2.4 truecm

<span>**Metamorphosis versus Decoupling in Nonabelian Gauge\
Theories at Very High Energies** </span>

0.7 truecm

Ruggero Ferrari$^{a,b}$ [^1]

$^a$ Center for Theoretical Physics\
Laboratory for Nuclear Science\
and Department of Physics\
Massachusetts Institute of Technology\
Cambridge, Massachusetts 02139\
and\
$^b$ Dip. di Fisica, Università degli Studi di Milano\
and INFN, Sez. di Milano\
via Celoria 16, I-20133 Milano, Italy\
(MIT-CTP 4275, IFUM-977-FT, June, 2011 )

> <span>Abstract:</span> In the present paper we study the limit of zero mass in nonabelian gauge theories both with Higgs mechanism and in the nonlinear realization of the gauge group (Stückelberg mass). We argue that in the first case the longitudinal modes undergo a metamorphosis process to the Goldstone scalar modes, while in the second we guess a decoupling process associated to a phase transformation.
>
> The two scenarios yield strikingly different behaviors at high energy, mainly ascribed to the presence of a massless Higgs doublet among the physical modes in the case of Higgs mechanism (i.e. not only the Higgs boson).
>
> The aim of this work is to show that the problem of unitarity at high energy in nonabelian gauge theory with no Higgs boson can open new perspectives in quantum field theory.

Introduction
============

The fate of the longitudinal mode of a vector boson at high energy is entangled with the problem of unitarity. On the basis of some well known papers in the late 70's and early 80's @Cornwall:1974km-@Chanowitz:1985hj people have acquired the conviction that the Higgs boson is necessary in order to ensure physical unitarity in nonabelian gauge theories. The heart of the argument is based on the behavior of the elastic scattering amplitude of the longitudinal modes $W_L~ W_L$ at high energy @Cornwall:1974km- @Denner:1997kq.

Thus the construction of Higgsless Elecroweak Models faces tremendously difficult theoretical problems dealing with basic principles as renormalization, unitarity, foundation of bound state quantum field theory, predictivity of the model (finite number of free parameters), etc. Ref. @referee updates a recent proposal for a Higgless scenario and provides a nice overview of most models.

The problem comes from the fact that the longitudinal polarization of a vector meson is a physical mode for any finite value of the mass ($M$). On the other side, for zero mass a vector meson has only two (transverse) polarizations. Thus either the mode decouples from the physical states in the massless limit (like in massive QED) or we face a singular behavior at zero vector boson mass.

In nonabelian gauge theories (we deal with $SU(2)$, with or without the $U(1)$ factor) the longitudinal polarization does not decouple in physical S-matrix elements at zero mass. A conundrum that shows up in really practical items as in the proof of physical unitarity and in phenomenology@Slavnov:1970tk [@Curci:1976kh]. The number of physical modes changes in the limit of zero mass, thus the cancellation of the unphysical modes in the proof of S-matrix unitarity must proceed with different patterns in the two regimes. On the other side, in phenomenology one must introduce a cut-off to mark the events region, where the longitudinal polarization of the vector meson can be established within the errors (at very high momentum one cannot distinguish between the longitudinal polarization and the spin zero wave function). Therefore the distinction between longitudinal mode for the spin one and a spin zero mode, in the limit of zero mass, has no operative meaning and, as said, conflicts with unitarity, due to the non-decoupling.

A na<span>ï</span>ve analysis, based on BRST transformations, indicates that in the massless limit the <span>*unphysical*</span> (at $M\not =0$) components of the Higgs field eventually describe <span>*physical*</span> massless scalars for $M =0$. Thus we suggest that the longitudinal polarization mode, in default of the decoupling, undergoes a metamorphosis to the mode of a massless scalar. Physical unitarity of the S-matrix is preserved with this assignment of the fields to physical and unphysical modes. The Equivalence Theorem (CLTCG) @Cornwall:1974km, @Chanowitz:1985hj, @Gounaris:1986cr, @Yao:1988aj-@Denner:1996gb supports this setting. [^2]

Of course the massless theory is plagued by infrared divergences; but we are going to ignore this difficulty, hoping that the two problems are not dangerously entangled. If instead infrared divergences are an insurmountable obstacle, as a last resource one can reverse the view point and consider the mass as the infrared regulator of an otherwise ill-defined field theory.

This possibility of a metamorphosis of states is suggested for nonabelian gauge theories where the mass $M$ is generated by the Higgs mechanism. The reason being that the limit of symmetry restoration (zero vacuum expectation value (v.e.v) of the Higgs boson field) seems doable on the effective action in a loop-wise perturbative expansion.

After the metamorphosis, the theory consists of a massless gauge field and a complex doublets of scalar fields (the fields used to induce the Higgs mechanism for $M\not =0$). According to the standard analysis the theory is not asymptotically free @Gross:1973ju-@Politzer:1973fx, due to the presence of scalars.

Thus one gets a consistent setting for studying the physics of the intermediate vector mesons at energies $E>>M_W,M_Z$. If needed, one can use the $M\not =0$ theory as the infrared regulated theory.

In the present paper we address the same question in the case where $M$ enters via a Stückelberg term. In this case the local gauge group is realized and hence there is no need of a Higgs boson in the perturbative spectrum.

Both theories obey the same set of equations used in the present work: Slavnov-Taylor Identity (STI), gauge fixing equation and anti-ghost equation [^3]. Moreover the CLTCG theorem takes the same form. However they are strikingly different in the zero mass limit. In the linear case the limit of v.e.v to zero in the Feynman amplitudes seems to be manageable. While in the nonlinear case the situation is fuzzier. In recent works @Ferrari:2005ii-@Ferrari:2009uj we proposed a divergences subtraction scheme for the nonlinear sigma model, for the massive Yang-Mills theory and for the Electroweak Model $SU(2)\otimes U(1)$. Locality and perturbative unitarity are obeyed. However the perturbative solution has a bad $M^{-1}$ behavior for $M=0$, essentially due to the nonlinear sigma model couplings.

Thus the scenario of a metamorphosis of the longitudinal modes for $M=0$ cannot be envisaged in the case of nonlinear realization of the gauge group.

From some considerations, based on the matching of the number of degrees of freedom and on the strong coupling limit of the lattice-regulated theory, we guess a $M\to 0$ behavior where both the longitudinal polarizations and the Goldstone bosons decouple. In order to support this scenario, we envisage the existence of two or more phases in the parameter space separated by some discontinuity. In particular we assume that the loop expansion cannot be continued to $M=0$. This would mark the difference with the linear case, where the Goldstone bosons survive as physical modes. For instance the nonlinear theory would be an asymptotically free theory in the limit.

The conjecture on the limit $M=0$ for the nonlinear case could be studied by lattice simulations. In particular one should make a survey of the phase diagram in the parameter space $(g^{-2},M^2)$ and look for possible singularities responsible for the bad behavior of the loop expansion for low mass [^4].

A further comparison of the two scenarios could come from high energy processes. However a quick analysis shows that this is pretty hard to achieve, as a simple example will show.

We work in the 't Hooft gauge.

The Classical Actions {#sec:action}
=====================

In this section we fix some notations. Matter fields are omitted in most part of the paper.

The work focuses on the gauge and scalar sectors of the following Yang-Mills classical actions written for the $SU(2)$ gauge group. We consider both cases of linear (Higgs) and nonlinear (Stückelberg) representation of the gauge group.

Yang Mills with Higgs Mechanism  {#sec:YM}
--------------------------------

We consider a $SU(2)$ Yang-Mills theory where the mass is generated through the Higgs mechanism $$\begin{aligned}
S_{\rm  H}  = \frac{\Lambda^{(D-4)}}{g^2} \int d^Dx \, 
\biggl( -\frac{1}{4} G_{a\mu\nu} G_a^{\mu\nu}
 +\Big [(\partial_\mu -i A_\mu) \Phi\Big]^\dagger
\Big[(\partial^\mu -i A^\mu) \Phi\Big] 
-\frac{\lambda}{4g^2}(\Phi^\dagger\Phi -2v^2g^2 )^2
\biggr) .
\label{YM.1}\end{aligned}$$ $\Lambda$ is a mass scale for the analytic continuation in $D$ dimensions. We use the short notation $$\begin{aligned}
\Lambda_g\equiv \frac{\Lambda^{(D-4)}}{g^2} .
\label{YM.1.1}\end{aligned}$$ We use the matrix notation $$\begin{aligned}
&&
A\mu= \frac{\tau_a}{2}A_{a\mu}
\nonumber \\&&
G_{\mu\nu}[A] = G_{a\mu\nu} \frac{\tau_a}{2} =
\partial_\mu A_\nu - \partial_\nu A_\mu -i[A_\mu,A_\nu]
\label{YM.2}\end{aligned}$$ and $\Phi$ is parametrized by $$\begin{aligned}
\Phi=
\frac{1}{\sqrt 2}
\left(
\begin{array}{l}
i\phi_1+\phi_2\\
\phi_0 -i\phi_3
\end{array}
\right).
\label{YM.3}\end{aligned}$$ The action (\[YM.1\]) is invariant under local $SU(2)_L$ left transformations $$\begin{aligned}
&& A'_\mu = U_L A_\mu U_L^\dagger + i U_L \partial_\mu U_L^\dagger 
\nonumber \\ &&
\Phi' = U_L\Phi
\label{YM.3.1}\end{aligned}$$ and under global $SU(2)_R$ right transformations $$\begin{aligned}
&& A'_\mu = A_\mu 
\nonumber \\ &&
\Omega' = \Omega \, U_R,
\label{YM.3.2}\end{aligned}$$ where $$\begin{aligned}
&& 
\Omega_{\alpha \beta} \simeq \sqrt 2 \Phi_\alpha \tilde\Phi_\beta
\nonumber \\ &&
\tilde\Phi = i \tau_2 \Phi^*.
\label{YM.3.3}\end{aligned}$$ Notice that in general $\Omega \not\in SU(2)$.

The spontaneous breakdown of the global $SU(2)_L\otimes SU(2)_R$ symmetry proceeds via the nonzero vacuum expectation value $$\begin{aligned}
\langle 0|\phi_0|0\rangle=\langle 0|(h+2vg)|0\rangle=2vg
\label{YM.4}\end{aligned}$$ and the global $SU(2)$ invariance is left over on the vector indexes.

The spontaneous breakdown induces a mass for the vector bosons ($M\equiv gv$), for the Higgs boson ($M^2_H \equiv \lambda v^2 $) and a mixing $A_\mu - \phi$ $$\begin{aligned}
&&
S_{\rm H\, Bilinear}  = \frac{\Lambda^{(D-4)}}{g^2} \int d^Dx \, 
\biggl( -\frac{1}{2}(\partial_\mu A_{a\nu} \partial^\mu A_a^\nu
-\partial_\mu A_{a\nu} \partial^\nu A_a^\mu)
\nonumber \\&&
 + \frac{M^2}{2}A_{a\nu} A_a^\nu 
+ \frac{1}{2}\partial_\mu h\partial^\mu h
 + \frac{1}{2}\partial_\mu\phi_a\partial^\mu\phi_a
- M^2A_{a\mu}\partial^\mu\phi_a - \frac{1}{2}M^2_{H} h^2\biggr).
\label{YM.5}\end{aligned}$$ We use the 't Hooft gauge in order to remove the mixing. In doing this we get a mass for the Goldstone bosons $\vec\phi$ $$\begin{aligned}
S_{\rm  H\,gf}  = \frac{\Lambda^{(D-4)}}{g^2} \int d^Dx \, 
\biggl( \frac{b^2}{2\xi }+\frac{ M}{\xi }b_a\phi_a +
b_a  \partial_\mu A_a^\mu \biggr). 
\label{YM.6}\end{aligned}$$ The Goldstone mass at the tree level is then $M_{G}^2\equiv  M^2 \xi^{-1 }$. In Appendix \[app:app1\] we give the complete effective action at zero loop with the necessary Faddeev-Popov ghosts.

The perturbation expansion is in the number of loops and the amplitudes are made finite by using na<span>ï</span>ve dimensional renormalization. Finite renormalization is not a relevant item for the content of the paper.

