Renormalization of Yang-Mills fields in the light-front without non-local terms
Alfredo T.Suzuki
%EndAName Instituto de F\'{\i}sica Te\'{o}rica-UNESP
Rua Pamplona
145
01405-900 - S\~{a}o Paulo
Brazil.
The study of renormalization of Yang-Mills fields in the light-front gauge has always been a delicate subject in that divergent {\em non-local} terms arise from the calculations of Feynman diagrams. In this short paper we show that this happened because of a deficiency in the gauge fixing procedure that results in an incorrect propagator and propose a cure for it by considering the {\em correct} propagator for the gauge potential. We explicitly show that the use of our correct propagator in the light-front leads to a vacuum polarization tensor at the one-loop level that is free of non-local terms.
hep-th0303129.tex

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\begin{document}

\title{\textbf{Renormalization of Yang-Mills fields in the light-front without non-local terms}}
 \author{Alfredo T.Suzuki  \\
%EndAName
 Instituto de F\'{\i}sica Te\'{o}rica-UNESP\\
Rua Pamplona, 145\\
01405-900 -  S\~{a}o Paulo, Brazil.}
\maketitle

\begin{abstract}
The study of renormalization of Yang-Mills fields in the light-front
gauge has always been a delicate subject in that divergent {\em non-local}
terms arise from the calculations of Feynman diagrams. In this short
paper we show that this happened because of a deficiency in the gauge
fixing procedure that results in an incorrect propagator and propose a
cure for it by considering the {\em correct} propagator for the gauge
potential. We explicitly show that the use of our correct propagator
in the light-front leads to a vacuum polarization tensor at the
one-loop level that is free of non-local terms.
\end{abstract}

\section{Introduction}

In the study of pure Yang-Mills fields in the light-front gauge one
faced the unwieldy emergence of divergent {\em non-local} terms
proportional to $(p\cdot n)^{-1}$ where $p^{\mu}$ is the external four
momentum and $n_\mu$ is the external, light-like $(n^2=0)$ and
constant four vector that defines the gauge choice
\cite{leibbrandt,basseto}. Note that we say $n_\mu$ defines the gauge, instead
of fixes the gauge, since the usual condition $n\cdot A=0$ is not
enough to fix the gauge properly \cite{atsjhos}. 

In the usual renormalization program we require that the counterterm
Lagrangian be of the same functional form as the original one, so that
at each level of perturbation expansion a limited number of infinities
are absorbed into the parameters defining the theory. It is clear,
however, that the {\em non-local} divergent terms that arise in the
computation of loop Feynman diagrams in the light-front gauge do not
satisfy this requirement: No {\em non-local} operator is present in
the original Lagrangian and therefore traditional renormalization
procedure seems to be non applicable. 

Therefore those {\em non-local} terms pose an additional burden in the
renormalization program since one has to deal with them answering the
relevant questions: how and to what extent do they affect
renormalization and are these effects still manageable? Otherwise the
whole renormalization of the theory may be hindered making the
light-front gauge quite limited and perhaps even rendered as
unrenormalizable, forcing us to the conclusion that maybe light-front
gauge is not an useful gauge choice after all.

To this end much research has been made and much effort has been set
forth into the solving of the questions \cite{leibbrandt,basseto}. The
first reference analyses within the framework of BRST invariance and
the latter, taking advantage of the decoupling of the Faddeev-Popov
sector in the light-front works from the Ward-Takahashi
identities. With many propositions and reasonings, it is argued that
the presence of {\em non-local''} divergent terms do not hinder
renormalization since they do not affect the effective Green's
functions. The sum of the matter has been that the Yang-Mills fields
in the light-front gauge is still renormalizable, thanks to the
following {\em ``fortunate''} characteristic properties that happen to
be valid in this gauge: The {\em non-local} term in the external
momenta of the polarization vector {\em ``does not contribute to the
corresponding Green's function thanks to the orthogonality of the free
propagator with respect to the gauge vector''} \cite{basseto}. And
{\em ``Although non-local terms do not contribute to Green functions,
they do generate factors with external $n^\mu$'s and also contribute
to higher-order vertex functions. Fortunately, however, non-local
terms do not generate higher-order gauge independent quantities''}
\cite{leibbrandt}.

