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{\hfill \parbox{6cm}{\begin{center} 
				    UG-FT-81/97
		      \end{center}}} 
	      
\vspace*{1cm}                               
\begin{center}
\large{\bf Constrained differential 
renormalization\footnote{Presented at the XXI International School of
Theoretical Physics ``Recent progress in theory and phenomenology
of fundamental interactions'', Ustro\'n, Poland, September 19-24,
1997.}} 
\vskip .3truein
\centerline {\sc F. del \'Aguila and M. P\'erez-Victoria}
\vspace{0.5cm}
\centerline {\it Dpto. de F\'{\i}sica Te\'orica y del Cosmos,} 
\centerline {\it Universidad de Granada, 18071 Granada, Spain}
\end{center}
\vspace{0.5cm}
\noindent We review the method of differential renormalization, 
paying special attention to a
new constrained version for symmetric theories.
  
\section{Introduction}
Popular regularization and renormalization methods work in momentum 
space. Typically, the divergences which appear in loop integrals at
large internal momenta (ultraviolet divergences) are first 
regulated---i.e., the integrals are modified so that they are finite,
but diverge in the limit of no regulator---and then substracted by
adding the necessary counterterms. For example, in dimensional
regularization~\cite{dimreg} the regulated integrals are defined 
in $n$ dimensions by analytical continuation, 
with $n$ an arbitrary complex number. 
Eventually, the poles appearing at $n=4$ 
are cancelled by appropriate counterterms and a finite (renormalized) 
result is obtained for $n \rightarrow 4$. 
Renormalization without intermediate
regularization is also possible, as in the BPHZ 
method~\cite{BPHZ}, where the
first terms of the Taylor expansion in external momenta of
the integrand are
substracted off before integrating. Although momentum space is more
natural for calculations of scattering amplitudes with fixed external
momenta, nothing prevents us from working in coordinate space and, if
required, perform a Fourier transform at the end. In coordinate space,
ultraviolet divergences correspond to a singular behaviour at short
distances.

In Ref.~\cite{FJL} a method of renormalization in coordinate space 
was proposed: differential renormalization (DR). 
It is based on the observation that primitively divergent
Feynman graphs are well defined in coordinate space for non-coincident
points, but too singular at coincident points to allow for Fourier
transform. In other words, the corresponding expressions are not
well-behaved distributions. The idea of DR is to substitute the
singular expressions by derivatives of well-behaved distributions, in
such a way that the former (`bare') and latter 
(`renormalized') expressions are equal almost
everywhere. These derivatives are prescribed to act formally by
parts in integrals such as Fourier transforms. In this
way, finite Green functions are obtained, without the
need of intermediate regularization. DR acts directly on
bare Feynman graphs and does not introduce explicit counterterms.
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\caption{One-loop diagram contributing to the 
four-point vertex in $\Phi^4$. \label{figphi}}
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The procedure is
best illustrated in terms of one example: the one-loop four-point bubble
graph of massless $\Phi^4$ (Fig.~\ref{figphi}). 
We work in euclidean
space, which leads to simpler functions.
The massless
propagator in position space is $\prop(x-y)=\frac{1}{4\pi^2} 
\frac{1}{(x-y)^2}$
and the vertex, $-\lambda \delta(x_1-x_4) \delta(x_2-x_4) \delta(x_3-x_4)$.
The bare expression for the amputated graph is
\begin{equation}
  \Gamma(x_1,x_2,x_3,x_4)  =  \frac{\lambda^2}{2} \frac{1}{16\pi^4}
  \delta(x_1-x_2)\delta(x_3-x_4) \frac{1}{(x_1-x_3)^4 } + {\mathrm 
  2~perms.}
\label{barephi}
\end{equation}
This involves
the singular function $\frac{1}{x^4}$, which has a logarithmically 
divergent Fourier transform. 
To renormalize it with DR, one must solve a differential 
equation and find $f(x^2)$ such that 
\begin{equation}
  \frac{1}{x^4} = \Box f(x^2)
\label{diffeq} 
\end{equation}
for $x \not = 0$. Actually, one derivative would be enough in
this case,
but one uses the D'Alambertian $\Box=\d_\mu \d_\mu$ to 
preserve manifest euclidean invariance. 