Yang-Mills with Stückelberg Mass
--------------------------------

The nonlinear sigma model field $\Omega$ is an element of the $SU(2)$ group, which is parametrized in terms of the coordinate fields $\phi_a$ as follows (compare with eq. (\[YM.3.3\])) [^5] $$\begin{aligned}
&& \Omega =  \phi_0 + i \tau_a \phi_a  \, , ~~~ \Omega^\dagger \Omega = 1 \,  , ~~~ {\rm det} \, \Omega = 1 \, ,
\nonumber \\
&& \phi_0^2 + \phi_a^2 = 1  \, .
\label{s2.2}\end{aligned}$$ The $SU(2)$ flat connection is $$\begin{aligned}
&& F_\mu = i \Omega \partial_\mu \Omega^\dagger 
= F_{a\mu} \frac{\tau_a}{2} \, , \nonumber \\
&& F_{a\mu} = 2 (\phi_0 \partial_\mu \phi_a -
\partial_\mu \phi_0 \phi_a + \epsilon_{abc} \partial_\mu \phi_b \phi_c ) \, .
\label{s2.1}\end{aligned}$$ The field strength of $F_\mu$ vanishes $$\begin{aligned}
G_{\mu\nu}[F] = 0 \, .
\label{flc.1}\end{aligned}$$ Under a local $SU(2)$ left transformation $U_L = \exp \Big ( i \alpha^L_a \frac{\tau_a}{2} \Big )$ one gets $$\begin{aligned}
&& \Omega' =  U_L \Omega \, , \nonumber \\
&& F'_\mu = U_L F_\mu U_L^\dagger + i U_L \partial_\mu U_L^\dagger \, , \nonumber \\
&& A'_\mu = U_L A_\mu U_L^\dagger + i U_L \partial_\mu U_L^\dagger \, .
\label{s2.4}\end{aligned}$$ The constraint in eq.(\[s2.2\]) implies that the gauge symmetry is nonlinearly realized on the fields $\phi_a$, whose infinitesimal transformations are $$\begin{aligned}
&& \delta \phi_a = \frac{1}{2} \phi_0 \alpha^L_a + \frac{1}{2} \epsilon_{abc} \phi_b \alpha^L_c \, , ~~~~
\phi_0 = \sqrt{1  - \phi_a^2} \, , \nonumber \\
&& \delta \phi_0 = -\frac{1}{2} \alpha^L_a \phi_a \, .
\label{s2.4.1}\end{aligned}$$ Under local $SU(2)_L$ symmetry the combination $A_\mu - F_\mu$ transforms in the adjoint representation of $SU(2)$. Hence one can construct out of $A_\mu - F_\mu$ and $\Omega$ invariants under $SU(2)_L$ local transformations. The Yang-Mills action in the presence of a Stückelberg mass term @Bettinelli:2007tq and in the 't Hooft gauge is $$\begin{aligned}
&&
S_{\rm S}  = \frac{\Lambda^{(D-4)}}{g^2} \int d^Dx \, \Big ( - \frac{1}{4} 
                              G_{a\mu\nu}[A] G^{\mu\nu}_a[A] + 
                    \frac{M^2}{2} (A_{a\mu} - F_{a\mu})^2  \Big ) \,
\nonumber\\&&
S_{\rm  S\,gf}  = \frac{\Lambda^{(D-4)}}{g^2} \int d^Dx \, 
\biggl( \frac{b^2}{2\xi }+2\frac{ M^2}{\xi }b_a\phi_a +
b_a  \partial_\mu A_a^\mu \biggr).
\label{stck.1}\end{aligned}$$ $S_S$ is invariant under local $SU(2)_L$ symmetry and also global $SU(2)_R$ symmetry. The choice of independent fields made in eq. (\[s2.4.1\]) fixed the direction of the spontaneous breakdown of the symmetry. The bilinear part of the action $S_S$ is essentially the same as in the Higgs mechanism (\[YM.5\]) apart from the absence of the Higgs boson terms.

Properties of the Two-point Functions (Higgs) {#sec:ptpf}
=============================================

In Appendix \[sec:tpf\] we derive the two-point connected functions of the unphysical bosonic sector. The solutions are given in terms of the 1PI two-point functions [^6] $$\Gamma_{\phi\phi}, \qquad ip^\nu\Gamma_{\phi A^\nu}, \qquad \Gamma_L
\label{ptpf.2}$$ which are related by the eq. (\[tpf.9.2\]) $$(p^\nu\Gamma_{A^\nu\phi})^2 +p^2\Gamma_L\Gamma_{\phi\phi}=0.
\label{ptpf.3}$$ The connected two-point functions involving the Lagrangian multiplier are (see eq. (\[basic.8\])) $$\begin{aligned}
&&
 W_{A^\mu b}=-\frac{i}{\Lambda_g}\frac{p^\mu}{
p^2-i  \frac{M}{\xi}\frac{p^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}
}
\nonumber\\&&
 W_{\phi b}
=\frac{i}{\Lambda_g}\frac{p^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}
\frac{1}{p^2-\frac{M}{\xi}\frac{ip^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}
}.
\label{ptpf.4}\end{aligned}$$ The two-point functions have the pole in the same position, i.e. the solution of $$\begin{aligned}
p^2=\frac{M}{\xi}\frac{ip^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}.
\label{basic.99}\end{aligned}$$

The connected two-point functions for unphysical modes involving $ \phi$ and $ A^\mu$ are given in eqs. (\[tpf.22\]), (\[tpf.23\]) and (\[tpf.24\]) $$\begin{aligned}
W_{\phi \phi}
 =-
\frac{p^2}{ \Gamma_{\phi \phi}}
\biggl(p^2-
\frac{1}{\Lambda_g\xi}\Gamma_L 
\biggr)
\frac{1}{\biggl(
p^2 -  \frac{M}{\xi}
\frac{ip^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}\biggr)^2
},
\label{ptpf.22}\end{aligned}$$ $$\begin{aligned}
  W_{A^\mu \phi}
=i \frac{1}{\xi\Gamma_{\phi\phi}}\biggl(\frac{i}{\Lambda_g}p^\nu\Gamma_{ A^\nu\phi}
+M p^2
\biggr)
\frac{p^\mu
}{\biggl(
p^2 -  \frac{M}{\xi} 
\frac{ip^\nu\Gamma_{ \phi A^\nu}}{\Gamma_{\phi\phi}}\biggr)^2
}
\label{ptpf.23}\end{aligned}$$ and $$\begin{aligned}
W_L 
=\frac{p^2}{\xi\Gamma_{\phi\phi}}
\frac{\frac{\Gamma_{\phi\phi}}{\Lambda_g} -\frac{M^2}{\xi}}{\biggl(
p^2 -  \frac{M}{\xi} \frac{ip^\nu\Gamma_{\phi A^\nu}}
{\Gamma_{\phi\phi}}\biggr)^2
}.
\label{ptpf.24}\end{aligned}$$ By direct computation one can derive the identity $$\begin{aligned}
\Gamma_L = \frac{p^2 W_{ b\phi}^2}{  \frac{1}{\xi}    W_{ b\phi}^2-\frac{1}{\Lambda_g}W_{\phi\phi}   } \, .
\label{ptpf.24.1}\end{aligned}$$ It is interesting to see the properties of the numerators in eqs. (\[ptpf.22\]), (\[ptpf.23\]) and (\[ptpf.24\]). They form a matrix (variables: $\phi,M^{-1}\partial^\mu A_\mu$) $$\begin{aligned}
{\cal G}\equiv
\frac{p^2}{ \xi\Gamma_{\phi \phi}}
\left(
\begin{array}{ll}-\xi p^2 +\frac{\Gamma_L}{\Lambda_g} &M^{-1}(\frac{i}{\Lambda_g}p^\nu\Gamma_{ A^\nu\phi}
+M p^2)\\
M^{-1}(\frac{i}{\Lambda_g}p^\nu\Gamma_{ A^\nu\phi}
+M p^2) & M^{-2}(p^2 \frac{\Gamma_{\phi\phi}}{\Lambda_g} -p^2 \frac{M^2}{\xi})
\end{array}
\right)
\label{ptpf.26}\end{aligned}$$ whose determinant is $$\begin{aligned}
\det ({\cal G})
=-\frac{1}{\Lambda_g}\frac{p^4}{M^2 \xi\Gamma_{\phi \phi}}\biggl( p^2 
- \frac{M}{\xi}\frac{ip^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi \phi}}\biggr)^2.
\label{ptpf.27}\end{aligned}$$ Thus the numerator-matrix has one vanishing eigenvalue on the double-poles. In fact the field given by the linear combination @Becchi:1974md $$\begin{aligned}
X_1 = \frac{\phi}{\xi} + \frac{\partial^\mu A_\mu}{M}
\label{ptpf.28}\end{aligned}$$ is shown (Appendix \[app:conv\] eq.(\[tpf.49\])) to be the eigenvector of the vanishing eigenvalue of the matrix in eq. (\[ptpf.26\]).

The Na<span>ï</span>ve $M=0$ Limit (Higgs) {#sec:limh}
==========================================

The paper is focused on extreme processes where one can neglect the mass parameters. In the Higgs case this is equivalent to the limit $v=0$, i.e. the limit of unbroken symmetry.

We do not consider skew limits as $M=0$ and $M_H \not =0$, which, although interesting in phenomenology @Denner:1996gb, requires a series resummation as $v\to 0$.

In the limit $v=0$ the symmetry $\Phi \to -\Phi$ is unbroken; as a consequence one has $$\begin{aligned}
\Gamma_{\phi A^\nu} =0
\label{ptpf.29}\end{aligned}$$ and therefore $$\begin{aligned}
&&
\Gamma_{L}=0
\nonumber\\&&
 W_{A^\mu b}=-i\frac{p^\mu}{\Lambda_g
p^2
}
\nonumber\\&&
 W_{\phi b}
=0.
\label{ptpf.31.0}\end{aligned}$$ Similarly the limit in the eqs. (\[ptpf.23\]), (\[ptpf.22\]) and (\[ptpf.24\]) yields $$\begin{aligned}
&&
 W_{A^\mu \phi}
=0
\nonumber\\&&
W_{\phi \phi}
 =-
\frac{1}{ \Gamma_{\phi \phi}}
\nonumber\\&&
W_L 
=\frac{1}{\Lambda_g\xi}
\frac{1}{p^2 }.
\label{ptpf.31.3}\end{aligned}$$ The vanishing of the two-point functions $ W_{A^\mu \phi}$ and $ W_{b \phi}$ shows that in the limit $\phi$ describes modes orthogonal to the unphysical modes. This fact is a preliminary condition for the realization of the metamorphosis of the vector meson longitudinal polarizations into the Goldstone bosons, as described in the next Sections. Moreover this suggests that the limit $v=0$ can be performed order by order in perturbation theory. This limit is possible on the amplitudes for generic external momenta, while for most S-matrix elements the limit cannot be performed due to infrared divergences. The amplitudes are those of massless nonabelian field coupled to a massless fields $\Phi$ belonging to the spinorial representation of the $SU(2)$ group of local left transformations.

The Longitudinal Polarization and its Fate for $M\to 0$ {#sec:long}
=======================================================

The longitudinal polarization [^7] $$\begin{aligned}
\epsilon_L = \frac{1}{{\tilde M} }\Big(|\vec p|,\frac{\vec p}{|\vec p|}E\Big), 
\qquad E=\sqrt{{\tilde M}\,^2+\vec p\,^2}
\label{long.1}\end{aligned}$$ can be written ($E_G\equiv \scriptstyle{\sqrt{\vec p\,^2+{\tilde M}_G\,\,^2}}$) $$\begin{aligned}
\epsilon_L = \frac{1}{{\tilde M}}\Big(E_G, \vec p\Big)
+  \Big ( - \frac{\tilde M_G\,^2}{\tilde M}\frac{1}{(p+E_G)},  \frac{\tilde M}{p(p+E)}\vec p\Big).
\label{long.2}\end{aligned}$$ Thus for large value of the energy ($E>>{\tilde M}$) we have $$\begin{aligned}
\epsilon_L = \frac{1}{{\tilde M}}\Big(E_G, \vec p\Big)
+ {\cal O}(\frac{{\tilde M}}{E}).
\label{basic.11.3}\end{aligned}$$ Equation (\[basic.11.3\]) has attracted the attention of many physicists. We briefly add our comments.

The Need of a Cut-off
---------------------

Equation (\[basic.11.3\]) shows that at very high energy one cannot experimentally distinguish the longitudinal mode of a vector field from a spin zero state described by a field like $\partial_\mu \phi$. Therefore a cut-off energy $E_C$ should be quoted in the experimental data saying for what energy $E<<E_C$ it is possible to distinguish the two states. For $E\geq E_C$ a statement about the spin content (spin one longitudinal versus spin zero) of the mode is void. The necessity of such cut-off energy is very relevant for our problem. In fact, if one is interested in the dynamics of the model at $M=0$, whose S-matrix elements are plagued by infrared divergences, the cut-off can be used in order to evaluate the physically relevant observables. Thus the model at $M=0$ can be traded with a massive nonabelian gauge theory with Higgs mechanism, provided that the mass $M$ is small enough for the given kinematic setup.