However, no matter the many propositions and reasonings, when it comes
to the essence of renormalization, one still needs to define the
proper counterterms. So, based on the virtue of such properties, the
pragmatic approach to renormalization of Yang-Mills fields in the
light-front gauge has been to choose an appropriate {\em non-local}
counterterm at each level in the perturbation expansion via
cumbersome {\em non-local} operators introduced in the bare Lagrangian
density \cite{leibbrandt}.

Our contribution in this paper is to show that none of the {\em ``ad
hoc''} maneuvers to circumvent the {\em non-local} divergent terms in
the renormalization program for non-Abelian fields in the light-front
gauge are in fact needed if we fix the gauge properly, leading to the
correct propagator for the gauge potential. A correct propagator in
the light-front can only be derived if no residual gauge freedom is
left, and $n\cdot A=0$ as stated earlier is a {\em necessary} but {\em
not sufficient} condition for the gauge fixing. The {\em necessary}
and {\em sufficient} condition is reached by imposing $n\cdot
A=\partial \cdot A=0$, where the second condition, $\partial \cdot
A=0$, accounts for the constraint in the $A^-$ component of the vector
potential, which relates to the so-called {\em ``zero-mode
problem''}. This is as fundamental as the condition $n\cdot A=0$ in
the light-front gauge.

The paper is organized as follows: in the next section we rederive the
one-loop gluon polarization tensor using the traditional two-term
propagator and leave the integrals to be evaluated to the very end to
see more clearly the lack of symmetry in the result and make prominent
the presence of a {\em non-local} term. Then in the following section
we calculate the same vacuum polarization tensor, now with a
three-term propagator with the crucial contact term in it and the
final section is for our concluding remarks.

\section{The way it was}

Using the traditional, incorrect propagator, namely, 
\begin{equation}  \label{wrongprop}
G^{\mu\nu}_{ab}(k) = \frac {-i\delta ^{ab}}{k^2}\left \{g^{\mu\nu}-\frac{
k^\mu n^\nu+k^\nu n^\mu}{k\cdot n}\right \}\,,
\end{equation}
where $a, b$ labels non-Abelian gauge group indices, $n_\mu$ is the
light-like vector that defines the gauge and $k$ is the momentum for
the gluon, the vacuum polarization tensor at one-loop level yields
\begin{equation}
\Pi^{\mu\nu}_{ab}(p) = \frac{1}{2}\,g^2\,f^{acd}f^{bcd}\int d^Dq\,\frac{N^{\mu\nu}(p,q,r)}{q^2\,r^2}
\end{equation}
where the overall $1/2$ accounts for the symmetry factor for the
Feynman diagram and $f^{abc}$ is the completely antisymmetric
structure constant of the gauge group. For simplicity, we omit the
mass parameter of the dimensional regularization which is
$(\mu)^{4-D}$. Also, for brevity we have defined $r\equiv q-p$ and the
numerator $N^{\mu\nu}(p,q,r)$ is given by
\begin{eqnarray}
\label{wrongtri}
N^{\mu\nu}(p,q,r)& = &g^{\mu\nu}\left(8p^2-q^2-r^2\right) \nonumber \\
& - & p^\mu p^\nu \left(10-D\right)-2\left(p^{\mu}q^{\nu}+q^{\mu}p^{\nu}-2q^{\mu}q^{\nu}\right)\left(D-2\right) \nonumber \\
& - & 4g^{\mu\nu}\left[(p^2-r^2)\frac{p\cdot n}{q\cdot n}-(p^2-q^2)\frac{p\cdot n}{r\cdot n}\right] \nonumber \\
& + & 2\left(p^\mu q^\nu+q^\mu p^\nu \right)\frac{p\cdot n}{q\cdot n}+2\left(p^\mu r^\nu+r^\mu p^\nu\right)\frac{p\cdot n}{r\cdot n}\nonumber  \\
& - & \frac{(q^\mu n^\nu+n^\mu q^\nu)}{q\cdot n}(2p^2-r^2)-\frac{(r^\mu n^\nu+n^\mu r^\nu)}{r\cdot n}(2p^2-q^2)\nonumber  \\
& + & (p^\mu n^\nu +n^\mu p^\nu) \left[\frac{(3p^2-q^2-3r^2)}{q\cdot n}\right.+\left. \frac{(3p^2-3q^2-r^2)}{r\cdot n}\right]\nonumber \\
& + & 2n^\mu n^\nu \frac{(p^2-q^2)(p^2-r^2)}{q\cdot n \; r\cdot n}.
\end{eqnarray}

