# The fate of infrared divergences in a finite formulation of field theory:
QED revisited

Jean-François Mathiot
Université Clermont Auvergne, Laboratoire de Physique de Clermont,
CNRS/IN2P3, BP10448, F-63000 Clermont-Ferrand, France e-mail: jean-
<EMAIL_ADDRESS>
###### Abstract

Within the framework of the recently proposed Taylor-Lagrange regularization
procedure, we reanalyze the calculation of radiative corrections in $QED$ at
next to leading order. Starting from a well defined local bare Lagrangian, the
use of this regularization procedure enables us to manipulate fully finite
elementary amplitudes in the ultra-violet as well as infra-red regimes, in
physical $D=4$ space-time dimensions and for physical massless photons, as
required by gauge invariance. We can thus separately calculate the
electromagnetic form factors of the electron and the cross-section for real
photon emission, each quantity being finite in these physical conditions. We
then discuss the renormalization group equations within this regularization
procedure. Thanks to the taming of infra-red divergencies, the renormalization
group equation associated to the (physical) effective charge exhibits an
ultra-violet stable fixed point at $\alpha^{*}=0$, showing an asymptotic
freedom type behavior. We finally consider the case of two mass scales, one
low and one heavy, paying particular attention to the natural decoupling
properties between heavy and light degrees-of-freedom. As a direct
consequence, the fine structure constant should be zero in the limit of
massless electrons.

##  1 Introduction

Following the recent development of a regularization procedure based on the
nature of quantum fields as operator valued distributions ($OPVD$) - the so-
called Taylor-Lagrange regularization scheme ($TLRS$) [1, 2] \- we shall
consider in this study the case of quantum electrodynamics ($QED$) in the one-
loop, next to leading order ($NLO$), approximation. This regularization
procedure originates from the observation that the divergences of bare
amplitudes can be traced back to the violation of causality due to the ill-
defined product of distributions at the same point [3, 4, 5, 6, 7, 8] (see
also Refs. [9, 10]). Since the Lagrangian we start from is constructed from
the product of fields or derivative of fields at the same point - it is thus
called local \- the calculation of any elementary amplitude must be done with
great care. The correct mathematical treatment for such a case is known since
a long time [11, 13, 12]. More recently, these considerations led to the
construction of $TLRS$. According to this procedure, physical fields are
constructed as $OPVD$, these distributions being applied on test functions
with well defined mathematical properties. Since this scheme is completely
finite - in a sense that will be defined below - it is not plagued by any
arbitrariness due to the way divergences in the ultra-violet ($UV$) as well as
infra-red ($IR$) regimes are cancelled. We can therefore concentrate on the
most important, physical, consequences of the finite renormalization of the
bare amplitudes, as in any interacting many-body system.

The main properties of $TLRS$ can be characterized by the following two
essential features:

  * •

$TLRS$ enables us to give a well defined meaning to the Lagrangian we start
from. This is of course also the case using dimensional regularization ($DR$).
Both regularization procedures are thus called a-priori in contrast to
a-posteriori regularization procedures, like for instance using a naïve cut-
off in momentum space. In this latter case, the regularization is done a
posteriori at the level of each elementary amplitude and not at the level of
the Lagrangian itself.

  * •

The calculation of any elementary amplitude in $TLRS$ is done in physical
conditions, i.e. in four space-time dimensions, with no additional non-
physical degrees of freedom like for instance (infinitely massive) Pauli-
Villars fields, and for massless photons. $TLRS$ is thus called an intrinsic
regularization procedure, in contrast to $DR$ for which elementary amplitudes
still diverge in physical conditions. This last procedure is thus called
extrinsic.

The construction of $TLRS$ enables us to treat at the same time $UV$ as well
as $IR$ singular operators [1, 2]. The $IR$ singularities do occur in
particular when massless degrees of freedom are involved. A textbook example
of such a case is given by one-loop corrections in $QED$. Using $DR$ for
instance, such calculation requires to give, in intermediate steps, a small
non-zero mass to the photon. The subsequent massless limit is then taken at
the very end of the calculation. These $IR$ singularities should be properly
taken care of before any physical consequences can be drawn from the
calculation of a given physical observable. We shall investigate in our study
the physical consequences of using $TLRS$ for the calculation of $NLO$
corrections in $QED$. Although these corrections are by now well known
examples, the use of this regularization procedure enables a completely new
analysis, free from any $UV$ and $IR$ divergences. This implies in particular
that no intermediate renormalization is necessary. In this sense, $TLRS$ is at
the same time a regularization procedure and a renormalization scheme, with
the same acronym. The only renormalization we should worry about is the field
strength renormalization for external, on-shell, particles. Thanks to the lack
of any $IR$ divergences, this renormalization factor is well defined for
massless photons.

The behavior of any elementary amplitude and any physical observable is
governed by two arbitrary scales, as already explained in Ref. [14]:

  * •

the regularization scale denoted by $\eta$. It is inherent to the
regularization procedure which is used to give a mathematical well defined
meaning to the local bare Lagrangian we start from. This scaling variable is
dimensionless in $TLRS$.

  * •

The energy scale $M$ at which an experiment is performed in order to fix the
value of the parameters of the Lagrangian. It is more precisely a set of
scales, like for instance in $\phi^{4}$ theories. There is however only one
scale in the case of $QED$. We call this scale the renormalization point since
it fixes the kinematical condition where the finite (physical) renormalization
of the bare parameters is performed.

These two arbitrary scales are the ones relevant for the calculation of the
running of the two universal coupling constants - the bare and the physical
ones - using the renormalization group ($RG$) equations [14]. The bare
coupling constant depends on the regularization scale $\eta$ only and is
denoted by $\alpha_{0}(\eta)$, while the physical one depends on the
renormalization point $M$ only and is denoted by $\alpha_{M}(M)$111When
$\alpha_{M}(M)$ is not directly accessible from an experiment at any value of
$M$ \- as it is the case for $QED$ \- we can consider equivalently an
effective charge directly related to a physical observable as we shall see in
Sec. 3.1.. These two coupling constants are universal in the sense that they
can be identified independently of the choice of any regularization procedure
or any renormalization scheme. Moreover, they can be defined both in the
perturbative as well as non-perturbative regimes. Note that the calculation of
the physical coupling constant $\alpha_{M}$, at $M\neq 0$, is only made
possible when $IR$ divergences are properly taken care of, as we shall see in
Sec. 3.

We would like to emphasize the very different nature of these two coupling
constants. On the one hand, the bare one - $\alpha_{0}(\eta)$ \- is defined at
the level of the bare Lagrangian, and knows nothing about the renormalization
scheme which will be used, if any, nor about the physical state which is
realized in Nature, like for instance in the presence of spontaneous symmetry
breaking. On the other hand, the physical coupling constant $\alpha_{M}(M)$ is
a definite property of this physical state, and is independent of the
regularization procedure which has been used. The running of these two
coupling constants is therefore governed by two separate $RG$ equations. The
one associated to the $\eta$-dependence of $\alpha_{0}$, called $RGE(\eta)$,
is mass-independent since it is associated to the local character of the
Lagrangian we start from, i.e. to the $UV$ limit of elementary amplitudes in
momentum space. The $RG$ equation associated to the $M$-dependence of
$\alpha_{M}$, called $RGE(M)$, is mass-dependent since the kinematical
condition $M$ is finite and not arbitrarily large.

Once elementary amplitudes have been calculated, like for instance the self-
energy of the electron, the polarization operator of the photon and the
electromagnetic vertex correction, one should consider physical observables.
Apart of course from the physical mass of the electron which is used to fix
its bare mass, or the fine structure constant which is used to fix the bare
coupling constant and its $\eta$-dependence, the first simple non-trivial
observable is the elastic $e^{-}-e^{\pm}$ scattering. As a direct consequence
of the unique properties of $TLRS$ recalled above, we shall see in Sec. 3 \-
in the one-photon-exchange approximation - that this scattering amplitude is
finite in physical conditions i.e. with a massless photon. We shall also check
that the cross-section for soft-photon bremstrahlung is finite in these
conditions. The calculation of the electromagnetic form factors of the
electron - for an arbitrary precision of the experimental apparatus in order
to separate real photon emission from virtual vertex corrections - is thus
made possible for the first time. As we shall see in Sec. 4, this has a non
trivial consequence in the high energy limit. In this limit, $QED$ exhibits an
$UV$ stable fixed point for the effective charge with $\alpha^{*}=0$, showing
an asymptotic freedom type behavior.

The plan of our article is the following. We calculate in Sec. 2 the
elementary amplitudes in $QED$ at $NLO$, and check the Ward identities. The
electromagnetic form factors of the electron together with soft-photon
bremstrahlung are calculated in Sec. 3. We discuss in Sec. 4 the use of the
$RG$ equations as well as the case of two mass scales and the limit of
massless electrons. Our conclusions are drawn in Sec. 5. We recall in A the
main physical properties of $TLRS$, while the calculation of all relevant
integrals is detailed in B.

##  2 Elementary amplitudes

For illustration purposes on how to use $TLRS$ in practice, we recall in this
section the calculation of the elementary amplitudes in $QED$ at $NLO$. For
simplicity, we restrict ourself to the Feynman gauge. The use of different
gauges is discussed in Ref. [15]. All the necessary integrals are detailed in
B.

###  2.1 Self-energy of the electron

The electron self-energy is written, with the appropriate test functions
$f_{\sigma}$ (see A),

$\Sigma(p)=-ie^{2}\lim_{\sigma\to
1^{-}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\mu}(\not\\!p\
-\not\\!k+m)\gamma_{\mu}}{k^{2}[(p-k)^{2}-m^{2}]}f_{\sigma}\left[\frac{k^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{(p-k)^{2}}{m^{2}}\right],$
(1)

where $e$ is the physical charge and $m$ the physical mass of the electron.
The $IR$ singularity at $k=0$ is taken care of by the first test function. We
thus get

$\Sigma(p)=2ie^{2}\left[(\not\\!p-2m)\overline{I}_{0}-\gamma_{\nu}\overline{I}_{1}^{\nu}\right],$
(2)

where the integrals $\overline{I}_{0}$ and $\overline{I}_{1}^{\mu}$ are given
by

$(\overline{I}_{0},\overline{I}_{1}^{\nu})(p)=\lim_{\sigma\to
1^{-}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{(1,k^{\nu})}{k^{2}[(p-k)^{2}+m^{2}]}f_{\sigma}\left[\frac{k^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{(k-p)^{2}}{m^{2}}\right].$
(3)

They are calculated in B. With the most general decomposition

$\Sigma(p)=mA(p^{2})+\not\\!pB(p^{2}),$ (4)

we have

$\displaystyle A(p^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\pi}\int_{0}^{1}dx\
\left[\mbox{Log}\frac{\eta^{2}}{\Delta_{p}}+cte\right],$ (5a) $\displaystyle
B(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{2\pi}\int_{0}^{1}dx(1-x)\
\left[\mbox{Log}\frac{\eta^{2}}{\Delta_{p}}+cte\right],$ (5b)

with

$\Delta_{p}=1-\frac{p^{2}}{m^{2}}(1-x),$ (6)

and $\alpha=\frac{e^{2}}{4\pi}$. In the above equations, and in all this
study, we have indicated by $cte$ a constant term, independent of any
kinematical variable, in order to remind us that the regularization scale
$\eta$ is defined up to a multiplicative constant (see A). Note that the
integrals in Eq. (5) do not involve any test function anymore since $A$ and
$B$ are finite. We recover here the standard result [16].