Default of Decoupling of Longitudinal Mode for Nonabelian Gauge Theories
------------------------------------------------------------------------

It might be tempting to neglect the ${\cal O}(\frac{{\tilde M}}{E})$ parts in eq. (\[basic.11.3\]) and to perform the replacement $$\begin{aligned}
\epsilon_L\, \to \,\frac{1}{{\tilde M}}\Big(E_G, \vec p\Big)
\label{long.3}\end{aligned}$$ in the Feynman amplitudes. Were it possible without ambiguity, then the problem of the decoupling of the longitudinal mode would be much easier. Unfortunately this procedure is not allowed since mixed terms in quadratic or higher-order forms give finite contributions, which cannot be neglected without further scrutiny. The reason is connected to the validity of the condition $$\begin{aligned}
E_G~{\mathfrak M}_0(E_G,\vec p)- p_i~{\mathfrak M}_i(E_G,\vec p)=0,
\label{long.4}\end{aligned}$$ as it will be illustrated here to some extent. For instance, let us consider the situation where $\epsilon_L^\mu$ multiplies some amplitude ${\mathfrak M}_\mu$ which depends on the momentum $p_\nu$. The limit $\tilde M =0$ might be performed by evaluating the difference $$\begin{aligned}
&&
\epsilon_L^\mu{\mathfrak M}_\mu(E,\vec p) - \frac{1}{{\tilde M}}\Big(
E_G~{\mathfrak M}_0(E_G,\vec p)- p_i~{\mathfrak M}_i(E_G,\vec p)\Big)
\label{long.5}\end{aligned}$$ which is of order ${\cal O}(\frac{{\tilde M}}{E})$ as a standalone expression. Let us expand around $E_G$, where it is much easier to use the STI. We get (the common dependence from $\vec p$ being suppressed) $$\begin{aligned}
&&
\epsilon_L^\mu{\mathfrak M}_\mu(E) - \frac{1}{{\tilde M}}p^\mu {\mathfrak M}_\mu(E_G)
=\frac{1}{{\tilde M}} \Bigg( [p-E_G]{\mathfrak M}_0(E_G)
- p_i (\frac{E}{p}-1){\mathfrak M}_i(E_G)
\nonumber\\&&
+(E-E_G)\Big\{ p\frac{\partial}{\partial E_G}{\mathfrak M}_0(E_G) - p_i \frac{E}{p}
\frac{\partial}{\partial E_G}{\mathfrak M}_i(E_G)
\Big \}
\Bigg )+ {\cal O}({\tilde M}\, ^3).
\label{long.5.1}\end{aligned}$$ Now we use the relations $$\begin{aligned}
&&
\frac{E_G}{p} = 1 + {\cal O}({\tilde M}\, ^2)
\nonumber \\&&
\frac{E}{p} = 1 + {\cal O}({\tilde M}\, ^2)
\nonumber \\&&
E_G\frac{\partial}{\partial E_G}{\mathfrak M}_0(E_G)= \frac{\partial}{\partial E_G}E_G{\mathfrak M}_0(E_G)
- {\mathfrak M}_0(E_G)
\label{long.5.1.1}\end{aligned}$$ and we get $$\begin{aligned}
&&
\epsilon_L^\mu{\mathfrak M}_\mu(E) - \frac{1}{{\tilde M}}p^\mu {\mathfrak M}_\mu(E_G)
=\frac{1}{{\tilde M}} \Big( [p-E]\,{\mathfrak M}_0(E_G)
- p_i (\frac{E}{p}-1){\mathfrak M}_i(E_G)
\nonumber\\&&
+(E-E_G)\Big\{ \frac{\partial}{\partial E_G}E_G{\mathfrak M}_0(E_G) - 
\frac{\partial}{\partial E_G}p_i {\mathfrak M}_i(E_G)
\Big \}
\Big )+{\cal O}({\tilde M}\, ^3). 
\nonumber\\&&
= - \frac{{\tilde M}}{2p}\Big( p \,{\mathfrak M}_0(E_G)
+ p_i {\mathfrak M}_i(E_G)\Big )
+ \frac{(E-E_G)}{{\tilde M}} \frac{\partial}{\partial E_G} [E_G{\mathfrak M}_0(E_G) -p_i {\mathfrak M}_i(E_G)] 
\nonumber\\&&
+ {\cal O}({\tilde M}\, ^3).
\label{long.5.1.2}\end{aligned}$$ Thus, if eq. (\[long.4\]) is valid, the derivative term can be neglected.

However the expression in eq. (\[long.5.1.2\]) may enter into some product with terms of order $\tilde M^{-1}$ (as for instance $\epsilon_L^\mu$ ) thus producing a non vanishing result. Typically this happens by evaluating the sum over final states as for instance $$\begin{aligned}
-\frac{d^3p}{2E}\,\,\, {\mathfrak M}^*_\mu \,\,\,\epsilon_L^{*\mu} \epsilon_L^{\nu} \,\,\,{\mathfrak M}_\nu,
\label{long.5.3}\end{aligned}$$ where the ${\cal O}(\tilde M)$ term in eq. (\[long.5\]) gives a finite contribution when multiplied by $\tilde M^{-1}$ in $\epsilon_L^{*\mu} $. Finally the problem consists in evaluating $$\begin{aligned}
\frac{1}{\tilde M}\Big(E_G~{\mathfrak M}_0(E_G,\vec p)- p_i~{\mathfrak M}_i(E_G,\vec p)\Big)
\label{long.5.3.1}\end{aligned}$$ for $\tilde M =0$. Then the terms in eq. (\[long.5.1.2\]) are expected to contribute, if the behavior of the expression in eq. (\[long.5.3.1\]) is like $\tilde M^{-1}$ (as in matrix elements with unphysical modes). Otherwise they yield vanishing products (as for matrix elements with only physical modes).

One can approach the problem of the decoupling of the longitudinal polarization in a different but nevertheless interesting way: by considering the sum of the contribution of eq. (\[long.5.3\]) and that of the spin zero part $$\begin{aligned}
\frac{d^3p}{2E_G}\,\,\,\frac{1}{{\tilde M}\,^2} \,\,\,{\mathfrak M}^*_\mu \,\,\,p^{\mu} p^{\nu} \,\,\,{\mathfrak M}_\nu.
\label{long.5.4}\end{aligned}$$ With an algebra similar to the one in eq. (\[long.5.3\]) one concludes that in the limit of ${\tilde M}=0$ the contributions of the longitudinal polarization and of the spin zero cancel if only physical states are present (i.e. eq. (\[long.4\]) is valid). A scholarly example in Appendix \[app:free\], based on the free fields, illustrates some of the features of the limit $\tilde M =0$, discussed in the present Section.

Now we compare the two quite different situations present in the abelian and the nonabelian gauge theories.

The Lagrange multiplier $b$, used for the gauge fixing, decouples from the physical modes, as can be seen by using the STI. In the abelian case this yields eq. (\[long.4\]). Consequently the contribution of the longitudinal polarization can be replaced according to eq. (\[long.3\]) in the zero mass limit, since the the expression in the second term of eq. (\[long.5\]) will never multiply a $\tilde M^{-1}$ factor.

In nonabelian gauge theories the decoupling does not happen. In fact the decoupling of the Lagrange multipliers does not bring to the eq. (\[long.4\]), but instead to a relation involving the Goldstone bosons as discussed later on. This is the source of many problems. For a single external gauge particle with longitudinal polarization the replacement (\[long.3\]) does yield the correct result. However already for two gauge particles with longitudinal polarization the replacement (\[long.3\]) might gives results that depend on the gauge or on the order of the replacements. The first replacement gives no problems because all other particles are physical. After the first replacement the $ {\cal O}(\frac{{\tilde M}}{E})$ term in eq. (\[long.3\]) might yield non zero contributions involving the Faddeev-Popov ghosts, since the spin zero part of the gauge boson ($\epsilon_\mu \simeq p_\mu, p^2= {\tilde M}_G\,^2$) is an unphysical mode.

This fact has further unpleasant consequences. For instance in the proof of physical unitarity the sum over final states is only on physical modes. After the replacement (\[long.3\]) (a physical mode by an unphysical one) this necessary property is lost, if no further condition is introduced to cancel out the spurious terms.

Metamorphosis in the Higgs Mechanism Scenario {#sec:cut}
=============================================

For ${\tilde M}=0$ only the transverse polarizations are physical, thus there is a problem in the limit. In the massless case the two unphysical modes of the vector fields conspire with the Faddeev-Popov ghosts in order to cancel out in the cutting rule, i.e. in the equation of perturbative unitarity. Instead, for every finite value of ${\tilde M}$ the net balance to zero involves the spin zero part of the vector mesons, the Goldstone bosons and the Faddeev-Popov ghosts. Therefore if the longitudinal polarization does not decouple, unitarity is violated in the limit. This has been noticed a long time ago @Slavnov:1970tk, @Curci:1976kh.

The conceptual difficulty of the limit disappears if we accept a scenario where the longitudinal mode transforms into the former Goldstone boson for zero vector meson mass. In fact the Goldstone boson field describes a physical mode at $\tilde M=0$. This scenario is in agreement with the discussion of Section \[sec:long\] about the impossibility to distinguish the modes at very high energy.

The metamorphosis scenario has a further advantage for practical calculations: one can use the limit theory in order to evaluate the amplitudes involving the longitudinal modes. This statement is very close to the CLTCG theorem which relates the S-matrix elements of the longitudinal modes to those of the Goldstone boson. This advantage, however, is limited by the infrared divergences, that eventually will emerge (for instance in self-energies). Our scenario provides a more flexible setup, where the objections on the zero mass limit are removed (metamorphosis versus decoupling) and a proper use of the theory is established (only at $M\not =0$ we have a <span>*bona fide*</span> theory and the S-matrix elements at $M=0$ can be evaluated by using $M$ as an infrared regulator). In the next Section we give some comments about the CLTCG theorem.

Comments on the CLTCG Theorem {#sec:equiv}
=============================

In this Section we provide the general formulation of the CLTCG Theorem in any covariant 't Hooft gauge.

Let $|\vec p L\rangle$ denotes an asymptotic state longitudinally polarized with momentum $\vec p$. Since it is a physical state then it must be annihilated by the operator $F$ that generates the BRST transformations on the fields of the nonabelian gauge theory (internal index is not displayed) @Curci:1976yb. $$\begin{aligned}
F |\vec p L\rangle =0.
\label{cut.1}\end{aligned}$$ However the state is also represented by any element of the equivalent class made of vectors like $$\begin{aligned}
 |\vec p L\rangle + F |X\rangle,
\label{cut.2}\end{aligned}$$ where $X$ is an arbitrary state. Due to the nilpotency of $F$, eq. (\[cut.1\]) is still valid $$\begin{aligned}
F\Bigl ( |\vec p L\rangle + F |X\rangle \Bigr )=0.
\label{cut.3}\end{aligned}$$ By the standard proof of physical unitarity, the states $|\vec p L\rangle$ and $ |\vec p L\rangle + F |X\rangle$ have the same $S-$ matrix elements.

We shall use this freedom in describing the physical modes of the gauge fields in order to evaluate the behavior for ${\tilde M}\to 0$, without using the replacement (\[long.3\]) and encountering some pitfalls. The recipe is the following: any longitudinal mode is replaced by $$\begin{aligned}
 |\vec p L\rangle +\frac{1}{{\tilde M}}~ F |\vec p \,\,\bar c\rangle,
\label{cut.4}\end{aligned}$$ where $|\vec p \,\,\bar c\rangle$ is a single-mode anti-ghost state and the relative weight is chosen in order to reproduce eq. (\[basic.11.3\]), when the wave functions are exhibited by the reduction formulas. We construct the in- and out-states in the Fock space by using the recipe in eq. (\[cut.4\]).

The $S-$matrix element for the longitudinal mode is constructed by using the amputated connected Green function defined by [^8] $$\begin{aligned}
W_{{A_\mu}} =   \sum_{\psi} W_{A_\mu \psi} W_{\widehat{\psi}}
\label{cut.5}\end{aligned}$$ The asymptotic states are described by the eigenvectors $\epsilon^{(r)}$ and eigenvalues $\lambda^{(r)}$ of the residuum matrix of the two-point connected function $-W(p)$ at the physical pole $p^2=m^2_r$. The construction of the $S-$matrix element where the state $|\vec p L\rangle$ appears as a factor in the final state proceed via the usual procedure (wave function renormalization factor and internal indexes are omitted) $$\begin{aligned}
S_{\vec p L\cdots} \simeq {\epsilon_{L\mu}}(\vec p)
~i~W_{\widehat{A_\mu(p)}***}\Big|_{p^2={\tilde M}^2} \,  .
\label{cut.5.1}\end{aligned}$$ With these notations the residuum of the $b-$ field (the Lagrange multiplier of the 't Hooft gauge) yields (see eq. (\[basic.8\])) $$\begin{aligned}
\lim_{p^2={\tilde M}_G^2}(p^2-{\tilde M}^2_G) W_{b(p) ***}=\biggl(
i\frac{p^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}W_{\widehat{\phi(p)} ***}
+ip^\mu W_{\widehat{A^\mu(p)} ***}\biggr)\Big |_{p^2={\tilde M}_G^2}
=0,
\label{cut.6}\end{aligned}$$ where $***$ denotes more $b$ and physical mode insertions. It's worth noticing that from eq. (\[basic.99\]) $$\begin{aligned}
i\frac{p^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}\Big |_{p^2={\tilde M}_G\,^2}
= \frac{\xi}{ M} {\tilde M}_G\,^2
\label{cut.6.1}\end{aligned}$$ and at the tree level $$\begin{aligned}
i\frac{p^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}\Big |_{p^2= M_G^2}
=  M.
\label{cut.6.2}\end{aligned}$$ In order to reproduce the pattern of eq. (\[long.2\]), the Feynman amplitude is replaced, according to (\[cut.4\]), $$\begin{aligned}
\epsilon_{L\mu }(p)
W_{\widehat{A_\mu(p)}***}\Big|_{p^2={\tilde M}\,^2} =
\lim_{p^2={\tilde M}\,^2}  \epsilon_{L\mu} W_{\widehat{A_\mu(p)} ***} 
+i\lim_{{p^2={\tilde M}_G\,^2}}(p^2-{\tilde M}_G\,^2)\frac{1}{{\tilde M}}W_{b(p) ***}
\label{cut.7}\end{aligned}$$ and by using eq. (\[cut.6\]) $$\begin{aligned}
&&
\epsilon_{L\mu }(p)
W_{\widehat{A_\mu(p)}***}\Big|_{p^2={\tilde M}\,^2} 
= 
\lim_{p^2={\tilde M}\,^2}  \epsilon_{L\mu} W_{\widehat{A_\mu(p)} ***} -\lim_{p^2={\tilde M}_G\,^2} \frac{1}{{\tilde M}}
\Bigl(p^\mu W_{\widehat{A^\mu(p)} ***} 
\nonumber\\&&
+
\frac{p^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}W_{\widehat{\phi(p)} ***}
\Bigr).
\label{cut.7.1}\end{aligned}$$ The above replacement (\[cut.4\]) may be repeated for every external gauge line with longitudinal polarization, since on both terms in eq. (\[cut.7\]) this replacement is allowed. In fact the external legs are either physical states or $b$-lines and therefore the STI (\[app2.7\]) guarantees the validity of eq. (\[cut.6\]).