It is clear from (\ref{wrongtri}) that the $N^{\mu\nu}$ does not
display the conspicuous symmetry apparent in the one-loop
diagram. Since we expect the result for $N^{\mu\nu}$ to reproduce the
symmetry of the diagram, the above result signals that something is
missing. Worst of all is not the manifest asymmetry of the above
result; it is the presence of the {\em non-local} term corresponding
to the last one in the expression, proportional to $n^\mu n^\nu$. We
shall give hereon a closer, more detailed look at it. Since it has
double, composite light-front singularity $(q\cdot n\;r\cdot n)^{-1}$
in it, first of all we split the denominator by partial fractioning
it, a standard procedure: 
\begin{equation}
2n^\mu n^\nu \frac{(p^2-q^2)(p^2-r^2)}{q\cdot n\;r\cdot n}=-2n^\mu n^\nu \frac{(p^2-q^2)(p^2-r^2)}{p\cdot n}\left[\frac{1}{q\cdot n}-\frac{1}{r\cdot n}\right].
\end{equation}

Now, it is a matter of straightforward evaluation of the 
momentum integrals which can be found tabulated in
\cite{leibbrandt}. This will inevitably lead to the awkward divergent
{\em non-local} term proportional to $n^\mu n^\nu (p^2 p\cdot
n^*)(p\cdot n\:\: n\cdot n^*)^{-1}$, which demands the {\em ``ad
hoc''} input of {\em non-local} divergent counterterms in the bare
Lagrangian, a non-standard procedure in renormalization to say the
least. The $n^{*\mu}$ vector is the light-like vector, dual to the
$n^\mu$, needed to span the entire four-dimensional space-time via
null vectors as basis, often found in the literature normalized to be
such that $n\cdot n^*=1$.

The complete result for the gluon polarization tensor, after some
minor algebraic manipulations such as transforming the vector and
tensor integrals into scalar ones, dropping of genuine tadpole
integrals, etc., and relevant momentum integral evaluations yields
\begin{eqnarray}
\Pi^{\mu\nu}_{ab}&=&\frac{1}{2}g^2\,f^{acd}\,f^{bcd}\left\{\frac{(7D-6)}{(D-1)}(g^{\mu\nu}p^2-p^\mu p^\nu)\right. \nonumber \\
&-&4(p^\mu n^\nu+n^\mu p^\nu)\frac{p\cdot n^*}{n\cdot n^*}\nonumber \\
&+&4(p^\mu n^{*\nu}+n^{*\mu} p^\nu)\frac{p\cdot n}{n\cdot n^*}\nonumber \\
&-&4(n^\mu n^{*\nu}+n^{*\mu} n^\nu)\frac{p^2}{n\cdot n^*}\nonumber \\
&+&\left. 8n^\mu n^\nu \frac{p^2}{p\cdot n}\frac{p\cdot n^*}{n\cdot n^*}\right\}\frac{i\,\pi^2}{(2-D/2)},
\end{eqnarray}
which agrees with the result quoted in \cite{leibbrandt} (see for
example (5.80) of chapter 5 and (7.15) of chapter 7). Note that the
last term in the above result is the conspicuous {\em non-local}
divergent term. Despite all the claims, propositions, remarks and
arguments stating that it is harmless for the renormalization program
because it does not affect the relevant Green's functions, nonetheless
the fact remains that one still needs an adequate {\em non-local}
counterterm in the Lagrangian to render the theory finite and
physically meaningful. This certainly does not satisfy one of the
basic tenets of the standard renormalization procedure.