The solution of Eq.~(\ref{diffeq}) is 
\begin{equation}
  f(x^2)= - \frac{1}{4} \frac{\log x^2 M^2}{x^2} ~,
\end{equation}
where $M$ is an arbitrary constant 
with dimensions of mass,  
required for dimensional reasons, and we have omitted a possible
but irrelevant additive constant. Although we shall not
discuss it here, it is worth mentioning that the constant
$M$ plays a central role in DR: the renormalized amplitudes satisfy
renormalization group equations, with $M$ the 
renormalization scale.
The renormalized expression of the singular function reads
\begin{equation}
  \left[ \frac{1}{x^4} \right]^R = - \frac{1}{4} \Box
  \frac{\log x^2 M^2}{x^2} ~.
\label{DRidentity} 
\end{equation}
With the formal integration by parts rule, this is a tempered
distribution which admits a finite Fourier transform:
\begin{eqnarray}
  \int {\mathrm d}^4 \!x \, e^{i p \cdot x} 
  \left[ \frac{1}{x^4} \right]^R & = &
  - p^2  \int {\mathrm d}^4 \!x \, e^{i p \cdot x} 
  (-\frac{1}{4}) \frac{\log x^2 M^2}{x^2}  \nonumber \\
  & = &- \pi^2 \log (\frac{p^2}{\bar{M}^2}) ~,
\label{DRFT}
\end{eqnarray}
where $\bar{M} = 2M/\gamma_E$,  and $\gamma_E=1.781...$ is  Euler's
constant.
Substituting Eq.~(\ref{DRidentity}) in Eq.~(\ref{barephi}), the 
renormalized vertex
graph is obtained:
\begin{eqnarray}
  \Gamma^R(x_1,x_2,x_3,x_4) & = & - \frac{\lambda^2}{128\pi^4} 
  \delta(x_1-x_2)\delta(x_3-x_4) \Box 
  \frac{\log (x_1-x_3)^2 M^2}{(x_1-x_3)^2} \nonumber \\
  && \mbox{} + {\mathrm 2~perms.}
\end{eqnarray}
The renormalized expression in momentum space follows  
from Eq.~(\ref{DRFT}):
\begin{eqnarray}
  \hat{\Gamma}^R(p_1,p_2,p_3,p_4) & = &
  - \frac{\lambda^2}{32\pi^2} \log \left[\frac{(q_1+q_2)^2
  (q_2+q_3)^2 (q_1+q_3)^2}{\bar{M}^6} \right] \nonumber \\
  && \mbox{} \times (2\pi)^4 \delta(\sum_i q_i).
\end{eqnarray}

DR has been successfully applied in different contexts:
the Wess-Zumino model~\cite{WZ}, 
lower-dimensional~\cite{Ramon} and non-abelian
gauge theories~\cite{nonabelian}, two-loop 
QED~\cite{QED}, a chiral model~\cite{chiral}, a 
non-relativistic anyon model~\cite{anyon}, 
curved space-time and finite temperature~\cite{curved}, 
the calculation of $(g-2)_l$ in supergravity~\cite{g2}, 
Chern-Simons theories~\cite{coreanos} and non-perturbative
calculations in supersymmetric gauge 
theories~\cite{nonperturbative}. Other formal aspects of the
method have been developed in 
Refs. \cite{counterterm,systematic,Nuria,massiveDR} and 
different versions of DR can be found in~\cite{Smirnov,Schnetz}.
 
When symmetries are an issue, it is important that
the renormalization program preserves the
corresponding Ward identities. In general, even when
the regularization procedure breaks some relevant symmetry,
one can still recover it by the addition of finite
counterterms\footnote{The
exception is called an anomaly: the
quantum renormalized theory does not have a symmetry
of the classical theory.}. In practical calculations and formal
proofs to all orders, it is nevertheless more convenient
to have a method that directly preserves the
Ward identities. The great success of dimensional
regularization is mainly due to the fact that it automatically
respects gauge invariance. It is known, however, that it has
problems in dimension-dependent theories like chiral
and supersymmetric theories. Its variant
dimensional reduction~\cite{dimred} is usually employed in
these cases, although inconsistencies may arise at high 
orders~\cite{dimredprob}.
DR does not change the space-time
dimension and it was expected to become a renormalization
procedure respecting gauge and chiral
symmetry. In its original version, however, this
is not automatic. The ambiguities inherent
to the manipulation of
singular functions are taken care of by introducing
arbitrary renormalization scales for different diagrams. 