In the limit $M=0$ the first two terms in the RHS of eq. (\[cut.7.1\]) cancel out $$\begin{aligned}
\lim_{p^2=\tilde M^2}  \epsilon_L^\mu W_{\widehat{A_\mu(p)} ***} 
-\lim_{p^2=\tilde M^2_G} \frac{1}{\tilde M}
p^\mu W_{\widehat{A^\mu(p)} ***}= 0 + {\cal O}(\frac{\tilde M}{E}).
\label{cut.8}\end{aligned}$$ Thus finally we get $$\begin{aligned}
&&
\epsilon_L^{\mu }(p)
W_{\widehat{A_\mu(p)}***}\Big|_{p^2={\tilde M}\,^2} 
=  
i~\frac{ p^\nu\Gamma_{\phi A^\nu}}{\tilde M\Gamma_{\phi\phi}}~W_{\widehat{\phi(p)} ***}
\Big |_{p^2=\tilde M^2_G}
+ {\cal O}(\tilde M).
\label{cut.9}\end{aligned}$$ The procedure can be repeated for every vector boson in the longitudinal mode. The feared occurrence of cross terms ${\cal O}(M) \times \frac{1}{M}$ vanishes since all external modes are either physical or $b-$insertions.

Few comments are in order on our proof of the CLTCG theorem in eq. (\[cut.9\]).

1.  The exact knowledge of the two-point functions in the unphysical sector, as displayed in Sec. \[sec:ptpf\], allows the correct formulation of the CLTCG theorem at any number of loops. The quantities needed are $\Gamma_{\phi\phi}$ and $p^\mu \Gamma_{\phi A^\mu}$.

2.  The CLTCG theorem as in eq. (\[cut.9\]) concerns the amputated connected amplitudes. In order to formulate the theorem for the $S-$matrix elements one has to introduce the wave-function normalization of the asymptotic states. In the limit $M=0$ the normalization of the longitudinal modes equals that of the Goldstone boson, since $W_L$ approaches the free-field value, as displayed in eq. (\[ptpf.31.3\]). Here we are not going into further details on this problem.

3.  After we introduce the necessary wave function renormalization factor in the LHS of eq. (\[cut.9\]) the $S$-matrix elements are gauge invariant. This is valid for any finite value of $\tilde M$. Thus also the limit, when it exists, is gauge invariant. The property has been verified in explicit calculations @Denner:1997kq.

4.  For generic $S$-matrix elements the limit of zero mass is expected to be infrared divergent. Then eq. (\[cut.9\]) is of no use. However one might consider quantities that are infrared finite as, for instance, some transition probabilities. On those quantities the theorem in eq. (\[cut.9\]) might apply. A further possibility is to work in generic dimension D, whenever it is possible.

5.  Physical unitarity is valid at every value of ${\tilde M}$. In case of the Higgs mechanism scenario we get the hint for considering a metamorphosis of the longitudinally polarized vector meson into a massless scalar at ${\tilde M}=0$.

6.  Within the Higgs mechanism scenario the limit ${\tilde M}=0$ can be performed on the perturbative expansion by providing a consistent picture of the metamorphosis of the longitudinal mode. Thus eq. (\[cut.9\]) can be read in the other way around: the ${\tilde M}\not = 0$ theory provides a doable infrared regulator for the $v=0$ theory (symmetric phase), when on-shell amplitudes are needed.

The CLTCG theorem as in eq. (\[cut.9\]) provides a tool for solving the problem associated to the longitudinal mode that does not decouple from physical states in the zero mass limit. The scenario of a metamorphosis of this mode into the Goldstone boson field, which can be consistently taken as a physical mode at zero mass, looks very promising for satisfying perturbative unitarity and the set of relations derived from the STI, gauge fixing equation and anti-ghost equation. This setting is compatible with the picture of a symmetry restoration through the limit $v=0$ on the perturbative series of the effective action. It is tempting to argue that this setting allows the limit $v=0$ in a nonabelian gauge theory coupled with scalars (the former, i.e. $v\not=  0$, Higgs field), i.e. the generating functionals are continuous in $v=0$. Such dynamical theory remains non asymptotically free, according to the classification of Refs. @Gross:1973ju-@Politzer:1973fx.

Zero Mass Limit with a Stückelberg Gauge Invariant Term {#sec:stuck}
=======================================================

The equations used to support the metamorphosis scenario are still valid in the case of a Yang-Mills theory with mass <span>*à la*</span> Stückelberg. In particular one has the same STI, gauge-fixing equation and anti-ghost equation. However the use of eq. (\[cut.9\]) is now in question: although the formal derivation is the same, the non-existence of a zero mass limit in the loop expansion makes the CLTCG theorem inapplicable.

For small $M$ many terms of perturbative expansion have singular $M^{-1}$ behavior. In fact, in the nonlinear theories the perturbative expansion in the loop number works for momenta small with respect to $M$. It is not known how this region is connected to the one around $M=0$. Consequently the extension of the CLTCG theorem to the massless limit becomes questionable.

The study of the zero-mass region implies a typical strong-coupling limit: the $M^{-2}$ factor in the $\phi$ propagator is responsible for the presence of many divergent terms in the perturbative expansion. This means that one cannot explore the $M=0$ region by starting from the series expansion in the number of loops.

One can make an educated guess on the small mass behavior of the theory on the basis of some na<span>ï</span>ve considerations.

i\) $M$ controls in some way the spontaneous breakdown of the global $SU(2)_L\otimes SU(2)_R$ transformations. In fact the Stückelberg mass term is the source of the interaction of $\phi$ with the rest of the fields. The order parameter $\langle \phi_0\rangle$ for the SBS has no effect at $M=0$. Thus a limit theory is expected to be symmetric.

ii\) BRST properties of the asymptotic fields (if they can be defined) indicate that the fields $\vec\phi$ remain unphysical for any value of $M$, since the vacuum expectation value of $\phi_0$ is expected to remain non-zero.

iii\) One might consider a resummation of the series, by performing first the integration over the $SU(2)$ group, thus taking $A_\mu$ as an external source. One gets an expansion in powers of $M$ where the coefficients are invariant under local gauge transformations. In this setting the path integral on the fields $\vec\phi$ is performed on a lattice, with spacing $a$, $$\begin{aligned}
&&
\int {\cal D}[\phi] \exp 
\Big(\int_E d^4 x  \frac{M^2}{2g^2} (A_{a\mu} - F_{a\mu})^2 \Big)
\nonumber\\&&
\simeq
\int {\cal D}\Omega
\exp \sum_{x\mu}\Biggl [
 \frac{ 2 M^2}{g^2}a^{2} {\mathfrak Re}
\sum_{x\mu}Tr\Bigl\{ \Omega(x)^\dagger U(x,\mu)\Omega(x+\mu)-1\Bigr\} \Biggr],
\label{stuck.1}\end{aligned}$$ where link variable is function solely of the classical field $A_\mu$, the remaining integration variable in the final expression of the generating functional. The path integral integration is over a compact set for each site, therefore we can expand in powers of $M$ $$\begin{aligned}
&&
\frac{1}{\int \prod_x {\cal D}[\Omega(x)]}
\int \prod_x {\cal D}[\Omega(x)]\Biggl[1
+\frac{1}{2} \Bigl (\frac{\beta M^2
  a^2}{2}\Bigr)^2 \Bigl ( \sum_{x\mu} Tr \{\Omega^\dagger(x)U(x,\mu)\Omega(x+\mu)\}\Bigr)^2
\nonumber\\&&
+\frac{1}{4!}(\frac{2 M^2 a^2}{g^2})^4\prod_{j=1}^4
\Bigl(
\sum_{x_j\mu_j}Tr\Bigl\{ \Omega(x_j)^\dagger U(x_j,\mu_j)\Omega(x_j+\mu_j)\Bigr\}
\Bigr)\Biggr]
\nonumber\\&&
=1+ \frac{1}{2} \Bigl (\frac{\beta M^2
  a^2}{2}\Bigr)^2 DN + \frac{1}{8} \Bigl (\frac{\beta M^2
  a^2}{2}\Bigr)^4 (DN)^2 + \frac{1}{4}(\frac{ M^2 a^2}{g^2})^4\sum_\Box Tr \Bigl\{ U_\Box \Bigr\},
\label{stuck.1.1}\end{aligned}$$ where $U_\Box$ is the $SU(2)$ matrix associated to the plaquette $\Box$ and $DN$ is the number of degrees of freedom of the gauge bosons times the number of sites. Subsequently we take into account the integration over the link variables. Finally we take the logarithm of the partition function $$\begin{aligned}
\!\!\! \ln Z = \ln Z_0 +  \frac{1}{2} \Bigl (\frac{\beta M^2
  a^2}{2}\Bigr)^2 DN  +\frac{1}{4}\Bigl(\frac{\beta M^2 a^2}{2}\Bigr)^4 \Big\langle \sum_\Box 
  Tr\{U_\Box \}\Big \rangle \Big |_{M^2a^2=0},
\label{stc.4.1}\end{aligned}$$ where $Z_0$ is the partition function of the massless theory. Thus the final result is a Yang-Mills theory with local insertions.

There is another point in favor of this guess and it comes from eq. (\[cut.9\]). From completely general consideration (i.e. no approximations are needed) we have argued that the limit of zero mass can be performed by keeping BRST invariance, gauge invariance of the $S$-matrix and perturbative unitarity. If the limit implied by the CLTCG theorem exists in the form of a well-defined local theory represented on a Fock space of asymptotic fields, then both the longitudinal polarization mode and the Goldstone boson must decouple in the limiting region. If not, perturbative unitarity is violated in default of cancellation of the unphysical modes. BRST transformations on the asymptotic fields show that $\vec\phi$ remains an unphysical mode, if not decoupled. Therefore the metamorphosis of a physical mode (longitudinal polarization) into an unphysical mode (the Goldstone boson) can not occur. This is an educated guess saying that the longitudinal polarization mode decouples in the zero-mass limit.

The resulting massless Yang-Mills theory is supposed to describe events where the mass is negligible (with respect to energies, momentum transfers and any other dimensionful quantity). This point should be made clear: no confinement is implied of any sort.

These arguments indicate that Yang-Mills with mass generated by the Higgs mechanism or introduced by the the Stückelberg term might be compared on phenomenological ground since at very high energy the number of degrees of freedom are different. Thus the two theories can be tested not only by the direct detection of the Higgs boson but also by this new very important difference in processes at high energy.

An Example {#sec:exam}
==========

The difference between nonabelian gauge theories with Higgs mechanism and with non linear realization can be shown in many realms. The one-loop corrections in the two theories have been discussed and analytically evaluated in Ref. @Bettinelli:2007cy. Moreover the two models show marked differences in the celebrated processes $WW$, $WZ$ and $ZZ$ elastic scattering. In fact, according to the previously presented arguments, the limit $M=0$ of the nonlinear theory is a pure nonabelian gauge model without Higgs scalars and vector meson longitudinal polarizations.

In the present Section we consider a process involving quarks or leptons in order to illustrate the metamorphosis and its consequences. In particular we focus on a process where no Higgs boson is mediating, in order to have a signature which is not directly connected to its existence @Barger:1987nn.

The present example is not intended as a quantitative argument for future measurements. For the last scope one needs more involved calculations including, for instance, the loop contributions of the top and the corrections due to the running of the constants. This part of the research is not considered in the present work.

We consider the following process @Altarelli:2000ye $$\begin{aligned}
 d + \bar u \to b + \bar t,
\label{exam.1}\end{aligned}$$ where the intervening quarks might be easily replaced by other constituents (e.g. $l,\bar\nu$). We follow the conventions of Ref. @Nakamura:2010zzi for the fermion sector. In case of Higgs mechanism we have a Drell-Yan process mediated by $W^-$ and $\phi^-$. In the Landau gauge we have (we consider only the $s-$channel graph) $$\begin{aligned}
&&
{\mathfrak M}(M) = \frac{g^2 V_{ud}V_{tb}^*}{2} \bar v_u \gamma_\mu\frac{1- \gamma_5}{2}u_d
\frac{g^{\mu\nu}-\frac{q^\mu q^\nu}{q^2}}{q^2-M^2}
\bar u_b \gamma_\nu \frac{1- \gamma_5}{2} v_{t}
\nonumber\\&&
+ V_{ud} V_{tb}^* \bar v_u \Big[f_u\frac{1- \gamma_5}{2}
- f_d \frac{1+ \gamma_5}{2}\Big]u_d \frac{1}{q^2}\bar u_b \Big[f_t\frac{1- \gamma_5}{2}
- f_b \frac{1+ \gamma_5}{2}\Big]v_t 
\label{exam.2}\end{aligned}$$ with $M=gv$ and $\sqrt 2 f_x v= m_x$. Unitarity is preserved on-shell; in fact at $q^2=M^2$ the only pole is in the propagator of the $W^-$ with a residuum that projects on the physical polarizations. Moreover there is no pole at $q^2=0$, since the Goldstone boson cancels the spin zero part of the vector meson.

