12 m^2 V_\mu^2
+\frac12 (\partial_\mu B)^2 -\frac12 m^2 B^2 \nn\\ && -\frac12 (\partial_\mu
C)^2 +\frac12 m^2 C^2 + {\cal L}_{int} (U_\mu, \phi) \label{U1}\ea The
non-vanishing commutation relations are \ba && \left[ V_\mu (x) , V_\nu (y)
\right]
= -i \left( g_{\mu\nu} +\frac1{m^2} \partial_\mu \partial_\nu \right)
\Delta_m(x-y) \nn\\ &&\left[ B (x) , B(y) \right] = i \Delta_m(x-y)
\label{U2}\\
&& \left[ C (x) , C (y) \right] = -i \Delta_m(x-y) \nn\ea and the interaction
lagrangian depends on some other ``matter'' fields $\phi(x)$ and on the vector
field \be U_\mu(x) = V_\mu (x) +\frac1m \partial_\mu(B(x) -C(x)) \label{U3}\ee

It follows from \eq{U1} and \eq{U3} that $E=B-C$ is a free massive field: \be
(\partial^2 + m^2 ) E(x) =0\label{U4}\ee
To ensure positivity of the physical Hilbert space, one can impose the
subsidiary
condition
\be E^{(-)}|{\bf phys}> =0 \label{U5}\ee for physical states in the Heisenberg
representation, since it is consistent with the field equations.
Define now $A_\mu$ by \be V_\mu(x) = A_\mu(x)  -\frac1m \partial_\mu C (x)
\label{U6}\ee
Then \eq{U3} implies that the interacting vector field is \be U_\mu(x) =
A_\mu(x) +
\frac1m \partial_\mu B (x) \label{U7}\ee Substituting \eq{U6} into \eq{U1} we
find,
after integrating by parts and dropping total derivatives, \be {\cal L}'_{UK}
= -\frac12 (\partial_\mu A_\nu)^2 +\frac12 m^2 A_\mu^2 +\frac12 (\partial_\mu
B)^2 -\frac12 m^2 B^2  -\frac12 D^2 + {\cal L}_{int} (U_\mu, \phi)
\label{U8}\ee where the auxiliary field \be D(x)= \partial_\mu A^\mu (x)+ mC
(x) \label{U9}\ee satisfies the algebraic equation of motion $D=0$. Using this
fact, the subsidiary condition \eq{U5} reads now \be (B + \frac1m \partial^\mu
A_\mu)^{(-)} |{\bf phys}> =0 \label{U10}\ee so that  both
Stueckelberg's lagrangian and Stueckelberg's subsidiary condition are
recovered.

Umezawa and Kamefuchi proceeded to extend the Stueckelberg lagrangian to
isovector fields. They decomposed the lagrangian into a free piece: \be {\cal
  L}_0 = -\frac12 (\partial_\mu \vec A_\nu)^2 +\frac12 m^2 \vec A_\mu^2
+\frac12
(\partial_\mu \vec B)^2 -\frac12 m^2 \vec B^2 \label{U12}\ee and an
interaction:
\be  {\cal L}_I = \frac12 (m^2 -m^{\prime2}) (V_3^\mu)^2 -\frac{g}2 \vec
{\cal A}_{\mu\nu} \cdot \vec V^\mu \times \vec V^\nu - \frac{g^2}4 (\vec V^\mu
\times
\vec V^\nu )^2 \label{U13}\ee where \be \vec V_\mu = \vec A_\mu -\frac1m
\partial_\mu \vec B \ee and the short-hand \be \vec {\cal A}_{\mu\nu} =
\partial_\mu \vec A_\nu - \partial _\nu \vec A_\mu \ee is {\sl not} the
non-Abelian field strength of $\vec A_\mu$.

Notice the astute first term in the interaction, with $m'$ a free parameter.
For $m=m'$, the full isospin symmetry is restored, and the isovector current
\be \vec j_\nu =\partial^\mu \vec {\cal A}_{\mu\nu} - m^2 \vec V_\nu
\label{U15}\ee is conserved, $\partial^\mu \vec j_\mu =0$.

The derivation of the equivalence theorems for the isovector lagrangians, for
$m=m'\not=0$ or for $m\not= m'$, is rather lengthy. It turns out that, if
$m\not=0$, the theory is not renormalizable, even when the current \eq{U15} is
conserved (\sl i.e. \rm when $m\not=m'$).




%\subsection{Salam}



In the classic paper \cite{Sal62} entitled {\sl Renormalizability of Gauge
Theories}, Salam gave a simpler and more general discussion of the
renormalizability condition, which we now summarize. Salam's paper is also
remarkably modern in its notation and outlook. Consider a set of spinor fields
$\psi$ on which acts a Lie
group with generators $T_i$: \be \psi(x) \to \psi'(x) = {\rm e}^{ig T_i b^i(x)
} \psi(x) \equiv U(x) \psi(x) \label{S1}\ee with  structure constants
defined
by \be [T_i, T_j] =i f_{ij}^{\quad k} T_k \label{S2}\ee and a set of vector
fields \be V_\mu(x) = T_i V_\mu^i (x) \label{S3}\ee transforming
inhomogeneously: \be V_\mu(x) \to V_\mu'(x) = U^{-1}(x) V_\mu(x) U(x)
+\frac{i}g
U^{-1}(x) \partial_\mu U(x) \label{S4}\ee

The following lagrangian is invariant under the gauge transformation $U(x)$:
\be
{\cal L}_S (\psi, V_\mu) = i \bar \psi  (\slash \partial  -ig \slash V )
\psi +M \bar\psi \psi -\frac14 {\rm tr}\, F_{\mu\nu} F^{\mu\nu} \label{S5}\ee
with \be F_{\mu\nu} = (\partial _\mu -ig V_\mu) V_\nu -(\partial _\nu -ig
V_\nu) V_\mu \label{S6} \ee the covariant field strength, which transforms
homogeneously: \be F_{\mu\nu}(x) \to F_{\mu\nu}'(x) =U^{-1}(x)F_{\mu\nu}(x)
U(x)
\label{S7}\ee
Add now a vector mass term, which is not invariant under the gauge
transformation  \eq{S4}, \be {\cal
L}_{mass} = -\frac12 m^2 \:{\rm tr}\, V_\mu^2 \label{S8}\ee

To study the renormalizability of this theory, Salam proposed two steps. First,
introduce the Stueckelberg fields $B^i$ through $V_\mu = A_\mu -\frac1m
\partial_\mu B$, with $A_\mu =T_i A_\mu^i$ and $B =T_i B^i$. Secondly, change
$\psi$ to $\psi'$ and $V_\mu$ to $V_\mu'$, using the gauge transformations
\eq{S1} and \eq{S4}, with the
gauge parameters chosen as $b^i=B^i$. Under this transformation, ${\cal
L}_S(\psi, V_\mu)$ is invariant, but ${\cal L}_{mass}$ is not.

On the other hand, it follows from \eq{S4} that $V^\prime_\mu = A_\mu +{\cal
O}(g)$, and hence, in the weak coupling limit $g\to0$ where asymptotic states
are defined, one obtains \be V_\mu ^{\prime in} =A_\mu^{in} \label{S10}\ee The
$S$--matrix
in the new variables has contributions from two pieces: those from ${\cal
L}_S(A^{in}_\mu , \psi^{in})$ yield only renormalizable infinities (the
derivative couplings of $B$ with $\psi$ have been eliminated), whereas
those from ${\cal L}_{mass}(A_\mu^{in}, B^{in})$  produce exponential
infinities {\sl unless} either
\be m=0 \label{S11}\ee
or the following two conditions hold:
\be {\rm tr} \left[ \frac{m^2}{g^2} (\partial_\mu U) (\partial^\mu U^{-1}) -
(\partial_\mu B^{in})^2 \right] =0 \label{S12} \ee
and
\be {\rm tr} \left[ A_\mu^{in} \left( U^{-1}\partial^\mu U - i\frac{g}{m}
\partial^\mu
B^{in}\right) \right] =0 \label{S13} \ee

This is a powerful theorem.
For a massive neutral vector field interacting with fermions, for example with
the nucleons, there is only one $B$ field, and $U$ is abelian. Then both
\eq{S12} and \eq{S13} are satisfied, and the theory is renormalizable even with
a massive vector.\footnote{This result can be understood in terms of the BRST
invariance of section \ref{sec:brs}.}

In general, \eq{S12} and \eq{S13} can be satisfied provided ${\rm tr}\, T_i
T_j=0$. However, for simple Lie groups ${\rm tr}\, T_i T_j=\lambda
\delta_{ij}$, with the normalization $\lambda\not=0$, and thus the last two
conditions are not satisfied.

The only term in the lagrangian considered which is not gauge invariant is the
vector mass term. Clearly, any other non-invariant term in the lagrangian, of
the generic form ${\cal L}(\psi)$, will transform into ${\cal L}(S\psi')$, with
$S$ containing
non-renormalizable exponentials of Stueckelberg's $B$ field.
This checks, for instance, with the fermion mass term $M\bar\psi \psi$ under
the transformation \eq{D1} above, which yielded the horrible \eq{D3}.

Salam concluded that renormalizability of a gauge theory requires vanishing
masses. He then developed ideas of \cite{Nam61} to get masses in a
self--consistent way. Later, he co-birthed the concept of broken symmetry
\cite{Gol62}, which gave mass to the vectors of a broken gauge invariance. The
first discussion of a
non-Abelian spontaneous symmetry breakdown is in \cite{Kib67}, whereas
\cite{Hoo71a,Hoo71b} provided the proof of renormalizability.

Earlier,  \cite{Kom60} had calculated explicitly to
lowest order the
self--energy and vertex correction of the isospin $SU(2)$
Yang--Mills theory, in agreement with the above result \cite{Sal62}. For a
generalization of the Stueckelberg formalism see \cite{Fuj64}. A later
discussion of these subjects can be found in \cite{Ito76}.







\subsection{\label{sec:hs}Hidden symmetries}

As mentioned above, it was widely assumed that giving a mass to the photon
would spoil gauge invariance. But   the Stueckelberg lagrangian
with a physical scalar field $B$ in addition to the massive photon
\cite{Stu38I,Stu38II} enjoys indeed gauge invariance \cite{Pau41} and even
BRST invariance \cite{Del88}. The Proca lagrangian does not have this symmetry,
so it is absolutely necessary to include the Stueckelberg field $B$ which does
not play, however, a dynamical role. One may call this state of affairs a
``hidden symmetry." As we shall see below, the same trick has been used by
several authors in more general contexts.

Glauber, who visited Pauli at Z\"urich in 1950, was the first to emphasize the
close relationship between Stueckelberg's
gauge--invariant scheme and QED
\cite{Gla53}. He also remarked that the \cite{Dys48} transformation was just a
gauge transformation, which eliminated the
$B$--field from the interaction. For
vanishing photon mass $m$, the Stueckelberg field $B$ disappears as well from
the
supplementary condition, which is then the same as in QED, and eliminates the
longitudinal polarization of the vector field $A_\mu$. For
non-vanishing mass $m\not=0$, this condition can be considered as a definition
of the
$B$--field,
while the massive photon is no longer restricted to be transverse.

Glauber then proceeds to calculate radiative corrections to the photon mass.
These can be separated in two parts, according to whether they are
gauge--invariant or not. Both pieces are formally divergent. As in
electrodynamics,
the non-gauge--invariant integrals must be presumed to vanish (to second
order, the contribution is identical to the one of QED). To renormalize the
theory, one must remember that the $B$--field is still present in the
free--field hamiltonian. To preserve gauge invariance of the corrections, one
has to
reintroduce $B$ through the supplementary condition in the last step. The
photon mass correction is logarithmitcally divergent, and vanishes when $m$
goes to zero.