We shall also need in Sec. 3 the electron field strength renormalization
factor $Z$. This factor is written as [16], at $NLO$,

$Z=1+\left.\frac{d\Sigma}{d\\!\\!\not\\!p}\right|_{\not
p=m}=2m^{2}\left[A^{\prime}(m^{2})+B^{\prime}(m^{2})\right]+B(m^{2}).$ (7)

The calculation of $A^{\prime}$ and $B^{\prime}$ requires some care since both
quantities involve $IR$ singular operators [15]. They are calculated in B. We
thus get

$Z\equiv 1+\delta=1-\frac{\alpha}{4\pi}\left[\mbox{Log}\ \eta^{2}+cte\right].$
(8)

This factor is free from any $IR$ divergences although it is calculated with a
massless photon.

###  2.2 Vacuum polarization of the photon

The calculation of the polarization operator of the photon proceeds similarly.
We have

$\Pi^{\mu\nu}(q)=ie^{2}\lim_{\sigma\to
1^{-}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{Tr[\gamma^{\mu}(\not\\!k+m)\
\gamma^{\nu}(\not\\!k-\not\\!q+m)]}{(k^{2}-m^{2})[(k-q)^{2}-m^{2}]}f_{\sigma}\left[\frac{k^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{(k-q)^{2}}{m^{2}}\right].$
(9)

We thus get

$\Pi^{\mu\nu}(q)=4ie^{2}\left[2\overline{J}_{2}^{\mu\nu}-g^{\mu\nu}\overline{J}_{2}+g^{\mu\nu}\overline{J}_{1}^{\rho}q_{\rho}-q^{\mu}\overline{J}_{1}^{\nu}-q^{\nu}\overline{J}_{1}^{\mu}+m^{2}g^{\mu\nu}\overline{J}_{0}\right].$
(10)

The various integrals entering in Eq. (10) are given by

$(\overline{J}_{0},\overline{J}_{1}^{\mu},\overline{J}_{2},\overline{J}_{2}^{\mu\nu})(q)=\lim_{\sigma\to
1^{-}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{(1,k^{\mu},k^{2},k^{\mu}k^{\nu})}{(k^{2}-m^{2})[(k-q)^{2}-m^{2}]}\\\
\times
f_{\sigma}\left[\frac{k^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{(k-q)^{2}}{m^{2}}\right].$
(11)

They are calculated in B. We have finally

$\Pi^{\mu\nu}(q^{2})=\Pi(q^{2})\left[g^{\mu\nu}q^{2}-q^{\mu}q^{\nu}\right],$
(12)

where

$\Pi(q^{2})=-\frac{2\alpha}{\pi}\int_{0}^{1}dx\
x(1-x)\left[\mbox{Log}\frac{\eta^{2}}{\Delta_{q}}+cte\right],$ (13)

with

$\Delta_{q}=1+\frac{Q^{2}}{m^{2}}x(1-x),$ (14)

and $Q^{2}=-q^{2}$. We can check explicitly here that the photon propagator
remains transverse as required by gauge invariance. It is instructive to
calculate the limiting cases $Q^{2}\ll m^{2}$ and $Q^{2}\gg m^{2}$. We get

$\displaystyle\Pi(Q^{2}\ll m^{2})$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{3\pi}\left[\mbox{Log}\
\eta^{2}-\frac{Q^{2}}{5m^{2}}+cte\right],$ (15a) $\displaystyle\Pi(Q^{2}\gg
m^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{3\pi}\mbox{Log}\frac{Q^{2}}{m^{2}}.$ (15b)

These results will be used in Sec. 3 for the calculation of the
electromagnetic form factors of the electron.

###  2.3 Electromagnetic vertex

For simplicity, we calculate here the electromagnetic vertex for external on-
shell electrons only. It is given, at $NLO$, by

$\Lambda^{\mu}(p,q)=-ie^{2}\bar{u}(p^{\prime})\lim_{\sigma\to
1^{-}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\rho}(\not\\!{p^{\prime}}-\not\\!k+m)\gamma^{\mu}(\not\\!{p}-\not\\!k+m)\gamma_{\rho}}{k^{2}[(p^{\prime}-k)^{2}-m^{2}][(p-k)^{2}-m^{2}]}\\\
\times
f_{\sigma}\left[\frac{k^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{(p-k)^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{p^{\prime}-k)^{2}}{m^{2}}\right]u(p),$
(16)

where $\bar{u}(p^{\prime})$ and $u(p)$ are the Dirac spinors, and
$q=p^{\prime}-p$. As usual, we decompose the electromagnetic vertex into two
parts. The first one, denoted by $\Lambda_{UV}^{\mu}$, is a divergent
contribution in the $UV$ domain in the absence of test functions, while the
second one, denoted by $\Lambda_{IR}^{\mu}$, is convergent in this domain but
has still $IR$ divergences which should be properly taken care of. The first
one depends explicitly on the regularization scale $\eta$ while the second one
does not. We get

$\Lambda^{\mu}_{UV}(p,q)=-ie^{2}\bar{u}(p^{\prime})\lim_{\sigma\to
1^{-}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\rho}\not\\!{k}\gamma^{\mu}\not\\!k\gamma_{\rho}}{k^{2}[(p^{\prime}-k)^{2}-m^{2}][(p-k)^{2}-m^{2}]}\\\
\times
f_{\sigma}\left[\frac{k^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{(p-k)^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{(p^{\prime}-k)^{2}}{m^{2}}\right]u(p).$
(17)

We thus can write

$\Lambda^{\mu}_{UV}(p,q)=2ie^{2}\bar{u}(p^{\prime})\left[2\overline{K}_{2}^{\mu\nu}\gamma_{\nu}-\overline{K}_{2}\gamma^{\mu}\right]u(p),$
(18)

where the integrals $\overline{K}_{2}$ and $\overline{K}_{2}^{\mu\nu}$ are
given by

$(\overline{K}_{2},\overline{K}_{2}^{\mu\nu})(p,q)=\lim_{\sigma\to
1^{-}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{(k^{2},k^{\mu}k^{\nu})}{k^{2}[(p^{\prime}-k)^{2}-m^{2}][(p-k)^{2}-m^{2}]}\\\
\times
f_{\sigma}\left[\frac{k^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{(p-k)^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{(p^{\prime}-k)^{2}}{m^{2}}\right].$
(19)

They are calculated in B. We finally have

$\Lambda^{\mu}_{UV}(p,q)=\Phi_{1}^{UV}(Q^{2})\
\bar{u}(p^{\prime})\gamma^{\mu}u(p),$ (20)

with

$\Phi_{1}^{UV}(Q^{2})=\frac{\alpha}{2\pi}\int_{0}^{1}dx\int_{0}^{1-x}dy\left[\mbox{Log}\frac{\eta^{2}}{\Delta}+cte\right],$
(21)

and $\Delta=(x+y)^{2}+\frac{Q^{2}}{m^{2}}xy$. By a standard change of variable
[16] with $w=x+y$ and $y=w\xi$, we get, after integration over $w$ and with
the change of notation $\xi\to x$,

$\Phi_{1}^{UV}(Q^{2})=\frac{\alpha}{4\pi}\int_{0}^{1}dx\left[\mbox{Log}\frac{\eta^{2}}{\Delta_{q}}+cte\right],$
(22)

with $\Delta_{q}$ given in Eq. (14). In the limits $Q^{2}\ll m^{2}$ and
$Q^{2}\gg m^{2}$ we have

$\displaystyle\Phi_{1}^{UV}(Q^{2}\ll m^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{4\pi}\left[\mbox{Log}\
\eta^{2}-\frac{Q^{2}}{6m^{2}}+cte\right],$ (23a)
$\displaystyle\Phi_{1}^{UV}(Q^{2}\gg m^{2})$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi}\mbox{Log}\frac{Q^{2}}{m^{2}}.$ (23b)

These results will be used in Sec. 3 for the calculation of the
electromagnetic form factors of the electron.

The contribution $\Lambda^{\mu}_{IR}$ is finite in the $UV$ domain but has
still singularities in the $IR$ domain in the absence of test functions, as
well known. We can write, using the on-shell conditions for the external legs,

$\Lambda^{\mu}_{IR}(p,q)=-4ie^{2}\bar{u}(p^{\prime})\left[\left[(p+p^{\prime})^{\mu}\gamma_{\nu}-(p+p^{\prime})_{\nu}\gamma^{\mu}\right]\overline{K}_{1}^{\nu}-m\overline{K}_{1}^{\mu}\right.\\\
+\left.\gamma^{\mu}\left(m^{2}+\frac{Q^{2}}{2}\right)\overline{K}_{0}\right]u(p),$
(24)

where the integrals $\overline{K}_{0}$ and $\overline{K}_{1}^{\lambda}$ are
given by

$(\overline{K}_{0},\overline{K}_{1}^{\lambda})(p,q)=\lim_{\sigma\to
1^{-}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{(1,k^{\lambda})}{k^{2}[(p^{\prime}-k)^{2}-m^{2}][(p-k)^{2}-m^{2}]}\\\
\times
f_{\sigma}\left[\frac{k^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{(p-k)^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{(p^{\prime}-k)^{2}}{m^{2}}\right].$
(25)

They are calculated in B. This gives

$\Lambda^{\mu}_{IR}(p,q)=\bar{u}(p^{\prime})\left[\gamma^{\mu}\Phi_{1}^{IR}+\frac{i}{2m}\sigma^{\mu\nu}q_{\nu}\
\Phi_{2}\right]u(p),$ (26)

with

$\displaystyle\Phi_{1}^{IR}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int_{0}^{1-x}dy\frac{1}{\Delta}\left[\frac{}{}\left[(x+y)^{2}+2(x+y)-2\right]\right.$
(27a)
$\displaystyle+\left.\frac{Q^{2}}{m^{2}}\left[(x+y-xy-1)\right]\right]F_{\sigma},$
$\displaystyle\Phi_{2}(Q^{2})$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{\pi}\int_{0}^{1}dx\int_{0}^{1-x}dy\frac{1}{\Delta}\left[(x+y)(x+y-1)\right].$
(27b)

We have kept in $\Phi_{1}^{IR}$ the relevant test functions, summarized by
$F_{\sigma}$, in order to take care of the $IR$ singularities. We recover here
of course the well known result for $\Phi_{2}(Q^{2})$ since it has no infrared
divergences. The continuum limit $\sigma\to 1^{-}$ can then be taken
immediately in this case. We thus have, using the results of B.4 and with the
same change of variables as above,

$\displaystyle\Phi_{1}^{IR}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{4\pi}\int_{0}^{1}\frac{dx}{\Delta_{q}}\left[5-2\mbox{Log}\Delta_{q}+\frac{Q^{2}}{m^{2}}\left[2-x(1-x)-\mbox{Log}\Delta_{q}\right]\right],$
(28a) $\displaystyle\Phi_{2}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\int_{0}^{1}\frac{dx}{\Delta_{q}}.$ (28b)

In the particular limits of very small or very large momentum transfer, we
have

$\displaystyle\Phi_{1}^{IR}(Q^{2}\ll m^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{4\pi}\left(5+\frac{2Q^{2}}{3m^{2}}\right),$ (29a)
$\displaystyle\Phi_{2}(Q^{2}\ll m^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\left(1-\frac{Q^{2}}{6m^{2}}\right),$ (29b)

and

$\displaystyle\Phi_{1}^{IR}(Q^{2}\gg m^{2})$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi}\left[\mbox{Log}\frac{Q^{2}}{m^{2}}\right]^{2},$
(30a) $\displaystyle\Phi_{2}(Q^{2}\gg m^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\pi}\frac{m^{2}}{Q^{2}}\mbox{Log}\frac{Q^{2}}{m^{2}}.$
(30b)

This completes the calculation of the electromagnetic vertex in $QED$, using
$TLRS$. As expected, all expressions are finite in physical conditions, i.e.
in four space-time dimensions and with a massless photon. Note the
$\left[\mbox{Log}\frac{Q^{2}}{m^{2}}\right]^{2}$ behavior of $\Phi_{1}^{IR}$
in the large $Q^{2}$ limit. We shall come back to this point in the next
Sections.