Eq. (\[exam.2\]) shows in detail what happens in the limit $v=0$: we perform the limit in two different ways. First we add the contributions of the Goldstone part to the vector meson propagator to obtain the unitary gauge amplitude. On that amplitude the limit is performed to discover that the longitudinal polarizations yield a finite result. Second the limit is taken separately on the two terms of eq. (\[exam.2\]). The gauge term $q^\mu q^\nu$ of the $W$-propagator vanishes in the limit, while the Goldstone contribution survives to match the longitudinal polarization's of the previous limit procedure, as in the mechanism described in eq. (\[long.5.1.2\]). the $q^\mu q^\nu$ term in the $W$-propagator vanishes via Dirac equation, while the "Goldstone" field contribution survives, as in the mechanism described in eq. (\[long.5.1.2\]). This exemplifies the metamorphosis of the longitudinal polarization into the physical massless scalar mode, originally associated to the unphysical Goldstone boson for $v\not =0$.

By taking the limit in the first fashion, for every value of $M$ the two terms add to $$\begin{aligned}
{\mathfrak M}(M) = \frac{g^2 V_{ud}V_{tb}^*}{2} \bar v_u \gamma_\mu\frac{1- \gamma_5}{2}u_d
\frac{g^{\mu\nu}-\frac{q^\mu q^\nu}{M^2}}{q^2-M^2}
\bar u_b \gamma_\nu \frac{1- \gamma_5}{2} v_{t}.
\label{exam.3}\end{aligned}$$ The $M^{-2}$ term does survive in the limit of zero mass, since the $q^\mu q^\nu$ produces a quadratic term in the quark or lepton masses and therefore the $v^2$ dependence disappears in the ratio. While in the second way we take the $v=0$ limit on the longitudinal part of the propagator in the Landau gauge ($q^\mu q^\nu/q^2$): the result is zero. But the Goldstone contribution is finite and adds to the total amplitude in eq. (\[exam.3\]) taken at $v^2=0$.

In the nonlinear theory, according to our guess, such terms are not present and the $W$-propagator in the Landau gauge is as usual $$\begin{aligned}
-i
\frac{g^{\mu\nu}-\frac{q^\mu q^\nu}{q^2}}{q^2},
\label{exam.3.1}\end{aligned}$$ where the $\frac{q^\mu q^\nu}{q^2}$ vanishes on the quark and lepton chiral currents, since the $M=0$ limit has been already taken. Finally, the difference between the two scenarios (linear versus nonlinear) are traced simply by the presence of a $\frac{1}{M^2}$ factor.

Now we look whether the difference is of some phenomenological relevance, in order to show the origin of the difficulty to find a measurable signature. The square modulus of the amplitude (\[exam.3\]) summed over the polarizations of the incoming and outgoing particles yields $$\begin{aligned}
&&
\sum_{\rm POL}|{\mathfrak M}(M)|^2
= \frac{1}{(q^2-M^2)^2}\Bigg\{
16 (p_bp_u)( p_tp_d)
\nonumber\\&&
- \frac{8}{M^2}\Big[
m^2_t m^2_u (p_bp_d)+m^2_t m^2_d (p_bp_u)+
m^2_b m^2_u (p_tp_d)+m^2_b m^2_d (p_tp_u)
\Big]
\nonumber\\&&
+\frac{4}{M^4}\Big[ 2 m^2_t m^2_b+ ( m^2_b+  m^2_t)(p_bp_t)  \Big]
\Big[ 2 m_u^2 m_d^2+ (m_u^2+ m_d^2) (p_up_d) \Big]
\Bigg\}.
\label{exam.8}\end{aligned}$$ It is clear that the $M^{-2}$ and $M^{-4}$ terms are negligible and therefore one cannot discriminate the linear model (with Higgs boson) from the nonlinear one (without Higgs boson) in this process.

Conclusions {#sec:concl}
===========

We consider the massive Yang-Mills theory at very high momenta both in the case of a Higgs mechanism generated mass and of a Stückelberg mass term. The kinematical set up is reproduced by the $M=0$ limit, by assuming that only one energy scale is present in the physical process. In this limit the number of degrees of freedom of vector mesons changes. This fact poses a problem for unitarity, since the longitudinal modes do not decouple in nonabelian gauge theories for $M=0$.

In the first case we suggest the metamorphosis of the longitudinal modes into the Goldstone scalar bosons when the limit $M=0$ is taken. The scenario is supported by the CLTCG theorem. <span>*In passing*</span> we present some improvements on the CLTCG theorem. According to this proposal the unitarity equation is consistently satisfied both for $M\not =0$ and $M=0$: no mismatch of degrees of freedom shows up and moreover the symmetric limit $v=0$ looks smooth. The limit theory consists of a massless gauge Yang-Mills in interaction with a doublet of physical complex scalar fields (the Higgs and the Goldstone bosons). The theory is expected to be non-asymptotically free.

In the Stückelberg mass case the limit $M=0$ on the perturbative series is not possible due to very singular terms. We suggest that the perturbative region is separated from the $M\sim 0$ region by some singularity line between different phases. We envisage the scenario where at $M\sim 0$ both the Goldstone bosons and the longitudinal modes decouple and the theory is realized in a confined phase, typical of a massless gauge theory. However the properties of the confinement phase are relevant only for extremely high momenta ($M\sim 0$) processes (e.g. asymptotic freedom) and not for low energy states.

The conjecture establishes a ground for developing experimental tests capable to distinguish a linearly (Higgs formalism)- from a nonlinearly (Stückelberg mass)-realized nonabelian massive gauge theory. However this aspect of the work is beyond the scope of the present paper.

In the present paper we use the covariant 't Hooft gauge. Several exact results are derived for the two-point functions in the unphysical sector. The limit at $M=0$ of these two-point functions is evaluated in the Higgs mechanism. This limit tells that the equivalence theorem (CLTCG) in its tree-level formulation is not modified by loop corrections.

Acknowledgments {#acknowledgments .unnumbered}
===============

I am honored to thank the warm hospitality of the Center for Theoretical Physics at MIT, Massachusetts, where the present work has been done. This work is supported by U.S. Department of Energy (D.O.E.) under cooperative research agreement DE-FG0205ER41360.

Free Field Example  {#app:free}
===================

In this Appendix we evaluate the contributions of the longitudinal mode and of the spin zero part of the vector meson to the unitarity sum in the free case. Consider the <span>*free*</span> vector meson propagator in the generic $\xi$ gauge $$\begin{aligned}
-\frac{g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2}}{p^2-M^2} - \frac{1}{\xi}\frac{1}{p^2-\frac{M^2}{\xi}}
\frac{p_\mu p_\nu}{p^2}.
\label{tpf.65}\end{aligned}$$ We use the residua of the poles both for $p^2=M^2$ and $p^2= \frac{M^2}{\xi}$ and at the end we take the limit $M^2=0$. The contribution of the longitudinal polarization on the pole $p^2=M^2$ as in eq. (\[long.1\]) is $$\begin{aligned}
\frac{1}{M^2}
\left (
\begin{array}{ll} p^2  &E p_j   \\
E p_i & \frac{E^2}{p^2}p_i p_j 
\end{array}
\right)
\label{tpf.66}\end{aligned}$$ while the spin zero at $p^2= \frac{M^2}{\xi}$ $$\begin{aligned}
-\frac{1}{M^2}
\left (
\begin{array}{ll} E_{\scriptscriptstyle G}^2  &E_{\scriptscriptstyle G}p_j  \\
E_{\scriptscriptstyle G} p_i &p_i p_j 
\end{array}
\right).
\label{tpf.67}\end{aligned}$$ The contribution of the two transverse polarizations is $$\begin{aligned}
\left (
\begin{array}{ll}0&0  \\
0 & \delta_{ij}-\frac{1}{p^2}p_i p_j 
\end{array}
\right).
\label{tpf.68}\end{aligned}$$ The expression in eq. (\[tpf.66\]) is multiplied by some quantity $M^{\mu\nu}(E,  \vec p)$, while that in eq. (\[tpf.67\]) by $M^{\mu\nu}(E_{\scriptscriptstyle G},  \vec
p)$.

Now we add the contributions with the front factors $1/E$ and $1/E_{\scriptscriptstyle G}$ originating from the Lorentz invariant measure . Thus we have to add the terms $$\begin{aligned}
\frac{1}{2EM^2}
\left ( p^2 M^{00}(E,  \vec p)+E p_j M^{0j}(E,  \vec p) +
E p_i M^{i0}(E,  \vec p) + \frac{E^2}{p^2}p_i p_j  M^{ij}(E,  \vec p)
\right)
\label{tpf.70}\end{aligned}$$ and $$\begin{aligned}
&&
-\frac{1}{2E_{\scriptscriptstyle G} M^2}
\left ( E_{\scriptscriptstyle G}^2 M^{00}(E_{\scriptscriptstyle G},  \vec p)
+E_{\scriptscriptstyle G} p_j M^{0j}(E_{\scriptscriptstyle G},  \vec p) +
E_{\scriptscriptstyle G} p_i M^{i0}(E_{\scriptscriptstyle G},  \vec p) 
+ p_i p_j  M^{ij}(E_{\scriptscriptstyle G},  \vec p)
\right)
\nonumber\\&&
=
-\frac{1}{2EM^2}(1-\frac{E_{\scriptscriptstyle G}-E}{E} )
\Big [\left(p^2+\frac{M^2}{\xi}\right)\Big( M^{00}(E)+(E_{\scriptscriptstyle
  G}-E)\frac{\partial}{\partial E} M^{00}(E)\Big)
\nonumber\\&&
+ p_j \Big(E  M^{0j}(E)+(E_{\scriptscriptstyle G}-E)(M^{0j}(E)
+E\frac{\partial}{\partial E} M^{0j}(E)\Big)
\nonumber\\&&
+ p_i \Big(E  M^{i0}(E)+(E_{\scriptscriptstyle G}-E)(M^{i0}(E)
+E\frac{\partial}{\partial E} M^{i0}(E)\Big)
\nonumber\\&&
+ p_i p_j\Big ( M^{ij}(E)
+(E_{\scriptscriptstyle G}-E)\frac{\partial}{\partial E} M^{ij}(E)\Big)
\Big ]
\label{tpf.71}\end{aligned}$$ Now we have $$\begin{aligned}
E_{\scriptscriptstyle G}-E
\simeq -  M^2\frac{1-\frac{1}{\xi} }{2p}
\label{tpf.72}\end{aligned}$$ and therefore we get $$\begin{aligned}
&&
\frac{1}{2p}\Bigg[
\frac{p}{2}(1-\frac{1}{\xi})\frac{\partial}{\partial E} M^{00}(E)
-\frac{1}{\xi} M^{00}(E)
\nonumber\\&&
+\frac{p_j}{2p}(1-\frac{1}{\xi})\Big(M^{0j}(E)
+E\frac{\partial}{\partial E} M^{0j}(E)\Big)
\nonumber\\&&
+\frac{p_i}{2p}(1-\frac{1}{\xi})\Big(M^{i0}(E)
+E\frac{\partial}{\partial E} M^{i0}(E)\Big)
\nonumber\\&&
+ \frac{p_ip_j}{p^2}  M^{ij}(E) 
+ \frac{p_ip_j}{2p}(1-\frac{1}{\xi})\frac{\partial}{\partial E} M^{ij}(E)\Bigg]
\nonumber\\&&
-\frac{1}{4p^3}(1-\frac{1}{\xi})\Bigg\{p^2 M^{00}+pp_jM^{0j}+p p_i M^{i0} +p_ip_jM^{ij}
\Bigg\}
\label{tpf.73}\end{aligned}$$ If we add the transverse part (\[tpf.68\]) we get $$\begin{aligned}
&&
\frac{1}{2p}\Bigg[
\frac{p}{2}(1-\frac{1}{\xi})\frac{\partial}{\partial E} M^{00}(E)
-\frac{1}{\xi} M^{00}(E)
\nonumber\\&&
+\frac{p_j}{2p}(1-\frac{1}{\xi})\Big(M^{0j}(E)
+E\frac{\partial}{\partial E} M^{0j}(E)\Big)
\nonumber\\&&
+\frac{p_i}{2p}(1-\frac{1}{\xi})\Big(M^{i0}(E)
+E\frac{\partial}{\partial E} M^{i0}(E)\Big)
\nonumber\\&&
+ \delta_{ij} M^{ij}(E) 
+ \frac{p_ip_j}{2p}(1-\frac{1}{\xi})\frac{\partial}{\partial E} M^{ij}(E)
\nonumber\\&&
-\frac{1}{2p^2}(1-\frac{1}{\xi})\Big\{p^2 M^{00}+pp_jM^{0j}+p p_i M^{i0} +p_ip_jM^{ij}
\Big\}
\Bigg]
\label{tpf.74}\end{aligned}$$ that can be written $$\begin{aligned}
&&
\frac{1}{2p}\Bigg(
-g_{\mu\mu} M^{\mu\nu}(E) +(1-\frac{1}{\xi})
\Bigg[ M^{00}(E)
+\frac{p}{2}\frac{\partial}{\partial E} M^{00}(E)
\nonumber\\&&
+\frac{p_j}{2p}\Big(M^{0j}(E)
+E\frac{\partial}{\partial E} M^{0j}(E)\Big)
\nonumber\\&&
+\frac{p_i}{2p}\Big(M^{i0}(E)
+E\frac{\partial}{\partial E} M^{i0}(E)\Big)
+ \frac{p_ip_j}{2p}\frac{\partial}{\partial E}
M^{ij}(E)
\nonumber\\&&
-\frac{1}{2p^2}\Big\{p^2 M^{00}+pp_jM^{0j}+p p_i M^{i0} +p_ip_jM^{ij}
\Big\}
\Bigg]\Bigg |_{E=|\vec p|} \Bigg)
\nonumber\\&&
=
\frac{1}{2p}\Bigg(
-g_{\mu\mu} M^{\mu\nu}(E) +(1-\frac{1}{\xi})
\Bigg[\frac{1}{2} M^{00}(E) -\frac{p_ip_j}{2p^2}M^{ij}
+\frac{p}{2}\frac{\partial}{\partial E} M^{00}(E)
\nonumber\\&&
+\frac{p_j}{2}\frac{\partial}{\partial E} M^{0j}(E)
+\frac{p_i}{2}\frac{\partial}{\partial E} M^{i0}(E)
+ \frac{p_ip_j}{2p}\frac{\partial}{\partial E}
M^{ij}(E)
\Bigg]\Bigg |_{E=|\vec p|} \Bigg)
.
\label{tpf.75}\end{aligned}$$ This agrees with the $M=0$ limit propagator $$\begin{aligned}
-\frac{g_{\mu\nu}-(1-\frac{1}{\xi})\frac{p_\mu p_\nu}{p^2}}{p^2}.
\label{tpf.76}\end{aligned}$$ In fact the positive frequency pole gives the first term in eq. (\[tpf.75\]) while the double pole gives $$\begin{aligned}
&&
(1-\frac{1}{\xi})
\frac{\partial}{\partial p_0}\frac{p_\mu
  p_\nu}{(p_0+p)^2}M^{\mu\nu}(p_0) |_{p_0=|\vec p|}
\nonumber\\&&
=(1-\frac{1}{\xi})\Bigg[\frac{1}{2p}M^{00}(p)
-\frac{1}{4p}M^{00}(p)+\frac{1}{4}\frac{\partial}{\partial
  p_0}M^{00}(p)
\nonumber\\&&
+\frac{p_j}{4p^2} M^{0j}-\frac{p_j}{4p^2}M^{0j}+\frac{p_j}{4p}\frac{\partial}{\partial
  p_0}M^{0j}(p)
\nonumber\\&&
+\frac{p_i}{4p}\frac{\partial}{\partial
  p_0}M^{i0}(p) -\frac{p_ip_j}{4p^3}M^{ij}(p)+\frac{p_ip_j}{4p^2}\frac{\partial}{\partial
  p_0}M^{ij}(p)
\Bigg]
\nonumber\\&&
=\frac{1}{2p}(1-\frac{1}{\xi})\Bigg[
\frac{1}{2}M^{00}(p)+\frac{p}{2}\frac{\partial}{\partial
  p_0}M^{00}(p)
+\frac{p_j}{2}\frac{\partial}{\partial
  p_0}M^{0j}(p)
\nonumber\\&&
+\frac{p_i}{2}\frac{\partial}{\partial
  p_0}M^{i0}(p) -\frac{p_ip_j}{2p^2}M^{ij}(p)+\frac{p_ip_j}{2p}\frac{\partial}{\partial
  p_0}M^{ij}(p)
\Bigg]
.
\label{tpf.77}\end{aligned}$$ The RHS of eq. (\[tpf.77\]) is zero in presence of the conservation laws $p^\mu M_{\mu\nu}= p^\nu M_{\mu\nu}=0$.