Different choices of the renormalization scales
give rise to different renormalization schemes and 
only a subset of these schemes corresponds to a symmetric
renormalized theory. Hence,
the scales must be fixed to enforce the relevant
Ward identities. A change of renormalization scales is equivalent
to the addition of finite counterterms, so the situation
does not differ much from the one with symmetry-breaking
regulators.

In Refs.~\cite{g2,CDR} a procedure of DR was proposed which
fixes all the manipulations and only introduces the
necessary renormalization group scale. This {\em constrained}
DR has been explicitly shown
to respect the one-loop Ward identities of abelian
gauge symmetry~\cite{CDR}\footnote{In Ref.~\cite{gaugeSmirnov}
Smirnov presented an abelian gauge invariant method within
his version of DR.} and to preserve supersymmetry
in a supergravity calculation~\cite{g2}.

In the following, we first describe the method of 
constrained DR and then use it to
calculate in detail the electron 
self-energy  and the vertex correction in QED, as an illustration. 
We also derive the corresponding momentum space
expressions and
check that the corresponding  Ward identity is 
automatically fulfilled. 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Constrained differential renormalization}
In this section we briefly describe the  constrained procedure of 
DR at one loop introduced in Ref.~\cite{CDR}. The idea is to find a
consistent way of performing the manipulations of singular
expressions carried out in the process of renormalization.
It turns out that a small set of formal rules is sufficient to 
completely fix  the renormalization
scales (except one, associated with the renormalization group
invariance). Furthermore, the resulting renormalized
amplitudes were explicitly shown in Refs.~\cite{CDR,g2}
to satisfy the one-loop Ward identities of abelian gauge symmetry
and to render a vanishing value for the magnetic moment of a 
charged lepton
in supergravity (which is the required value if supersymmetry
is respected~\cite{Ferrara}). 

The set of rules contains the two basic DR rules: the use of DR 
identities like Eq.~(\ref{DRidentity}) (always with the same 
renormalization scale!) ({\em rule 1}) and the formal integration by
parts prescription ({\em rule 2}). In addition, we need another two
rules.
One is ({\em rule 3}): 
\begin{equation}
  [F(x,x_1,...,x_n) \delta(x-y)]^R =  [F(x,x_1,...,x_n)]^R
  \delta(x-y)~,
\end{equation}
where $F$ is an arbitrary function. The other one 
requires
the general validity of the propagator equation
({\em rule 4}):
\begin{equation} 
  F(x,x_1,...,x_n) \Box \prop(x) = 
  F(x,x_1,...,x_n) (- \delta(x)) ~,
  \label{propeq}
\end{equation}
where $\prop(x) = \frac{1}{4\pi^2} \frac{1}{x^2}$ is
the massless propagator (the massive case is
analogous).
This is a valid mathematical identity between tempered
distributions if F is well-behaved enough. 
This rule formally extends its range of applicability 
to an arbitrary function.
The main point of the constrained method is to 
require consistency
of renormalization with these rules. Such requirement 
fixes all the ambiguities in DR at least to one loop.
Let us explain how the rules are used in practice 
with some simple examples. 
The results will be used in the next section. 
We first introduce some convenient notation: we
define the {\em bubble} and {\em triangular} basic functions as
\begin{eqnarray}
  \B[\O] & \equiv & \prop(x) \O \prop(x) ~, \label{B}\\
  \T[\O] & \equiv & \prop(x) \prop(y) \O^x \prop(x-y)~, \label{T} 
\end{eqnarray}
where $\O$ is a differential operator. The significance of
this kind of functions is that any one-loop bubble or triangular
Feynman diagram 
can be expressed in terms of them (and their
derivatives) using only algebraic
manipulations and the Leibnitz rule for derivatives.
Hence, the problem of renormalization reduces (at this
order) to finding the renormalized expressions of
the singular basic functions. 
Note that $\B[\O]$ is singular\footnote{In this paper the term 
`singular' should always be undertood as `too singular to allow for
a Fourier transform'.}
at $x=0$ for any $\O$,
whereas $\T[\O]$ is only singular (at $x=y=0$) when
$\O$ contains two or more derivatives.