###  2.4 Ward-Takahashi identity

With our notations, the Ward-Takahashi identity is written as

$\Lambda^{\mu}(p,0)=-\bar{u}(p)\left[\frac{\partial}{\partial
p_{\mu}}\Sigma(p)\right]u(p).$ (31)

From the expression (1) for $\Sigma(p)$ we have

$\displaystyle\bar{u}(p)\left[\frac{\partial}{\partial
p_{\mu}}\Sigma(p)\right]u(p)$ $\displaystyle=$
$\displaystyle-\Lambda^{\mu}(p,0)$
$\displaystyle-4ie^{2}\bar{u}(p)(\gamma^{\mu}p_{\nu}-p^{\mu}\gamma_{\nu})\overline{u}(p)K_{1}^{\nu}(p,p)$
$\displaystyle-ie^{2}\bar{u}(p)\left[\lim_{\sigma\to
1^{-}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\mu}(\not\\!p\
-\not\\!k+m)\gamma_{\mu}}{k^{2}[(p-k)^{2}-m^{2}]}\frac{\partial}{\partial
p_{\mu}}F_{\sigma}\right]u(p),$

with the integral $\overline{K}_{1}^{\nu}$ calculated in B. With the on-shell
condition, the second term in the r.h.s. of Eq. (2.4) is identically zero.
Moreover, the third term in the r.h.s. of this equation is also zero in the
continuum limit since the derivative of the test functions is zero everywhere
except in the asymptotic region where it goes to zero more rapidly than any
power of the momentum as a rapidly decreasing function (see A). This insures
the conservation of the Ward identities at that order. Note that this is only
true in the continuum limit.

##  3 Physical observables

From the above results, it is easy to anticipate that the use of $TLRS$
enables us to calculate, for the first time, physical observables free from
any $IR$ divergences. We shall concentrate in this first study on the
electromagnetic form factors of the electron. Within $TLRS$, these form
factors are unambiguous since they are $IR$ finite and do not depend on any
regularization scale. They can be extracted from a combination of
$e^{-}-e^{-}$ and $e^{-}-e^{+}$ elastic scattering cross-sections. It is
commonly said that $IR$ divergences in the calculation of these cross-sections
are cancelled once soft-photon bremstrahlung - which is not separated out
experimentally below a given energy threshold $\Delta E$ of the photon - is
considered (the well known Bloch-Nordsieck mechanism [17]). Note that,
strictly speaking, these $IR$ divergences in first order perturbation theory
have not disappeared anyhow in this case but they have just been reinterpreted
in terms of $IR$ divergences when $\Delta E$ tends to $0$ in any Gedanken
experiment. This is indeed the price to pay, when using $DR$ for instance, for
not having treated properly these $IR$ divergences from the start, in physical
conditions.

We shall show in Sec. 3.2 how the use of $TLRS$ enables us to calculate the
cross-section for the emission of real massless photons. We first concentrate
on the calculation of the electromagnetic form factors.

###  3.1 The electromagnetic form factors of the electron

The physical amplitude for elastic $e^{-}-e^{-}$ scattering is written as, in
the Feynman gauge and in the one-photon exchange approximation,

${\cal
M}_{ee}=i\frac{e^{2}}{q^{2}}\left[\bar{u}(p^{\prime}_{1})\Gamma_{\mu}u(p_{1})\right]\times\left[\bar{u}(p^{\prime}_{2})\Gamma^{\mu}u(p_{2})\right],$
(33)

with

$\Gamma_{\mu}=\gamma_{\mu}F_{1}(Q^{2})+\frac{i}{2m}\sigma_{\mu\nu}q^{\nu}F_{2}(Q^{2}).$
(34)

The form factor $F_{1}$ is normalized to $F_{1}(Q^{2}=0)=1$ by definition of
the physical electric charge $e$ of the electron. The various contributions to
${\cal M}_{ee}$ at $NLO$ are indicated on Fig. 1 in the one-photon-exchange
approximation.

Figure 1: Elementary $e-e$ amplitude at $NLO$ in the one-photon exchange
approximation. The dots indicate similar contributions with self-energy
corrections on any of the external legs.

All these contributions have been calculated in Sec. 2. We thus get, in terms
of the bare coupling constant $\alpha_{0}$,

$\alpha
F_{1}^{2}(Q^{2})=\alpha_{0}Z^{2}+\alpha\Pi(Q^{2})+2\alpha\Phi_{1}(Q^{2}),$
(35)

with $\Phi_{1}=\Phi_{1}^{UV}+\Phi_{1}^{IR}$, while $F_{2}$ is simply identical
to $\Phi_{2}$ at that order. The value of $\alpha_{0}$, and its
$\eta$-dependence, is then fixed from the calculation of the fine structure
constant at $Q^{2}=0$, with

$\alpha_{0}=\alpha\left[1-\Pi(0)-2\delta-2\Phi_{1}(0)\right],$ (36)

where $\delta$ is defined in Eq. (8). We can then calculate the form factor
$F_{1}$ of the electron, with

$F_{1}(Q^{2})=1+\frac{1}{2}\left[\Pi(Q^{2})-\Pi(0)\right]+\left[\Phi_{1}(Q^{2})-\Phi_{1}(0)\right].$
(37)

While this expression is of course not new and refers to the early days of
$QED$ [18], it is calculated here with a massless photon, as demanded by gauge
invariance, and is free from any $IR$ divergences. As expected $F_{1}(Q^{2})$
is $\eta$-independent, as it should.

It is instructive to calculate $F_{1}(Q^{2})$ in the two limiting kinematical
conditions $Q^{2}\ll m^{2}$ and $Q^{2}\gg m^{2}$. We get immediately, from the
results of Sec. 2,

$\displaystyle F_{1}(Q^{2}\ll m^{2})$ $\displaystyle=$ $\displaystyle
1+\frac{\alpha}{4\pi}\frac{Q^{2}}{m^{2}}\frac{19}{30}+{\cal{O}}(\alpha^{2}),$
(38a) $\displaystyle F_{1}(Q^{2}\gg m^{2})$ $\displaystyle=$ $\displaystyle
1-\frac{\alpha}{4\pi}\left[\mbox{Log}\frac{Q^{2}}{m^{2}}\right]^{2}+{\cal{O}}(\alpha^{2}).$
(38b)

The value of $F_{1}$ in the large momentum region is of particular interest.
While it shows the usual $\left[\mbox{Log}\ Q^{2}\right]^{2}$ behavior, this
contribution is finite although the calculation is done, from the start, with
a massless photon. This is a direct consequence of using $TLRS$ which enables
us to tame both $UV$ and $IR$ divergences in physical conditions.

###  3.2 Soft photon emission

The elementary cross-section for the emission of a single soft photon is well
known [19]. It is given, in first order perturbation theory, by

$d\sigma(p\to p^{\prime}+\gamma)=d\sigma(p\to p^{\prime})I,$ (39)

with

$I=e^{2}\lim_{\sigma\to 1^{-}}\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}2\omega}\left[\frac{2p\makebox[2.27626pt]{$\cdot$}p^{\prime}}{p\makebox[2.27626pt]{$\cdot$}k\
p^{\prime}\makebox[2.27626pt]{$\cdot$}k}-\frac{m^{2}}{(p\makebox[2.27626pt]{$\cdot$}k)^{2}}-\frac{m^{2}}{(p^{\prime}\makebox[2.27626pt]{$\cdot$}k)^{2}}\right]f_{\sigma}\left[\frac{\omega^{2}}{m^{2}}\right],$
(40)

where $\omega=|{\bf k}|$. We have kept explicitly in Eq. (40) the appropriate
test function for the outgoing photon [15]. For large momentum transfer, and
with an upper limit $\Delta E$ for the energy of the outgoing real photon, we
have

$I=\frac{2\alpha}{\pi}\mbox{Log}\frac{Q^{2}}{m^{2}}\lim_{\sigma\to
1^{-}}\int_{0}^{\Delta
E}\frac{d\omega}{\omega}f_{\sigma}\left[\frac{\omega^{2}}{m^{2}}\right]\equiv\frac{2\alpha}{\pi}\mbox{Log}\frac{Q^{2}}{m^{2}}J.$
(41)

With $X=\omega^{2}/m^{2}$, we can write

$J=\frac{1}{2}\lim_{\sigma\to 1^{-}}\int_{0}^{\frac{(\Delta
E)^{2}}{m^{2}}}\frac{dX}{X}f_{\sigma}(X).$ (42)

From the properties of $TLRS$, and by definition of the Pseudo-function (see
A), we get

$J=\frac{1}{2}\int_{0}^{\frac{(\Delta
E)^{2}}{m^{2}}}dXPf\left[\frac{1}{X}\right]=\frac{1}{2}\mbox{Log}\left[\frac{(\Delta
E)^{2}}{m^{2}}\right].$ (43)

The elementary cross-section for the emission of a soft-photon is thus

$d\sigma(p\to p^{\prime}+\gamma)=d\sigma(p\to p^{\prime})\ \frac{\alpha}{\pi}\
\mbox{Log}\frac{Q^{2}}{m^{2}}\ \mbox{Log}\left[\frac{(\Delta
E)^{2}}{m^{2}}\right].$ (44)

This cross-section does not show any $IR$ divergences associated to the zero
mass of the photon anymore. One can thus treat separately virtual corrections
to the electromagnetic form factors of the electron from the emission of soft
real photons. This contribution should further be summed up to all orders in
order to account for the emission of an arbitrary number of real photons [19].
This gives the usual Sudakov-type correction, for large $Q^{2}$, with

$I\to\mbox{exp}[I]=\mbox{exp}\left[-\frac{\alpha}{\pi}\
\mbox{Log}\frac{Q^{2}}{m^{2}}\ \mbox{Log}\left(\frac{m^{2}}{(\Delta
E)^{2}}\right)\right].$ (45)

This correction tends to $0$ when $\Delta E$ gets very small, leaving only the
virtual photon contribution embedded in the electromagnetic form factors of
the electron, independently of the ability of the experimental apparatus in
disantangling the emission of soft real photons from elastic $e-e$ scattering.