The attentive reader will recognize in this {\em non-locality} the
remnant of the forgotten constraint on $A^-$ to eliminate the residual
gauge freedom, for
\begin{equation}
A^-=\frac{\partial^{\perp}A^{\perp}}{\partial^+}\Rightarrow
\frac{p^{\perp}A^{\perp}}{p^+}.
\end{equation}

\section{The way it should be} 

We have argued that (\ref{wrongprop}) is not the correct propagator
for the light-front gauge because its derivation is based on an
incomplete fixing of the gauge choice \cite{atsjhos}. The light-front
gauge is defined by the following conditions $n\cdot A=\partial \cdot
A=0$ which can be implemented in the Lagrangian density through a
Lagrange multiplier of the form $(n\cdot A)(\partial \cdot A)$. This
will fix the gauge properly and lead to the correct propagator, given
by
\begin{equation}
G_{ab}^{\mu \nu }(k)=\frac{-i\delta ^{ab}}{k^{2}}\left\{ g^{\mu \nu }-\frac{
k^{\mu }n^{\nu }+k^{\nu }n^{\mu }}{k\cdot n}+\frac{k^{2}n^{\mu }n^{\nu }}{
(k\cdot n)^{2}}\right\} ,  \label{correct}
\end{equation}
where the third term is often referred to as the {\em contact term},
which plays a crucial role in the calculations. With this propagator, we have
\begin{eqnarray}
N^{\mu\nu}(p,q,r) & = & 8g^{\mu\nu}p^2 - (10-D)p^\mu p^\nu \nonumber \\
& - & 2(D-2)(p^\mu q^\nu+q^\mu p^\nu-2q^\mu q^\nu)\nonumber \\
& - & 4g^{\mu\nu}\left[(p^2+q^2-r^2)\frac{p\cdot n}{q\cdot n}-(p^2-q^2+r^2)\frac{p\cdot n}{r\cdot n}\right] \nonumber \\
& + & 2(p^\mu q^\nu+q^\mu p^\nu)\frac{p\cdot n}{q\cdot n}+2(p^\mu r^\nu+r^\mu p^\nu)\frac{p\cdot n}{r\cdot n}\nonumber \\
& + & 3(p^\mu n^\nu+n^\mu p^\nu)\left[\frac{(p^2+q^2-r^2)}{q\cdot n}-\frac{(p^2-q^2+r^2)}{r\cdot n}\right] \nonumber \\
& - & 2(q^\mu n^\nu+n^\mu q^\nu)\frac{(p^2+q^2-r^2)}{q\cdot n}-2(r^\mu n^\nu+n^\mu r^\nu)\frac{(p^2-q^2+r^2)}{r\cdot n} \nonumber \\
& - & 4(p^\mu n^\nu+n^\mu p^\nu)(p\cdot n)\left[\frac{q^2}{(q\cdot n)^2}+\frac{r^2}{(r\cdot n)^2}\right]\nonumber \\
& + & 4 n^\mu n^\nu p^2 \left[ \frac{q^2}{(q\cdot n)^2}+\frac {r^2}{(r\cdot n)^2}\right]\nonumber \\
& + & 2 n^\mu n^\nu p^2 \left[\frac{q^2+r^2}{q\cdot n\;r\cdot n}\right]\nonumber \\
& - & 2 n^\mu n^\nu \frac{(q^4+r^4-q^2r^2)}{q\cdot n\;r\cdot n}
\end{eqnarray}
where the symmetry of the diagram is clearly reproduced here. 