These basic functions are easily renormalized using
the set of rules described above. For example,
\begin{eqnarray}
  \B^R[1] & = & \left[ \prop(x) \prop(x) \right]^R \nonumber \\
     & = & \frac{1}{(4\pi^2)^2} 
           \left[ \frac{1}{x^4} \right]^R \nonumber \\
     & \stackrel{\mathrm Rules~1,2}{=} & 
           -\frac{1}{4} \frac{1}{(4\pi^2)^2} 
           \Box \frac{\log x^2 M^2}{x^2}
\label{renB}
\end{eqnarray}
and
\begin{eqnarray}
\T^R[\Box]  & = & \left[ \prop(x) \prop(y) \Box^x \prop(x-y) 
                  \right]^R \nonumber \\
   & \stackrel{\mathrm Rule~4}{=} & 
         - \left[ (\prop(x))^2 \delta(x-y) \right]^R \nonumber \\
   & = & - \left[ \B[1](x) \delta(x-y) \right]^R \nonumber \\
   & \stackrel{\mathrm Rule~3}{=} & 
         - \B^R[1](x) \delta(x-y) \nonumber \\
   & \stackrel{\mathrm Eq.~(\ref{renB})}{=} &  
         \frac{1}{4} \frac{1}{(4\pi^2)^2} \Box
         \frac{\log x^2 M^2}{x^2} \delta(x-y) ~. 
\label{renTbox}
\end{eqnarray}
For basic functions with non-trivial tensor structure
the procedure is more involved and can be found
in Ref.~\cite{CDR}. 
In particular,  we shall need in the following the identity:
\begin{eqnarray}
\T^R[\d_\mu\d_\nu] & = & 
    \T[\d_\mu\d_\nu - \frac{1}{4} 
    \delta_{\mu\nu}\Box] \nonumber \\
    && \mbox{} + (\frac{1}{16} \frac{1}{(4\pi^2)^2} 
    \Box \frac{\log x^2 M^2}{x^2} \delta(x-y) 
    -  \frac{1}{32} \frac{1}{4\pi^2} 
    \delta(x) \delta(y) ) \delta_{\mu\nu}
\label{renTdd}
\end{eqnarray}
where the first term is finite thanks to its tracelessness.
The local term proportional to $\delta(x) \delta(y)$
was not considered in the earlier literature (before Ref.~\cite{CDR},
where Eq.~(\ref{renTdd}) is worked out in detail) 
and comes from imposing consistency with the propagator equation
(rule~4).
Notice that
\begin{equation}
   \delta_{\mu\nu} \T^R[\d_\mu\d_\nu] \not =
   \left[ \delta_{\mu\nu} \T[\d_\mu\d_\nu] \right]^R ~.
\end{equation}
This might seem strange, but in fact also occurs in
other schemes like dimensional regularization
or Pauli-Villars. Differentially renormalized expressions
of basic functions appearing in one-, two- and three-point
one-loop Green functions can be found in the
tables of Ref.~\cite{CDR}.
The treatment of four-point one-loop Green functions
will be presented in Ref.~\cite{SQED}. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Simple applications}
We now show how the method works in practice with
two detailed examples:
the renormalization of the one-loop 
1PI electron self-energy and electron-electron-photon vertex
in massless QED.
The Feynman rules of massless QED in euclidean coordinate
space are gathered in Fig.~\ref{Feynrules}, with
$\{ \g_\mu , \g_\nu \} = 2 \delta_{\mu\nu}$.
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\caption{Feynman rules of massless QED
in euclidean coordinate space. \label{Feynrules}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
We work in the Feynman gauge. The case of an arbitrary Lorentz
gauge was discussed in Ref.~\cite{CDR}.
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\caption{One loop diagrams contributing to the
electron self-energy and the electron-electron-photon vertex
in QED \label{diagrams}}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us first
calculate the electron self-energy, given by the first Feynman
graph in Fig~\ref{diagrams}. The bare expression is
\begin{equation}
  \Sigma(x) = e^2 \g_\alpha \prop(x) \dsl \prop(x) \g_\alpha ~,
\label{sigma}
\end{equation}
where $x=x_1-x_2$. Due to translation invariance,
$\Sigma$ only depends on $x$. Notice that
Eq.~(\ref{sigma}) involves no integration, in contradistinction
with the corresponding expression in momentum 
space\footnote{In
general, coordinate space calculations involve one integral
less than the momentum space ones. If one is interested in
quantities defined at  determined momenta
(like scattering amplitudes), the missing
integral must be done at the end as a Fourier transform
(without regulator!). One example when this Fourier transform is
not needed is the calculation of beta functions.}.  