###  3.3  Comparison with dimensional regularization

It is particularly interesting to compare our results with those using $DR$
for instance, as far as $IR$ divergences are concerned. In this latter
approach, the only meaningful contribution to consider in order to get an $IR$
finite physical observable is the sum of the $IR$ divergent contributions for
both the virtual vertex correction and the contribution from (non-detected)
soft-photon emission below a given photon energy $\Delta E$. This sum is
simply given [20, 21], for the differential cross-section at large $Q^{2}$,
by:

$d\sigma(p\to p^{\prime})\ \frac{2\alpha}{\pi}\
\mbox{Log}\frac{Q^{2}}{m^{2}}\left[-\mbox{Log}\frac{m}{\lambda}+\mbox{Log}\frac{2\Delta
E}{\lambda}\right],$ (46)

where $\lambda$ is a small finite mass of the photon. In $TLRS$, the only
($IR$-finite) contribution to compare with comes from soft-photon emission, as
calculated in Eq. (44). It is rewarding to check that both contributions (44)
and (46) are identical for large values of $\Delta E/m$. This insures that the
use of $TLRS$ should be compatible, at $NLO$ at least, with all $QED$-related
experimental results already analyzed in the framework of $DR$. In all these
calculations, $\Delta E$ should be fixed from the exact threshold for (non-
detected) soft-photon emission according to the characteristics of each
experimental apparatus. Contrarily to $DR$, the use of $TLRS$ enables us to
unambiguously define the electromagnetic form factors of the electron
independently of any experimental considerations, as expected from general
arguments for a well-defined theoretical framework. As a direct consequence,
we can calculate the effective charge of the electron for an arbitrary value
of the energy scale, as we shall see below.

##  4 The renormalization group equations

###  4.1 Decoupling equation

As already mentioned in the Introduction, we must consider separately two $RG$
equations. The first one, $RGE(\eta)$, is associated to the independence of
physical observables on the dimensionless regularization scale $\eta$. It
concerns the bare parameters and is mass-independent. The second one,
$RGE(M)$, is associated to the independence of physical observables on the
dimensionful renormalization point $M$. It concerns the effective charge
$\alpha_{M}$ and is mass-dependent. These two $RG$ equations can be obtained
simultaneously from the calculation of the physical coupling constant, or
effective charge, in terms of the bare one, and similarly for the bare mass.
By definition of $\alpha_{M}$, we can write

$\alpha_{M}(M)\equiv\alpha
F_{1}^{2}(Q^{2}=M^{2})=\alpha_{0}(\eta)Z(\eta)^{2}+\alpha\Pi(\eta,M^{2})+2\alpha\Phi_{1}(\eta,M^{2}),$
(47)

with $Z$ given by Eq. (7). For completeness, we have indicated in the above
equation the various $\eta$\- and $M$-dependences. From the results of Sec. 2,
we can see that the $\eta$-dependence of $\Pi(\eta,M^{2})$ and
$\Phi_{1}(\eta,M^{2})$ can be easily separated out, with

$\displaystyle\Pi(\eta,M^{2})$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{3\pi}\left(\mbox{Log}\
\eta^{2}+cte\right)+\overline{\Pi}(M^{2}),$ (48a)
$\displaystyle\Phi_{1}(\eta,M^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{4\pi}\left(\mbox{Log}\
\eta^{2}+cte\right)+\overline{\Phi}_{1}^{UV}(M^{2})+\overline{\Phi}_{1}^{IR}(M^{2}),$
(48b)

and

$\displaystyle\overline{\Pi}(M^{2})$ $\displaystyle=$
$\displaystyle\frac{2\alpha}{\pi}\int_{0}^{1}dx\
x(1-x)\mbox{Log}\left[1+\frac{M^{2}}{m^{2}}x(1-x)\right],$ (49a)
$\displaystyle\overline{\Phi}_{1}^{UV}(M^{2})$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi}\int_{0}^{1}dx\
\mbox{Log}\left[1+\frac{M^{2}}{m^{2}}x(1-x)\right].$ (49b)

We have defined in these equations
$\overline{\Phi}_{1}^{IR}=\Phi_{1}^{IR}(M^{2})-\Phi_{1}^{IR}(M^{2}=0)$ with
$\Phi_{1}^{IR}$ given in Eq. (28a). For convenience, we have normalized
$\overline{\Phi}_{1}^{IR}$ to $\overline{\Phi}_{1}^{IR}(M^{2}=0)=0$ by
including $\Phi_{1}^{IR}(M^{2}=0)$ into the constant $cte$ in Eq. (48b). This
simply corresponds to a finite on-mass-shell renormalization condition. We can
thus write Eq. (47) as

$\alpha_{M}(M)-\alpha\overline{\Pi}(M^{2})-2\alpha\overline{\Phi}_{1}^{UV}(M^{2})-2\alpha\overline{\Phi}_{1}^{IR}(M^{2})=\alpha_{0}(\eta)-\frac{\alpha^{2}}{3\pi}\left(\mbox{Log}\
\eta^{2}+cte\right)\equiv\alpha.$ (50)

Thanks to the Ward identity, the $\eta$-dependence of $\alpha_{0}(\eta)$ is
given only by the vacuum polarization of the photon, as well known, while the
energy scale dependence of the physical coupling constant includes in addition
the contribution from the electromagnetic vertex. As shown in Ref.[14], this
decoupling property persists when these radiative corrections are summed up to
all orders.

Note that the common value of Eq. (50) - which should be independent of both
$\eta$ and $M$ \- is just the fine structure constant $\alpha$ since it is by
construction the value of $\alpha_{M}(M=0)$. With the results of Sec. 2, we
thus have in both limits $M\ll m$ and $M\gg m$,

$\displaystyle\alpha_{M}(M\ll m)$ $\displaystyle=$
$\displaystyle\alpha+\frac{\alpha^{2}}{4\pi}\frac{19}{15}\frac{M^{2}}{m^{2}}+{\cal{O}}(\alpha^{3}),$
(51a) $\displaystyle\alpha_{M}(M\gg m)$ $\displaystyle=$
$\displaystyle-\frac{\alpha^{2}}{2\pi}\left(\mbox{Log}\frac{M^{2}}{m^{2}}\right)^{2}+{\cal{O}}(\alpha^{3}).$
(51b)

###  4.2 $\beta$ functions

The decoupling equation (50) is instructive from many points of view.

i) The behavior of $\alpha_{0}$ as a function of $\eta$ should be compared
with the behavior of $\alpha_{R}(\mu)$ in $DR$ in the minimal subtraction
scheme ($MS$) as a function of the unit of mass [22] $\mu$ of $DR$. They both
give the same mass-independent $\beta$ function with, in $TLRS$,

$\beta_{\eta}\equiv\eta\frac{\partial\alpha_{0}(\eta)}{\partial\eta}=\frac{2\alpha_{0}^{2}}{3\pi}+{\cal
O}(\alpha_{0}^{3}).$ (52)

This behavior should not be identified with any physical pattern. It is just
the remnant of the scaling properties associated to the local character of the
Lagrangian we start from, independently of the relevance of this Lagrangian to
describe the physical reality in a given energy domain.

ii) The behavior of the physical coupling constant $\alpha_{M}$ as a function
of $M$ is given by its $\beta$ function which is written as

$\beta_{M}\equiv M\frac{\partial\alpha_{M}(M)}{\partial
M}\equiv\alpha_{M}^{2}b_{M}(M).$ (53)

It involves three different contributions easily calculated from Eqs. (48-50).
The first one is associated to $\overline{\Pi}(M^{2})$ and is equal, in the
limit of large $M$, to $\beta_{\eta}$ as expected. The second one is
associated to $\overline{\Phi}_{1}^{UV}(M^{2})$ and has no equivalence in
$\beta_{\eta}$. The third one is associated to
$\overline{\Phi}_{1}^{IR}(M^{2})$, with also no equivalence in $\beta_{\eta}$.
It is $IR$ finite. To get some insight into these contributions, let us
investigate $\beta_{M}$ in two different limits:

  * •

In the limit of small energy scale, $M\ll m$, or equivalently in the limit of
heavy electron mass, we have

$\beta_{M}(M\ll
m)=\frac{\alpha_{M}^{2}}{2\pi}\frac{19}{15}\frac{M^{2}}{m^{2}}+{\cal
O}(\alpha_{M}^{3}).$ (54)

It goes therefore to zero in the limit of infinitely large electron mass. This
insures the decoupling of very heavy degrees-of-freedom (d.o.f.) from light
ones, as expected from a mass-dependent $RG$ equation.

  * •

In the limit of large momentum scale, $M\gg m$, we get

$\beta_{M}(M\gg
m)=-\frac{2\alpha_{M}^{2}}{\pi}\mbox{Log}\frac{M^{2}}{m^{2}}+{\cal
O}(\alpha_{M}^{3}).$ (55)

Remarkably enough, the $\beta_{M}$ function in this limit is negative and
mass-dependent. It is discussed in more details below.

iii) The above decoupling equation between $\eta$\- and $M$-dependences is
also important in order to understand how the requirement for a perturbative
calculation to remain valid should be understood. The only relevant (physical)
coupling constant is $\alpha_{M}$, expressed in terms of the physical
parameter $M$. This coupling constant should be small compared to $1$ in order
to be able to perform a meaningful perturbative calculation. This constraint,
however, does not imply any constraint on $\eta$ since the behavior of
$\alpha_{0}$ as a function of $\eta$ is decoupled from the behavior of
$\alpha_{M}$ as a function of $M$. In other words, $\eta$ can be chosen in
principle to be very large, with $\alpha_{0}(\eta)$ also very large, while
maintaining a well defined perturbative calculation in terms of $\alpha_{M}$.
From a practical point of view however, $\eta$ should be chosen in order to
avoid large numerical cancellations between $\alpha_{0}(\eta)$ and terms
explicitly dependent on $\eta$, as shown in Eq. (50), order by order in
perturbation theory. This argument translates also to $DR$ with the
identification $\mu=\eta m$ [14]. We emphasize again that this particular
choice of $\eta$, or equivalently of $\mu$, should not lead to any physical
interpretation.

###  4.3 Asymptotic behavior

Figure 2: The effective charge $\alpha_{M}$ divided by the fine structure
constant - or equivalently the square of the $F_{1}$ form factor of the
electron - as a function of $M$.

From integration of the $\beta_{M}$ function of Eq. (53), we immediately get

$\alpha_{M}(M)=\frac{\alpha}{1-\alpha\int_{0}^{M}d\nu\frac{b_{M}(\nu)}{\nu}},$
(56)

where $b_{M}$ is defined in Eq. (53). This effective charge is indicated on
Fig. 2. It shows two immediate and far reaching consequences.

  * •

The physical coupling constant does not show any Landau pole. This is at
variance with the bare coupling constant which, as expected from Eq. (52),
exhibits a Landau pole at a critical value of the regularization scale $\eta$.
As already emphasized in Ref. [14], this Landau pole for $\alpha_{0}(\eta)$
should not have any physical interpretation.

  * •

The physical coupling constant shows an asymptotic freedom type behavior at
very large energies. This is a direct consequence of the mass-dependent
contribution to $\beta_{M}$ at large $M^{2}$ originating from the taming of
$IR$ divergences in $TLRS$ for the calculation of the electromagnetic vertex
function.