This Appendix shows that terms in eq. (\[long.5\]), otherwise neglected on-shell, give essential contributions for the limit $M=0$.

BRST transformations and STI (Higgs) {#app:app1}
====================================

We discuss the BRST transformations for both models. We derive the STI and the relations that are needed for the CLTCG theorem. First we consider the case where the mass is generated by the Higgs mechanism.

The necessity to maintain physical unitarity after the introduction of a gauge fixing term requires the use of the Faddeev-Popov ghosts. This is easily done by imposing BRST invariance of the action under the transformations $$\begin{aligned}
&&
{\mathfrak s} A_{a\mu} = (D_\mu [A]c)_a, \qquad 
{\mathfrak s} \phi_0 = -
\frac{1}{2} \phi_a c_a
 \nonumber \\&& 
{\mathfrak s} \phi_a = 
\frac{1}{2} \phi_0 c_a + \frac{1}{2} \epsilon_{abc} \phi_b c_c 
 \nonumber \\&& 
{\mathfrak s} \bar c_a = b_a , \qquad  {\mathfrak s}b_a = 0. 
\label{app1.1}\end{aligned}$$ In the above equation $D_\mu[A]$ denotes the covariant derivative w.r.t. $A_{a\mu}$: $$\begin{aligned}
(D_\mu[A])_{ac} = \delta_{ac} \partial_\mu + \epsilon_{abc} A_{b\mu} \, .
\label{app1.2}\end{aligned}$$ The BRST transformation of $c_a$ then follows by nilpotency $$\begin{aligned}
{\mathfrak s} c_a = - \frac{1}{2} \epsilon_{abc} c_b c_c \, .
\label{app1.3}\end{aligned}$$ Now we can easily obtain a BRST invariant action by making invariant the gauge fixing term. This is achieved by using the nilpotency of $ {\mathfrak s}$. For the generic covariant 't Hooft gauge we have $$\begin{aligned}
S_{\rm H\,_{gf}} \to
S_{\rm  H\, _{GF}}  = \frac{\Lambda^{(D-4)}}{g^2} \int d^Dx \,
 {\mathfrak s} \,
\biggl[ \bar c_a\,\Big( \frac{b_a}{2\xi }+\frac{ M}{\xi }\phi_a +
 \partial_\mu A_a^\mu\Big )\biggr]. 
\label{app1.4}\end{aligned}$$ The STI associated to the above BRST transformations, in the notations of Batalin and Vilkovisky @Batalin:1981jr [@Batalin:1984jr] can be easily derived. By introducing the external sources $$\begin{aligned}
\int d^Dx \, (A_{a\mu}^* ~ {\mathfrak s} A_a^\mu 
+ \phi_a^* ~{\mathfrak s} \phi_a+ \phi_0^* ~{\mathfrak s} \phi_0 
 + c_a^* ~{\mathfrak s} c_a ) \, .
\label{app1.5}\end{aligned}$$ for the 1-PI functional one gets $$\begin{aligned}
&&
\int d^Dx \, \Big (
 \Gamma_{ A^*_{a\mu}}  \Gamma_{ A_a^\mu}
+
 \Gamma_{ \phi_a^*}  \Gamma_{ \phi_a}
+
 \Gamma_{ \phi_0^*}  \Gamma_{ \phi_0}
+ 
 \Gamma_{ c_a^*} \Gamma_{ c_a}
+ b_a  \Gamma_{ \bar c_a} 
\Big ) = 0 \, .
\label{app1.6}\end{aligned}$$ While for the generating functional of the connected amplitudes one has $$\begin{aligned}
&&
\int d^Dx \, \Big (
- W_{ A^*_{a\mu}}J_{a\mu} 
-
 W_{ \phi_a^*} K_a
-  W_{ \phi_0^*}K_0  
+ 
 W_{ c_a^*}\bar\eta_a
- W_{  b_a} \eta_a
\Big ) = 0\, . 
\label{app1.8}\end{aligned}$$ The equation associated to the gauge fixing gives $$\begin{aligned}
\Gamma_{b_a}
=\Lambda_g \Big(
 \frac{b_a}{\xi }+\frac{ M}{\xi }\phi_a +
 \partial_\mu A_a^\mu \Big)
\label{app1.11}\end{aligned}$$ $$\begin{aligned}
-J_{b_a}
=\Lambda_g \Big(
 \frac{1}{\xi }W_{b_a}+\frac{ M}{\xi }W_{\phi_a} +
 \partial_\mu W_{ A_a^\mu} \Big).
\label{app1.12}\end{aligned}$$ The anti-ghost equation $$\begin{aligned}
\Gamma_{\bar c_a}
=\Lambda_g \Big[
-\frac{ M}{2\xi }(c_a \phi_0 + \epsilon_{abc}c_c \phi_b)-
 \partial_\mu \left(D^\mu[A]c\right)_a \Big]
 =\Lambda_g \Big[
-\frac{ M}{\xi }\Gamma_{\phi_a^*}-
 \partial_\mu \Gamma_{A_{a\mu}^*}\Big]
\label{app1.13}\end{aligned}$$ $$\begin{aligned}
\eta_a
=\Lambda_g \Big[
\frac{ M}{\xi }W_{\phi_a^*}+
 \partial_\mu W_{A_{a\mu}^*}\Big].
\label{app1.14}\end{aligned}$$

BRST transformations and STI (Stückelberg) {#sec:st}
==========================================

In the case where the mass of the Yang-Mills fields comes from a Stückelberg term, the same BRST apply as in eqs. (\[app1.1\]) and (\[app1.3\]). Moreover the STI and the gauge fixing equation are akin to those of Yang-Mills with Higgs mechanism. Here we illustrate this property, which is not trivial due to the fact that Stückelberg's mass term makes the theory nonrenormalizable.

Eq. (\[stck.1\]) becomes $$\begin{aligned}
S_{\rm S\,_{gf}} \to
S_{\rm  S\, _{GF}}  = \frac{\Lambda^{(D-4)}}{g^2} \int d^Dx \,
 {\mathfrak s} \,
\biggl[ \bar c_a\,\Big( \frac{b_a}{2\xi }+2\frac{ M^2}{\xi }\phi_a +
 \partial_\mu A_a^\mu\Big )\biggr]. 
\label{app2.1}\end{aligned}$$ The process of removal of the divergences of this nonrenormalizable field theory @Bettinelli:2007tq, requires a much wider set of external sources ($K_0,V_{a\mu},z_0,z_a $) than in eq. (\[app1.5\]) $$\begin{aligned}
&&
\int d^Dx \, (A_{a\mu}^* ~ {\mathfrak s} A_a^\mu 
+ \phi_a^* ~{\mathfrak s} \phi_a+ \phi_0^* ~{\mathfrak s} \phi_0 
 + c_a^* ~{\mathfrak s} c_a 
+V_{a\mu}{\mathfrak s} D^\mu[A]_{ab}\bar c_b
\nonumber\\&&
+ z_0\chi_0+z_a\chi_a
+K_0\phi_0 ) \, ,
\label{app2.2}\end{aligned}$$ where $$\begin{aligned}
&&
\bar c\equiv  \bar c_a \frac{\tau_a}{2}
\nonumber\\&&
\chi_0 + i \tau_a \chi_a \equiv  2i\, {\mathfrak s}\,
\bar c \, \Omega = 2i( b~\Omega -\, \bar c ~{\mathfrak s}~\Omega).
\label{app2.3}\end{aligned}$$ By construction, the fields $$\begin{aligned}
&&
\chi_0 = - b_a\phi_a + \frac{1}{2}\bar c_a c_a \phi_0+ \frac{1}{2}\epsilon_{abc}\bar c_b c_a \phi_c
\nonumber\\&&
\chi_c = \phi_0 b_c - \epsilon_{abc}b_a \phi_b 
+ \frac{1}{2}\epsilon_{abc}\bar c_b c_a \phi_0
+ \frac{1}{2}\bigl(
\bar c_c c_a\phi_a + \bar c_a c_a \phi_c - \bar c_a \phi_a c_c
\bigr) .
\label{app2.4}\end{aligned}$$ transform like $\phi_0, \phi_a$. The source $K_0$ is needed in order to perform the insertion of the composite operator $\phi_0$ @Ferrari:2005ii, $\phi_0^*$ and $V_{a\mu}$ are the external sources for the BRST transform of $\phi_0$ and for some operator entering in the gauge fixing @Bettinelli:2007tq and finally $z_0,z_a$ are necessary in order to deal with the 't Hooft gauge @Ferrari:2010ge.