In order to do the actual computations, we take advantage of 
the following symmetry property of the relevant scalar integrand, namely, 
\begin{equation}
\int \frac{d^Dq}{q^2 r^2},
\end{equation}
which is invariant under the intercahnge $(-r \leftrightarrow q)$. Thus
\begin{eqnarray}
N^{\mu\nu}(p,q)& = &8g^{\mu\nu}p^2-(10-D)p^\mu p^\nu \nonumber \\
& - & 2(D-2)(p^\mu q^\nu+q^\mu p^\nu-2q^\mu q^\nu)\nonumber \\
& - & 8g^{\mu\nu}(p^2+q^2-r^2)\frac{p\cdot n}{q\cdot n}+8g^{\mu\nu}(p\cdot n)^2\frac{q^2}{(q\cdot n)^2}\nonumber \\
& + & 4(p^\mu q^\nu+q^\mu p^\nu)\frac{p\cdot n}{q\cdot n}+6(p^\mu n^\nu+n^\mu p^\nu)\frac{(p^2+q^2-r^2)}{q\cdot n}\nonumber \\
& - & 4(q^\mu n^\nu+ n^\mu q^\nu)\frac{(p^2+q^2-r^2)}{q\cdot n}-8(p^\mu n^\nu+n^\mu p^\nu)(p\cdot n)\frac{q^2}{(q\cdot n)^2}\nonumber \\
& + & 8 n^\mu n^\nu p^2 \frac {q^2}{(q\cdot n)^2}+4 n^\mu n^\nu p^2 \frac {q^2}{q\cdot n \; r\cdot n} \nonumber \\
& - & 4 n^\mu n^\nu \frac {q^4}{q\cdot n\;r\cdot n}+2 n^\mu n^\nu \frac {q^2r^2}{q\cdot n\; r\cdot n}.
\end{eqnarray}

Now, again we shall focus our attention on the potentially troublesome
pieces of the above result, that is, the three last terms in the
expression above
\begin{equation}
\label{NL}
{\cal (NL)}^{\mu\nu}=2n^\mu n^\nu \frac{(2p^2 q^2-2q^4+q^2 r^2)}{q\cdot n\:r\cdot n}.
\end{equation}

The last term in (\ref{NL}), proportional to $q^2 r^2$ corresponds to
genuine tadpoles, so we can drop them straight away. The other two,
after splitting of denominators reads:
\begin{eqnarray}
{\cal (NL)}^{\mu\nu} & = & -4n^\mu n^\nu\frac{p^2}{p\cdot n}\frac{(q^2+r^2)}{q\cdot n} \nonumber \\
&& +4n^\mu n^\nu\frac{q^2}{p\cdot n}\frac{(q^2+r^2)}{q\cdot n}.
\end{eqnarray}

Again, terms proportional to $r^2$ corresponds to genuine tadpoles
which we drop straight away. So the relevant {\em non-loal} divergent terms
come from
\begin{equation}
{\cal (NL)}^{\mu\nu}=-4n^\mu n^\nu \frac{(p^2 q^2-q^4)}{p\cdot n\:q\cdot n}.
\end{equation}

The momentum integral proportional to $p^2 q^2$ yields
\begin{equation}
T_1^{\mu\nu} = -8n^\mu n^\nu \frac{p^2}{p\cdot n}\frac{p\cdot n^*}{n\cdot n^*}I_2^{(\tt div)},
\end{equation}
where 
\begin{eqnarray}
I_2^{(\tt div)}& \equiv & {\mbox (\tt divergent\:\:\: part\:\:\: of)} \int \frac{d^Dq}{q^2(q-p)^2}\nonumber \\
& = & \frac{i\pi^2}{(2-D/2)},
\end{eqnarray} 
whereas the one proportional to $q^4$ yields
\begin{eqnarray}
T_2^{\mu\nu} & = & 8n^\mu n^\nu \frac{p^2}{p\cdot n}\frac{p\cdot n^*}{n\cdot n^*}I_2^{(\tt div)}\nonumber \\
& - & 24n^\mu n^\nu \frac {(p\cdot n^*)^2}{(n\cdot n^*)^2}I_2^{(\tt div)},
\end{eqnarray}
so that $T_1^{\mu\nu}$, which is a {\em non-local} divergent
contribution that comes from the second term in the propagator, is
cancelled by the first term of $I_2^{\mu\nu}$, which comes from the
third term in the propagator. Thus, the {\em non-local} divergent term
is exactly cancelled by the contribution that comes from the contact
term in the propagator.