 Aware of Glauber's preprint, \cite{Ume52} generalized these
results to tensor representations,   studied the transition when the photon's
mass vanishes [see also \cite[pp. 113 and 204]{Ume56}], and  gave a
classification of the renormalizable and non-renormalizable interactions of
neutral and charged particles of spin 0, 1/2 and~1.  \cite{Bon63} considers the
gauge invariance of massive vector theories in a five--dimensional formalism
and shows the connection between Stueckelberg's formalism and that of
\cite{Ogi61}. The latter start from the lagrangian for the real field $A_\mu$
interacting with a conserved (Dirac) current
\be {\cal L}= -\frac12 \partial_\mu A_\nu \partial^\mu A^\nu -\frac{m^2}2
A_\mu^2 +j_\mu A^\mu \ee
This is invariant under the transformation $\delta A_\mu=\partial_\mu
\Lambda(x)$ subject to $(\partial^2-m^2)\Lambda=0$. The $A_\mu$ field can be
split into an invariant spin--1 part, and a non-invariant spin--zero part. They
show that the scalar has no interactions, so one can forget it. Furthermore,
the total energy operator  is positive definite up to an irrelevant constant.
Hence, the supplementary condition \eq{216} imposed by Stueckelberg to ensure
positivity is no longer required.

Many other papers deal with the relation of massive to massless QED, focusing
on a variety of questions, independently of Stueckelberg's $B$ field. For
example, \cite{Coe51} and \cite{Stu57} find that
after a suitable canonical transformation, the contributions of the scalar and
longitudinal components of the vector field to the
$S$--matrix compensate each other in the
limit $m\to0$. \cite{Sch62,Sch62b} exhibits
a gauge invariant massive field theory which has no continuous limit to QED
when $m\to0$. \cite{Bou62} invent a soluble field theory in which they then
carry out the limit as the bare mass vanishes, but in which the vector particle
remains massive. This toy model is gauge invariant, since   a
Stueckelberg massless scalar field is also introduced. \cite{Fel63} also show
that ``gauge
invariance does not require the bare photon mass to be zero''.
\cite{Kam64} use the original formalism of
\cite{Stu38I,Stu38II,Stu38III} to show that the representation of gauge
transformations for massive vector fields is inequivalent to that for zero
mass. As  discussed below, the limit $m\to0$ was also studied by
\cite{Dam70} and \cite{Sla71}.

\cite{Ram86}  applied Stueckelberg's scheme to a completely new
domain, in order  ``to obtain the fully covariant and gauge
invariant
field theory for free open bosonic strings in [the critical] 26 dimensions.
[This]
approach [$\ldots$] is based on very simple analogies with local field theory.
[$\dots$] Stueckelberg fields arise naturally and are shown to be unrestricted
for the most general gauge transformations."
 
 
 
Ramond remarks that ``in any theory which  is known in a specific gauge, one
can always reconstruct the original gauge invariant theory provided one knows
the form of the gauge transformations and the gauge conditions."  This was
precisely the situation for the first quantized string, where the gauge
symmetry is given by (half of) the Virasoro algebra.

In massless QED, the gauge transformation is of course $\delta A_\mu(x) =
\partial_\mu \Lambda(x)$, and the gauge condition is $\partial^\mu A_\mu=0$.
From this, and the equation of motion $\partial^2 A_\mu=0$, one can deduce the
Lorentz invariant and gauge invariant equation $\partial^\mu(\partial_\mu A_\nu
- \partial_\nu A_\mu)=\partial^\mu F_{\mu\nu}=0$.

In the Proca theory, one would start with \be (\partial^2 +m^2 ) A_\mu(x) =0
\label{R1}\ee and \be \partial^\mu A_\mu =0 \label{R2}\ee One could try the
gauge
transformation \be \delta A_\mu = \partial_\mu \Lambda(x) \label{R3}\ee The
variation of \eq{R2} gives \be \partial^\mu A_\mu + \partial^\mu \partial_\mu
\Lambda =0 \label{R4}\ee However, the variation of \eq{R1} implies \be
(\partial^2 +m^2 ) A_\mu + \partial_\mu (\partial^2 +m^2) \Lambda=0
\label{R5}\ee which is not compatible with \eq{R4}. Now, Ramond rewrites
\eq{R4} as \be \partial^\mu A_\mu + (\partial^2 +m^2) \Lambda -m^2 \Lambda =0
\label{R6}\ee and interprets the last term as the variation of the Stueckelberg
scalar field $-mB$.  Equation \eq{R6} then becomes \be \partial^\mu
A_\mu + mB + (\partial^2 +m^2) \Lambda =0 \label{R7}\ee On the mass-shell,
$(\partial^2 + m^2 )\Lambda=0$, hence the supplementary condition is now \be
<{\bf phys'}| \partial^\mu A_\mu + mB |{\bf phys}> =0 \label{R8}\ee which is
gauge invariant provided one completes \eq{R3} with \be \delta B = m \Lambda
\label{R9}\ee Substituting \eq{R7} into \eq{R5}, one gets the covariant
equation
of motion \be \partial^\mu F_{\mu\nu} +m^2 A_\nu -m \partial_\nu B =0
\label{R10}\ee ``which is the Stueckelberg equation for a massive vector
field."

Ramond proceeds ``to apply these tricks" to the string equation of motion \be
(L_0-1) \Phi =0 \label{R11}\ee with gauge conditions \be L_n \Phi =0 \qquad
(n\ge1) \label{R12}\ee and gauge transformation \be \delta \Phi = \sum_{n\ge1}
L_{-n} \Lambda^{(n)} \label{R13}\ee The Virasoro operators $L_n$ satisfy the
algebra \be \left[ L_n, L_m \right] = (n-m) L_{n+m} +{D\over12} n(n^2-1)
\delta_{n,-m} \label{R14}\ee where $D=26$ is the number of spacetime
dimensions. He then introduces the Stueckelberg fields $\Phi_n^{(p)}$  with
variations \be \delta \Phi_n^{(p)} = -L_n \Lambda^{(p)} + (2n+p)
\Lambda^{(n+p)}
\label{R15}\ee and equations of motion \be L_0 \Phi_n^{(p)} = - L_0 \left( L_n
\Lambda^{(p)} + (2n+p) \Lambda^{(n+p)} \right) \label{R16}\ee

We leave to the reader the pleasure of exploring the rest of the paper, which
concludes as follows: ``It should be clear that the subsidiary (Stueckelberg)
fields lead to much simpler looking expressions." For technical details, see
\cite{Pfe86} and its superpartner \cite{Kle89}.





The ten--dimensional ``superstring'' \cite{Gre84} is equivalent to the
``fermionic string'' \cite{Ram71,Nev71}. Its covariant quantization turns out
to be tricky, so people started by quantizing the
ten--dimensional superparticle
\cite{Cas76,Bri81}, which describes the dynamics of the
zero--modes of the
ten--dimensional superstring. As shown by \cite{Ber90a,Ber90b}, Stueckelberg
symmetries appear also in this context.

The lagrangian of the classical superparticle in first order formalism is
given by \be {\cal L}_{cl} = P_\mu \dot{X}^\mu - \Theta \Gamma^\mu P_\mu
\dot\Theta -\frac12 g P^2 \label{B1}\ee Here, $(X^\mu, \Theta)$ are the
classical coordinates of the superparticle, with $X^\mu$ a vector of the
ten--dimensional Lorentz group and $\Theta$ a
16--component
Majorana--Weyl spinor of
positive chirality, $P_\mu$ is the canonical momentum conjugate to $X^\mu$, $g$
is the einbein, and the dot denotes a time derivative. In order to quantize
\eq{B1}, \cite{Ber90a} introduce an infinity of ghosts, antighosts, and
Lagrange multipliers. They propose a new lagrangian which is BRST invariant.
Besides, it is also invariant under Stueckelberg transformations.

To illustrate these symmetries, we write down the transformation of the
coordinate $X^\mu$: \be \delta_{St} X^\mu = \sum_{p\ge0} \theta_{p+1,0}
\Gamma^\mu \epsilon^{p,0} \label{B2}\ee where $\theta_{p,0}$ are some of the
ghosts, and $\epsilon^{p,0}$ are local parameters with have the same
commutation properties and reality and chirality conditions as the antighosts
$\bar\theta^{p,0}$.

The Stueckelberg symmetries show that, effectively, the antighosts
$\bar\theta^{p,0}$ do not occur in the new lagrangian. Hence, they can be
eliminated by field redefinitions.

Commenting on a previous paper on the quantization of the superparticle,
\cite{Ber90a,Ber90b} showed that ``the mysterious gauge symmetry found by
\cite{Fis89} is a Stueckelberg symmetry \eq{B2}."

The final result of \cite{Ber90b} is a free quadratic lagrangian, BRST
invariant without any constraints. The Noether BRST charge $Q$ is nilpotent
($Q^2=0$)
off--shell.


\subsection{\label{subsec:nohiggs}Massive gauge theories without Higgs}

\subsubsection{Successes and problems of the standard theory}

The standard theory of electroweak interactions has many virtues
\cite{Gla61a,Sal68,Wei67}. It is a gauge theory with BRST invariance
\cite{Bec74,Bec75,Tyu75} and its gauge group, $SU(2)\times U(1)$, is
spontaneously broken through the  Higgs mechanism to the $U(1)$ invariance of
quantum electrodynamics
\cite{Hig64,Eng64,Gur64,Kib67}. The theory is unitary and renormalizable
\cite{Hoo71b,Bec76,Bec81}.   The massive gauge vector bosons $W^\pm$ and $Z$
corresponding to the broken symmetries have been discovered
\cite{Arn83a,Arn83b,Arn83c,Ban83,Bag83} and have been abundantly produced, and
of course the $U(1)$ gauge boson is the massless photon. The standard theory is
well suited for perturbative computations, allowing detailed calculations of
cross--sections and decay rates, in remarkable agreement with experiment; a
good review is  \cite{Alt00}.

In spite of its extraordinary and complete success, the standard theory has
some weaknesses, though what they are is somewhat a matter of taste. The
remarkable agreement of all known data with the standard theory has prompted
theoreticians to look for alternatives of it which preserve such valuable
virtue and overcome its shortcomings.

To begin with, one should point out that the
spin--zero Higgs particle has not
yet been discovered, although indications exist of  $M_{\rm Higgs}=115 \,{\rm
GeV}$  \cite{Ale02,Del01,Opa01,L301}. Even if this result is falsified, this is
not worrisome, and the experimental results available to date can be
reinterpreted as bounding $M_{\rm Higgs}>114\,{\rm GeV}$. The
high--energy
measurements are of such precision that they verify the radiative corrections
of the standard theory, and thus bound the  Higgs mass (through its logarithm)
to around $  M_{\rm   Higgs}<215 \,{\rm GeV})  $  at 95\% C.L. \cite{PDBook},
so not finding the Higgs at the Tevatron would not be catastrophic, in sharp
contrast to what would happen if it were not found at the LHC. At any rate, the
theory does not predict the Higgs mass, which is quite an independent parameter
in the theory (subject to more or less educated bounds, less stringent than the
experimental ones). It is important to stress that there is a logical
difference between the Higgs mechanism and the existence of the Higgs particle:
the latter provides an elegant and simple implementation of the former. In
terms of parameters, the
non-zero vacuum expectation value is independent of
the (fundamental or effective) scalar field's mass. Nevertheless, it is
important to stress right away that all efforts to implement the Higgs
mechanism of the standard model without a physical Higgs boson have failed so
far.