The corresponding $\beta_{M}$ function is indicated on Fig. 3. As expected, it
exhibits both an $IR$ stable fixed point at $\alpha_{M}(M=0)=\alpha$, and an
$UV$ stable fixed point at $\alpha_{M}(M\to\infty)=0\equiv\alpha^{*}$. Note
that for the calculation of the physical coupling constant, the limit of high
energy scale is identical to the limit of small electron mass. This implies
immediately that the physical coupling constant at finite $M$ should tend to
zero when the electron mass tends to zero.

Figure 3: The $\beta_{M}$ function as a function of $\alpha_{M}$, showing an
ultra-violet stable fixed point at $\alpha^{*}=0$. The inserted figure shows a
zoom on $\beta_{M}$ for $M\leq 50$ GeV.

###  4.4 The case of two mass scales

To complete our discussion, let us consider the case of two fermionic degrees
of freedom, one with a low mass $m_{<}$, the other one with a high mass
$m_{>}$, with the hierarchy $m_{<}\ll m_{>}$. We shall concentrate for this
discussion on the bare and physical coupling constants.

The running of the bare coupling constant is entirely given by the vacuum
polarization of the photon. It is mass independent, as recalled above. The
only change when considering these two d.o.f. is thus just to multiply the
$\beta_{\eta}$ function by two, without any consideration of threshold effect.
The running of the bare coupling constant just counts the number of (charged)
d.o.f. present in the Lagrangian we start from, independently of their mass.
Since any physical observable is independent of the regularization scale
$\eta$, this change of the $\eta$ dependence of the bare coupling constant has
absolutely no influence on the calculation of physical observables.

We should thus concentrate only on the behavior of the physical coupling
constants as a function of the physical energy scale $M$, or in other words on
the behavior of the electromagnetic form factors for the light or heavy
d.o.f.. We denote by $\alpha_{M}^{<}$ and $\alpha_{M}^{>}$ the physical
coupling constants of the light and heavy d.o.f. respectively. The only new
contribution to consider as compared to the calculation of the electromagnetic
form factor in Sec. 3, with $m=m_{<}$ or $m=m_{>}$, corresponds to the
contribution of the vacuum polarization of the photon. We can identify three
characteristic kinematical conditions:

  * •

$M\ll m_{<}\ll m_{>}$
Since the vacuum polarization of the photon behaves in this limit like
$M^{2}/m^{2}$, both contributions from light or heavy d.o.f. are negligible
and the corresponding form factors are close to 1. This insures that both
coupling constants $\alpha_{M}^{<}$ and $\alpha_{M}^{>}$ are equal to $\alpha$
for $M=0$, as they should.

  * •

$m_{<}\ll m_{>}\ll M$
In this case, the contribution from the vacuum polarization of the photon is
given by

$\frac{\alpha_{M}^{<}}{6\pi}\mbox{Log
}\frac{M^{2}}{m_{<}^{2}}+\frac{\alpha_{M}^{>}}{6\pi}\mbox{Log}\frac{M^{2}}{m_{>}^{2}}.$
(57)

This contribution is however subdominant as compared to the $IR$ contributions
in the large $M$ kinematical region, as detailed in the preceeding subsection.

  * •

$m_{<}\ll M\ll m_{>}$
This case is very similar to the above one, since the contribution from the
heavy d.o.f. behaves in this condition like $M^{2}/m_{>}^{2}$ which is
subdominant.

According to the above discussion, the physical coupling constant
$\alpha_{M}^{<}$ is thus almost identical to the physical coupling constant
with one mass scale only, as discussed in the preceding sections. This is in
complete agreement with the decoupling theorem [23]. On the contrary, the
physical coupling constant $\alpha_{M}^{>}$ is almost equal to $\alpha$ except
in the very far $UV$ domain $M\gg m_{>}$ where it tends to zero like
$\left[\mbox{Log}\frac{M^{2}}{m_{>}^{2}}\right]^{-1}$.

###  4.5 The limit of massless electrons

The discussion of the limit of massless electrons, and the appearance of
associated $IR$ divergences, is usually done in terms of exceptional or non-
exceptional momenta [24]. Momenta are said exceptional if any partial sum of
external momenta is zero. This classification, however, does not make any
reference to whether the amplitude under consideration is a physical one or
not. It should thus be clarified.

Let us first recall the three different objects we have to manipulate in any
calculation of a cross-section. An elementary amplitude is a single diagram
which contributes to this cross-section, as investigated for instance in Sec.
2. Its calculation does not make any a-priori assumption about the external
legs. When these legs correspond to physical, real, particles, they are on
their mass-shell: the corresponding amplitude is thus called a physical
amplitude. Finally, a physical observable corresponds to the sum of all
physical amplitudes contributing to a physical process at a given order of
perturbation theory.

From the above classification, it is clear that the relevant amplitudes to
worry about when calculating a physical process are therefore physical
amplitudes and not elementary ones. This will be our guiding line for
discussing the limit of massless electrons. What happens, however, for
elementary amplitudes with off-shell external legs? This case corresponds to
the calculation of diagrams beyond $NLO$ in perturbation theory. The off-shell
self-energy $\Sigma(p)$ for instance will be attached in this case to an
internal line of a more complex physical amplitude. For this more complex
amplitude, the external legs of $\Sigma(p)$ contribute to internal propagators
with appropriate test functions, according to the use of $TLRS$. These
additional test functions will prevent any new $UV$ as well as $IR$
singularities in such a way that only the final, on-shell, physical observable
is independent of the regularization scale $\eta$ without any new $IR$
singularities. The (apparent) singularities appearing for exceptional momenta
will thus be taken care of in $TLRS$ thanks to the presence of the test
functions in all internal propagators.

According to our discussion in Sec 4, the calculation of the massless limit of
the electron for the different $\beta$ functions is immediate. On the one
hand, the case of $\beta_{\eta}$ is trivial but also particularly instructive.
Since it is mass-independent, its value for a massless electron is given by
Eq. (52). It is therefore non-zero, with the bare coupling constant given by
(50). This would not be the case, however, if the regularization scale $\eta$
would have been dimensionful, as it is the case in the standard formulation of
$DR$ for instance. In this scheme indeed, the regularization scale is
identified with the unit of mass [22] $\mu$ of $DR$. In absence of any other
mass scale at the level of the $QED$ Lagrangian, the dimensionless
renormalized coupling constant in the $MS$ scheme cannot depend on a single
dimensionful variable only. It should therefore be a constant independent of
$\mu$. This would imply a zero $\beta$ function for $QED$ at $NLO$, in
contradiction with the mass-independence of the (non-zero) $\beta$ function in
$DR+MS$.

On the other hand, the case of $\beta_{M}$ is not trivial. Since $\alpha_{M}$
depends on the dimensionful variable $M$ through the ratio $M^{2}/m^{2}$, it
should be independent of $M$ in the limit of massless electrons for obvious
dimensional arguments. Since this limit is also equivalent to the large $M$
limit, this implies that, by construction, one should have an $UV$ stable
fixed point at a given value $\alpha^{*}$ of the physical coupling constant.
This is precisely what we get from the analysis of the $\beta_{M}$ function in
Sec. 4.3, with $\alpha^{*}=0$. In the limit of massless electrons, the
physical coupling constant at finite $M$ should therefore also tend to zero.
Note that this is only true in the absence of any other mass scale in the
physical world, i.e. in absence of spontaneous symmetry breaking. This is not
true for instance for quantum chromodynamics ($QCD$).

##  5 Conclusions

We have reanalyzed in this study first order radiative corrections to $QED$ in
the light of the recently proposed regularization procedure called $TLRS$.
While these corrections are by now standard textbook exercises, the use of
$TLRS$ enables a completely new insight into our understanding of quantum
corrections: it enables a direct calculation of physical observables in
physical conditions starting from the bare Lagrangian itself, without
encountering any $UV$ nor $IR$ divergences. The only renormalization we should
worry about is the finite, physical, renormalization needed to calculate
physical parameters in terms of bare ones. These features refer to the unique
properties of $TLRS$: as an a-priori regularization procedure it enables us to
give a mathematically well defined meaning to the local Lagrangian we start
from, and as an intrinsic regularization procedure, all intermediate
calculations are done in physical conditions, i.e. in four space-time
dimensions with massless photons, as required by gauge invariance.

The analysis of any physical observable should be done in terms of two
different sets of parameters: the bare coupling constants and bare masses
defined at the level of the bare Lagrangian, and the physical ones - physical
coupling constants and physical masses - which are measured experimentally and
which are used to fix the value of the bare parameters. These two sets of
parameters do depend on two different running variables. On the one hand, the
bare coupling constant depends on the regularization scale $\eta$. This scale
is inherent to the local character of the Lagrangian we start from, which is
constructed from the product of fields or derivative of fields at the same
space-time point. It is therefore associated to the scaling invariance of the
$UV$ limit since for any internal momentum $k\to\infty$, we also have
$\eta~{}k\to\infty$, for any finite scaling variable $\eta$. This
regularization scale is therefore dimensionless. This is at variance with the
usual dimensionful unit of mass of $DR$ for instance. Note that using $DR$ we
can also identify a corresponding dimensionless variable, as explained in Refs
[14, 15]. On the other hand, the physical coupling constant depends on the
kinematical conditions chosen to measure it. This is the so-called
renormalization point $M$. By construction, $M$ is dimensionful.

The relationship between these two coupling constants, $\alpha_{0}$ and
$\alpha_{M}$, is governed by a decoupling equation, as given by Eq. (50). This
equation can also be used to understand how the decoupling between heavy and
light d.o.f. is at work. From its mass-dependence, the physical coupling
constant exhibits explicitly the decoupling between these d.o.f., as well
known. This behavior however is in complete agreement with the mass-
independence of the $\eta$-dependence of $\alpha_{0}$, as shown by this
decoupling equation. This is indeed expected from general physical arguments
since $\eta$ is associated to the scaling properties of elementary amplitudes
in the $UV$ regime. These ones are therefore the same for any d.o.f. of finite
mass, light or heavy. It is just associated to the local character of the
Lagrangian we start from. As a direct consequence, the running of the bare
coupling constant as a function of $\eta$ should thus include all (charged)
physical d.o.f. present in the Lagrangian we start from. This should also be
the case for the renormalized coupling constant in $DR$: the running of the
renormalized coupling constant in $DR+MS$ should include all (charged)
physical d.o.f. present in the Lagrangian of the Standard Model, with no
threshold effects depending on the energy scale under consideration.

Physical observables, when calculated in $TLRS$, are free from any $IR$
divergences. This is the case for the electromagnetic form factors of the
electron which can be calculated directly with massless photons. The $F_{1}$
form factor is then independent of the characteristics of the detector and of
its ability to discriminate a single electron from an electron-photon state.
This unique property of $TLRS$ opens the way for a direct unambiguous
measurement of this form factor and at the same time could provide a direct
non-trivial test of the validity of $TLRS$. The calculation of this form
factor may also have important consequences for precision experiments
involving $QED$, like for instance the calculation of the charge radius of the
proton from electron scattering experiments [25]. It also immediately implies
the presence of an $UV$ stable fixed point for the physical coupling constant
at $\alpha^{*}=0$. Note that the $UV$ behavior of the physical coupling
constant is entirely dominated by the behavior of the vertex function in the
$IR$ domain, and is thus $UV$ complete.

