# Modified Abelian and SU(2) Wilson theories on a lattice from a non-compact
regularization

D. Babusci and F. Palumbo  INFN, Laboratori Nazionali di Frascati, 00044
Frascati, Italy

###### Abstract

Multiflavour gauge theories of matter systems on a three-dimensional lattice
have recently been widely investigated especially in connection with a
possible symmetry enlargement at a continuous phase transition. Abelian models
were studied both with compact gauge fields and in a mixed formulation in
which the coupling with matter fields is in compact form while the gauge
fields Lagrangian is written in terms of non-compact gauge fields, getting
quite different results. Such a mixed formulation is not permissible for non-
Abelian theories, for which however there exists an entirely non-compact
formulation (in which exact gauge invariance is enforced by help of auxiliary
fields), that for SU(2) was shown to yield in the scaling window a larger
physical volume than Wilson’s one. The corresponding U(1) non-compact
regularization is derived in the present work. In both Abelian and SU(2) cases
there is only one auxiliary field which for large mass has a linear coupling
with the other fields and it can be integrated out. The coupling with the
auxiliary field might make the inversion of the fermion matrix easier.

##  I Introduction

The initial motivation of the present study was the recent interest in
multiflavor gauge theories in three space-time dimensions [Peli1, ; Bona1, ;
higher-charge, ; Bona2, ], but we think that the results have a wider
perspective. Multiflavor gauge theories appear in the investigation of
condensed matter [Klei, ] and critical phenomena, especially in connection
with a possible symmetry enlargement at a second order phase transition [Sach,
]. The relative non-perturbative investigations [Peli1, ; Bona1, ; higher-
charge, ; Bona2, ] have been mostly performed by lattice Monte Carlo
simulations and in several Abelian models the phase diagrams turned out to
depend on the way, compact or non-compact, the gauge fields are defined on a
lattice.

The compact theory was formulated long ago by Wilson [Wils, ] who, in order to
enforce gauge invariance in the presence of discrete derivatives, replaced the
gauge fields in the algebra of a group by the elements of the group, the
Wilson variables, which are compact and decompactify in the continuum limit.

Later on a new regularization of GL($N_{c}$) theories on a lattice was defined
[Palu1, ], in which the gauge fields are maintained in the algebra of the
group, namely they are non-compact, and in order to enforce gauge invariance
auxiliary fields are introduced which decouple in the continuum limit. Since
also the auxiliary fields are non-compact, such a regularization can be called
non-compact. For $N_{c}=2$ the irreducible representation has an SU(2)
invariance [Becc1, ], while the general definition for SU($N_{c}>2$) symmetry
has been given in a subsequent work [Palu2, ]. The construction of such non-
compact regularization was done with the idea of being closer to the continuum
at finite lattice spacing, an expectation supported by a Monte Carlo
simulation in four dimensions which showed that in the scaling window it
generates for SU(2) a physical space larger than the Wilson’s one [DiCa, ].

In the study of Abelian theories another regularization [Bona1, ] is used, in
which the gauge fields Lagrangian is written as a discretization of the
continuous one, namely it is non-compact, while the coupling to matter fields
is written in terms of Wilson variables. For this reason, and to distinguish
it from the entirely non-compact regularization [Palu1, ] that we will use, we
will call it half-compact, even though in the literature it is generally
referred to as non-compact, hoping that this will not generate any confusion.

Compact and half-compact regularizations in three dimensions give quite
different phase diagrams. In the first case the flavor symmetry is broken
while the gauge fields are in the confined phase; in the second case there are
three phases: the Higgs and molecular phases, in which the flavor symmetry is
broken and the electromagnetic correlations are gapped and ungapped,
respectively, and the Coulomb phase in which the flavor symmetry is unbroken
and the electromagnetic correlations are long range. In view of such striking
difference it seems worth while to compare them with the entirely non-compact
one [Palu1, ] which might also introduce further physical behaviors.

The half-compact regularization cannot be used for non-Abelian theories,
because it breaks the gauge invariance of gauge fields self-interactions and
therefore a comparison between compact and half-compact was restricted to the
Abelian case. Also for non-Abelian theories, however, a comparison can be made
with the non-compact regularization, for which there is the additional
expectation of a faster convergence to the continuum.

As a first approach instead of studying entirely non-compact models, one might
investigate the corrections they give to the Wilson Lagrangian at small
lattice spacing, in analogy to the modifications obtained by adding some extra
terms chosen to cancel the leading lattice artifacts or using the
Renormalization Group transformations [Syma, ; Nied, ]. It is the purpose of
the present work to derive such corrections.

SU(2) and U(1) theories, which are the ones investigated in the quoted
multiflavour models [Peli1, ; Bona1, ; higher-charge, ; Bona2, ] have in their
non-compact form one feature in common: they have only one auxiliary field
which in polar variables is a gauge invariant function of the sites. For large
mass this field has linear couplings with all the other (gauge and matter)
fields. The resulting Lagrangian is analogous to the gauged Nambu–Jona-Lasinio
model [Kogu2, ] after linearization: the auxiliary field can then be
integrated out leaving a Lagrangian expressed in terms of Wilson variables and
matter fields only. Of course, in the presence of fermion fields the
integration over the auxiliary field has to be performed after that on the
Grassmann variables, but in analogy with the gauged Nambu–Jona-Lasinio model,
the presence of the auxiliary field might make easier the inversion of the
fermion matrix (but see later on this point).