There are other theoretical misgivings about the standard theory. Paramount is
the hierarchy problem. What stabilizes the energy scale of electroweak symmetry
breaking, $m\sim O(10^2)$~GeV, with that of gravity, $M\sim O(10^{18})$~GeV?
Equivalently, the likely  unification of the electroweak and strong couplings,
should  take place at a comparably remote energy \cite{Geo74}. Why should it be
so different from the electroweak scale? Typically, the radiative corrections
to the $Z$ mass would be of the order of log$M/m$, so in order to reach
phenomenological agreement and keep $m<<M$, one needs a careful and unnatural
fine tuning of the mass parameters in the theory, order by order in
perturbation theory. The hierarchy problem is, thus, the wide difference
between the Higgs vacuum expectation value and the superstring (or quantum
gravity, or unification) scale. Supersymmetry stabilizes the hierarchy problem
but does not solve it. It could well be that the hierarchy problem is related
to the cosmological constant problem, the solution to which has not yet been
found, even in the framework of string theory.

Another problem, called the infrared catastrophe, arises in quantum
chromodynamics, the
non-Abelian gauge theory of strong interactions. Its gauge
group, $SU(3)$, is unbroken, and its massless vector bosons, the gluons,
interact with colored particles, namely quarks and themselves. When the energy
of the scattering process becomes small, the number of gluons emitted by a
colored particle diverges. The same problem occurs in quantum electrodynamics,
where the number of photons of low energy emitted by a charged particle
diverges. In QED, a remedy to this situation has been found, since the photon
is neutral. But in QCD, the gluons carry color and therefore interact with
themselves. This makes the infrared problem untractable, and the divergences
are laboriously removed only at the very end of the computation of suitable
observables \cite{Wei65}, see however \cite{Sla81}. On the other hand, a theory
of massive vector bosons would not be plagued by these infrared divergences
(the Higgs mechanism does not help since it would break asymptotic freedom
\cite{Hoo78,Nie95,Oji82}).

\subsubsection{Alternative models}

For the above reasons, among others, models have been proposed where the vector
boson
mass is put in by hand, in a variety of more or less astute ways. One could
call them ``genuine" massive
non-Abelian gauge theories. All such models
proposed so far are either not unitary or not renormalizable. This could well
be the end of the story. Nevertheless, the history of physics is full of
no--go
theorems which turned out to be wrong. Hence the continued theoretical interest
in these theories.  On the other hand,  massive vector
boson models which are renormalizable but not unitary could still provide
valuable hints towards a solution of the infrared problem.

On the more theoretical side, one might ask whether a
``non-renormalizable''
theory is really useless. For instance, it is conceivable that the divergences
in the Green's functions would cancel in the
on--shell
$S$--matrix elements
\cite{Nie95}: this happy situation does not seem to be realized in any of the
models constructed so far. A quite unconventional proposal is  to
 work in the Euclidean region of
space--like external momenta, to consider a massive scalar theory with an
exponential self--interaction which seems
non-renormalizable, and then to suggest a method for constructing an
$S$--matrix finite to all orders in perturbation theory \cite{Efi65}. See also
\cite{Fra63,Del69,Gho72, Gin75}. This method was developed in the
Yang--Mills
case by \cite{Sal71,Tay71,Leh71} and later by \cite{Fuk81,Fuk82,Fuk83}.

 In a different vein,      \cite{Geo93} discussed  composite Higgs fields,
 chiral symmetry, and  technicolor, whereas \cite{Nie96} discussed how massive
 Yang--Mills could arise from massless
 Yang--Mills coupled to a  topological
 field theory.

As emphasized, among others, by \cite{Vel68,Rei69,Vel70} and \cite{Sla72b}, the
problem of unitarity and renormalizability is very delicate without having at
our disposal a consistent and
parameter--free method of regularizing the
perturbative expansion. Also, as epitomized by the neutral vector boson theory,
hidden symmetries may be responsible for  the cancellation of  divergences in
individual graphs. Let us just emphasize in passing that the standard theory is
indeed unitary and renormalizable in the dimensionally regularized perturbative
expansion \cite{Bol72,Hoo72}, see also \cite{Bre77}.

Our aim now is not to provide an exhaustive history of ``genuine" massive
vector models, but only to focus on the influence of Stueckelberg's seminal
papers \cite{Stu38I,Stu38II,Stu38III} in this general line of research. His
crucial contribution was the introduction of a \sl bona fide \rm auxiliary
scalar field with positive metric and positive energy, modifying Proca's
original model.

We shall limit ourselves to sketching the context in which Stueckelberg's idea
was generalized or applied to various domains. In the following, we rely mostly
on the lucid reviews by \cite{Del88} and by \cite{Nie95}. Let us recall that
\cite{Del88} proved that the original Stueckelberg theory for neutral massive
vector fields (with the addition of
Fadeev--Popov ghosts and the
Nakanishi--Lautrup Lagrange multiplier) was invariant under nilpotent BRST
transformations. This ensures unitarity and renormalizability. The upshot of
both reviews for existing massive
Yang--Mills theories without a Higgs
mechanism is the following: those which use a suitable generalization of the
Stueckelberg mechanism are unitary order by order, but not perturbatively
renormalizable, due to the
non-polynomial interaction, whereas those which do not use the
Stueckelberg mechanism are renormalizable but not unitary, because of physical
ghosts \cite{Cur76,Boe96}.


\subsubsection{1962--1986: generalized Stueckelberg trick}

What is the generalization of the Stueckelberg trick to
non-Abelian massive
gauge theories? Remember that \cite{Sal62} and \cite{Ume61} kept the
 substitution $A_\mu\to A_\mu - m^{-1} \partial_\mu B$ of the Abelian case,
letting  $A_\mu=A_\mu^i T_i $, $B= B^i T_i$ with $T_i$ the generators of a Lie
algebra. The new lagrangian was then
gauge--invariant. The divergences due to the Proca vector propagator were then
explicit in the $\partial_\mu B^i$ terms. The latter was transformed away by a
unitary redefinition of the fields which yielded a
non-polynomial (actually, exponential) term in the Stueckelberg field $B$.

\cite{Kun67} proceeded in a slightly different way, starting from
\be {\cal L}= -
\frac14 \left( \partial_\mu A_\nu^i -\partial_\nu A_\mu^i + g f_{ijk} A_\mu^j
A_\nu^k \right)^2 +\frac{m^2}2 {\rm Tr}\, \left[ A_\mu^i T^i -\frac{i}g U^{-1}
\partial_\mu U \right]^2 \label{Kuku}\ee with \be U= {\rm exp} \left( i
\frac{g}m B^i T_i\right) \ee The lagrangian \eq{Kuku} is invariant under the
gauge transformation \ba &&\delta A_\mu^i = \left(D_\mu \Lambda\right)^i =
\partial_\mu \Lambda^i + g f^{ijk} A_\mu^j \Lambda^k \nn\\ && \delta B^i = m
\Lambda^i \ea The end result is very similar to that of \cite{Sal62} and
\cite{Ume61}.
A common weakness of these schemes is the absence
of ghosts  in the lagrangian \eq{Kuku}, as emphasized by  \cite{Sla72b}: in the
language of path integrals,  the  ghosts compensate the propagation of
unphysical states of the gauge  fields \cite{Fad67}. It turns out that the same
 $S$--matrix is obtained with or without the
 Stueckelberg fields. Indeed, using
 Pauli--Villars regularization and the
 preprint \cite{Hoo71b}, Slavnov showed that the
 $S$--matrix is independent of
 the longitudinal part of the Green's function.  He then proceeded to point out
 that the arguments for the renormalizability of the massive neutral vector
 theory do not apply to the massive
 Yang--Mills case. The symmetry of the
 theory, however, ensures a partial cancellation of divergences. Indeed, as
 shown by \cite{Vel68, Sla71}, the
 one--loop diagrams of the massive
 Yang-- Mills theory do not generate any divergences other than the usual ones
 associated with mass, charge, and
 wave--function renormalization.

What about higher orders? Since the massless
Yang--Mills field theory is
renormalizable, one could expect that the massive theory is also
renormalizable, if the $m\to0$ limit exists. Alas, this limit is sick
\cite{Sla71,Bou70,Dam70}: the matrix elements of  massive
Yang--Mills theories
are discontinuous in the limit $m\to0$. The reason for this singularity is easy
to understand from counting physical fields: a massive vector particle has
three physical degrees of freedom whereas a massless one has only two. If the
vector field happens to be neutral, as in QED, all the matrix elements which
are not diagonal in the number of longitudinal photons vanish as $m\to0$, and
thus the massless limit is
well--defined. In the
non-Abelian case, however,
where the gauge fields interact with themselves, this is not so, and the limit
contains a charged massless scalar field in addition to the tranverse vector
modes.

One would be tempted to conclude that the massive
Yang--Mills theory is not
renormalizable in the usual sense, and indeed \cite{Rei69} found new
divergences at two loops. To settle this issue satsifactorily, an invariant
regularization is required. Attempts have been made \cite{Del69} to apply the
method of \cite{Efi65,Fra63}, but they have been criticized by Slavnov:
ambiguity in the summation procedure, unreliable transition to the
pseudo--Euclidean region, and the fact that the solutions obtained do not, in
general,
have the symmetry built into the original lagrangian.

The extremely clear paper \cite{Sal70} precedes \cite{Sla72b} and covers
roughly the same ground as it. After recalling the Stueckelberg formalism for
neutral vector
fields, they  recast it in the language of path integrals. The advantage is
that field redefinitions can be tracked more carefully, including
non-trivial
Jacobians in the measure \cite{Fad67}. The Stueckelberg substitution \be V_\mu
\to A_\mu
=V_\mu +\frac1m \partial_\mu B \ee of the neutral Proca field $V_\mu$ by a
vector field $A_\mu$ and a scalar $B$ yields the generating functional \ba& Z
\left[ I^\mu,\eta ,\bar\eta\right] =&\ \int {\cal D} A^\mu{\cal D} B {\cal D}
\bar\psi{\cal D} \psi \; {\rm exp}  i \int \Biggl\{\frac12 A^\mu \left(
\partial^2 +m^2 \right) A_\mu -\frac12 B \left( \partial^2 +m^2 \right) B \nn
\\ &&+ {\cal L}_f + I^\mu \left( A_\mu -\frac1m \partial_\mu B \right)
+ \bar\psi\eta+\bar\eta\psi \Biggr\} \ea where ${\cal L}_f$ contains all the
terms involving fermions. One can now introduce a Lagrange multiplier $C$ to
get an equivalent expression, using the functional identity \be \int {\cal D}C
\;\delta \left( \partial_\mu V^\mu +\frac1m \partial^2 B \right) =1 \ee
Dropping, for the sake of notational convenience, the fermionic fields and
their  sources, the above generating functional is equivalent to \be Z \left[
I^\mu\right] = \int {\cal D} A^\mu{\cal D} B {\cal D} C \; {\rm exp}  i \int
\left\{{\cal L}[A^\mu -\frac1m \partial^\mu B ] +C \partial_\mu A^\mu  + I^\mu
\left( A_\mu -\frac1m \partial_\mu B \right) \right\} \ee The propagators are
now \ba &&\left< T A_\mu(x) A_\nu (y) \right> = -\left( g_{\mu\nu} +
\frac{\partial_\mu \partial_\nu}{\partial^2} \right) \Delta_m (x-y) \nn\\
&&\left< T A_\mu(x) B (y) \right> = 0 \\ &&\left< T B(x) B (y) \right> =
\Delta_0 (x-y) \nn\ea and thus \be \left< T (A_\mu(x)-\frac1m \partial_\mu B(x)
)
(A_\nu(y)-\frac1m \partial_\nu B(y) )\right> = -\left( g_{\mu\nu} +
\frac{\partial_\mu \partial_\nu}{\partial^2} \right) \Delta_m (x-y) \ee