This finding has remarkable consequences. It implies both the absence of a
Landau pole for the physical coupling constant, as well as an asymptotic
freedom type behavior. This $UV$ fixed point at $\alpha^{*}=0$ is also
required by dimensional analysis since the limit of very high energy scale is
identical to the limit of zero electron mass. In this limit, the physical
coupling constant - which is dimensionless - should be independent of any
energy scale, hence its $\beta$ function should be zero. This implies
therefore from Eq. (55) that $\alpha_{M}$ should be $0$ in this limit, in
first order perturbation theory. This may explain why the fine structure
constant is small, since the electron mass is also small. The physical
coupling constant $\alpha_{M}$ at very large energy scale $M$ thus shows an
asymptotic freedom type behavior, similarly to the effective coupling constant
of $QCD$ extracted for instance from the Bjorken sum rule [26]. This behavior
should completely change our understanding of a possible unification of the
physical coupling constants at very high energy [27].

## Acknowledgement

Preprint of an article published in Int. J. Mod. Phys. A, 2250204 (2022),
doi:10.1142/S0217751X22502049 © World Scientific Publishing Company.

##  Appendix A Main properties of $TLRS$

The general mathematical properties of $TLRS$ have been detailed in Refs. [2]
and [15]. Several applications have already been considered: application to
light-front dynamics [28], interpretation of the fine-tuning of the Higgs mass
in the Standard Model [29], the recovery of the axial anomaly [30], the fate
of the trace anomaly [31] or the study of conformal field theories in two-
dimensions [32]. For completness, we shall briefly recall in this Appendix the
main properties of $TLRS$.

As already mentioned in the Introduction, this procedure originates from the
well known observation that the divergencies of bare amplitudes can be traced
back to the violation of causality due to ill-defined products of
distributions at the same point [3, 4]. It requires therefore the whole
apparatus of the theory of distribution [33] to correctly define any local
Lagrangian. We consider here for simplicity the case of a scalar field.

As detailed in Ref. [2], the physical field $\varphi$ is constructed in $TLRS$
as a functional of the original quantum field $\phi(x)$, considered as a
distribution, according to

$\varphi[\rho](x)\equiv\int d^{4}y\phi(y)\rho(x-y),$ (58)

where the reflection symmetric test function $\rho$ belongs to the Schwartz
space $\mathscr{S}$ of rapidly decreasing functions [33]. The physical
interest to use the test function $\rho$ is to smear out the original
distribution in a space-time domain of typical extension $a$. The test
function can thus be characterized by $\rho_{a}(x)$ and the physical field by
$\varphi_{a}(x)\equiv\varphi[\rho_{a}](x)$.

For practical calculations, it is convenient to construct the physical fields
in momentum space. If we denote by $f_{\sigma}$ the Fourier transform of the
test function $\rho_{a}$, we can write ${\varphi_{a}}$ in terms of creation
and annihilation operators, leading to [2]

$\varphi_{a}(x)\\!=\\!\\!\int\\!\frac{d^{3}{\bf
p}}{(2\pi)^{3}}\frac{f_{\sigma}(\varepsilon_{p}^{2},{\bf
p}^{2})}{2\varepsilon_{p}}\left[a^{\dagger}_{\bf p}e^{i{p.x}}+a_{\bf
p}e^{-i{p.x}}\right],$ (59)

with $\varepsilon^{2}_{p}={\bf p}^{2}+m^{2}$. Each propagator being the
contraction of two fields is proportional to $f_{\sigma}^{2}$. This test
function in momentum space is a dimensionless quantity. It should therefore be
expressed in terms of dimensionless arguments. To do that, one shall introduce
an arbitrary scale $M_{0}$ to ”measure” all momenta. In practical
calculations, $M_{0}$ can be any of the non-zero physical mass of the theory
under consideration 222For purely massless theories, $M_{0}$ corresponds for
instance to the scale fixed to measure any non-zero momentum.. It is taken
equal to the electron mass $m$ in this study. Note that a change in the value
of $M_{0}$ is just equivalent to a redefinition of $\eta$, without any
consequences on physical observables which should anyhow be independent of
$\eta$ and $M_{0}$.

As shown in Ref. [2], it is appropriate to choose $f_{\sigma}$ as a partition
of unity. A simple example of such function, with $f_{\sigma}$ constructed
from the sum of two elementary functions only, is discussed in Ref. [28]. It
is equal to $1$ almost everywhere and is $0$ outside a finite domain of
$\mathbb{R}^{+4}$, along with all its derivatives (super-regular function).
The parameter $\sigma$, chosen for convenience between 0 and 1, controls the
lower and upper limits of the support of $f_{\sigma}$. Note that for any
partition of unity, the product of two partitions of unity is also a partition
of unity. We shall therefore identify $f_{\sigma}^{2}$ by $f_{\sigma}$ when
needed. As we shall see in B, we do not need to know the precise form of the
test function as a partition of unity, we just rely on its asymptotic
properties. Note that the construction of the test function as a partition of
unity is essential in order to relate its $IR$ and $UV$ properties.

Requiring locality for the bare Lagrangian we start from implies considering
the subsequent limit $a\to 0$, dubbed the continuum limit. In this process, it
is essential to preserve the scaling properties

$\rho_{a}(x)\to^{{}^{\\!\\!\\!\\!\\!\\!\\!\\!\\!a\to 0}}\rho_{\eta}(x)\ ;\
\varphi_{a}(x)\to^{{}^{\\!\\!\\!\\!\\!\\!\\!\\!\\!a\to 0}}\varphi_{\eta}(x),$
(60)

where $\eta$ is an arbitrary, dimensionless, scaling variable since in the
limit $a\to 0$, we also have $a/\eta\to 0$, for any finite $\eta$. This
arbitrary scale just governs the “spead” at which the continuum limit is
reached. Any physical observable should of course be independent of this
dimensionless scaling variable, also called regularization scale in order to
stick to the common denomination, although this denomination may be misleading
when using $TLRS$ since this regularization scale is dimensionless in this
case. From the choice of parametrization of $f_{\sigma}$, the continuum limit
corresponds to $\sigma\to 1^{-}$.

As we shall see in the next Appendix, any amplitude associated to a singular
operator $T(X)$ is written schematically as

${\cal A}_{\sigma}=\int_{0}^{\infty}dX\ T(X)\ f_{\sigma}(X).$ (61)

We consider here for simplicity a one-loop amplitude, with a single
dimensionless variable $X$. It is easy to check that a naïve implementation of
the continuum limit for the test function, with a constant boundary condition
like for instance $X\leq H_{\sigma}=1/(1-\sigma)$, will result in a divergent
amplitude $\lim_{\sigma\to 1^{-}}A_{\sigma}$, as expected from the calculation
of the amplitude in terms of a cut-off in momentum space. However, from the
scaling properties (60), we should get

$A_{\eta}\equiv\lim_{\sigma\to 1^{-}}A_{\sigma}.$ (62)

To achieve this, we should rather consider a “running” boundary condition on
$f_{\sigma}$ defined by

$f_{\sigma}(X\geq H_{\sigma}(X))=0,$ (63)

with

$H_{\sigma}(X)\equiv\eta^{2}Xg_{\sigma}(X)+(\sigma-1),$ (64)

where $\eta$ is the dimensionless regularization scale in $TLRS$ 333The square
of $\eta$ in (64) is just for convenience since $X$ is usually identified with
the square of a momentum, as shown in B. with $\eta^{2}>1$. The function
$g_{\sigma}(X)$ is constructed in such a way that the boundary on $X$, as
defined by Eq. (63), is finite and tends to $1^{-}$ when $\sigma\to 1^{-}$. A
typical (but not unique) simple form for $g_{\sigma}(X)$ is given by

$g_{\sigma}(X)=X^{\sigma-1}.$ (65)

The conditions (63) and (64) amount to an infinitesimal drop-off of the test
function in the $UV$ region, with the drop-off rate governed by the
regularization scale $\eta$. It also preserves the super-regular properties of
the test function in the continuum limit, with the test function and all its
derivatives being zero at infinity.

Remarkably enough, this boundary condition defines at the same time the $UV$
and $IR$ boundaries once $f$ is constructed from a partition of unity [28].
The explicit calculation of standard one-loop integrals using $TLRS$ is thus
straightforward, as recalled in B. It relies on the identification of the test
function $f_{\sigma}$ with its Taylor remainder in the $UV$ as well as $IR$
domains - thanks to its asymptotic properties - and the subsequent use of the
Lagrange formula, hence the name Taylor-Lagrange regularization scheme [2].
The calculation of elementary amplitudes in the $UV$ and $IR$ domains is thus
immediate. In the $UV$ domain the continuum limit (62) is taken after
integration by part. The extension of $IR$ singular operators [2, 15] involves
the Pseudo-function [33, 32], denoted by $Pf$, of $1/X^{n}$, with $n\geq 1$.
This gives, for $n>1$,

$\int_{0}^{X_{0}}dX\mbox{Pf}\left[\frac{1}{X^{n}}\right]=\lim_{\epsilon\to
0}\left[\int_{\epsilon}^{X_{0}}\frac{dX}{X^{n}}-\frac{1}{1-n}\frac{1}{\epsilon^{n-1}}\right],$
(66)

and for $n=1$

$\int_{0}^{X_{0}}dX\mbox{Pf}\left[\frac{1}{X}\right]=\lim_{\epsilon\to
0}\left[\int_{\lambda\epsilon}^{X_{0}}\frac{dX}{X}+\mbox{Log}(\epsilon)\right],$
(67)

where $\lambda$ is an arbitrary scale variable [33]. The value of $\lambda$ is
fixed from the choice of gauge [15]. In the Feynman gauge we have $\lambda=1$.

##  Appendix B Relevant integrals

For completness, and as an illustration of the use of $TLRS$ in practical
calculations, we detail in this Appendix all the relevant integrals needed in
our study.