The half-compact and non-compact regularizations differ from each other also
in another point: only the first one needs a gauge fixing to get rid of the
pure gauge degrees of freedom which have an infinite volume in the partition
function. Some gauge fixings, however, are not compatible with some boundary
conditions (see for instance Ref. [Bona3, ]),while the choice of boundary
conditions may have physical consequences, a point illustrated by a few
examples in Appendix A. In the non-compact regularization [Palu1, ] instead no
gauge fixing is necessary because the mentioned gauge invariant potential
damps all the gauge fields, physical and auxiliary, so that all boundary
conditions can be handled on the same footing.

Because our modified Lagrangian is valid in any number of dimensions it seems
appropriate to review the comparisons made so far in four dimensions.

Abelian theories have been compared in the compact and half-compact forms. It
was soon established that the first one does not yield a second order phase
transition in the bare coupling constant and therefore it cannot define a
continuum theory [Kogu1, ]. Several investigations have been performed
afterwards using the half-compact formulation. After the result that at small
coupling the theory is trivial also in this case, it was searched whether a
continuum theory could be defined at finite coupling [Azco1, ], until it was
established that models of Abelian gauge fields coupled to scalars [Baig, ]
_or_ fermions with four flavours in the chiral limit [Kogu2, ] are indeed
trivial. Later on, however, evidence was reported that in a large-$N$
expansion an ultraviolet fixed point exists in a theory of gauge fields
coupled to _both_ fermions and scalars [Liti, ].

Non-Abelian theories instead have been studied in compact and non-compact
form, the latter one however restricted to SU(2) and only in the absence of
matter fields. Firstly it was established that the non-compact formulation
gives an evidence for confinement [Poli1, ] comparable to that of Wilson. This
was a prerequisite, because non-compact gauge fields with a simple
discretization of the continuum theory [Patr, ] or a non-renormalizable gauge
fixing [Poli2, ] gave a vanishing string tension. Next a perturbative study of
the scaling properties determined the renormalization scale parameter and
showed that the theory has the same fixed point as Wilson’s one [Becc2, ].
This result was confirmed by a calculation [Bora, ] in the Hamiltonian
formalism [Dick, ] using Lüscher’s effective Hamiltonian [Lusc, ]. Lastly, as
already said, the physical volume evaluated in the scaling region [DiCa, ]
turned out to be larger than in Wilson’s regularization.

In conclusion, in four dimensions there is agreement between half-compact and
compact regularizations for Abelian theories coupled to matter fields which
however turn out to be both trivial, and between compact and non-compact
regularizations of non-Abelian theories restricted to SU(2) and in the absence
of matter fields.

The paper is organized in the following way. In Section II we derive the
definition of gauge transformations for Abelian fields and the relative
Lagrangian including interactions with matter fields both in Cartesian and in
polar representation, the latter being more convenient for a comparison with
Wilson’s theory. Such results can be straightforwardly obtained from the
general definition for non-Abelian theories [Palu1, ] but they have not been
spelled out before. In Section III we derive the U(1) modified Lagrangian. In
Section IV we report for the convenience of the reader the known results about
models with gauge fields in the irreducible representation of SU(2) and
construct the corresponding modified Lagrangians. All the above results are
valid in four-dimensional Euclidean space. In Section V we briefly indicate
how they should be modified to get the three-dimensional case. In Section VI
we give our conclusions. We set everywhere $\hbar=c=1$.

##  II Non-compact U(1) gauge theories on a lattice

We will derive the formulation of non-compact U(1) gauge theories on a lattice
in four dimensions from the GL$(N_{c})$ ones [Palu1, ], which is
straightforward but it has not been spelled out so far. We do it for a twofold
motivation that it will be _per sé_ interesting if, unlike the compact and
half-compact regularization, it will turn out to be not trivial, and because
it will serve to construct the three dimensional case in Section V.

###  II.1 Gauge fields transformations

The basic ingredient is the complex function of the lattice site

$D_{\mu}(x)=\frac{1}{a}-g\,[W_{\mu}(x)-i\,A_{\mu}(x)]$ (1)

where $a$ is the lattice spacing, $g$ the gauge coupling constant, $W_{\mu}$
an auxiliary field and $A_{\mu}(x)$ the gauge field. Under the gauge
transformation

$g(x)=\exp(-i\,\theta(x))$ (2)

it transforms according to

$D^{\prime}_{\mu}(x)=g(x)D_{\mu}(x)g^{*}(x+\mu),$ (3)

where $\mu$ in the argument of $g^{*}(x+\mu)$ represents the unit vector in
the $\mu$-direction and $\theta(x)$ is the parameter of the transformation.
For small $\theta(x)$ the gauge fields transform can be obtained from those of
SU(2) [Becc1, ]

$\displaystyle A^{\prime}_{\mu}(x)$
$\displaystyle=A_{\mu}(x)+\nabla_{\mu}^{+}\theta(x)-a\,W_{\mu}(x)\,\nabla_{\mu}^{+}\theta(x)$
$\displaystyle W^{\prime}_{\mu}(x)$
$\displaystyle=W_{\mu}(x)+a\,A_{\mu}(x)\,\nabla_{\mu}^{+}\theta(x)$ (4)

where

$\nabla_{\mu}^{\pm}f=\pm\,\frac{1}{a}[f(x\pm\mu)-f(x)]$ (5)

are the right/left discreet derivatives of a function $f$. In the continuum
limit $W_{\mu}$ becomes invariant (and decouples) while $A_{\mu}$ transforms
as the vector potential. For this reason $W_{\mu}$ was called auxiliary, even
though at finite lattice spacing the massless gauge field to be identified
with the physical vector potential is a linear superposition of
$A_{\mu},W_{\mu}$ and then the actual auxiliary field is the orthogonal
superposition [Becc1, ; Palu2, ].