This language was then generalized by \cite{Sal70} to charged vector fields.
The first example is provided by an isotriplet $V_\mu^i$ of vector fields. Let
$A_\mu$ and $\Omega$ denote the transverse and longitudinal parts of $V_\mu$,
defined by \be V_\mu^i \tau^i =A_\mu^i \Omega \tau^i \Omega^{-1} + \frac{i}{g}
\Omega \partial_\mu \Omega^{-1} \label{slauni}\ee  with $\tau^i$ are the Pauli
matrices and \be \partial^\mu
A_\mu^i=0\ee Equation \eq{slauni} can be viewed as a
non-Abelian gauge
transformation, where $\Omega $ is a
$2\times2$ unitary matrix, conveniently expanded  as \be \Omega(x) = \frac{g}m
\left[ \sigma(x) -i \tau^i B^i(x) \right] \ee in terms of the constrained field
variables subject to \be \sigma(x)^2 + \vec B(x)^2 =m^2 /g^2 \ee
The change of field
variables is done according to \be {\cal D}V=\int {\cal D}A {\cal D} \Omega
J(A^\Omega) \exp \left\{-\frac{i}2 \int (\partial A + mB )^2 \right\} \ee where
the Jacobian is given by \cite{Fad67} \be (J(A^\Omega))^{-1} = \int {\cal D}
\Omega \exp
\left\{-\frac{i}2 \int (\partial A^\Omega + mB )^2 \right\}\ee and $A^\Omega$
is
defined by the
right--hand side of \eq{slauni}. The generating functional is
then \ba Z[I] &&= \int {\cal D} V \exp i\int \left( {\cal L}[V] +I V \right)
\nn\\ &&= \int {\cal D} A {\cal D} \Omega J(A^\Omega) \exp i \int \left(
L[A^\Omega] -\frac12 (\partial A +m B)^2 + I A^\Omega \right)\ea and the
chronological pairings read as follows:   \ba &&\left< T V^i_\mu(x) V^j_\nu (y)
\right> = -\left( g_{\mu\nu} + \frac{\partial_\mu \partial_\nu}{\partial^2}
\right) \Delta_m (x-y) \delta^{ij} \nn\\ &&\left< T A^i_\mu(x) A^j_\nu (y)
\right> = - g_{\mu\nu}  \Delta_m (x-y) \delta^{ij} \nn\\ &&\left< T A^i_\mu(x)
B^j (y) \right> = 0 \\ &&\left< T B^i(x) B ^j(y) \right> =  \Delta _0(x-y)
\delta^{ij}\nn\ea From this, \cite{Sal70} developed a perturbation theory.

Note that the above procedure can be applied to a subset of the charged vector
fields. For example, if $V_\mu^3$ is not present, then merely replace ${\cal
D}V \to {\cal D} V \delta (V^3)$.




\cite{Fuk81} describe a quantum theory of massive
Yang--Mills fields. They
start from the lagrangian of \cite{Kun67}, to which they add
Faddeev--Popov
ghosts and the
Nakanishi--Lautrup Lagrange multiplier. The latter is quantized
in a Hilbert space of indefinite metric, following \cite{Kug78,Kug79}. Their
scalar field $\xi$ is in fact the generalized Stueckelberg field $B$ defined by
\cite{Kun67} and used by \cite{Sla72b}. \cite{Fuk81} introduce, however, a
different subsidiary condition, again following \cite{Kug78}. The resulting
theory is claimed to be invariant under a nilpotent BRST transformation and
unitary; this statement is questioned by \cite[p. 442]{Del88}.  Its
lagrangian contains an exponential in the scalar field, just like \cite{Kun67}
and hence, it is not renormalizable in the conventional sense, although
\cite{Fuk81} claim, quoting \cite{Sal71}, that it is renormalizable in the
 sense of \cite{Efi65}.


Based on \cite{Kun67} and \cite{Fuk81,Fuk82,Fuk83}, \cite{Son84} introduced a
Stueckelberg
scalar for the $U(1)_Y$ vector boson in the usual way, plus an isovector
Stueckelberg field for the $SU(2)_L$ sector. They found, interestingly, that
the ratio of the masses of these scalars had to be proportional to the Weinberg
angle, that is to $g'/g$. The theory is, however, not renormalizable.


%\subsection{\label{subsec:burn}Burnel}

\cite{Bur86a} formulates the Abelian theory of massive vector bosons in a gauge
invariant way, without Higgs fields, but using the theory of constrained
systems \cite{Dir64}. The Proca and Stueckelberg formulations appear as
particular gauges, unitarity being obvious in the former, and renormalizability
in the latter \cite{Mat49a,Mat49b}. He also discusses BRST invariance after
introducing
Faddeev--Popov ghosts.

 \cite{Bur86b} extends this method to the
 non-Abelian case, constructing a
 gauge invariant lagrangian, following \cite{Kun67,Sla71,Fuk81}. There exists a
 gauge with only physical particles, which can be  called the unitary gauge.
However,
 it is not power counting renormalizable. In a different gauge, there are
 Faddeev--Popov ghosts and a Stueckelberg scalar field. Due to the niloptent
 BRST and
 anti--BRST invariance, the fundamental
 Ward--Takahashi--Slavnov--Taylor  identities are satisfied
\cite{War50,Tak57,Sla72a,Tay71b}, and thus
 unitarity is ensured, but not renormalizability. In yet another gauge, the
 theory is renormalizable but the BRST transformation is not nilpotent, and
 unitarity is not satisfied \cite{Cur76a,Cur76,Cur79}; see also \cite{Car88}.
This emphasizes
the importance  of the nilpotence of the BRST invariance. \cite{Cur79} apply
this model (renormalizable but not unitary) to the infrared problem.

  \cite{Bur86b} then abandons the  lagrangian formalism and extends the
  Abelian  field equations \ba & & \partial^\mu F_{\mu\nu} + m^2 A_\nu +
  \partial_\nu B =0 \\ && \partial^\mu A_\mu = \alpha B \ea to the
  non-Abelian
  case as follows: \ba & & D_{ij}^\mu F^j_{\mu\nu} + m^2 A^i_\nu + D^{ij}_\nu
  B_j =0 \\ && \partial^\mu A_\mu^i = \alpha B ^i \ea With canonical
commutation
  relations, the gauge field propagator is now \be \Delta_{\mu\nu}^{ij}= -
  i\delta^{ij} \left\{ \frac{g_{\mu\nu} -k_\mu k_\nu /m^2 }{k^2 -m^2
  +i\epsilon} + \frac{k_\mu k_\nu /m^2 }{k^2 -\alpha m^2 +i\epsilon} \right\}
  \ee Although it is possible to maintain gauge invariance, unitarity cannot be
  satisfied. The conclusion, therefore, is rather pessimistic. \cite{Bur86b},
  however, ends the paper with the following remark: ``We have also
  emphasized that the nonrenormalizable couplings always involve unphysical
fields. Since, in general, ghosts of the type used here do not contribute at
all to physical amplitudes, it is quite plausible that the physical sector of
massive
Yang--Mills theory be renormalizable although it is not by power
counting and although it is not unitary at each order of the perturbation
expansion. This is particular to massive
Yang--Mills theory and the existence
of such a renormalizability is still an open question which merits further
attention expecially if Higgs bosons remain experimentally undetected."



\subsubsection{\label{subsec:delb}1988: unitarity versus renormalizability
reassessed}


We now present the important review \cite{Del88}. We have already mentioned
several times their treatment of the Stueckelberg theories of the Abelian
massive vector field. In their discussion of a variety of massive
gauge--invariant
non-Abelian models without Higgs mechanism, the themes are unitarity,
renormalizability and BRST invariance. They also provide a rich bibliography up
to 1988.

Their first example is the
non-Abelian generalization of the Stueckelberg
formalism by \cite{Kun67}, supplemented by \cite{Sla71, Sla72b}. Following the
latter,
they establish unitarity for the gauge propagator to one loop, working in the
Landau gauge. They show explicitly that one half of the contribution of the
Faddeev--Popov ghosts to the imaginary part of this propagator is compensated
by the
spin--zero part of the gauge vector field, whereas the other half is
compensated by the Stueckelberg scalar field. This means that in the
zero--mass
limit one does not recover the massless
Yang--Mills theory, as already
emphasized by \cite{Sla71}. Repeating their analysis of unitarity for
fermion--antifermion scattering to order $g^4$, \cite{Del88} find again that
the
Stueckelberg scalar contributes the essential factor of $1/2$ with the correct
sign. They then turn to the
high--energy behavior of longitudinally polarized
vector bosons, computing their elastic scattering. In a theory with Higgs
bosons, the amplitude is bounded, in agreement with unitarity. In the
Stueckelberg case, even if $S^\dagger S=1$ is satisfied order by order in
$g^2$, it turns out that the amplitude in increasing  orders of $g^2$ scales
with an increasing power of $E^2/m^2$ (they show this explicitly up to
$E^4/m^4$). They conclude that renormalizability is not satisfied
perturbatively in the generalized Stueckelberg scheme. Nevertheless,
\cite{Del88} point out that \cite{Shi75,Shi75a,Shi77} established in a
non-conventional manner the renormalizatiblity of
two--dimensional massive
Yang--Mills, elaborated upon by   \cite{Bar78}.

A more complete discussion of unitarity bounds can be found in
\cite{Cor73,Cor74,Lle73}. They introduce the concept of ``tree unitarity'',
holding when the
$N$--particle
$S$--matrix elements in the tree approximation
diverge no more rapidly than $E^{4-N}$ in the
high--energy limit, and discuss
its relation to gauge invariance and  renormalizability.  They remark  that
``a big advantage of the Stueckelberg formalism is that all bad behavior is now
isolated in the vertices." Curiously, they quote \cite{Stu57} instead of
\cite{Stu38I,Stu38II}. They conclude that tree unitarity is only
satisfied in models with spontaneously broken symmetry.

The second model discussed by \cite{Del88} was proposed by \cite{Fra69} and
\cite{Cur76} as a possible candidate for a theory of massive
Yang--Mills
fields; see also \cite{Oji80} and the particularly clear \cite{Oji82}. The
lagrangian involving no Stueckelberg fields but including
Faddeev--Popov ghosts
is  \be {\cal L}= -\frac14 F^2 + \frac{m^2}2 A^2 -\frac1{2\alpha} (\partial
\cdot A)^2 + \bar\omega \partial \cdot D \omega + \alpha m^2 \bar\omega \omega
+ \frac\alpha8 (\bar\omega \times \omega)^2 \label{uuyy}\ee where all fields
carry $SU(2)$ indices. In \cite{Fra69}, the Landau gauge    $\alpha=0$ is
chosen. This lagrangian is
invariant under the extended ``BRST transformation'' \ba && \delta A_\mu =
D_\mu
\omega \nn\\ && \delta \omega = \frac12 \omega \times \omega \\ && \delta
\bar\omega = -\frac1\alpha \partial\cdot A + \bar\omega \times \omega \nn \ea
which is not nilpotent: \be \delta^2 \not= 0 \ee The theory is gauge invariant
and has a good
high--energy behavior, albeit only  in the Landau gauge. Indeed,
thanks to the
gauge--fixing term, the vector propagator has a $k_\mu k_\nu/k^2$
term, instead of $k_\mu k_\nu/m^2$. Hence, the model is power counting
renormalizable. It is not unitary, for at least three reasons:

 1) The proof of unitarity for massive gauge theories in \cite{Kug79} rests on
 the nilpotency of the BRST charge $Q$, to whose cohomology physical states
 belong. Nilpotency could be enforced using the
 Nakanishi--Lautrup Lagrange
 multiplier $b$, such that $\delta \bar\omega= b+ \bar\omega \times \omega$ and
 $\delta b= b\times \omega$. This modification would spoil the invariance,
 however.

2) The ghost and
gauge--fixing terms, that is the last four terms of
\eq{uuyy},  are not the $\delta$ of something, and the physical lagrangian is
not by itself gauge invariant. Hence, the ghosts cannot be eliminated from the
physical
$S$--matrix.