After some straightforward (four-dimensional) diracology
and the use of Leibnitz rule to extract the derivative, one
obtains
\begin{eqnarray}
  \Sigma(x) & = & - e^2 \dsl (\prop(x))^2 \nonumber \\
            & = & - e^2 \dsl \B[1] ~. 
\end{eqnarray}
The renormalized value is, from Eq.~(\ref{renB}),
\begin{equation}
  \Sigma^R(x) = \frac{1}{64\pi^4} e^2 \dsl \Box
  \frac{\log x^2 M^2}{x^2} ~. 
\label{sigmar}
\end{equation}
Let us now deal with the vertex correction (see
Fig.~\ref{diagrams}). Reading
directly from the Feynman rules,
\begin{equation}
   V_\mu(x,y) = (-ie)^3 \g_\alpha \dsl^x \prop(x) \g_\mu
   (- \dsl^y) \prop(y)
   \g_\alpha \prop(x-y) ~,
\label{v1}
\end{equation}
with $x=x_1-x_3$ and $y=x_2-x_3$.
Simplifying the Dirac algebra and using systematically
the Leibnitz rule to rearrange derivatives,
$V_\mu(x,y)$ can be expressed in terms of triangular basic
functions:
\begin{eqnarray}
V_\mu(x,y) & = & ie^3 \{  -2 \g_b \g_\mu \g_a (
  \d_a^x \d_b^y \T[1] + \d_a^x \T[\d_b]
  - \d_b^y \T[\d_a])  \nonumber \\
  & & \mbox{} -2 \g_\mu \T[\Box] + 
  4 \g_a \T[\d_a\d_\mu] \}~.
\label{v2}
\end{eqnarray}
The renormalized expression is obtained directly from
Eqs. (\ref{renTbox}) and~(\ref{renTdd}):
\begin{eqnarray}
V_\mu^R(x,y) & = & ie^3 \{  -2 \g_b \g_\mu \g_a (
  \d_a^x \d_b^y \T[1] + \d_a^x \T[\d_b] \nonumber \\
  && \mbox{} - \d_b^y \T[\d_a])  
  + 4 \g_a \T[\d_a\d_\mu - 
  \frac{1}{4}\delta_{a\mu} \Box] \nonumber \\
  && \mbox{} - \frac{1}{4}
  \frac{1}{(4\pi^2)^2} \g_\mu \Box \frac{\log x^2 M^2}{x^2}
  \delta(x-y)
  - \frac{1}{8} \frac{1}{4 \pi^2} 
  \g_\mu \delta(x) \delta(y)\} ~.
\label{vr}
\end{eqnarray}
Once the graphs have been renormalized in coordinate
space, one can perform a Fourier transform (without any 
regulator)
to obtain the corresponding finite expressions in 
momentum space. 
We need the Fourier transforms of the basic
functions in Eqs. (\ref{sigmar}) and~(\ref{vr}).
The latter are more involved, so let us see in some
detail how to calculate them.
The Fourier transform of a distribution ``of
two variables'', $f(x,y)$, is
\begin{equation}
  \hat{f}(p,p') = \int {\mathrm d}^4 \! x \, e^{ix\cdot p}
  e^{iy \cdot p'} f(x,y)~.
\end{equation}
With the integration by parts prescription, total
derivatives in $f(x,y)$ yield
\begin{equation}
  \d_\mu^x \rightarrow -i p_\mu ~;~\d_\mu^y \rightarrow 
  -i p_\mu^\prime ~. 