One can define the gauge field strength

$F_{\mu\nu}(x)=\frac{1}{ig}\left[D_{\mu}(x)\,D_{\nu}(x+\mu)-D_{\nu}(x)\,D_{\mu}(x+\nu)\right]$
(6)

which transforms according to

$F_{\mu\nu}^{\prime}(x)=g(x)\,F_{\mu\nu}(x)\,g^{*}(x+\mu+\nu)$ (7)

and to first order in the lattice spacing it becomes

$F_{\mu\nu}=\nabla_{\mu}^{+}A_{\nu}-\nabla_{\nu}^{+}A_{\mu}\,.$ (8)

For a field $\phi$ transforming as

$\phi^{\prime}(x)=g(x)\,\phi(x)$ (9)

one can define right/left covariant derivatives

$\displaystyle(\mathcal{D}^{+}_{\mu}\phi)(x)$
$\displaystyle=D_{\mu}(x)\,\phi(x+\mu)-\frac{1}{a}\,\phi(x)$
$\displaystyle(\mathcal{D}^{-}_{\mu}\phi)(x)$
$\displaystyle=\frac{1}{a}\,\phi(x)-D_{\mu}^{*}(x-\mu)\,\phi(x-\mu)$ (10)

and the symmetric covariant derivative

$(\mathcal{D}_{\mu}\phi)(x)=\frac{1}{2}\,[D_{\mu}(x)\,\phi(x+\mu)-D_{\mu}^{*}(x-\mu)\phi(x-\mu)].$
(11)

They all transform according to Eq. (9).

###  II.2 The non-compact gauge fields Lagrangian

The Yang-Mills Lagrangian is

$\displaystyle\mathcal{L}_{\rm YM}(x)$
$\displaystyle=\frac{1}{4}\,\sum_{\mu\nu}F_{\mu\nu}^{*}(x)F_{\mu\nu}(x)$ (12)
$\displaystyle=\frac{\beta}{2}\,\sum_{\mu\nu}\Big{[}D^{*}_{\mu}(x+\nu)D^{*}_{\nu}(x)D_{\nu}(x)D_{\mu}(x+\nu)$
$\displaystyle\qquad-\,D^{*}_{\nu}(x+\mu)D^{*}_{\mu}(x)D_{\nu}(x)D_{\mu}(x+\nu)\Big{]}\,$

where $\beta=1/g^{2}$. One must avoid, however, contributions with
$W_{\mu}\approx 1/a$ for which the covariant derivative does not contain an
ordinary derivative. To this end one can include a potential which must
necessarily be a function of the invariant

$\rho_{\mu}^{2}=D_{\mu}^{*}D_{\mu}=\left(\frac{1}{a}-gW_{\mu}\right)^{2}+g^{2}A_{\mu}^{2}\,.$
(13)

A natural choice for the potential is

$\mathcal{L}_{\rm
m}=\frac{\gamma^{2}}{2}\,\sum_{\mu}\left(\rho_{\mu}^{2}-\frac{1}{a^{2}}\right)^{2}\,.$
(14)

It gives to the auxiliary field $W_{\mu}$ the mass $4\gamma^{2}/a^{2}$ which
will play the role of an inverse expansion parameter. Moreover it damps all
the gauge fields, including their zero-modes, so making unnecessary a gauge
fixingaaaIt must be noticed that the potential $\mathcal{L}_{\rm m}$ has two
minima: for $A_{\mu}=0$ $W_{\mu}=\frac{1\mp 1}{a}\,$ which should both be
accounted for. The problem disappears using polar fields, as noted for SU(2)
in Ref. [Becc2, ].

We notice that the theory so defined is strictly local, being a polynomial in
the fields. Arbitrary polynomials in $\rho^{2}_{\mu}$ can be added for
particular purposes.