3) In the Landau gauge, $\cal L$ is just the effective action in
\cite{Kun67,Sla71}, without the Stueckelberg terms, which were shown to be
crucially necessary for unitarity at one loop.

A pretty variant of the Stueckelberg model was presented by two of the authors
in \cite{Del86}. Using the field equations, the scalar Stueckelberg field $B$
was eliminated in favor of a
gauge--fixing functional of the vector field in
such a way that the gauge invariance of the mass terms was preserved; the
inherent nonpolynomiality could then be disregarded in some particular gauge.
Whereas the renormalizability of the scheme was not cast in doubt, it turns out
that unitarity is violated \cite{Kub87,Kos87}. For example, \cite{Kub87} shows
that the
one--loop correction to the imaginary part of the gauge boson
self--energy is the same as in the spontaneously broken theory in the Landau
gauge,
except that the
would--be Goldstone boson is missing. But its contribution is
necessary to compensate the
Faddeev--Popov ghosts.

\cite{Del88} reexamine this model in an equivalent formulation. They find that
it is invariant under a nilpotent BRST transformation. It is also power
counting renormalizable and gauge covariant (that is to say, the
$S$--matrix
does not depend on the gauge). Nevertheless, unitarity is violated. According
to the authors, ``we learn that gauge invariance and gauge covariance of a
theory are not strong enough conditions to ensure unitarity." This result is
paradoxical since it comes into conflict with the proof of unitarity by
\cite{Kug79}. \cite{Del88} resolve this contradiction by pointing out that,
contrary to the Stueckelberg model \cite{Kun67}, the model in \cite{Del86} does
not tend to the massive
Yang--Mills theory when the Stueckelberg field is taken
to zero. They conclude that ``the original Stueckelberg model is just right to
ensure (one loop) unitarity, and any tampering leads to
non-unitarity.''

Finally, \cite{Del88} describe the completely different model of \cite{Bat74},
which avoids several of the problems of the previous ones. The lagrangian is
\be {\cal L} = -\frac14 F^2 + \frac{m^2}2 A^2 + \frac\xi2 (\partial \cdot A)^2
+ {\cal G}(\xi, A) \ee with ${\cal G}(0,A)=0$. Under a gauge transformation one
has in particular \ba && \xi \to \xi + \delta \xi \nn \\ && {\cal L} \to {\cal
L}
+ \frac{\delta\xi}2 (\partial \cdot A)^2 + {\cal G}(\xi+\delta\xi, A)  - {\cal
G}(\xi, A)\ea Batalin gives a procedure to calculate $\cal G$, but this model
is
not perturbatively renormalizable and, as far as we know, its unitarity has not
been  established either.

 In conclusion, \cite{Del88} notice that renormalizability and unitarity seem
 to be competing qualities of massive non Abelian theories. ``The original
 Stueckelberg formulation, with its inherent non-polynomiality, is unitary but
 not renormalizable. This is in itself quite interesting, implying that the
 naive massive
 Yang--Mills action is of the correct form to ensure unitarity,
 and as we have seen any tampering with this leads us astray."

 \cite{Bur86b} has argued that in the Stueckelberg model one can find
 gauges in which ultraviolet divergences are confined to vertices that always
 involve unphysical fields, so that these divergences may cancel in the
 physical sector. ``Finally, it must be admitted that the Higgs mechanism
 remains the most complete method for giving mass to the vector bosons."



\subsubsection{\label{subsec:vann}1995--2002: new viewpoints}

Van Nieuwenhuizen and collaborators  have reexamined several aspects of the
review of \cite{Del88} in a series of papers.

\cite{Nie95} emphasizes the importance of Veltman's and Slavnov's work,
comprising explicit computations of one-loop \cite{Vel68} and two-loop
\cite{Rei69} divergences, the relation between massive and massless Yang-Mills
theories \cite{Dam70,Sla71}, and the generalized Ward identity  \cite{Vel70}.
He then  rederives in detail the \cite{Cur76} model by requiring a (not
nilpotent) BRST invariance.  Renormalizability holds, but not unitarity,
because one cannot enforce nilpotent BRST invariance.

\cite{Boe96} confirm \cite{Nie95}: the BRST transformation leaving invariant
the \cite{Cur76} model cannot be made nilpotent by the addition of a scalar
field Lagrange multiplier. Due to nilpotency, the Ward identity of non-Abelian
gauge theories acquires an additional term which calls for a more careful
enquiry into renormalizability. This is carried out using the full beauty of
the BRST formalism, with or without Lagrange multipliers. They find again that
the model is indeed renormalizable, with five multiplicative renormalization
$Z$ factors. This checks with an explicit one-loop computation. The authors
then ``determine the physical states, extending the work of \cite{Oji82}. Many
of these states have, for arbitrary values of the parameters of the theory, a
negative norm, and from this we conclude that the model is not unitary".  This
statement contradicts the curious claim of \cite{Per95} that the theory is
unitary and renormalizable, with three $Z$ factors. \cite{Boe96} conclude that
the \cite{Cur76} model might be useful as a regularization scheme for infrared
divergences, in particular in superspace.

 \cite{Hur97} analyzes both the \cite{Cur76} and the non-Abelian Stueckelberg
 models of \cite{Kun67,Sla72b}. He uses the causal Epstein-Glaser approach to
 quantum field theory \cite{Eps73,Eps76}. This allows to consider only the
 asymptotic (linear) BRST symmetry. Furthermore, the technical details
 concerning the well-known ultraviolet and infrared problems in field theory
 are separated and reduced to mathematically well-defined problems: the LSZ
 reduction formalism is not necessary. Let us sketch a few of the salient
points
 of the \cite{Hur97} analysis.

In the well-defined Fock space of the free asymptotic fields,  the
$S$--matrix
is constructed directly as a formal power series: \be S(g) = 1 +
\sum_{n=1}^\infty \frac1{n!} \int d^4x_1 \cdots\int d^4x_n \quad
T_n(x_1,\ldots,x_n) g(x_1) \cdots g(x_n) \label{vn1} \ee where $g(x)$ is a
tempered test function which switches on the interaction. The theory is defined
by the fundamental (anti)commutation relations of the free field operators,
their dynamical equations, and the specific coupling of the theory $T_1$.  The
$n$--point distributions $T_n$ ($n>1$) in \eq{vn1} are then constructed
inductively, making sure that they are compatible with causality and Poincar\'e
invariance.

The formalism can be applied to Yang-Mills theories. Non-Abelian gauge
invariance is introduced by a linear operator condition, separately  at every
order of perturbation theory. This is done with the help of the generator $Q$
of the  BRST transformation of the free asymptotic field operators, an operator
which is both linear and Abelian, by requiring that
\be \left[ Q,  T_n(x_1,\ldots,x_n) \right]   \label{vn2}\ee be the derivative
of a local operator.  Physical unitarity,
the decoupling of the unphysical degrees of freedom, is a direct consequence of
\eq{vn2}and the nilpotency $Q^2=0$.

Normalizability of the theory means, in the Epstein-Glaser approach, that the
number of finite constants to be fixed by physical conditions stays the same at
any order of perturbation theory. If a theory can be normalized in a gauge
invariant way, it is called renormalizable.

A normalizable theory can be established by a suitable choice of defining
equations. For example, the massive non-Abelian gauge potentials in a general
linear
$\xi$--gauge, transforming according to the adjoint representation of
$SU(N)$, satisfy \be  \left( \partial^2 +m^2 \right) A_\mu^i  - (1-\alpha^{-1})
\partial_\mu \partial^\nu A_\nu^i =0 \ee \be \left[ A_\mu^i(x) , A_\nu ^j (y)
\right] = i\, \delta^{ij} \left( g_{\mu\nu} + \frac{\partial_\mu
\partial_\nu}{m^2} \right)  \Delta_m(x-y)  - i\, \delta^{ij}
\frac{\partial_\mu \partial_\nu}{m^2}   \Delta_M(x-y) \ee where $\Delta_m(x)$
is  the massive
Pauli--Jordan commutation distribution \eq{27}  and \be M^2=
\alpha\; m^2 \ee The
Faddeev--Popov ghost fields are required to fulfill \be
\left\{ \omega^i (x) , \bar\omega^j(y)  \right\} = - i \delta_{ij} \Delta_M(x-
y) \ee \be (\partial^2 +M^2) \omega^i(x) =  (\partial^2 +M^2) \bar\omega^i(x)
=0\ee Already \cite{Cur76} and \cite{Oji82} noticed that if one takes over the
formula for $Q$ from the massless case in the general
$\alpha$--gauge \be
Q_{CF}= \int \frac{\partial_\nu A^\nu}\alpha \partial_0 \omega \; {\rm d}^3x
\ee one gets an operator which is not nilpotent in the massive case. But, as
noted above,  nilpotency is a necessary condition for unitarity. So a different
$Q$ is needed.

The generalization in \cite{Kun67}, \cite{Fuk81,Fuk82,Fuk83} of the
Stueckelberg
formalism is to add  scalar fields $B^i(x)$ satisfying \ba  && \left[ B^i(x) ,
B^j (y) \right] =-i \delta^{ij} \Delta_M(x-y)  \\ && (\partial^2 + M^2 ) B^i(x)
= 0 \ea The BRST generator is then \be Q_S= \int \eta (x) \partial_0 \omega (x)
\; {\rm d}^3x \ee with the short-hand \be \eta^i(x) = \alpha^{-1} \partial^\mu
A^i_\mu (x) + m B^i(x) \ee The corresponding BRST transformation looks
familiar:
\ba && \left[ Q_S, A^i_\mu \right]= i \partial_\mu \omega^i \\ && \left[
Q_S,B^i
\right]= i \,m \omega^i \\ &&  \left[ Q_S,  \omega^i \right] =0\\ &&  \left[
Q_S,
\partial^\mu A^i_\mu \right] = -i \, M^2 \omega^i \\ &&  \left[ Q_S,
\bar\omega^i \right] =-i \,\eta^i \\ &&  \left[ Q_S,  \eta^i \right] =0\ea

\cite{Hur97} proceeds to construct the most general
gauge--invariant coupling
$T_1$ such that  $[ Q_S,  T_1] $ is a total derivative as in \eq{vn2},
Lorentz-- and
$SU(N)$--invariant, with ghost number zero, and with maximal mass dimension
equal to four (for normalizability). He finds a certain number of trilinear
couplings in the fields $A_\mu^i$, $B^i$, $\omega^i$, $\bar\omega^i$ and their
derivatives. He has thus defined a manifestly normalizable theory which is
gauge invariant to first order in perturbation theory, and respects further
symmetry conditions. Can one prove the condition of gauge invariance, namely
that
$[Q_S,T_n]$  be a total derivative, inductively to all orders in perturbation
theory? By
explicit calculation, \cite{Hur97} shows that this condition fails already at
second order. The constraint of normalizability is essential for this
conclusion. Hence, the Stueckelberg generalization for
non-Abelian massive
gauge theories is not perturbatively renormalizable.