###  B.1 Self-energy of the electron

####  B.1.1 Calculation of $\overline{I}_{0}$

We recall here the various steps of the calculation of this simple integral.
More details can be found in the Appendix of Ref. [30]. We calculate the
relevant integrals in Euclidian space, using Feynman representation. The
integral $\overline{I}_{0}$ is written

$\overline{I}_{0}(p)=\frac{i}{(2\pi)^{4}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int d^{4}{\bf K}\frac{1}{[{\bf
K}^{2}+m^{2}x\Delta_{p}]^{2}}F_{\sigma},$

where $F_{\sigma}$ is a simplified notation for the product of the two test
functions, with

$F_{\sigma}=f_{\sigma}\left[\frac{({\bf
K}+xp)^{2}}{M_{0}^{2}}\right]f_{\sigma}\left[\frac{({\bf
K}-(1-x)p)^{2}}{M_{0}^{2}}\right].$

For a non zero electron mass, it is convenient to choose $M_{0}\equiv m$. In
the absence of test functions, this integral is divergent in the $UV$ regime
only. We can thus safely concentrate on the behavior of the test functions for
large ${\bf K^{2}}$. The use of test functions when $IR$ singular operators
are involved is detailed in B.4. In the $UV$ domain, the arguments of the two
test functions are both equivalent to ${\bf K}^{2}/m^{2}\equiv\Delta_{p}X$
with $\Delta_{p}\neq 0$. We extract here from the running variable $X$ the
scale $\Delta_{p}$ which depends on the kinematical conditions. This insures
that the integrand $X/(X+x)^{2}$ is independent of any momentum-dependent
scale so that the scaling variable $\eta$ is also (implicitely) independent of
any momentum-dependent scale. We thus have, with the identification
$f_{\sigma}^{2}\sim f_{\sigma}$ valid for a partition of unity,

$\overline{I}_{0}(p)=\frac{i}{(4\pi)^{2}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int_{0}^{\infty}dX\frac{X}{(X+x)^{2}}f_{\sigma}\left[\Delta_{p}X\right].$

From the properties of the test function [2], we can write a Lagrange formula
for $f_{\sigma}$, at fixed support [30], with

$f_{\sigma}\left[\Delta_{p}X\right]=-X\int_{\Delta_{p}}^{\infty}\frac{dt}{t}\frac{\partial}{\partial
X}f_{\sigma}\left[Xt\right].$

We can then write $\overline{I}_{0}$ in the following form, after integration
by part,

$\overline{I}_{0}(p)=\frac{i}{(4\pi)^{2}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int_{0}^{\infty}dX\frac{\partial}{\partial
X}\left[\frac{X^{2}}{(X+x)^{2}}\right]\int_{\Delta_{p}}^{\infty}\frac{dt}{t}f_{\sigma}[Xt].$

From the boundary condition (63), the argument of $f_{\sigma}$ under the
integral is bounded from above by the support of the test function given by
$H_{\sigma}(X)$, so that

$Xt\leq\eta^{2}Xg_{\sigma}(X)\ \ \mbox{and}\ \ \Delta_{p}\leq
t\leq\eta^{2}g_{\sigma}(X).$

Since the integral over $X$ is now finite thanks to the derivative, we can
safely take the continuum limit $\sigma\to 1^{-}$ which gives $g_{\sigma}\to
1$ and $f_{\sigma}\to 1$. We finally get

$\overline{I}_{0}(p)=\frac{i}{(4\pi)^{2}}\int_{0}^{1}dx\
\mbox{Log}\frac{\eta^{2}}{\Delta_{p}}.$

A different choice for $M_{0}$ will just induce a multiplicative factor at
$\eta^{2}$. This shall induce a finite additive constant on top of any
contribution in $\mbox{Log}\ \eta^{2}$, as indicated in the final results for
the elementary amplitudes calculated in Sec. (2), with no consequences for any
physical observables. This is reminiscent of the flexibility in choosing the
unit of mass $\mu$ of $DR$ [22], like for instance using either the $MS$ or
the $\overline{MS}$ schemes.

####  B.1.2 Calculation of $\overline{I}_{1}^{\mu}$

The integral $\overline{I}_{1}^{\mu}$ is written as

$\overline{I}_{1}^{\mu}(p)=I_{1}^{\mu}(p)+{\bf
p}^{\mu}\frac{i}{(2\pi)^{4}}\lim_{\sigma\to 1^{-}}\int_{0}^{1}dx\int d^{4}{\bf
K}\frac{x}{[{\bf K}^{2}+m^{2}x\Delta_{p}]^{2}}F_{\sigma},$

with

$I_{1}^{\mu}(p)=\frac{i}{(2\pi)^{4}}\lim_{\sigma\to 1^{-}}\int_{0}^{1}dx\int
d^{4}{\bf K}\frac{{\bf K}^{\mu}}{[{\bf
K}^{2}+m^{2}x\Delta_{p}]^{2}}F_{\sigma},$

The term in ${\bf p}^{\mu}$ is calculated similarly to $\overline{I}_{0}$. The
calculation of $I_{1}^{\mu}$ should be done with care since the test functions
do depend on all the relevant momenta of the system. Following the
calculations of Ref. [30], we start from the identity

$\frac{\partial}{\partial{\bf K}_{\mu}}\frac{1}{{\bf
K}^{2}+m^{2}x\Delta_{p}}=-2\frac{{\bf K}^{\mu}}{({\bf
K}^{2}+m^{2}x\Delta_{p})^{2}}.$

We can thus write immediately

$I_{1}^{\mu}=-\frac{i}{2(2\pi)^{4}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int\frac{d^{4}{\bf
K}}{(2\pi)^{4}}\frac{\partial}{\partial{\bf K}_{\mu}}\left[\frac{1}{{\bf
K}^{2}+m^{2}x\Delta_{p}}\right]F_{\sigma}.$

By integration by part, the surface term is a 3-dimensional integral
orthogonal to the $\mu$-direction. It should be taken at ${\bf
K}_{\mu}\to\pm\infty$. Thanks to the presence of the test functions, this term
is identically zero. The remaining integral involves the derivative of
$F_{\sigma}$, with

$F_{\sigma}=f_{\sigma}\left[\frac{({\bf K}+x{\bf
p})^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{({\bf K}-(1-x){\bf
p})^{2}}{m^{2}}\right].$

One thus gets

$\displaystyle I_{1}^{\mu}=$
$\displaystyle\frac{i}{m^{2}(2\pi)^{4}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int\frac{d^{4}{\bf K}}{(2\pi)^{4}}\frac{1}{{\bf
K}^{2}+m^{2}x\Delta_{p}}$ $\displaystyle\times$ $\displaystyle\left[({\bf
K}^{\mu}+x{\bf p}^{\mu})f_{\sigma}^{\prime}\left[\frac{({\bf K}+x{\bf
p})^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{({\bf K}-(1-x){\bf
p})^{2}}{m^{2}}\right]\right.$ $\displaystyle+$ $\displaystyle\left.({\bf
K}^{\mu}-(1-x){\bf p}^{\mu})f_{\sigma}\left[\frac{({\bf K}+x{\bf
p})^{2}}{m^{2}}\right]f_{\sigma}^{\prime}\left[\frac{({\bf K}-(1-x){\bf
p})^{2}}{m^{2}}\right]\right].$

In this equation $f^{\prime}_{\alpha}$ denotes $\frac{d}{dX}f_{\sigma}(X)$.
The integral $I_{1}^{\mu}$ is a-priori non-zero only in the $UV$ region where
$f^{\prime}\neq 0$. In this region, all test functions are equivalent to
$f_{\sigma}\left[\frac{{\bf K}^{2}}{m^{2}}\right]$. By symmetry arguments, the
integral over ${\bf K}_{\mu}$ is strictly zero and it remains to calculate

$I_{1}^{\mu}=\frac{i}{(2\pi)^{4}}\frac{{\bf p}^{\mu}}{m^{2}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int\frac{d^{4}{\bf K}}{(2\pi)^{4}}\\\ \frac{2x-1}{{\bf
K}^{2}+m^{2}x\Delta_{p}}f_{\sigma}\left[\frac{{\bf
K}^{2}}{m^{2}}\right]f_{\sigma}^{\prime}\left[\frac{{\bf
K}^{2}}{m^{2}}\right].$

With $\Delta_{p}X={\bf K}^{2}/m^{2}$ we have

$I_{1}^{\mu}=\frac{i}{2(4\pi)^{2}}{\bf p}^{\mu}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx(2x-1)\Delta_{p}\int_{0}^{X_{max}}dX\frac{X}{X+x}\left[f_{\sigma}^{2}(\Delta_{p}X)\right]^{\prime}.$

By integration by part, we have

$I_{1}^{\mu}=\frac{i}{2(4\pi)^{2}}{\bf p}^{\mu}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx(2x-1)\Delta_{p}\\\
\left[\left.\frac{X}{X+x}f_{\sigma}^{2}(X)\right|_{0}^{\infty}-\int_{0}^{\infty}dX\left[\frac{X}{X+x}\right]^{\prime}f_{\sigma}^{2}(\Delta_{p}X)\right].$
(68)

Both contributions are finite in the absence of the test functions, so that we
can safely take the continuum limit $f_{\sigma}\to 1^{-}$ and we finally get

$I_{1}^{\mu}=0.$

We recover here rotational invariance. Note that this property is only true in
the continuum limit. We thus have finally for $\overline{I}_{1}^{\mu}$

$\overline{I}_{1}^{\mu}(p)=\frac{i}{(4\pi)^{2}}\ p^{\mu}\int_{0}^{1}dx\ x\
\mbox{Log}\frac{\eta^{2}}{\Delta_{p}}.$

###  B.2 Vacuum polarization of the photon

The calculation of $\overline{J}_{0}$ and $\overline{J}_{1}^{\mu}$ is very
similar to the calculation of $\overline{I}_{0}$ and $\overline{I}_{1}^{\mu}$
detailed above, with $\Delta_{p}$ replaced by $\Delta_{q}$. We thus get
immediately

$\displaystyle\overline{J}_{0}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{(4\pi)^{2}}\int_{0}^{1}dx\
\mbox{Log}\frac{\eta^{2}}{\Delta_{q}},$
$\displaystyle\overline{J}_{1}^{\mu}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{(4\pi)^{2}}\ p^{\mu}\int_{0}^{1}dx\ x\
\mbox{Log}\frac{\eta^{2}}{\Delta_{q}}.$

####  B.2.1 Calculation of $\overline{J}_{2}$

Following the calculation of $\overline{I}_{0}$, the integral
$\overline{J}_{2}$ is written as

$\overline{J}_{2}(p)=J_{2}(p)-p^{2}\frac{i}{(2\pi)^{4}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int d^{4}{\bf K}\frac{x^{2}}{[{\bf
K}^{2}+m^{2}\Delta_{p}]^{2}}F_{\sigma},$

with

$J_{2}(p)=-\frac{i}{(2\pi)^{4}}\lim_{\sigma\to 1^{-}}\int_{0}^{1}dx\int
d^{4}{\bf K}\frac{{\bf K}^{2}}{[{\bf K}^{2}+m^{2}\Delta_{p}]^{2}}F_{\sigma}.$

The term in $p^{2}$ is calculated similarly to $\overline{I}_{0}$, and we have
for $J_{2}$

$J_{2}(p)=-\frac{i}{(4\pi)^{2}}m^{2}\Delta_{q}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int_{0}^{\infty}dX\frac{X^{2}}{(X+1)^{2}}f_{\sigma}\left[\Delta_{p}X\right].$

The integrand over $X$ can be written as

$\frac{X^{2}}{(X+1)^{2}}=1-\frac{1}{X+1}-\frac{X}{(X+1)^{2}}.$

It is easy to check that the contribution of the first term to $J_{2}$ is
strictly zero, with

$\lim_{\sigma\to
1^{-}}\int_{0}^{\infty}dXf_{\sigma}\left[\Delta_{p}X\right]=\lim_{\sigma\to
1^{-}}\int_{0}^{\infty}\frac{dY}{Y^{2}}f_{\sigma}\left[\frac{\Delta_{p}}{Y}\right]=\int_{0}^{\infty}dY\mbox{Pf}\left[\frac{1}{Y^{2}}\right]\equiv
0.$

It remains

$J_{2}(p)=\frac{i}{(4\pi)^{2}}m^{2}\Delta_{q}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int_{0}^{\infty}dX\left[\frac{1}{X+1}+\frac{X}{(X+1)^{2}}\right]f_{\sigma}\left[\Delta_{p}X\right].$

Using the Lagrange formula for $f_{\sigma}$, and after integration by part, we
get, in the continuum limit