###  II.3 Matter fields Lagrangians

Let us now write down the interactions with matter fields. The Lagrangian of a
scalar field $\phi$ transforming as in Eq. (9) is

$\mathcal{L}_{\phi}=\frac{1}{2}\,\sum_{\mu}|{\mathcal{D}}_{\mu}\phi|^{2}+\frac{1}{2}\,m_{\phi}^{2}|\phi|^{2}+{\mathcal{V}}(|\phi|)$
(15)

where ${\mathcal{V}}(|{\phi}|)$ is a potential and

$\left({\mathcal{D}}_{\mu}\phi^{f}\right)(x)=\frac{1}{2}\,\left[D_{\mu}(x)\phi^{f}(x+\mu)-D_{\mu}^{*}(x-\mu)\phi^{f}(x-\mu)\right]$
(16)

is the covariant derivative of the scalar field with flavor quantum number
$f$. Here and in analogous expressions we adopt the summation convention

$|\phi|^{2}=\sum_{f}\big{(}\phi^{f}\big{)}^{*}\phi^{f}.$ (17)

The Lagrangian of a spinor field $\psi$ transforming as
$\psi^{\prime}(x)=g(x)\psi(x)$ is

$\mathcal{L}_{\psi}=i\,\sum_{\mu}\bar{\psi}\gamma_{\mu}{\mathcal{D}}_{\mu}\psi+m_{\psi}\bar{\psi}\psi\,.$
(18)

This Lagrangian is usually called naive because it is plagued by the so called
doubling problem, which has been tackled in various ways [Mont, ; Kaplan, ].
We will show in Sec. III how it can be handled in the non-compact formulation.

###  II.4 Polar fields

We derive the explicit expression of the non-compact Abelian Lagrangian with
polar fields, which makes the comparison with Wilson’s theory easier.

The covariant derivative $D_{\mu}(x)$ in polar variables reads

$D_{\mu}=\rho_{\mu}\,U_{\mu}$ (19)

where the polar radius $\rho_{\mu}$ is the auxiliary field (which is obviously
gauge invariant) and $U_{\mu}(x)$ are the Wilson variables parametrized in
terms of the angular fields $\alpha_{\mu}(x)$

$U_{\mu}=\exp(i\,a\,g\,\alpha_{\mu}).$ (20)

They are related to the Cartesian fields according to

$\rho_{\mu}=\sqrt{D_{\mu}^{*}D_{\mu}}\,,\qquad
U_{\mu}=\frac{1}{\rho_{\mu}}\,\left(\frac{1}{a}-gW_{\mu}+i\,gA_{\mu}\right)\,.$
(21)

With a little abuse we use the same notation $D_{\mu}$ for the covariant
derivative in polar as well as Cartesian fields.

The potential $\mathcal{L}_{\rm m}$ is already written in Eq.(14) while the
non-compact Yang-Mills Lagrangian density (12) in polar form is

$\displaystyle\mathcal{L}_{\rm YM}(x)$
$\displaystyle=\frac{\beta}{2}\,\sum_{\mu,\nu}\left[\rho_{\mu}^{2}(x+\nu)\rho_{\nu}^{2}(x)-\rho_{P\mu\nu}(x)\right.$
$\displaystyle\left.\qquad\qquad+\,a^{4}\rho_{P\mu\nu}(x)\,U_{P\mu\nu}(x)\right]\,$
(22)
$\rho_{P\mu\nu}(x)=\rho_{\nu}(x)\rho_{\mu}(x+\nu)\rho_{\mu}(x)\rho_{\nu}(x+\mu)\,,$
(23)

and, taking into account the symmetry of $\rho_{P\mu\nu}$ under
$\mu\leftrightarrow\nu$

$U_{P\mu\nu}(x)=\frac{1}{a^{4}}\,\left\\{1-\frac{1}{2}\,\left[U_{\mu\nu}(x)+U_{\nu\mu}(x)\right]\right\\}$
(24)

with

$U_{\mu\nu}(x)=U_{\mu}^{*}(x+\nu)U_{\nu}^{*}(x)U_{\mu}(x)U_{\nu}(x+\mu)\,.$
(25)

The above equations can be directly used in numerical calculations. However,
to compare with the compact and half-compact regularizations and the continuum
it is convenient to split the Yang-Mills Lagrangian according to

$\mathcal{L}_{\rm YM}=\mathcal{L}_{\rm G}^{\rm W}+\mathcal{L}_{\rm
int}+\mathcal{L}_{\rho}\,,$ (26)

where

$\mathcal{L}_{\rm G}^{\rm W}=\frac{\beta}{2}\,\sum_{\mu,\nu}U_{P\mu\nu}$ (27)

is the Wilson Lagrangian,

$\mathcal{L}_{\rm
int}(x)=\frac{\beta}{2}\,\sum_{\mu,\nu}\left[a^{4}\rho_{P\mu\nu}(x)-1\right]\,U_{P\mu\nu}(x)$

is the interaction between the auxiliary field $\rho_{\mu}$ and the gauge
fields, and

$\mathcal{L}_{\rho}(x)=\frac{\beta}{4}\,\sum_{\mu,\nu}\left[\rho_{\mu}(x)\rho_{\nu}(x+\mu)\,-\,\rho_{\nu}(x)\rho_{\mu}(x+\nu)\right]^{2}\,.$
(28)

is the $\rho_{\mu}$ Lagrangian. Notice that while $\mathcal{L}_{\rm YM}$ is
local in Cartesian coordinates, it has been separated in non-local polar terms
(non-polynomial when expressed in terms of Cartesian fields).