In a standard way one gets the STI $$\begin{aligned}
&&
\int d^Dx \, \Big (
 \Gamma_{ A^*_{a\mu}}  \Gamma_{ A_a^\mu}
+
 \Gamma_{ \phi_a^*}  \Gamma_{ \phi_a}
-K_0
 \Gamma_{ \phi_0^*}  
+ 
 \Gamma_{ c_a^*} \Gamma_{ c_a}
+ b_a  \Gamma_{ \bar c_a} 
\Big ) = 0 \, ,
\label{app2.5}\end{aligned}$$ and the gauge fixing equation $$\begin{aligned}
\Gamma_{b_a}
= \Lambda_g\Big[
 \frac{b_a}{\xi }+2\frac{ M^2}{\xi }\phi_a +
 \partial^\nu A_a^\mu \Big] - z_0 \phi_a +z_c (\Gamma_{K_0} -\epsilon_{abc} \phi_b).
\label{app2.6}\end{aligned}$$

For the connected amplitudes we have $$\begin{aligned}
\int d^Dx \, \Big (
- J_{a\mu}  W_{ A^*_{a\mu}}
-
 K_a W_{ \phi_a^*} 
- K_0   W_{ \phi_0^*} 
+ 
 \bar\eta_a W_{ c_a^*}
-\eta_a W_{  b_a} 
\Big ) = 0 
\label{app2.7}\end{aligned}$$ and the gauge-fixing equation $$\begin{aligned}
-J_{b_a}
=\Lambda_g\Big[
 \frac{1}{\xi }W_{b_a}+2\frac{ M^2}{\xi }W_{\phi_a} +
 \partial^\nu W_{ A_a^\mu} \Big] - z_0 W_{\phi_a} 
+z_c (W_{K_0} -\epsilon_{abc} W_{\phi_b}).
\label{app2.8}\end{aligned}$$ The presence of the sources $z_a$ upsets the anti-ghost equation, in fact $\Gamma_{\bar c_a}$ contains the insertion of composite operators that are not associated to the listed external sources. The use of the sources $K_0,V_{a\mu},z_0,z_a $ is necessary for the subtraction of the infinities @Ferrari:2010ge. In this work we evaluate only the two-point function of $b, A_\mu, \phi$, then we can put to zero all the external source $K_0,V_{a\mu},z_0,z_a $ and consequently the usual anti-ghost equation (see eqs. (\[app1.13\]) and (\[app1.14\])) is at our disposal.

In this subsection we have shown that in the massive Yang-Mills theory the two-point function $W$ and $\Gamma$ in the unphysical sector obey the same equations with the Higgs mechanism as well as with the Stückelberg term, apart from an unessential rescaling of the field $\phi\to 2M \phi$. Also the results of Appendix \[app:conv\] apply, with the same rescaling.

Properties of the Two-point Functions (Higgs) {#sec:tpf}
=============================================

The results of this section apply to both Higgs and Stückelberg scenario, since the STI (eqs. (\[app1.6\]), (\[app1.8\]), (\[app2.5\]) and (\[app2.7\]) ) and the gauge fixing equations (eqs. (\[app1.11\]), (\[app1.12\]), (\[app2.6\]) and (\[app2.8\]) ) are the same after rescaling the $\phi$ field.

Now we derive some consequences of the above equations.

From eq. (\[app1.11\]) we get $$\begin{aligned}
\Gamma_{bb} = \Lambda_g\frac{1}{\xi}, \qquad
\Gamma_{b\phi} = \Lambda_g\frac{M}{\xi}, \qquad
\Gamma_{A^\mu b} =   i \Lambda_g p_\mu
\label{app1.14.1}\end{aligned}$$ From the STI in eq. (\[app1.8\]) $$\begin{aligned}
&&
W_{b_a b_b} = 0
\nonumber\\&&
W_{A_{a\mu}^* \bar c_b } =W_{A_{a\mu} b_b}
\nonumber\\&&
W_{\phi_a^* \bar c_b } =W_{\phi_a b_b}
\label{app1.15}\end{aligned}$$ and from eq. (\[app1.12\]) $$\begin{aligned}
\Lambda_g\Big[
\frac{M}{\xi}W_{\phi_a^* \bar c_b } -ip^\mu W_{A_{a\mu}^* \bar c_b }\Big] =-\delta_{ab}.
\label{app1.15.1}\end{aligned}$$ From the STI in eq. (\[app1.13\]) $$\begin{aligned}
 \Gamma_{c_a A^*_{{a'}\mu}}  \Gamma_{ b_b A_{a'}^\mu}+\Gamma_{c_a \phi^*_{{a'}}}  \Gamma_{ b_b \phi_{a'}}
+\Gamma_{c_a \bar c_b}  = 0 
\label{app1.16}\end{aligned}$$ i.e. $$\begin{aligned}
\Lambda_g\Big[
 - i p_\mu\Gamma_{c_a A^*_{b\mu}}  + \frac{M}{\xi} \Gamma_{c_a \phi^*_{b}} \Big]
=-\Gamma_{c_a \bar c_b}
\, .
\label{app1.17}\end{aligned}$$ Moreover from the STI in eq. (\[app1.6\]) we get (we drop unnecessary indexes) $$\begin{aligned}
&&
\Gamma_{c A^*_{\mu}} \Gamma_{ A^\mu \phi}+ \Gamma_{c \phi^*} \Gamma_{\phi \phi}=0
\label{app1.18}
\\&&
\Gamma_{c A^*_{\mu}} \Gamma_{ A^\mu A^\nu}  + \Gamma_{c \phi^*} \Gamma_{\phi A^\nu} = 0 
\, .
\label{app1.19}\end{aligned}$$ Eqs. (\[app1.18\]) and (\[app1.19\]) are compatible if the Jacobian is zero $${
(p^\nu\Gamma_{A^\nu\phi})^2 +p^2\Gamma_L\Gamma_{\phi\phi}=0.
}
\label{tpf.9.2}$$ Now we solve the linear system given by eqs. (\[app1.17\]) and (\[app1.18\]) $$\begin{aligned}
&&
\Lambda_g p_\mu \Gamma_{c A^*_{\mu}}=- \frac{\Gamma_{c \bar c}\Gamma_{\phi \phi}}{i\Gamma_{\phi \phi}-
 \frac{M}{\xi p^2}p^\nu \Gamma_{ A^\nu \phi}}
= i p^2 \frac{\Gamma_{c \bar c}}{  p^2+i
 \frac{M}{\xi\Gamma_{\phi \phi}}p^\nu \Gamma_{ A^\nu \phi}}
\nonumber\\&& 
\Lambda_g \Gamma_{c\phi^*} = - \frac{p^\nu \Gamma_{ A^\nu \phi}}{ p^2\Gamma_{\phi \phi}}p_\mu \Gamma_{c A^*_{\mu}}
=  - i\frac{p^\nu \Gamma_{ A^\nu \phi}}{ \Gamma_{\phi \phi}}   \frac{\Gamma_{c \bar c}}{  p^2+i
 \frac{M}{\xi\Gamma_{\phi \phi}}p^\nu \Gamma_{ A^\nu \phi}}.
\label{app1.20}\end{aligned}$$ Thus eq. (\[app1.15\]) gives $$\begin{aligned}
&&
 W_{A^\mu b}=-\frac{i}{\Lambda_g}\frac{p^\mu}{
p^2-i  \frac{M}{\xi}\frac{p^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}
}
\nonumber\\&&
 W_{\phi b}
=\frac{i}{\Lambda_g}\frac{p^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}
\frac{1}{p^2-\frac{M}{\xi}\frac{ip^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}
}.
\label{basic.8}\end{aligned}$$ The two-point functions have the pole in the same position, i.e. the solution of $$\begin{aligned}
p^2=\frac{M}{\xi}\frac{ip^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}.
\label{basic.99p}\end{aligned}$$

Two-point Functions
-------------------

From the STI eq. (\[app1.8\]) one gets ($k>1$) $$\begin{aligned}
W_{b_1\cdots b_k ***}=0,
\label{basic.1}\end{aligned}$$ where $\scriptstyle{ ***}$ indicates any reduction formula operator (on-shell) for modes.

From eqs. (\[app1.12\]) and (\[basic.1\]) we get $$\begin{aligned}
&&
\frac{M}{\xi} W_{\phi b}  - i p_\mu W_{A^\mu b} = -\frac{1}{\Lambda_g}
\label{basic.6.1}
\\&&
\frac{1}{\xi} W_{b\phi }  +\frac{M}{\xi} W_{\phi\phi }
 - i p_\mu W_{A^\mu \phi} = 0
\label{basic.6.2}
\\&&
\frac{1}{\xi} W_{b A^\nu } +\frac{M}{\xi} W_{\phi A^\nu }
 - i p_\mu W_{A^\mu A^\nu} = 0.
\label{basic.6.3}\end{aligned}$$ Now we use $$\begin{aligned}
\Gamma W = - I\!\!I.
\label{basic.7}\end{aligned}$$ and eqs. (\[basic.1\])-(\[basic.6.3\]) in order to derive the two-point function $W$ in terms of the two-point function $\Gamma$. We explicit write some elements of the matrix in eq. (\[basic.7\]) $$\begin{aligned}
&&
\bigl(\Gamma W\bigr)_{\phi b}=0\Longrightarrow \Gamma_{\phi\phi}W_{\phi b}
+\Gamma_{\phi A^\mu}W_{A^\mu b}=0
\label{tpf.8.1}
\\&&
\bigl(\Gamma W\bigr)_{A^\nu b}=0\Longrightarrow \Gamma_{A^\nu\phi}W_{\phi b}
+\Gamma_{A^\nu A^\mu}W_{A^\mu b}=0
\label{tpf.8.2} \\&&
\bigl(\Gamma W\bigr)_{\phi \phi}=-1\Longrightarrow \Gamma_{\phi b}W_{b \phi}
+\Gamma_{\phi\phi}W_{\phi \phi} +\Gamma_{\phi A^\mu}W_{A^\mu \phi}=-1
\label{tpf.8.3} \\&&
\bigl(\Gamma W\bigr)_{\phi A^\nu }=0\Longrightarrow \Gamma_{\phi b}W_{ bA^\nu}+
\Gamma_{\phi\phi}W_{\phi A^\nu}+\Gamma_{\phi A^\mu}W_{A^\mu A^\nu}=0
\label{tpf.8.4} \\&&
\bigl(\Gamma W\bigr)_{ A^\nu \phi }=0\Longrightarrow \Gamma_{A^\nu b}W_{ b\phi}+
\Gamma_{ A^\nu\phi}W_{\phi\phi}+\Gamma_{ A^\nu A^\mu}W_{A^\mu\phi}=0
\label{tpf.8.5} \\&&
\bigl(\Gamma W\bigr)_{ A^\nu A^\mu }=-g_{\mu\nu}
\nonumber\\&&
\Longrightarrow \Gamma_{A^\nu b}W_{ bA^\mu}+
\Gamma_{ A^\nu\phi}W_{\phi A^\mu}+\Gamma_{ A^\nu A^\sigma}W_{A^\sigma A^\mu}
=-g_{\mu\nu}
\label{tpf.8.6}\end{aligned}$$

Now we solve the linear system given by the eqs.(\[basic.6.2\]) and (\[tpf.8.3\]). The jacobian of the homogeneous system is $$\begin{aligned}
J_\Delta = p^\nu\Gamma_{\phi A^\nu}\frac{M}{\xi p^2} + i\Gamma_{\phi\phi}
=i\frac{\Gamma_{\phi\phi}}{ p^2}\Bigl( p^2-ip^\nu\Gamma_{\phi A^\nu}\frac{M}{\xi 
\Gamma_{\phi\phi}}\Bigr)
\label{tpf.23.-1}\end{aligned}$$ The straightforward solutions are $$\begin{aligned}
  W_{A^\mu \phi}
=i \frac{1}{\xi\Gamma_{\phi\phi}}\biggl(\frac{i}{\Lambda_g}p^\nu\Gamma_{ A^\nu\phi}
+M p^2
\biggr)
\frac{p^\mu
}{\biggl(
p^2 -  \frac{M}{\xi} 
\frac{ip^\nu\Gamma_{ \phi A^\nu}}{\Gamma_{\phi\phi}}\biggr)^2
}
\label{tpf.23}\end{aligned}$$ and $$\begin{aligned}
W_{\phi \phi}
 =-
\frac{p^2}{ \Gamma_{\phi \phi}}
\biggl(p^2-
\frac{1}{\Lambda_g \xi}\Gamma_L 
\biggr)
\frac{1}{\biggl(
p^2 -  \frac{M}{\xi}
\frac{ip^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}\biggr)^2
}.
\label{tpf.22}\end{aligned}$$ A similar calculation for the linear system given by the eqs. (\[basic.8\]), (\[basic.6.3\]) and (\[tpf.23\]) yields $$\begin{aligned}
W_L 
=\frac{p^2}{\xi\Gamma_{\phi\phi}}
\frac{\frac{1}{\Lambda_g}\Gamma_{\phi\phi} -\frac{M^2}{\xi}}{\biggl(
p^2 -  \frac{M}{\xi} \frac{ip^\nu\Gamma_{\phi A^\nu}}
{\Gamma_{\phi\phi}}\biggr)^2
}.
\label{tpf.24}\end{aligned}$$ The presence of double poles is the source of some technical problems in dealing with the proof of Physical Unitarity.

For comparison, we evaluate the relevant quantities for the free fields $$\begin{aligned}
&&
\Gamma_{bb}= \frac{\Lambda_g}{\xi}, \quad
\Gamma_{\phi b }=\Lambda_g \frac{M}{\xi}, \quad
\Gamma_{ A^\nu b}=i\Lambda_g p_\nu, \quad
\nonumber\\&&
\Gamma_{ A^\nu\phi}=i\Lambda_g Mp_\nu, \quad
\Gamma_{\phi\phi}=\Lambda_g p^2, \quad
\Gamma_L=\Lambda_g M^2.
\label{tpf.25}\end{aligned}$$ Then $$\begin{aligned}
&&
 W_{A^\mu\phi }=0
\label{tpf.26.1}\end{aligned}$$ and $$\begin{aligned}
&&
 W_{A^\mu b}=-i\frac{p^\mu}{\Lambda_g(
p^2-  \frac{M^2}{\xi})
},\qquad
 W_{\phi b}=\frac{M}{\Lambda_g(
p^2 -  \frac{M^2}{\xi})
}
\nonumber\\&&
 W_L=\frac{1}{\Lambda_g\xi}\frac{1}{
p^2 -  \frac{M^2}{\xi}
},\qquad
W_{\phi \phi}
 =- \frac{1}{\Lambda_g(
p^2 -  \frac{M^2}{\xi})
}.
\label{tpf.27.1}\end{aligned}$$ Only simple poles appear in the free field approximation.