% \subsection{\label{subsec:dragon}dragons}

\cite{Dra97} start with the \cite{Kun67} generalization of the Stueckelberg
model to a massive
Yang--Mills theory: the vector fields $A_\mu^i$ and scalar
fields $B^i$ belong to  the adjoint representation of a Lie  group $G$. The
kinetic and mass terms of the lagrangian are separately
gauge--invariant. To
compensate the unphysical degrees of freedom of $A_\mu$ and $B$, they introduce
Faddeev--Popov ghosts $\omega^i$ and $\bar\omega^i$, also in the adjoint.  The
lagrangian, including a
gauge--fixing term, is invariant under a BRST operator
which is nilpotent if one adds a further
Nakanishi--Lautrup Lagrange multiplier
to the set of fields. The
gauge--fixing term is such that the propagators of
$A_\mu$ and $B$ fall off like $k^{-2}$ for large momenta $k^\mu$.
Unfortunately, the exponential of $B$ appears in the lagrangian. They then
redefine the vector field $A_\mu$ into a new $\hat A_\mu$ such that the
$S$--matrix is unchanged \cite{Col68}. The BRST transformation $\bf s$ is then
given by
\ba && {\bf s} \hat A_\mu = 0\\ &&{\bf s}B= m \bar\omega\\ &&{\bf
s}\bar\omega=0\\ &&{\bf s}w=b\\  &&{\bf s}b=0\ea
and the
BRST--invariant lagrangian is of the form \be {\cal L} ={\cal L}_{\rm
phys} (\hat A_\mu, \partial^\nu \hat A_\nu) + s( \omega X)\ee where one can
choose \be X= \frac1{2} \left[ b- \partial^\mu \hat A _\mu - (\partial^2 +m^2)
\frac{B}{m} \right] \ee so that the
Faddeev--Popov ghosts $\omega $ and
$\bar\omega$ are
free and the
Nakanishi--Lautrup auxiliary field $b$ can be solved from $X=0$.
Redefining again $\bar A_\mu = \hat A_\mu + m^{-1} \partial_\mu B$, the
physical lagrangian containing  the gauge  and the Stueckelberg fields becomes
\be {\cal L}_{\rm phys} = -\frac{1}{4} \left[ F_{\mu\nu}^i (\bar A -m^{-1}
\partial B) \right] ^2 \label{dddD1}\ee
with the usual notation $ F_{\mu\nu}^i (Y) = \partial_\mu Y_\nu^i -
\partial_\nu Y_\mu^i +gf^i_{\; jk} Y_\mu^j Y_\nu^k $.

The result of all these redefinitions is to replace the original exponential in
$B$ by a polynomial in $\partial_\mu B^i$, of mass dimension eight. The
propagators of $\bar A_\mu$ and $B$ are
well--behaved at high energies, but the
derivative interaction of $B$ is still not
power--counting renormalizable.  Of
course, \eq{dddD1} reminds us very much of expressions in \cite{Ume61,Sal62}.


\subsection{\label{sec:pubel}Related applications}

We end this chapter mentioning a variety of applications of the Stueckelberg
formalism which fall outside our main line of exposition, but might be of
interest to the reader.

Early phenomenological applications of charged vector mesons in the Proca and
Stueckelberg theories are \cite{You63},\cite{Bai64a,Bai64b,Bai65} and also
\cite{Abe69,San68}.

\cite{Wat67} applied the Stueckelberg formalism to the
Rarita--Schwinger (spin~$3/2$) field, whereas \cite{Del75b} did it for
the spin~2 field.

%\cite{Rat02} have found how to formulate a massive spin--two
% field using $D$--brane technology.

\cite{Zim68} used Stueckelberg's lagrangian and its invariance under the Pauli
gauge transformations to study the renormalization of masses of real vector
fields,

 The Stueckelberg theory has been analyzed in the
 Batalin--Fradkin--Vilkovisky formalism
\cite{Day93,Saw95,Saw97},  in axiomatic field theory
\cite{Mor86,Mor87} and in new quantization procedure in algebraic field theory
\cite{Wie96}.


 The Stueckelberg theory was supersymmetrized ($N=1$) in
the early days of supersymmetry \cite{Del75a}. It was found that the condition
for the superStueckelberg mechanism to work was that $\bar D D J=0$,    with
$J$ the external supercurrent coupled to the superphoton. In the following, we
use Wess-Bagger notation (i.e. two-component fermions) but with our usual
metric $+---$.
 The standard kinetic term (in terms of $W^\alpha = \bar D^2 D^\alpha V$, with
$V=V^\dagger$ a vector superfield)
\be {\cal L}_0 =\frac14 \int {\rm d}^2\theta W^2
+\frac14\int {\rm d}^2\bar \theta \bar W^2 \ee
which in the Wess-Zumino gauge reduces to
\be {\cal L}_0 =-\frac14 F_{\mu\nu}^2 -i\lambda \sigma^\mu \partial_\mu
\bar\lambda +\frac12 D^2\ee
is supplemented with
\be {\cal L}_m = -m^2 \int {\rm d}^2\bar\theta {\rm d}^2\theta [V
+\frac{i}{m}(\Phi -\Phi^\dagger)]^2 \ee
mimicking the (bosonic) Stueckelberg starting point. Note that we must
introduce
a chiral and an antichiral superfield. The mass term above  gives a mass to the
vector, yields the kinetic terms for the complex Stueckelberg scalar fields $a$
and $a^*$ and their spin 1/2  superpartners $\psi$ and $\bar\psi$, induces a
mixed mass term between the photino $\lambda$  and the Stueckelberg fermion
$\psi$, and provides the cross term $mA^\mu \partial_\mu (a+a^*)$. When the
auxiliary field $D$ is eliminated, a mass term for $(a-a^*)$ comes out as well.
 
We still must add a gauge-fixing term. In the massless case, it is of  the form
$\int {\rm d}^4\theta  (\bar D^2 V)(D^2 V) $, whereas now it is better to take
\be {\cal L}_{gf} = \xi \int {\rm d}^2\bar\theta {\rm d}^2  \theta (\bar D^2 V
+\frac1{m^2} \Phi)(D^2 + \frac1{m^2}\Phi^\dagger) \ee
Problem: twice the kinetic term for the chiral fields.
 
 
 
 
The Stueckelberg mechanism has been
used extensively in the pedagogical presentations of \cite{Gat83,Sie99} where the  Stueckelberg
or ''compensator fields are introduced as a simple example of Goldstone fields, and numerous
applications to supersymmetry are discussed.
See also \cite{Gue91}.
 
 
 
The non-renormalizability of the non-Abelian version
turned out to be crucial to
establish the non-renormalizability of the $N=2$ massive
Yang--Mills theory
\cite{Khe91}, and useful for its analysis in harmonic superspace \cite{Vol94}.

Stueckelberg's trick has been used to reformulate chiral
\cite{Kul94a,Kul94b,Kul98,Kul01a,Kul01c,Kul02a}, and
other two-dimensional models \cite{Kul01b,Kul01d,Kul02b,Kul02c},
as well as  the massive
three-dimensional gauge theory \cite{Sch81,Dil95}.
\cite{Ban96}  established a three-dimensional duality similar to the
well--known duality between sine--Gordon and Thirring \cite{Col75} exploiting
the Stueckelberg formulation.
Attempts to extend the three-dimensional  topological Yang--Mills
mass  \cite{Des82}
to four dimensions
used also the Stueckelberg trick \cite{All91,Hwa97}, as did a powerful no-go
theorem that pretty much  ended such attempts \cite{Hen97}.
 


 The two--dimensional Stueckelberg theory has been studied in   a
Robertson--Walker background, in a black hole metric, and in a Rindler wedge
\cite{Chi92,Chi93,Chi94}.
 
\cite{Jan87} formulated the  Stueckelberg theory in
anti--de Sitter space.
 
 
 \cite{Deg94,Deg95a,Deg95b} derived  Stueckelberg's construction
starting from loop space, and extended it to a  certain classical non-Abelian
 version for
 rank--two tensor fields. \cite{Deg95c} found, in particular,
that only null strings interact with massive vector fields, and no strings
interact with massive
third--rank tensor fields at the classical level.

The idea of inventing new fields in order to
uncover or make manifest hidden symmetries has been applied  in many contexts.
  The extension of the Stueckelberg formalism for a massive antisymmetric field
\cite{Kal74} has been the subject of intensive research
\cite{Saw95,Bar96,Deg99a,Sma01,Kuz02}, approaching a clear understanding
\cite{Dia01}. The generalization to
 $p$--forms ($p=2$ for the
Kalb--Ramond field) has met with success \cite{Biz96,Biz98,Biz99}. \cite{Deg97}
 show that a $U(1)$ gauge theory defined in the
 configuration space for closed
 $p$--branes yields the
 gauge theory of a massless  $(p + 1)$
 antisymmetric tensor field and the Stueckelberg
 formalism for a massive vector field.  The $p$--form extension of the
Stueckelberg formalism has been
used to establish dualities between  field theories
\cite{Fre81,Saw96,Saw97,Sma00},  to study symmetry breaking in
$D$--brane theories \cite{Ans00}.





\cite{Ban97} consider the (classical) hamiltonian formulation of a
Higgs--free
massive
Yang--Mills theory with the Stueckelberg trick, and
\cite{Bar97} apply it to the standard model.
\cite{Deg99b} interpolates between various classical lagrangians, with or
without Higgs and Stueckelberg fields.


\section{\label{sec:conc}Conclusions and outlook}

Stueckelberg's original idea was to introduce a physical
scalar field $B(x)$ into the Abelian
massive vector field lagrangian to make the theory as similar as possible to QED.
It was shown by several authors that this proposal facilitated the discussion of the
renormalizability of massive vector field theories and made manifest some hidden
symmetries. The neutral massive Abelian vector field theory is gauge invariant
and BRST invariant, because the transformation of the  Stueckelberg $B$ field
compensates the transformation of the mass term.
 This explains the renormalizability of this theory. The field $B$ plays a role
 similar to that of the Goldstone boson in spontaneously broken theories. Charged
 or non-Abelian theories without the Higgs mechanism
 have not been shown to be renormalizable, because the
 derivative couplings of the Stueckelberg field can only be eliminated at the
  expense  either of exponential couplings or of unitarity.
 Work on these issues has not been completely abandoned, however.
 
 Stueckelberg's mechanism, simple as it seems today and always elegant, inspired
 numerous imitations, ranking from non-Abelian massive vector theories without
 Higgs fields to   supersymmetric, topological and  string    theories.  In many cases,
 new symmetries were discovered.
 
Additionally, in this paper we have pursued an idea of Raymond Stora to construct a
standard electroweak theory with a massive photon, preserving the $SU(2)_L\times U(1)_Y$
BRST symmetry. The neutral scalar Stueckelberg field $B$ appears together with
a massive hypercharge  vector field, and the photon inherits a mass after
the spontaneous symmetry breaking.  This can be interpreted in two ways.
The photon mass
can be considered as an infrared cut-off, a mere calculational trick, allowing
one to deal cleanly and separately with the  infrared divergences.
   This would require,
 of course, the zero mass limit to be smooth.  A less conservative point of view
 calls for taking the photon mass seriously, albeit limited by empirical data to
 a very small value. In this case, new phenomena appear, proportional to the photon's mass
  (squared): neutrino photon couplings, parity violation of the electron photon
  couplings, slightly different $Z$ mass, etc.  None of them seem comparable,
 in precision,
 to the direct limits on the photon mass, but more research in this direction seems
 warranted, in particular for neutrino cosmology. Finally, the physical definition
  of the electric current appears now in a new light.




\begin{acknowledgments} We are very grateful to Raymond Stora for lending us
the transparencies of his talk at Leipzig 2000, where he suggested to use
the Stueckelberg mechanism in the standard model, and for many illuminating
discussions, in particular on the distinction between asymptotic fields and
local fields perturbatively coupled to conserved currents. Any
errors, conceptual or otherwise, are ours and not his.  We also thank
Tobias Hurth for making his bibliography available to us.
\end{acknowledgments}


 
% Specify following sections are appendices. Use \appendix* if there
% only one appendix.
%\appendix
%\section{}
  \appendix
           \section{} %{Electroweak conventions and long formulas}
 
We have collected here some longish formulae on the Stueckelberg
modification of the standard model, section \ref{sec:sm}.
 
             \subsection{\label{a:nota}Notation}
             We use the metric $(+1,-1,-1,-1)$. It is worth pointing out that
\cite{Stu38I,Stu38II,Stu38III} used a real metric but with the opposite sign,
while Pauli and others used Euclidean metric with an imaginary fourth
component.