The gauge fields partition function in polar coordinates is

$T_{\rm G}=\int_{0}^{\infty}d\rho_{\mu}\rho_{\mu}^{3}\int
dU_{\mu}\exp\left[-\sum_{x}a^{4}(\mathcal{L}_{\rm YM}+\mathcal{L}_{\rm
m}+\mathcal{P})\right]$ (29)

where $\mathcal{P}$ is an arbitrary polynomial in $\rho_{\mu}^{2}$. We will
use also another expression obtained through the following steps. We first
introduce $\rho_{\mu}^{2}$ as integration variable so that neglecting here and
in the following a constant factor the partition function can be rewritten

$T_{\rm G}=\int_{0}^{\infty}d\rho^{2}_{\mu}\int
dU_{\mu}\exp\left(-\sum_{x}a^{4}\mathcal{L}_{\rm G}\right)$ (30)

where

$\mathcal{L}_{\rm G}=\mathcal{L}_{\rm YM}+\mathcal{L}_{\rm
m}+\mathcal{L}_{l}+\mathcal{P}$ (31)

with

$\mathcal{L}_{l}=-\frac{1}{a^{4}}\sum_{\mu}\ln(a^{2}\rho_{\mu}^{2})\,.$ (32)

Next we perform the change of variables

$\rho^{2}_{\mu}=\frac{1}{a^{2}}\,\left(1+\frac{1}{\gamma}\,s_{\mu}\right)\,,$
(33)

so that the partition function becomes

$T_{\rm G}=\int_{-\gamma}^{\infty}\,ds_{\mu}\int
dU_{\mu}\exp\left(-\sum_{x}a^{4}\mathcal{L}_{\rm G}\right)$ (34)

where now $\mathcal{L}_{\rm G}$ must be expressed in terms of
$s_{\mu},U_{\mu}$. In particular

$\mathcal{L}_{\rm m}=\frac{1}{2}\,\sum_{\mu}\frac{1}{a^{4}}s^{2}_{\mu}\,.$
(35)

We achieve a simplification if we can choose

$\mathcal{P}=-\mathcal{L}_{\rho}-\mathcal{L}_{l}\,.$

If we want to stay strictly local this is not possible because $\mathcal{P}$
must be a polynomial in $\rho_{\mu}^{2}$. However, if we set

$\displaystyle\mathcal{P}$ $\displaystyle=-\frac{\beta
a^{2}}{16}\,\sum_{\mu,\nu}\left(\nabla^{+}_{\mu}\rho_{\nu}^{2}-\nabla^{+}_{\nu}\rho_{\mu}^{2}\right)^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2a^{4}}\sum_{\mu}\left(a^{4}\rho_{\mu}^{4}-4a^{2}\rho_{\mu}^{2}+3\right)\,,$
(36)

in the approximation we will use
$\mathcal{P}+\mathcal{L}_{\rho}+\mathcal{L}_{l}=0$, and thus:

$\mathcal{L}_{\rm G}\approx\mathcal{L}_{\rm G}^{\rm W}+\mathcal{L}_{\rm
int}+\mathcal{L}_{\rm m}\,.$ (37)

A remark is in order here. The Yang-Mills Lagrangian $\mathcal{L}_{\rm YM}$ is
positive definite, but $\mathcal{L}_{\rm G}+\mathcal{L}_{\rm int}$ is not.
However $\mathcal{L}_{\rm m}$ makes $\mathcal{L}_{\rm G}$ bounded from below.

##  III Modified U(1) Lagrangians from the non-compact one

We firstly observe that we get exactly the compact regularization by making
the coupling constant $\gamma$ to diverge for vanishing lattice spacing
[Palu1, ]

$\gamma=\left(\frac{\lambda}{a}\,\right)^{\eta}\,,\qquad\eta>0$ (38)

where $\lambda$ is an arbitrary parameter with the dimension of a length. The
potential term $\mathcal{L}_{\rm m}$ generates in such a case the
$\delta$-function $\delta(\rho_{\mu}(x)-1/a)$ which eliminates the auxiliary
field $\rho_{\mu}$ from the partition function.

But, based on the results [DiCa, ], which showed that the non-compact
regularization generates for SU(2) a physical space larger than the compact
one, we evaluate also for U(1) the contribution arising from the auxiliary
field for small lattice spacing, and therefore we assume $\eta=0$.

###  III.1 Modified compact Lagrangian

In order to estimate the effective Lagrangian, we perform the approximation

$|s_{\mu}|\ll\gamma\,,\qquad\rho_{\mu}\approx\frac{1}{a}\,\left(1+\frac{1}{2\gamma}\,s_{\mu}\right)\,.$
(39)

It is easy to check that in this approximation
$\mathcal{L}_{\rho}+\mathcal{L}_{l}+\mathcal{P}=0$, and

$\mathcal{L}_{\rm
int}\approx\frac{\beta}{2\gamma}\,\sum_{\mu,\nu}(s_{\mu}+s_{\nu})\,U_{P\mu\nu}=\frac{\beta}{\gamma}\,\sum_{\mu}s_{\mu}\,\sum_{\nu}U_{P\mu\nu}\,,$
(40)

so that (37) becomes

$\mathcal{L}_{\rm G}\approx\mathcal{L}_{\rm G}^{\rm
W}+\sum_{\mu}\left(\frac{1}{2a^{4}}\,s_{\mu}^{2}+\frac{\beta}{\gamma}s_{\mu}\,\sum_{\nu}U_{P\mu\nu}\right)\,,$
(41)

which shows that $\gamma$ (the mass of $s_{\mu}$-field, as it can be seen by
the rescaling $s_{\mu}\rightarrow\gamma s_{\mu}$) plays the role of an inverse
coupling constant. Therefore