\end{equation}
For the finite triangular functions we have:
\begin{eqnarray}
  \hat{\T}[\O] & = & \int {\mathrm d}^4 \! x {\mathrm d}^4 \! y \, 
  e^{ix\cdot p}
  e^{iy \cdot p'} \prop(x) \prop(y) \O^x \prop(x-y) \nonumber \\
  & = &  \int {\mathrm d}^4 \! x {\mathrm d}^4 \! y \, e^{ix\cdot p}
  e^{iy \cdot p'} \nonumber \\
  && \mbox{} \times \int \frac{{\mathrm d}^4 \! k_1}{(2\pi)^4} 
  \frac{{\mathrm d}^4 \! k_2}{(2\pi)^4} 
  \frac{{\mathrm d}^4 \! k}{(2\pi)^4} \,
  e^{-ix \cdot k_1} e^{-iy \cdot k_2} e^{-i(x-y) \cdot k}
  \frac{\hat{\O}(k)}{k_1^2 \, k_2^2 \, k^2} \nonumber \\
  & = & \int \frac{{\mathrm d}^4 \! k_1}{(2\pi)^4} 
  \frac{{\mathrm d}^4 \! k_2}{(2\pi)^4} 
  \frac{{\mathrm d}^4 \! k}{(2\pi)^4} \,
  \frac{\hat{\O}(k)}{k_1^2 \, k_2^2 \, k^2} \nonumber \\
  && \mbox{} \times (2\pi)^4 \delta(p-k_1-k) (2\pi)^4 \delta(p'-k_2+k) 
  \nonumber \\
  & = & \int \frac{{\mathrm d}^4 \! k}{(2\pi)^4} \,
  \frac{\hat{\O}(k)}{(p-k)^2 (p'+k)^2 k^2} ~,
\label{finiteFT}
\end{eqnarray}
where $\hat{\O}(k)$ is obtained from $\O^x$ by the
replacement $\d^x \rightarrow -ik$. 
The integrals in Eq.~(\ref{finiteFT}) appear 
(with a regularization which is not
present here) in standard one-loop calculations in momentum
space and can be evaluated
with standard techniques. We shall do that later on, in the limit
$p' \rightarrow 0$.
On the other hand, the Fourier transform of the renormalized basic 
function in Eq.~(\ref{vr}) reduces to 
Eq.~(\ref{DRFT}): 
\begin{eqnarray}
  \hat{\T}^R[\Box] & = & 
  \int {\mathrm d}^4 \!x {\mathrm d}^4 \!y
  \, e^{i x \cdot p} e^{i y \cdot p'}
  \frac{1}{64\pi^4} \Box \frac{\log x^2 M^2}{x^2} 
  \delta(x-y)  \nonumber \\
  &=& \int {\mathrm d}^4 \!x \, e^{i x \cdot (p+p')} 
  \frac{1}{64\pi^4} \Box \frac{\log x^2 M^2}{x^2} \nonumber \\
  & = & \frac{1}{16\pi^2} \log (\frac{(p+p')^2}{\bar{M}^2})~.
\end{eqnarray} 
With all these formulae, we get the renormalized 
vertex correction in momentum space: 
\begin{eqnarray}
  \hat{V_\mu}^R(p,p') & = & ie^3 \{-\g_b \g_\mu \g_a (
  -p_a p'_b \hat{\T}[1] - i p_a \hat{\T}[\d_b] + 
  i p_b' \hat{\T}[\d_a])  \nonumber \\
  && \mbox{}  + 4\g_a \hat{\T}[\d_a \d_\mu 
  - \frac{1}{4} \delta_{a\mu}\Box] \nonumber \\
  && \mbox{} - \frac{1}{16\pi^2} \g_\mu 
  ( \log(\frac{(p+p')^2}{\bar{M}^2}) + \frac{1}{2} ) \}~.
\label{momvr}
\end{eqnarray}
The Fourier transform (in one variable) of the self-energy
in Eq.~(\ref{sigmar}) is directly given by Eq.~(\ref{DRFT}):
\begin{equation}
  \hat{\Sigma}^R(p) = -i \frac{e^2}{16\pi^2} \not \! p 
  \log (\frac{p^2}{\bar{M}^2})~.
\label{momsigmar}
\end{equation}


\section{The vertex Ward identity}
Finally, let us verify the Ward identity between the renormalized 
vertex correction and electron self-energy. 
For 1PI Green functions it reads
\begin{equation}
(\d_\mu^x + \d_\mu^y) V_\mu^R(x,y) =
  i e \Sigma^R(x-y) (\delta(x) - \delta(y)) ~,
\label{WI}
\end{equation}
where $\d^{x_3} f(x_1-x_3,x_2-x_3) = - (\d^x + \d^y) f(x,y)$ 
has been used to express it
in the translated variables $x$ and $y$.