 
 We use throughout the notations
\be A_\pm=\frac1{\sqrt 2} \left(A_1 \pm i A_2\right) \ee
\be \vec A \cdot \vec B = \sum_{i=1}^3 A_i B_i = A_+B_-+A_-B_+ +A_3B_3 \ee
and
\be (\vec A \times \vec B)_i = \varepsilon_{ijk} A_j B_k \ee
with $\varepsilon_{123}=1$ and cyclic; in the $\{+,-,3\}$ basis the non-zero
elements of the $\varepsilon_{ijk}$ are
\be \varepsilon_{3+-}=-\varepsilon_{3-+}
=\varepsilon_{+3+}=-\varepsilon_{++3}=\varepsilon_{--3}=-\varepsilon_{-3-}=i
\ee
Sometimes we use the short-hand $\vec A^2=\vec A \cdot \vec A$.
 
 \subsection{\label{a:scal}Scalar lagrangian}
Explicitly,
\ba & \left| D_\mu \Phi\right|^2 =& \frac12 (\partial_\mu H)^2  +
\frac12 (\partial_\mu \phi_3)^2 +\partial _\mu \phi_+ \partial^\mu \phi_- \nn\\
&&+\frac12 \partial_\mu H (g W_3^\mu -g'V^\mu) \phi_3
-\frac12 \partial_\mu \phi_3 (g W_3^\mu -g'V^\mu) (H+f) \nn\\
&& +\frac{g}2 (\partial_\mu H -i\partial_\mu \phi_3) W_-^\mu \phi_+
 + \frac{g}2 (\partial_\mu H +i\partial_\mu \phi_3) W_+^\mu \phi_- \nn\\
 && -\frac{g}2 \partial_\mu  \phi_+ W_-^\mu (H+f-i\phi_3)
 -\frac{g}2 \partial_\mu  \phi_- W_+^\mu (H+f+i\phi_3) \nn\\
 &&+\frac14 \left[ (H+f)^2 +\phi_3^2 \right]
  \left[ g^2 W_+^\mu W_{-\mu} +\frac12 (gW_3^\mu -g'V^\mu)^2 \right] \nn \\
 && -i\frac{gg'}2 (H+f-i\phi_3) \phi_+ V_\mu W_3^\mu
 +i\frac{gg'}2 (H+f+i\phi_3) \phi_- V_\mu W_3^\mu \nn\\
 && +\frac12 \phi_+\phi_-
  \left[ g^2 W_+^\mu W_{-\mu} +\frac12 (gW_3^\mu +g'V^\mu)^2 \right]
 \ea
 
 \subsection{\label{a:quad}Quadratic lagrangian}
Using the above choices of
gauge--fixing functions, and
dropping three total derivatives, the quadratic part of the lagrangian is
\ba & {\cal L}_{2} =& -\frac12 ( \partial_\mu \vec W_\nu )^2
+ \frac{f^2g^2}8  \vec  W_\mu^2
+\frac12\left(1-\frac1{\alpha}\right)  (\partial_\mu \vec W^\mu)^2
 - \frac{gg'f^2}4 W_3^\mu V_\mu \nn\\
&&-\frac12 (\partial_\mu V_\nu )^2 +
\frac12 \left( m^2 + \frac{f^2g^{\prime 2}}4 \right) V_\mu^2
+\frac12\left( 1-\frac1{\alpha'} \right) (\partial_\mu V^\mu)^2 \nn\\
&&  +\frac12 (\partial_\mu \vec \phi  )^2
- \frac{\alpha f^2g^2}8  \vec \phi^2
  - \frac{\alpha' f^2g^{\prime 2}}8  \phi_3^2  \nn\\
  &&+\frac12 (\partial_\mu B)^2
  - \frac{\alpha' m^2}2 B^2
  + \frac {\alpha' g' mf }2  \phi_3 B\nn\\
  && +\frac12 (\partial_\mu H)^2 -\lambda f^2 H^2 \nn\\
  && -\omega^* \left (\partial^2 + \alpha' m^2
  + \frac{\alpha' f^2g^{\prime 2}}4 \right) \omega
  + \frac{ f^2g g^{\prime }}4 \left( \alpha' \omega^* \omega_3
        + \alpha  \omega_3^* \omega \right) \nn\\
&& -\vec\omega^* \left (\partial^2  + \frac{\alpha f^2g^{ 2}}4 \right) \vec
\omega
\label{lacuadra}\ea
 
 
 It might be necessary to expand to second order in $\epsilon$ for some
computations, but for the time being we restrict ourselves to first order.
The various rotations of $g$ and $g'$ to order $\epsilon$ are
\be  g {\rm c}_w
+g'{\rm s}_w \simeq \sqrt{g^2 +g^{\prime2}} +{\cal    O}(\epsilon^2)
\label{giol1}\ee
\be
g {\rm c}_w - g'{\rm s}_w \simeq \frac{g^2 -g^{\prime2}}{\sqrt{g^2
+g^{\prime2}}} \left(
1-2 \frac{g^2g^{\prime2}}{ g^4-g^{\prime4}} \epsilon   \right)
+{\cal    O}(\epsilon^2)  \ee
\be
 g{\rm s}_w - g'{\rm c}_w \simeq
 \epsilon \frac{gg'}{\sqrt{g^2 +g^{\prime2}}}   +{\cal    O}(\epsilon^2)  \ee
\be
g {\rm s}_w +g' {\rm c}_w \simeq
\frac{2gg'}{\sqrt{g^2 +g^{\prime2}}} \left(
1+\frac12 \frac{g^2-g^{\prime2}}{ g^2+g^{\prime2}} \epsilon   \right)
+{\cal    O}(\epsilon^2)  \label{giol4}\ee
 
 \subsection{\label{a:mass}Mass formulas}
 
The exact mass eigenvalues of the neutral vectors, scalars and ghosts are
\be M^2_{Z\atop A} = \frac{f^2}8 \left(g^2
+g^{\prime2}\right) \left(1 +\epsilon \pm \sqrt{1  -2\epsilon \frac{g^2
-g^{\prime2} }{g^2 +g^{\prime2}}
+\epsilon^2} \right)\label{mma} \ee

 \ba & M^2_{\chi_Z\atop
\chi_A}
=M^2_{G\atop S} = &\frac{\alpha' f^2}8 \Bigl(  \frac\alpha{\alpha'} g^2
+g^{\prime2}
+\epsilon  \left(  g^2 +g^{\prime2}\right)
\nn \\ && \pm
\sqrt{ \left ( \frac\alpha{\alpha'}  g^2 +g^{\prime2}\right)^2 -
2\epsilon\left(\frac\alpha{\alpha'} g^2 -g^{\prime2}\right)
\left( g^2 +g^{\prime2}\right) +
\epsilon^2 \left( g^2 +g^{\prime2}\right)^2}
\Bigr) \ea This formula is the same as \eq{mma} with the substitutions
$g^2\to\alpha\,g^2$, $ g^{\prime2} \to\alpha'\,g^{\prime2} $, and $\epsilon\to
\alpha'\,\epsilon$.
In general, to lowest order in $\epsilon$, the masses of the neutral
longitudinal scalars and the neutral ghosts are
\be M^2_{G} =  M^2_{\chi_Z} \simeq \frac{ f^2 }{4}
\left( \alpha g^2 +\alpha' g^{\prime2} \right) \left(1+
+\epsilon \frac { g^{\prime2}\left( g^2 +g^{\prime2}\right) }
{\left ( \frac\alpha{\alpha'}  g^2 +g^{\prime2}\right)^2 } \right)
\ee
\be M^2_{ S} = M^2_{\chi_A} \simeq \epsilon \frac{\alpha f^2 g^2}{4}
\frac{ g^2 +g^{\prime2} } {  \frac\alpha{\alpha'}  g^2 +g^{\prime2} }
\ee




 
\subsection{\label{a:inte}Interaction lagrangian}

We turn now to the interaction lagrangian ${\cal L}_{\rm int}$, which
added to the quadratic lagrangian \eq{lacuadra} makes up the full physical
gauge--fixed (but still matter--free) lagrangian useful for quantum
computations. We
quote it in terms of the original variables first: these fields do not have a
well--defined propagator!



\ba & {\cal L}_{\rm int} =&
-ig \left[ \partial_\mu W_\nu^+
 \left( W^{\mu -} W^{\nu 3}-W^{\mu 3}W^{\nu -} \right)
+{\rm cyclic} \right]
\nn\\  &&
+\frac12 \left( g W_\mu^3 -g' V_\mu \right)
  \left( \phi_3 \partial^\mu H -H \partial^\mu \phi_3\right)
\nn\\  &&
+\frac{g}2 W^-_\mu \left[ \left( \phi_+ \partial^\mu H -H \partial^\mu \phi_+
\right)
  +i \left( \phi_3 \partial^\mu \phi_+ - \phi_+ \partial^\mu \phi_3 \right)
\right]
\nn\\  &&
+\frac{g}2 W^-+\mu
\left[
\left( \phi_- \partial^\mu H -H \partial^\mu \phi_- \right)
  -i \left( \phi_3 \partial^\mu \phi_- -\phi_- \partial^\mu \phi_3\right)
\right]
\nn\\  &&
+ f H \left[
\frac{g^2}2 \left( W^+ W^- \right) +\frac14 \left(gW^3-g'V\right)^2
-\lambda \left( 2\phi_+\phi_- +H^2 +\phi_3^2 \right) \right]
\nn\\  &&
+i \frac{gg'f}2 \left(\phi_- -\phi_+\right) \left( V W^3 \right)
\nn\\ &&
+g^2 \left[
\left(W^3 W^+\right) \left( W^3 W^-\right)
-\left(W^3\right)^2 \left( W^+W^-\right)
-\frac12 \left(W^+W^-\right)^2
+\frac12 \left(W^+\right)^2 \left(W^- \right)^2
\right]
\nn\\ &&
+\frac{g^2} 2 \left( \phi_+\phi_- +\frac12 H^2 +\frac12 \phi_3^2 \right)
\left( W^+W^-\right)
\nn\\ &&
-\lambda \left( \phi_+\phi_- +\frac12 H^2 +\frac12 \phi_3^2 \right) ^2
\nn\\ &&
+\frac18 \left( H^2 +\phi_3^2 \right) \left(gW^3-g'V\right)^2
+\frac14  \phi_+\phi_-  \left(gW^3+g'V\right)^2
\nn\\ &&
-\frac{g'g}2 \left[ \left( \phi_3 +i H \right) \phi_+ +
\left( \phi_3 -i H \right) \phi_- \right] \left( V W^3 \right)
\ea
 







Of course, to do actual computations of $S$--matrix elements, this has to be
rewritten in terms of the
mass eigenstates. The general expressions are quite unwieldy, even in the
simplified case $\alpha'=\alpha$, whereby $\tilde\theta_w
=\theta_w$.