$T_{\rm G}\approx\int_{-\gamma}^{\infty}ds_{\mu}\int
dU_{\mu}\exp\left(-\sum_{x}a^{4}\mathcal{L}_{\rm G}\right)$ (42)

Now the field $s_{\mu}$ can be integrated out and for large $\gamma$

$T_{\rm G}\approx\int dU_{\mu}\exp\left(-\sum_{x}a^{4}\mathcal{L}^{\rm
mod}_{\rm G}\right)$ (43)

with an error of the order of $e^{-\gamma^{2}}$ and

$\mathcal{L}^{\rm mod}_{\rm G}=\mathcal{L}_{\rm G}^{\rm
W}-\frac{\beta^{2}a^{4}}{2\gamma^{2}}\,\sum_{\mu}\left(\sum_{\nu}U_{P\mu\nu}\right)^{2}\,.$
(44)

$\mathcal{L}^{\rm mod}_{\rm G}$ is the modified gauge fields Lagrangian, which
for $\gamma\to\infty$ coincides with that of Wilson. For a first investigation
one might just evaluate this contribution to the Wilson action at the phase
transition.

Let us go to scalar fields, whose covariant derivative (omitting the flavour
index) is

$\displaystyle\left({\mathcal{D}}_{\mu}\phi\right)(x)$
$\displaystyle=a\,\rho_{\mu}(x)\,\left({\mathcal{D}}^{\rm
W}_{\mu}\phi\right)(x)$
$\displaystyle\qquad+\frac{a}{2}\,(\nabla^{-}_{\mu}\rho_{\mu})\,U^{*}_{\mu}(x-\mu)\,\phi(x-\mu)$
(45)

where

$\left({\mathcal{D}}^{\rm
W}_{\mu}\phi\right)(x)=\frac{1}{2a}\,\left[U_{\mu}\,\phi(x+\mu)-U_{\mu}^{*}(x-\mu)\,\phi(x-\mu)\right]$
(46)

is the Wilson covariant derivative.

The scalar field Lagrangian to leading order in the lattice spacing, after a
by parts integration in the action, is therefore

$\mathcal{L}_{\phi}\approx\mathcal{L}_{\phi}^{\rm
W}+\frac{1}{\gamma}\,\sum_{\mu}s_{\mu}\,\delta\mathcal{L}^{\phi}_{\mu}$ (47)

where

$\mathcal{L}_{\phi}^{\rm W}=\frac{1}{2}\,\sum_{\mu}|\left({\mathcal{D}}^{\rm
W}_{\mu}\phi\right)|^{2}+\frac{1}{2}\,m_{\phi}^{2}|\phi|^{2}+{\mathcal{V}}(|{\phi}|)$
(48)

is the Wilson Lagrangian of the scalar field and

$\delta\mathcal{L}^{\phi}_{\mu}=\frac{1}{2}\left\\{|\left({\mathcal{D}}^{\rm
W}_{\mu}\phi\right)|^{2}-\frac{1}{4}\,\nabla^{+}_{\mu}\left[\left({\mathcal{D}}^{\rm
W}_{\mu}\phi\right)^{*}\phi+\left({\mathcal{D}}^{\rm
W}_{\mu}\phi\right)\phi^{*}\right]\right\\}$ (49)

is its correction. After integration over the auxiliary field we get the
modified Lagrangian

$\mathcal{L}^{\rm mod}=\mathcal{L}_{\rm G}^{\rm W}+\mathcal{L}_{\phi}^{\rm
W}-\frac{a^{4}}{2\gamma^{2}}\,\sum_{\mu}\left(\delta\mathcal{L}^{\phi}_{\mu}+\beta\,\sum_{\nu}U_{P\mu\nu}\right)^{2}\,.$
(50)

The naive spinor field Lagrangian is

$\mathcal{L}_{\psi}=ia\rho_{\mu}\sum_{\mu}\bar{\psi}\gamma_{\mu}\mathcal{D}^{\rm
W}_{\mu}\psi\,+\,m_{\psi}\bar{\psi}\psi+\frac{i}{2}a\nabla^{-}_{\mu}\rho_{\mu}(\bar{\psi}\gamma_{\mu}\psi).$
(51)

Performing a by part integration in the action and expanding the $\rho$-field,
we get to first order in the lattice spacing

$\mathcal{L}_{\psi}\approx\mathcal{L}_{\psi}^{\rm
W}+\frac{1}{\gamma}\,\sum_{\mu}s_{\mu}\,\delta\mathcal{L}^{\psi}_{\mu}$ (52)

where

$\displaystyle\mathcal{L}_{\psi}^{\rm W}$
$\displaystyle=i\,\sum_{\mu}\bar{\psi}\gamma_{\mu}{\mathcal{D}}^{\rm
W}_{\mu}\psi+m_{\psi}\bar{\psi}\psi$
$\displaystyle\delta\mathcal{L}^{\psi}_{\mu}$
$\displaystyle=\frac{i}{2}\left[\bar{\psi}\gamma_{\mu}{\mathcal{D}}^{\rm
W}_{\mu}\psi-\frac{1}{2}\,\nabla^{+}_{\mu}(\bar{\psi}\gamma_{\mu}\psi)\right]\,.$
(53)

In order to see how can we account for the doubling problem in the present
formalism we observe that we can perform the reverse transformation from
compact to non-compact form by multiplying the Wilson variables $U_{\mu}$ by
$a\rho_{\mu}$. Therefore, in all the cases in which the doubling problem is
solved by addition of an appropriate gauge invariant term to the Lagrangian,
one must make in such a term the replacement
$U_{\mu}\,\to\,a\,\rho_{\mu}\,U_{\mu}$ and expand $\rho_{\mu}$ according to
Eq. (39).