At points away from the origin this identity must
hold, since the bare Green functions
have not been modified there. A possible disagreement 
can only arise from the contact terms at $x=y=0$. In fact,
both sides of Eq.~(\ref{WI}) are distributions, and to compare
them one must either use formal properties of delta functions,
etc, or integrate with an arbitrary test function $\phi(x,y)$. 
In particular, one can  perform a Fourier transform
($\phi(x,y)=e^{ix\cdot p}e^{iy\cdot p'}$), which retains all the
information. In other words, we can check the Ward identity in
momentum space, 
\begin{equation}
  -i(p_\mu + p_\mu') \hat{V}_\mu^R(p,p') =
  ie [\hat{\Sigma}^R(-p') - \hat{\Sigma}^R(p)] ~,
\label{momWI}
\end{equation}
using the momentum space renormalized Green functions in 
Eqs. (\ref{momvr}) and~(\ref{momsigmar}).
For simplicity, we consider the
limit $p' \rightarrow 0$ ({\it i.e.\/}, the Fourier transform
in $y$ reduces to an integral without any weight). 
In this limit the relevant integrals
reduce to
\begin{eqnarray}
  \hat{\T}[\d_\alpha] & \stackrel{p'\rightarrow 0}{\rightarrow} &
  \frac{i}{16\pi^2} \frac{p_\alpha}{p^2} ~, \\
  \hat{\T}[\d_\alpha \d_\beta - \frac{1}{4} \delta_{\alpha\beta} \Box]
  & \stackrel{p'\rightarrow 0}{\rightarrow} &
  - \frac{1}{32\pi^2}
  \frac{p_\alpha p_\beta - \frac{1}{4} \delta_{\alpha\beta} p^2}{p^2}~,
\end{eqnarray}
whereas $\hat{\T}[1]$ is  logarithmically infrared divergent and
$p_\alpha' \hat{\T}[1] \stackrel{p'\rightarrow 0}{\rightarrow}0$. 
With these values, we obtain 
for both sides
of the Ward identity in Eq.~(\ref{momWI}), in the limit
$p' \rightarrow 0$, the same result:
\begin{displaymath}
    -\frac{e^3}{16\pi^2} \not \! p 
    \log \frac{p^2}{\bar{M}^2} ~.
\end{displaymath}
Since both sides are equal, 
the Ward identity is indeed satisfied: constrained DR
has preserved it automatically, {\it i.e.\/}, without
any {\it a posteriori\/} adjustment.
  
\section{Conclusions}
DR is a renormalization method which works in 
coordinate space and does not introduce any intermediate
regulator~\cite{FJL}. We have shown how it
can be easily applied to the calculation and
renormalization of one-loop Feynman diagrams. 
In many cases, worked out calculations at higher orders are 
also simpler in DR than in other methods~\cite{FJL,systematic}. 
Some nice features of the method are the following:
\begin{itemize}
\item It is minimal, in the sense that  
the Green functions are never modified except at the singular
points.
\item It does not change the space-time dimension, easing
the diracology and tensor manipulation, and the treatment
of chiral theories~\cite{chiral,WZ,g2}.
\item One integration less has to be performed, unless
one is interested in some quantity defined for fixed
external momenta. In such case, one has to Fourier transform the
renormalized expressions, but without any regularization.
\item Some overlapping divergencies disentangle in
coordinate space.
\item It is better suited for theories which are
naturally defined in coordinate space, like
theories with conformal invariance~\cite{nonabelian}, 
in curved space or at finite 
temperature~\cite{curved}.
\end{itemize}
In the constrained procedure of DR~\cite{CDR} 
all the local terms are fixed. This determines
a renormalization scheme which turns out to be symmetric
in all known examples~\cite{CDR,g2}. Here, we have used constrained DR 
to calculate two one-loop Green functions in QED and 
we have verified that the Ward identity relating 
them  is automatically fulfilled after renormalization. 
We have dealt with massless theories for simplicity. 
The treatment of massive theories in (constrained) DR is
worked out in Ref.~\cite{massiveDR} (Ref.~\cite{CDR}).
\section*{Acknowledgements}
It is a pleasure to thank A. Culatti and R. Mu\~noz Tapia for a 
fruitful collaboration when developing this subject.
This work has been supported by CICYT, contract number AEN96-1672,
and by Junta de Andaluc\'{\i}a, FQM101. MPV thanks MEC for financial
support.

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\end{document}