Zero Eigenvalue Mode {#app:conv}
====================

We construct the mode corresponding to the zero eigenvalue of the matrix ${\cal G}$ in (\[ptpf.26\]) when on-shell.

First consider $$\begin{aligned}
\Upsilon _a\equiv \frac{1}{\xi}b_a + \frac{M}{\xi}\phi_a + \partial_\mu A_a^\mu.
\label{tpf.41}\end{aligned}$$ From eq. (\[app1.12\]) we see that the field decouples from every other $$\begin{aligned}
W_{\Upsilon b}=-1, \qquad W_{\Upsilon \phi}=0, \qquad W_{\Upsilon A_\mu}=0.
\label{tpf.42}\end{aligned}$$ A next interesting local field is (used in Ref. @Becchi:1974md) $$\begin{aligned}
X_1\equiv \frac{1}{\xi} \phi
 + \frac{1}{M}\partial_\mu A^\mu. 
\label{tpf.48}\end{aligned}$$ From eqs. (\[basic.6.2\]), (\[basic.6.3\]) and (\[basic.8\]) we have $$\begin{aligned}
&&
W_{X_1\phi} = - \frac{1}{\xi M}W_{b\phi}
=-\frac{i}{\Lambda_g }\frac{p^\nu\Gamma_{\phi A^\nu}}{\xi M\Gamma_{\phi\phi}}
\frac{1}{p^2-\frac{M}{\xi}\frac{ip^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}
}
\nonumber\\&&
W_{X_1 A^\nu} = - \frac{1}{\xi M}W_{b A^\nu}
=-\frac{i}{\Lambda_g \xi  M}\frac{p^\nu}{
p^2-i  \frac{M}{\xi}\frac{p^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}
}.
\label{tpf.49}\end{aligned}$$ From eq. (\[tpf.49\]) we see that $X_1$ is the mode corresponding to the zero eigenvalue of $\cal G$ when taken on-shell. Finally one has $$\begin{aligned}
&&
W_{X_1X_1}= \frac{1}{\xi}W_{X_1\phi}+i\frac{1}{M}p^\nu W_{X_1 A^\nu}
\nonumber\\&&
=-\frac{1}{\Lambda_g\xi  M^2}\biggl(i\frac{M}{\xi}\frac{p^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}
-p^2
\biggr)
\frac{1}{p^2-\frac{M}{\xi}\frac{ip^\nu\Gamma_{\phi A^\nu}}{\Gamma_{\phi\phi}}
}
= \frac{1}{\Lambda_g\xi M^2}.
\nonumber\\&&
\label{tpf.50}\end{aligned}$$

<span>99</span>

J. M. Cornwall, D. N. Levin and G. Tiktopoulos, Phys. Rev.  D <span>**10**</span>, 1145 (1974) \[Erratum-ibid.  D <span>**11**</span>, 972 (1975)\].

B. W. Lee, C. Quigg and H. B. Thacker, Phys. Rev.  D <span>**16**</span>, 1519 (1977). H. A. Weldon, Phys. Rev.  D <span>**30**</span>, 1547 (1984).

M. S. Chanowitz and M. K. Gaillard, Nucl. Phys.  B <span>**261**</span>, 379 (1985).

A. Denner, S. Dittmaier and T. Hahn, Phys. Rev.  D <span>**56**</span>, 117 (1997) \[arXiv:hep-ph/9612390\].

A. Denner and T. Hahn, Nucl. Phys.  B <span>**525**</span>, 27 (1998) \[arXiv:hep-ph/9711302\].

E. Accomando, D. Becciolini, S. De Curtis, D. Dominici and L. Fedeli, Phys. Rev. D <span>**84**</span>, 115014 (2011) \[arXiv:1107.4087 \[hep-ph\]\].

A. A. Slavnov and L. D. Faddeev, Theor. Math. Phys.  <span>**3**</span>, 312 (1970) \[Teor. Mat. Fiz.  <span>**3**</span>, 18 (1970)\]. G. Curci and R. Ferrari, Nuovo Cim.  A <span>**35**</span>, 1 (1976) \[Erratum-ibid.  A <span>**47**</span>, 555 (1978)\].

G. J. Gounaris, R. Kogerler and H. Neufeld, Phys. Rev.  D <span>**34**</span>, 3257 (1986).

Y. P. Yao and C. P. Yuan, Phys. Rev.  D <span>**38**</span>, 2237 (1988).

J. Bagger and C. Schmidt, Phys. Rev.  D <span>**41**</span>, 264 (1990).

H. J. He, Y. P. Kuang and X. y. Li, Phys. Rev. Lett.  <span>**69**</span>, 2619 (1992). H. J. He, Y. P. Kuang and X. y. Li, Phys. Lett.  B <span>**329**</span>, 278 (1994) \[arXiv:hep-ph/9403283\].

H. J. He, Y. P. Kuang and X. y. Li, Phys. Rev.  D <span>**49**</span>, 4842 (1994). A. Denner and S. Dittmaier, Phys. Rev.  D <span>**54**</span>, 4499 (1996) \[arXiv:hep-ph/9603341\].

F. J. Dyson, <span>*Phys. Rev.*</span> <span>**73**</span> (1948) 929. K. M. Case, <span>*Phys. Rev.*</span> <span>**76**</span> (1949) 1. K. M. Case, <span>*Phys. Rev.*</span> <span>**76**</span> (1949) 14. L. L. Foldy, <span>*Phys. Rev.*</span> <span>**84**</span> (1951) 168. R. E. Kallosh and I. V. Tyutin, <span>*Yad. Fiz.*</span> <span>**17**</span> (1973) 190 \[<span>*Sov. J. Nucl. Phys.*</span> <span>**17**</span> (1973) 98\]. S. Kamefuchi, L. O'Raifeartaigh and A. Salam, <span>*Nucl. Phys.*</span> <span>**28**</span>, 529 (1961). I. V. Tyutin, Phys. Atom. Nucl.  <span>**65**</span>, 194 (2002) \[Yad. Fiz.  <span>**65**</span>, 201 (2002)\] \[arXiv:hep-th/0001050\]. <span>A. Blasi, N. Maggiore, S. P. Sorella and L. C. Vilar, <span>*Phys. Rev.*</span> <span>**D59**</span> (1999) 121701 \[arXiv:hep-th/9812040\].</span> <span>Y.-M. P. Lam, <span>*Phys. Rev.*</span> <span>**D7**</span> (1973) 2943–2949.</span> <span>M. C. Bergere and Y.-M. P. Lam, <span>*Phys. Rev.*</span> <span>**D13**</span> (1976) 3247.</span> R. Ferrari, M. Picariello and A. Quadri, JHEP <span>**0204**</span>, 033 (2002) \[arXiv:hep-th/0203200\]. D. Chicherin, V. Gorbenko and V. Vereshagin, arXiv:1104.5681

D. J. Gross and F. Wilczek, Phys. Rev.  D <span>**8**</span>, 3633 (1973). D. J. Gross and F. Wilczek, Phys. Rev. Lett.  <span>**30**</span>, 1343 (1973).

H. D. Politzer, Phys. Rev. Lett.  <span>**30**</span>, 1346 (1973).

R. Ferrari, JHEP <span>**0508**</span>, 048 (2005) \[arXiv:hep-th/0504023\]. R. Ferrari and A. Quadri, Int. J. Theor. Phys.  <span>**45**</span>, 2497 (2006) \[arXiv:hep-th/0506220\].

R. Ferrari and A. Quadri, JHEP <span>**0601**</span>, 003 (2006) \[arXiv:hep-th/0511032\]. D. Bettinelli, R. Ferrari and A. Quadri, Int. J. Theor. Phys.  <span>**46**</span> (2007) 2560 \[arXiv:hep-th/0611063\]. D. Bettinelli, R. Ferrari and A. Quadri, Int. J. Mod. Phys.  A <span>**23**</span>, 211 (2008) \[arXiv:hep-th/0701197\]. D. Bettinelli, R. Ferrari and A. Quadri, JHEP <span>**0703**</span> (2007) 065 \[arXiv:hep-th/0701212\].

D. Bettinelli, R. Ferrari and A. Quadri, Phys. Rev.  D <span>**77**</span> (2008) 045021 \[arXiv:0705.2339 \[hep-th\]\].

D. Bettinelli, R. Ferrari and A. Quadri, Phys. Rev.  D <span>**77**</span> (2008) 105012 \[arXiv:0709.0644 \[hep-th\]\].

D. Bettinelli, R. Ferrari and A. Quadri, Int. J. Mod. Phys.  A <span>**24**</span> (2009) 2639 \[arXiv:0807.3882 \[hep-ph\]\].

D. Bettinelli, R. Ferrari and A. Quadri, Acta Phys. Pol.  B <span>**41**</span>, 597 (2010) \[arXiv:0809.1994 \[hep-th\]\].

D. Bettinelli, R. Ferrari and A. Quadri, Phys. Rev.  D <span>**79**</span>, 125028 (2009) \[arXiv:0903.0281 \[hep-th\]\].

R. Ferrari, J. Math. Phys.  <span>**51**</span>, 032305 (2010) \[arXiv:0907.0426 \[hep-th\]\].

R. Ferrari, "On the Phase Diagram of Massive Yang-Mills," arXiv:1112.2982 \[hep-lat\].

C. Becchi, A. Rouet and R. Stora, Commun. Math. Phys.  <span>**42**</span>, 127 (1975).

G. Curci and R. Ferrari, Nuovo Cim.  A <span>**35**</span>, 273 (1976).

I. A. Batalin and G. A. Vilkovisky, Phys. Lett.  B <span>**102**</span>, 27 (1981). I. A. Batalin and G. A. Vilkovisky, Phys. Rev.  D <span>**28**</span>, 2567 (1983) \[Erratum-ibid.  D <span>**30**</span>, 508 (1984)\].

R. Ferrari and A. Quadri, JHEP <span>**0411**</span>, 019 (2004) \[arXiv:hep-th/0408168\].

G. Curci and R. Ferrari, Nuovo Cim.  A <span>**32**</span>, 151 (1976).

V. D. Barger and R. J. N. Phillips, <span>*"COLLIDER PHYSICS,"*</span> Redwood City, USA: Addison-Wesley (1987) 592 P. (Frontiers in Physics, 71)

G. Altarelli (Ed.) and M. L. Mangano (Ed.), "Standard model physics (and more) at the LHC." Proceedings, Workshop, Geneva, Switzerland, May 25-26, October 14-15, 1999, Pag. 142. CERN-2000-004

K. Nakamura <span>*et al.*</span> \[Particle Data Group\], J. Phys. G <span>**37**</span>, 075021 (2010). Sect. 10.1

R. Ferrari, J. Math. Phys.  <span>**52**</span>, 092303 (2011) arXiv:1008.2779 \[hep-th\].

[^1]: e-mail: <span>vernonthompson@example.com</span>

[^2]: After the work of J. M. Cornwall, D. N. Levin, G. Tiktopoulos, M. S. Chanowitz and M. K. Gaillard the relation between the S-matrix elements for longitudinal modes of the gauge fields and those of the Goldstone bosons has developed to a somewhat more complex result, than the one implied by the theorem on the point transformations of fields in scattering theory @et. The present work adds more consequences to the discovery of the above mentioned physicists. Thus we choose to denote the theorem by their names.

[^3]: The Local Functional Equation @Ferrari:2005ii, employed in the subtraction strategy of the ultraviolet divergences in nonlinear theories, is not used here.

[^4]: Recent lattice simulations @Ferrari:2011ef support this scenario

[^5]: In the nonlinear case we use dimensionless fields $\phi_a$.

[^6]: We drop internal indexes whenever there is no ambiguity. Moreover we use the notation $$\Gamma_{ A^\mu A^\nu}= \Gamma_T (p^2)(g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2})+
    \Gamma_L (p^2) \frac{p_\mu p_\nu}{p^2}.
    \label{ptpf.1}$$

[^7]: In this Section, and in the sequel, ${\tilde M}$ and ${\tilde M}_G$ are fixed by the poles of the transverse and longitudinal tensors of the vector mesons after radiative corrections.

[^8]: $\psi$ is an irreducible set of fields. Throughout the paper we use the notation $$\begin{aligned}
    W_{ A^*_{a\mu}\dots}\equiv  \frac{\delta^n W}{\delta A^*_{a\mu} \dots}
    =  i^{n-1}\langle 0|T((D^\mu [A]c)_a\dots)|0\rangle_C
    \label{app1.9}\end{aligned}$$ for composite fields, while for elementary fields $$\begin{aligned}
    W_{ \underbrace{b_a\dots}_n}\equiv  i^{n-1}\langle 0|T(b_a\dots)|0\rangle_C.
    \label{app1.10}\end{aligned}$$ For the effective action we use a similar short notation $$\begin{aligned}
    \Gamma_X\equiv  \frac{\delta \Gamma}{\delta X}.
    \label{app1.7}\end{aligned}$$

{'source': 'arXiv_out/1106'}