The final correction to $\mathcal{L}^{\rm W}_{\psi}$ has the same form as that
of the scalar fields, and it can be handled the same way which yields,
however, a quartic term in the fermion fields. Usually it is preferable
instead to first integrate over the fermion Grassmann variables. We notice in
this connection that the correction term might make easier the inversion of
the fermion matrix in the chiral limit even though, at variance with the
linearized gauge Nambu–Jona-Lasinio model [Kogu2, ], the help would not come
from a mass-like term but from the fermion current.

###  III.2 Modified half-compact Lagrangian

The half-compact regularization can be obtained from the compact one by
expanding the link variables in the pure gauge fields Lagrangian with respect
to the lattice spacing

$\sum_{\mu,\nu}U_{P\mu\nu}\approx\,\frac{1}{2}\,g^{2}\,\sum_{\mu,\nu}\left(\nabla^{+}_{\mu}\alpha_{\nu}-\nabla^{+}_{\nu}\alpha_{\mu}\right)^{2}$
(54)

which yields the gauge invariant kinetic term of the vector potential. In the
coupling to matter fields instead such an expansion would break gauge
invariance and for this reason the Wilson variables are maintained in the
half-compact regularization in the coupling to matter fields.

Again one can look at the difference between the present non-compact
regularization and the half-compact one performing an expansion of the
interaction term to leading order in the lattice spacing. The result is found
inserting Eq.(54) in Eq.(44)

$-\frac{a^{4}}{8\gamma^{2}}\,\sum_{\mu}\left[\sum_{\nu}(\nabla^{+}_{\mu}\alpha_{\nu}-\nabla^{+}_{\nu}\alpha_{\mu})^{2}\right]^{2}\,,$
(55)

where we took into account that $\beta=1/g^{2}$. The Lagrangians of the matter
fields get obviously the same corrections as in the non-compact
regularization.

##  IV Non-compact SU(2) gauge theories on a lattice

The formulation of non-compact SU(2) gauge theories has been treated in detail
[Becc1, ; Becc2, ], and most of the results of this Section about the
modification of the compact one can simply be derived from the Abelian case
introducing color indices and the relative trace in the proper places.
Nevertheless we will report more equations than strictly necessary to make
this Section somewhat selfcontained for a reader interested in the SU(2) case
only.

The basic ingredient for the SU(2) theory is the quaternion

$D_{\mu}(x)=\frac{1}{a}-gW_{\mu}(x)+i\,gA_{\mu a}(x)T_{a}$ (56)

where a is the lattice spacing, $W_{\mu}$ an auxiliary field, $A_{\mu a}(x)$
the gauge field and $T_{a}=\sigma_{a}/2$ the generators of the gauge group,
$\sigma_{a}$ being the Pauli matrices. We assume a convention of summation
over repeated color indices. Under the gauge transformations

$g(x)=\exp(-i\,\theta_{a}(x)\,T_{a})$ (57)

$D_{\mu}(x)$ transforms according to

$D^{\prime}_{\mu}(x)=g(x)D_{\mu}(x)g^{\dagger}(x+\mu),$ (58)

where $\theta_{a}(x)$ are the parameters of the transformation. For small
$\theta_{a}(x)$ the transformations of the gauge fields can be found in
Ref.[Becc1, ] (where the group generators have a different normalization)

$\displaystyle A^{\prime}_{\mu a}$ $\displaystyle=A_{\mu
a}+\left(1-aW_{\mu}\right)\nabla_{\mu}^{+}\theta_{a}-\epsilon_{abc}A_{\mu
b}\left(1+\frac{a}{2}\nabla_{\mu}^{+}\right)\theta_{c}$ $\displaystyle
W^{\prime}_{\mu}$ $\displaystyle=W_{\mu}+\frac{a}{4}\,A_{\mu
a}\nabla_{\mu}^{+}\theta_{a}\,.$ (59)

As in the Abelian case $W_{\mu}$ becomes invariant in the continuum (and
decouples) while $A_{\mu}$ transforms as the vector potential, and at finite
lattice spacing the massless gauge field to be identified with the vector
potential is a linear superposition of $A_{\mu a},W_{\mu}$ and then the actual
auxiliary field is the orthogonal superposition [Becc1, ; Palu2, ].

One can define the gauge field strength

$F_{\mu\nu}(x)=\frac{1}{ig}\left[D_{\mu}(x)\,D_{\nu}(x+\mu)-D_{\nu}(x)\,D_{\mu}(x+\nu)\right]$
(60)

which to first order in the lattice spacing becomes

$F_{\mu\nu}=\left(\nabla_{\mu}^{+}A_{\nu a}-\nabla_{\nu}^{+}A_{\mu
a}-g\,\epsilon_{abc}A_{\mu b}A_{\nu c}\right)\,T_{a}\,.$ (61)

It transforms according to

$F_{\mu\nu}^{\prime}(x)=g(x)\,F_{\mu\nu}(x)\,g^{\dagger}(x+\mu+\nu).$ (62)

For a field $\phi$ transforming as in Eq. (9) one can define right/left
covariant derivatives and the symmetric covariant derivative as in Eqs.
(II.1), (11), resp. They all transform according to Eq. (9).