$J_{2}(p)=\frac{i}{(4\pi)^{2}}m^{2}\Delta_{q}\int_{0}^{1}dx\int_{0}^{\infty}dX\frac{\partial}{\partial
X}\left[\frac{X}{X+1}+\frac{X^{2}}{(X+1)^{2}}\right]\int_{\Delta_{q}}^{\eta^{2}}\frac{dt}{t},$

so that

$J_{2}(p)=\frac{2i}{(4\pi)^{2}}\ m^{2}\Delta_{q}\int_{0}^{1}dx\
\mbox{Log}\frac{\eta^{2}}{\Delta_{q}}.$

We thus get

$\overline{J}_{2}(p)=\frac{i}{(4\pi)^{2}}\
m^{2}\Delta_{q}\int_{0}^{1}dx\left(2+\frac{x^{2}}{\Delta_{q}}\frac{p^{2}}{m^{2}}\right)\mbox{Log}\frac{\eta^{2}}{\Delta_{q}}.$

####  B.2.2 Calculation of $\overline{J}_{2}^{\mu\nu}$

Following the calculation of $\overline{I}_{0}$ and $\overline{I}_{1}^{\mu}$,
the integral $\overline{J}_{2}^{\mu\nu}$ is written as

$\overline{J}_{2}^{\mu\nu}=J_{2}^{\mu\nu}+p^{\mu}p^{\nu}\frac{i}{(2\pi)^{4}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int d^{4}{\bf K}\frac{x^{2}}{[{\bf
K}^{2}+m^{2}\Delta_{p}]^{2}}F_{\sigma},$

with

$J_{2}^{\mu\nu}=\frac{i}{(2\pi)^{4}}\lim_{\sigma\to 1^{-}}\int_{0}^{1}dx\int
d^{4}{\bf K}\frac{{\bf K}^{\mu}{\bf K}^{\nu}}{[{\bf
K}^{2}+m^{2}\Delta_{p}]^{2}}F_{\sigma},$

The term in $p^{\mu}p^{\nu}$ is calculated similarly to $\overline{I}_{0}$.
From symmetry arguments, and in the absence of any external momentum in the
continuum limit, we can write

$J_{2}^{\mu\nu}=Ag^{\mu\nu}.$

By contraction with $g_{\mu\nu}$, we get immediately

$A=\frac{1}{4}g_{\mu\nu}I_{2}^{\mu\nu}.$

Note that, due to the presence of the test functions, the contraction
$g_{\mu\nu}I_{2}^{\mu\nu}$ is not a-priori equal to $\tilde{J}_{2}$ written as

$\tilde{J}_{2}(p)=-\frac{i}{(2\pi)^{4}}\lim_{\sigma\to
1^{-}}g_{\mu\nu}\int_{0}^{1}dx\int d^{4}{\bf K}\frac{{\bf K}^{\mu}{\bf
K}^{\nu}}{[{\bf K}^{2}+m^{2}\Delta_{p}]^{2}}F_{\sigma}.$

The difference, if any, should come from the asymptotic behavior of the test
functions in the continuum limit. This prevents to reverse the order of taking
the continuum limit $\sigma\to 1^{-}$ with the contraction by $g_{\mu\nu}$. It
cannot therefore depend on any mass scale. Since $J_{2}$ and $J_{2}^{\mu\nu}$
have a dimension of a mass squared, this difference is thus zero. This is
however not the case for the integrals $K_{2}$ and $K_{2}^{\mu\nu}$, as we
shall see below, since these integrals are dimensionless. We thus get

$J_{2}^{\mu\nu}=\frac{1}{4}g^{\mu\nu}\tilde{J}_{2}.$

where $\tilde{J}_{2}$ can be deduced easily from $J_{2}$ with

$\tilde{J}_{2}(p)=\frac{2i}{(4\pi)^{2}}\ m^{2}\Delta_{q}\int_{0}^{1}dx\
\mbox{Log}\frac{\eta^{2}}{\Delta_{q}}.$

###  B.3 Electromagnetic vertex

The integrals involved in the calculation of the electromagnetic form factor,
with three propagators, have already been detailed in Ref. [30] for the
calculation of the triangular diagrams leading to the axial anomaly.

####  B.3.1 Calculation of $\overline{K}_{0}$

The integral $\overline{K}_{0}$ is written as

$\overline{K}_{0}(p,q)=\frac{2i}{(2\pi)^{4}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int_{0}^{1-x}dy\int d^{4}{\bf K}\frac{1}{({\bf
K}^{2}+m^{2}\Delta)^{3}}F_{\sigma},$

with

$F_{\sigma}=f_{\sigma}\left[\frac{({\bf
K+P})^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{({\bf
K+P-p^{\prime}})^{2}}{m^{2}}\right]f_{\sigma}\left[\frac{({\bf
K+P-p})^{2}}{m^{2}}\right],$

and $P=xp^{\prime}+yp$. Since this integral is finite, we can safely take the
continuum limit with $f_{\sigma}\to 1$ and get

$\overline{K}_{0}(p,q)=-\frac{i}{(4\pi)^{2}}\int_{0}^{1}dx\int_{0}^{1-x}dy\frac{1}{\Delta}.$

####  B.3.2 Calculation of $\overline{K}_{1}^{\lambda}$

The integral $\overline{K}_{1}^{\lambda}$ is written as

$\overline{K}_{1}^{\lambda}(p,q)=K_{1}^{\lambda}(p,q)+\frac{2i}{(2\pi)^{4}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int_{0}^{1-x}dy\int d^{4}{\bf K}\frac{{\bf
P}^{\lambda}}{({\bf K}^{2}+m^{2}\Delta)^{3}}F_{\sigma},$

with

$K_{1}^{\lambda}(p,q)=\frac{2i}{(2\pi)^{4}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int_{0}^{1-x}dy\int d^{4}{\bf K}\frac{{\bf
K}^{\lambda}}{({\bf K}^{2}+m^{2}\Delta)^{3}}F_{\sigma},$

In the continuum limit, ${K_{1}}^{\lambda}$ is strictly zero as shown in Ref.
[30] so that we get immediately, from the calculation of $\overline{K}_{0}$,

$K_{1}^{\lambda}(p,q)=-\frac{i}{(4\pi)^{2}}\int_{0}^{1}dx\int_{0}^{1-x}dy\frac{{\bf
P}^{\lambda}}{m^{2}\Delta}.$

####  B.3.3 Calculation of $\overline{K}_{2}$

Similarly, the integral $\overline{K}_{2}$ is written as

$\overline{K}_{2}(p,q)=K_{2}(p,q)+\frac{2i}{(2\pi)^{4}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int_{0}^{1-x}dy\int d^{4}{\bf K}\frac{{\bf P}^{2}}{({\bf
K}^{2}+m^{2}\Delta)^{3}}F_{\sigma},$

with

$K_{2}(p,q)=\frac{2i}{(2\pi)^{4}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int_{0}^{1-x}dy\int d^{4}{\bf K}\frac{{\bf K}^{2}}{({\bf
K}^{2}+m^{2}\Delta)^{3}}F_{\sigma},$

From the results of Ref. [30] we get immediately

$\overline{K}_{2}(p,q)=\frac{i}{(4\pi)^{2}}\int_{0}^{1}dx\int_{0}^{1-x}dy\left[2\mbox{Log}\frac{\eta^{2}}{\Delta}-\frac{{\bf
P}^{2}}{m^{2}\Delta}\right].$

####  B.3.4 Calculation of $\overline{K}_{2}^{\mu\nu}$

$\overline{K}_{2}^{\mu\nu}(p,q)=K_{2}^{\mu\nu}(p,q)+\frac{i}{(2\pi)^{4}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int_{0}^{1-x}dy\int d^{4}{\bf K}\frac{{\bf P}^{\mu}{\bf
P}^{\nu}}{({\bf K}^{2}+m^{2}\Delta)^{3}}F_{\sigma},$

with

$K_{2}^{\mu\nu}(p,q)=\frac{i}{(2\pi)^{4}}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\int_{0}^{1-x}dy\int d^{4}{\bf K}\frac{{\bf K}^{\mu}{\bf
K}^{\nu}}{({\bf K}^{2}+m^{2}\Delta)^{3}}F_{\sigma}.$

Following the above discussion for the calculation of $J_{2}^{\mu\nu}$, and
according to the results of Ref. [30], we have

$K_{2}^{\mu\nu}(p,q)=\frac{1}{4}g^{\mu\nu}\left[K_{2}(p,q)+m^{2}\Delta\overline{K}_{0}(p,q)\right].$

###  B.4 Infra-red divergences

####  B.4.1 Calculation of $A^{\prime}$ and $B^{\prime}$

The calculation of $A^{\prime}(m^{2})$ and $B^{\prime}(m^{2})$ involves a
singular integral in $1/x$. This singularity corresponds to a pole at ${\bf
K}=0$. In this kinematical domain, the relevant test function is written as
$f_{\sigma}\left(\frac{x^{2}p^{2}}{m^{2}}\right)$. We thus have, for
$p^{2}=m^{2}$,

$\displaystyle A^{\prime}(m^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\pi m}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\frac{1-x}{x}f_{\sigma}(x^{2})=\frac{\alpha}{\pi
m}\left[\frac{1}{2}\int_{0}^{1}dX\mbox{Pf}\left(\frac{1}{X}\right)-1\right]$
$\displaystyle=$ $\displaystyle-\frac{\alpha}{\pi m},$

and

$B^{\prime}(m^{2})=-\frac{\alpha}{2\pi m}\lim_{\sigma\to
1^{-}}\int_{0}^{1}dx\frac{(1-x)^{2}}{x}f_{\sigma}(x^{2})=\frac{3\alpha}{4\pi
m}.$

####  B.4.2 Calculation of $\Phi_{1}^{IR}$

The $IR$ singularities in the calculation of $\Phi_{1}^{IR}$ originates from
the poles in $w=x+y=0$, i.e. for $x=y=0$. This pole occurs for ${\bf K}=0$ and
is taken care of by the test function
$f_{\sigma}\left[\frac{k^{2}}{m^{2}}\right]$. This test function is written
as, in this limit,

$f_{\sigma}\left[\frac{k^{2}}{m^{2}}\right]\to f_{\sigma}\left[\frac{{\bf
P}^{2}}{m^{2}}\right]=f_{\sigma}\left[w^{2}\left(1+\frac{Q^{2}}{m^{2}}\xi(1-\xi)\right)\right],$

with the variables $w$ and $\xi$ introduced in Eq. (22). The a-priori singular
part of the relevant integrals is thus written as

$\displaystyle I$ $\displaystyle=$ $\displaystyle\lim_{\sigma\to
1^{-}}\int_{0}^{1}\frac{dw}{w}f_{\sigma}\left(w^{2}\Delta_{q}\right)=\frac{1}{2}\lim_{\sigma\to
1^{-}}\int_{0}^{\Delta_{q}}\frac{dX}{X}f_{\sigma}(X)=\frac{1}{2}\int_{0}^{\Delta_{q}}dX\mbox{Pf}\left(\frac{1}{X}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\mbox{Log}\Delta_{q}.$

Thanks to the presence of the test function, this contribution is finite,
eventhough we have considered a massless photon. In a calculation using $DR$
with a finite mass $\delta$ of the photon, such integral will have an
additional contribution in $\mbox{Log}\left(\frac{\delta^{2}}{m^{2}}\right)$.

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