  \subsection{\label{a:ghos}Ghost lagrangian}

The interaction part of the ghost lagrangian is
\ba
  && -g \vec \omega ^*
\cdot
\partial_\mu (\vec W _\mu \times \vec \omega) -\frac{\alpha' f g^{\prime2}}4 H
\omega^* \omega  -\frac{\alpha f g^{2}}4  \vec \omega^*\cdot ( H \vec \omega +
\vec \phi \times \vec\omega ) \nn\\ && +
 \frac {f g g'}4 \left\{ \alpha'
\omega^*
[i(\phi_+ \omega_- - \phi_- \omega_+) + H \omega_3]  + \alpha [i(\omega_-^*
\phi_+
- \omega_+^* \phi_-) + \omega_3^* H] \omega \right\} \ea
whereas the full ghost lagrangian in the mass eigenstate basis, with the
tree--level simplification $\alpha'=\alpha$ is
\ba &{\cal L}_{gh}=& \omega_+^* (\partial^2 +\alpha
M_W^2
) \omega_- + \omega_-^* (\partial^2 +\alpha M_W^2 ) \omega_+ \nn\\ &&+\chi_Z^*
(\partial^2 +\alpha M_Z^2 ) \chi_Z +\chi_A^* (\partial^2 +\alpha M_A^2 ) \chi_A
\nn\\
&& + i \,\omega_+^* \partial_\mu \left[W_-^\mu ({\rm c}_w \; \chi_Z +
{\rm s}_w \; \chi_A) -({\rm c}_w \; Z^\mu + {\rm s}_w \; A^\mu)
\omega_-\right] \nn\\
&& -i\, \omega_-^* \partial_\mu \left[W_+^\mu ({\rm c}_w
\; \chi_Z + {\rm s}_w \; \chi_A) -({\rm c}_w \; Z^\mu + {\rm s}_w \;
A^\mu) \omega_+ \right] \nn\\ &&
+ i \left({\rm c}_w \; \chi_Z^* + {\rm s}_w \; \chi^*_A \right)\; \partial_\mu
(W_+^\mu \omega_- -W_-^\mu \omega_+) \nn\\ &&
+\frac{\alpha\, f\, g^2}4 \; \Bigl[ i (\cos\beta \; G + \sin
\beta \; S) \left(\omega_-^* \omega_+ -\omega_+^* \omega_- \right) +H
\left(\omega_+^* \omega_- +\omega_-^* \omega_+ \right) \nn\\
&&\qquad +
({\rm c}_w \; \chi_Z^* + {\rm s}_w \; \chi_A^* ) i \left(\omega_- \phi_+
- \omega_+ \phi_- \right)\Bigr] \nn\\
&& -i\frac{\alpha\, f\, g}{8} \;
\left(\phi_+ \omega_-^* -\phi_- \omega_+^* \right ) \left[ (g\,{\rm c}_w -
g'\, {\rm s}_w )\; \chi_Z +(g\,{\rm s}_w +g'\, {\rm c}_w )\; \;
\chi_A\right] \nn\\
&& +\frac{\alpha\, f\, g}4 \; H \; \Biggl[ (g\,{\rm c}_w
+g'\, {\rm s}_w ) {\rm c}_w\; \chi_Z^*\chi_Z +(g\,{\rm s}_w -g'\,
 {\rm c}_w ){\rm s}_w \; \chi_A^* \chi_A \nn\\
 &&\qquad (g\,{\rm c}_w
+g'\, {\rm s}_w ){\rm s}_w\; \chi_Z^*\chi_A +(g\,{\rm s}_w -g'\,
{\rm c}_w ){\rm c}_w \; \chi_Z^* \chi_A \Biggr] \ea


 \subsection{\label{a:coup}Neutral couplings}

 
   The neutral current lagrangian is
   \be {\cal L}_{nc} =\sum_\psi \bar \psi \left(
n_\psi^A \slash A +n_\psi ^Z \slash Z \right) \psi \ee where the
sum runs over all the fermionic fields with non-zero isospin,
$\psi\in\{\nu_L,e_L,e_R,d_L,d_R,u_L,u_R\}   $.
The couplings are  \ba & n_\nu^A & = -\frac12 g' {\rm c}_w +\frac12 g {\rm
s}_w \simeq {\frac\epsilon2} \frac{gg'}{\sqrt{g^2+g^{\prime2}}}
\label{nnnu}\\ & n_\nu^Z & = \frac12 g' {\rm s}_w +\frac12 g
{\rm c}_w \simeq \frac12 \sqrt{g^2+g^{\prime2}} \\ & n_{e_L}^A &
= -\frac12 g' {\rm c}_w -\frac12 g {\rm s}_w \simeq -
\frac{gg'}{\sqrt{g^2+g^{\prime2}} } \left(1+\frac\epsilon2
\frac{g^2-g^{\prime2}}{g^2+g^{\prime2}} \right) \\ & n_{e_R}^A &
= -g' {\rm c}_w \simeq - \frac{gg'}{\sqrt{g^2+g^{\prime2}}}
\left(1-\epsilon \frac{g^{\prime2}}{g^2+g^{\prime2}} \right) \\
& n_{e_L}^Z & = \frac12 g' {\rm s}_w -\frac12 g {\rm c}_w \simeq
-\frac12 \frac{g^2-g^{\prime2}}{\sqrt{g^2+g^{\prime2}}}
\left(1-2\epsilon \frac{g^2g^{\prime2}}{g^4-g^{\prime4}} \right)
\\ & n_{e_R}^Z & = g' {\rm s}_w \simeq
\frac{g^{\prime2}}{\sqrt{g^2+g^{\prime2}}} \left(1+\epsilon
\frac{g^{2}}{g^2+g^{\prime2}} \right) \\ & n_{u_L}^A & = \frac16
g' {\rm c}_w +\frac12 g {\rm s}_w \simeq \frac23
\frac{gg'}{\sqrt{g^2+g^{\prime2}} } \left(1+\frac\epsilon4
\frac{3g^2-g^{\prime2}}{g^2+g^{\prime2}} \right) \\ & n_{u_R}^A
& = \frac23 g' {\rm c}_w \simeq \frac23
\frac{gg'}{\sqrt{g^2+g^{\prime2}}} \left(1-\epsilon
\frac{g^{\prime2}}{g^2+g^{\prime2}} \right) \\ & n_{u_L}^Z & =
-\frac16 g' {\rm s}_w +\frac12 g {\rm c}_w \simeq \frac16
\frac{3g^2-g^{\prime2}}{\sqrt{g^2+g^{\prime2}}}
\left(1-4\epsilon \frac{g^2g^{\prime2}}{(g^2+g^{\prime2})(3g^2
-g^{\prime2})} \right) \\ & n_{u_R}^Z & = -\frac23 g' {\rm s}_w
\simeq - \frac23 \frac{g^{\prime2}}{\sqrt{g^2+g^{\prime2}}}
\left(1+\epsilon \frac{g^{2}}{g^2+g^{\prime2}} \right) \\ &
n_{d_L}^A & = \frac16 g' {\rm c}_w - \frac12 g {\rm s}_w \simeq
-\frac13 \frac{gg'}{\sqrt{g^2+g^{\prime2}}}
\left(1+\frac\epsilon4 \frac{3g^2+ g^{\prime2}}{g^2+g^{\prime2}}
\right) \\ & n_{d_R}^A & = -\frac13 g' {\rm c}_w \simeq -\frac13
\frac{gg'}{\sqrt{g^2+g^{\prime2}}} \left(1-\epsilon
\frac{g^{\prime2}}{g^2+g^{\prime2}} \right) \\ & n_{d_L}^Z & =
-\frac16 g' {\rm s}_w - \frac12 g {\rm c}_w \simeq - \frac16
\frac{3 g^2+g^{\prime2}}{\sqrt{g^2+g^{\prime2}}}
\left(1-2\epsilon \frac{g^2g^{\prime2}}{(g^2+g^{\prime2}) (3g^2
+g^{\prime2})} \right) \\ & n_{d_R}^Z & = \frac13 g' {\rm s}_w
\simeq \frac13 \frac{g^{\prime2}}{\sqrt{g^2+g^{\prime2}}}
\left(1+\epsilon \frac{g^{2}}{g^2+g^{\prime2}} \right)
\label{nnnv}\ea
 
 

These neutral currents can be rewritten in Dirac spinor notation as follows:
\be
{\cal L}_{nc} = \sum_\psi \left\{
\bar \psi \slash  A (v_\psi^A +a_\psi^A \gamma_5) \psi
+  \bar \psi \slash  Z (v_\psi^Z +a_\psi^Z \gamma_5) \psi
\right\}  \ee
where the sum runs over $\psi\in\{\nu,e,u,d\}$, and we recall that the
left--handed projector is $(1+\gamma_5)/2$. The various couplings are the
following:
\ba
&&v_\nu^A= a_\nu^A=-a_e^A=a_u^A=-a_d^A =
\frac{1}{4}\left( g'{\rm s} _w- g'{\rm c}_w  \right)
\simeq \frac\epsilon4 \frac{g g'}{\sqrt{g^2 +g^{\prime2}}} \\
&&v_\nu^Z= a_\nu^Z=-a_e^Z=a_u^Z=-a_d^Z=
\frac{1}{2}\left(g{\rm c} _w + g'{\rm s}_w \right)
\simeq \frac12 \sqrt{g^2 +g^{\prime2}} \\
&&v_e^A= -\frac14 g{\rm s}_w -\frac34 g' {\rm c}_w
\simeq  -\frac{gg'}{\sqrt{g^2 +g^{\prime2} } } \left(1 +
\frac\epsilon4 \frac{ g^2 -3g^{\prime2} } {g^2 +g^{\prime2} } \right) \\
&&v_u^A= \frac{g}{4}\sin\theta_w +\frac{5g'}{12}\cos\theta_w
\simeq \frac23 \frac{gg'}{\sqrt{g^2 +g^{\prime2} }} \left( 1+
\frac\epsilon8 \frac {3g^2-5g^{\prime2} }{g^2 +g^{\prime2} } \right) \\
&&v_d^A= -\frac{g}{4}\sin\theta_w -\frac{g'}{12}\cos\theta_w  \simeq
 - \frac13 \frac{gg'}{\sqrt{g^2 +g^{\prime2} }} \left( 1+
 \frac\epsilon4 \frac {3g^2-g^{\prime2} }{g^2 +g^{\prime2} } \right) \\
&&v_e^Z=  -\frac14 g{\rm c}_w +\frac34 g' {\rm s}_w
\simeq  \frac{1}{4 \sqrt{g^2 +g^{\prime2} } } \left(-g^2 +3g^{\prime2}
+  4\epsilon \frac{ g^2 g^{\prime2} } {g^2 +g^{\prime2} } \right)  \\
&&v_u^Z=\frac{g}{4}\cos\theta_w -\frac{5g'}{12}\sin\theta_w
\simeq   \frac{ 1 }{12 \sqrt{g^2 +g^{\prime2} }}
\left( 3g^2 -5g^{\prime2} -8\epsilon \frac {g^2g^{\prime2} }
{ g^2 +g^{\prime2} } \right) \\
&&v_d^Z= -\frac{g}{4}\cos\theta_w +\frac{g'}{12}\sin\theta_w
\simeq   \frac{ 1}{12 \sqrt{g^2 +g^{\prime2} }}
\left( -3g^2 +g^{\prime2} +4\epsilon \frac {g^2g^{\prime2} }
{  g^2 +g^{\prime2}  }\right)
\ea




\vfill\eject 
 
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\begin{document}

\preprint{hep-th/0304245}
\preprint{UGVA-DPT-2003-04-1106}

\title{The Stueckelberg Field}

\author{Mart\1 Ruiz-Altaba}%
\email{ruiz@kalymnos.unige.ch}

\author{Henri Ruegg}%
\email{henri.ruegg@physics.unige.ch}

\affiliation{%
D\'epartement de Physique Th\'eorique\\ Universit\'e de Gen\`eve \\
24 Quai Ernest Ansermet,
1211 Gen\`eve 4, Switzerland}

\date{\today}

\begin{abstract}
We first review the Stueckelberg field in the massive Abelian gauge theory.
  We then extend  this idea to the standard model, stueckelberging the hypercharge
 $U(1)$ and thus giving a mass to the photon.
 This introduces an infrared regulator for the photon in the
standard electroweak theory, along with a modification of the
 Weinberg angle
 accompanied by a plethora of new effects.
 Finally, we  review  the historical influence  of  Stueckelberg's 1938 idea,
 which led notably to the discovery of the Higgs mechanism in spontaneous symmetry
 breaking as the only presently known
 way to give masses to non-Abelian vector fields.
 

\end{abstract}

%\pacs{numbers}

\maketitle
\tableofcontents

\input stext

 
%\nocite{*}
\bibliography{stuck}
 
% {\bf More bibliography ???}

%\input morebib

\end{document}

\end{document}