Finally, for completeness we quote the gluon vacuum polarization tensor
\begin{equation}
\label{gvpt}
\Pi^{\mu\nu}_{ab}(p)=\frac{1}{2}g^2 f^{acd}f^{bcd}(C^{\mu\nu}+L^{\mu\nu})I_2^{(\tt div)}
\end{equation}
where
\begin{eqnarray}
C^{\mu\nu} & = & \frac{(7D-6)}{(D-1)}(g^{\mu\nu} p^2-p^\mu p^\nu)=\frac{22}{3}(g^{\mu\nu} p^2-p^\mu p^\nu), \\
L^{\mu\nu} & = & -16g^{\mu\nu}\frac{(p\cdot n\:p\cdot n^*)}{n\cdot n^*}+4(p^\mu n^\nu+n^\mu p^\nu)\frac{p\cdot n^*}{n\cdot n^*} \nonumber \\
&& + 4(p^\mu n^{*\nu}+n^{*\mu} p^\nu)\frac{p\cdot n}{n\cdot n^*}-16n^\mu n^\nu\frac{(p\cdot n^*)^2}{(n\cdot n^*)^2}\nonumber \\
&& + 4(n^\mu n^{*\nu}+n^{*\mu} n^\nu)\frac{(2p\cdot n\;p\cdot n^*-p^2 n\cdot n^*)}{(n\cdot n^*)^2}.
\end{eqnarray}

We can make a special choice for the light-like vectors, namely
$2n^{\mu}=\sqrt{2}(1,0,0,1)$ and $2n^{*\mu}=\sqrt{2}(1,0,0,-1)$, in
such a way to have
\begin{eqnarray}
L^{\mu\nu} & = & -16\,g^{\mu\nu}p^+\,p^-\:+\:4\,(p^\mu n^\nu+n^\mu p^\nu)\,p^- \nonumber \\
&& +\:4\,(p^\mu n^{*\nu}+n^{*\mu} p^\nu)\,p^+\:-\:16\,n^\mu n^\nu (p^-)^2\nonumber \\
&& + 4\,(n^\mu n^{*\nu}+n^{*\mu} n^\nu)\,(p^\perp)^2.
\end{eqnarray}



\section{Conclusions}

We have shown that using the correct propagator for the light-front
gauge the vacuum polarization tensor at the one-loop level for the
Yang-Mills fields are {\em local}. Therefore, the renormalization
program for the theory is enhanced in that there is no need to define
{\em non-local} operators to add into the Lagrangian density as
counterterms. Moreover, this result brings a better feeling for the
users of light-front gauge since one of its oddities -- namely,
renormalization in the presence of non-local terms -- becomes a thing
of the past. Moreover, our result enhances the possibilities of
light-front as a good choice for non-Abelian gauge fields encouraging
those who were reluctant to use it because of its so odd and peculiar
properties that have emerged along the way.

\vspace{.3cm}
\noindent{\bf Acknowledgements:} 

\noindent Work partially supported by CNPq under process 303848/2002-2

\begin{thebibliography}{99}

\bibitem{leibbrandt} G.Leibbrandt, \emph{Noncovariant Gauges:
Quantization of Yang-Mills and Chern-Simons Theory in Axial-type
Gauges}, World Scientific Publishing Co. Pte. Ltd., Singapore
(1994). See especially chapter 7: \emph{Renormalization in the
presence of nonlocal terms} and references therein.

\bibitem{basseto} A.Bassetto, G.Nardelli and R.Soldati, {\em
Yang-Mills Theories in Algebraic Non-covariant Gauges}, World
Scientific Publishing Co. Pte. Ltd., Singapore (1991).

\bibitem{atsjhos} Alfredo T.Suzuki and J.H.O.Sales, eprint nucl-th/0303016.

 
\end{thebibliography}

\end{document}



hep-th0303129.extracted_bib
\begin{thebibliography}{99}

\bibitem{leibbrandt} G.Leibbrandt, \emph{Noncovariant Gauges:
Quantization of Yang-Mills and Chern-Simons Theory in Axial-type
Gauges}, World Scientific Publishing Co. Pte. Ltd., Singapore
(1994). See especially chapter 7: \emph{Renormalization in the
presence of nonlocal terms} and references therein.

\bibitem{basseto} A.Bassetto, G.Nardelli and R.Soldati, {\em
Yang-Mills Theories in Algebraic Non-covariant Gauges}, World
Scientific Publishing Co. Pte. Ltd., Singapore (1991).

\bibitem{atsjhos} Alfredo T.Suzuki and J.H.O.Sales, eprint nucl-th/0303016.

 
\end{thebibliography}