The Yang-Mills Lagrangian is

$\displaystyle\mathcal{L}_{\rm YM}(x)$
$\displaystyle=\frac{\beta}{8}\,\sum_{\mu\nu}\mbox{Tr}\left[F_{\mu\nu}^{\dagger}(x)F_{\mu\nu}(x)\right]$
(63)
$\displaystyle=\frac{\beta}{4}\,\sum_{\mu\nu}\mbox{Tr}\left[D^{\dagger}_{\mu}(x+\nu)D^{\dagger}_{\nu}(x)D_{\nu}(x)D_{\mu}(x+\nu)\right.$
$\displaystyle\qquad\qquad\left.-\,D^{\dagger}_{\nu}(x+\mu)D^{\dagger}_{\mu}(x)D_{\nu}(x)D_{\mu}(x+\nu)\right]$

where $\beta=4/g^{2}$ and Tr is the trace over colours.

Also with SU(2), in order to avoid contributions of $W_{\mu}\approx 1/a$ for
which the covariant derivative does not contain an ordinary derivative, one
can include in the Lagrangian a potential which must necessarily be a function
of the invariant

$\rho_{\mu}^{2}=D_{\mu}^{\dagger}D_{\mu}=\left(\frac{1}{a}-gW_{\mu}\right)^{2}+\frac{1}{4}\,g^{2}\sum_{a}A_{\mu
a}^{2}\,.$ (64)

Notice that $D_{\mu}^{\dagger}D_{\mu}$ is not a quaternion but it is a real
function of the lattice sites as in U(1), so that the potential has exactly
the form (14), which gives to the auxiliary field $W_{\mu}$ the mass
$4\gamma^{2}/a^{2}$. Moreover it damps all the gauge fields, including their
zero-modes, so making unnecessary a gauge fixing (see Appendix A for details).

As in the Abelian case an arbitrary polynomial of $\rho^{2}_{\mu}$ can be
added for particular purposes. In conclusion, the total gauge fields
Lagrangian is

$\mathcal{L}_{\rm G}=\mathcal{L}_{\rm YM}+\mathcal{L}_{\rm m}+\mathcal{P}\,.$
(65)

We notice that the theory so defined, being a polynomial, is strictly local
also for SU(2).

###  IV.1 Polar coordinates

In polar coordinates

$D_{\mu}=\rho_{\mu}U_{\mu}$ (66)

where

$U_{\mu}=\exp(ia\,g\alpha_{\mu a}T_{a}).$ (67)

The Yang-Mills Lagrangian density in polar form is

$\displaystyle\mathcal{L}_{\rm YM}(x)$
$\displaystyle=\frac{\beta}{4}\sum_{\mu,\nu}\mbox{Tr}\left[\rho^{2}_{\nu}(x+\mu)\rho^{2}_{\mu}(x)-\rho_{P\mu\nu}(x)\right.$
$\displaystyle\qquad\qquad\qquad\left.+\,a^{4}\rho_{P\mu\nu}(x)\,U_{P\mu,\nu}(x)\right]$
(68)

where

$\displaystyle\rho_{P\mu\nu}(x)$
$\displaystyle=\rho_{\mu}(x)\rho_{\nu}(x+\mu)\rho_{\nu}(x)\rho_{\mu}(x+\nu)$
$\displaystyle U_{P\mu\nu}(x)$
$\displaystyle=\frac{1}{a^{4}}\,\left\\{1-\frac{1}{2}\,\left[U_{\mu\nu}(x)+U_{\nu\mu}(x)\right]\right\\}$
(69)

with

$U_{\mu\nu}(x)=U_{\mu}^{*}(x+\nu)U_{\nu}^{*}(x)U_{\mu}(x)U_{\nu}(x+\mu)\,.$
(70)

are the same as Eq. (23), but now the ordering is in $U_{\mu\nu}$.

$\mathcal{L}_{\rm YM}$ can be decomposed as in Eq.(26) where now Wilson
Lagrangian density and interaction of auxiliary and gauge fields are

$\mathcal{L}_{\rm G}^{\rm W}=\frac{\beta}{4}\,\sum_{\mu,\nu}{\rm
Tr}\,U_{P\mu\nu}$ (71) $\mathcal{L}_{\rm
int}(x)=\frac{\beta}{4}\,\sum_{\mu,\nu}\left[a^{4}\rho_{P\mu\nu}(x)-1\right]\,{\rm
Tr}\,U_{P\mu\nu}(x)\,,$ (72)

while $\mathcal{L}_{\rho}$ remains unchanged. The gauge fields partition
function in polar coordinates is

$T_{\rm G}=\int_{0}^{\infty}d\rho_{\mu}\rho_{\mu}^{3}\int
dU_{\mu}\exp\left[-\sum_{x}a^{4}(\mathcal{L}_{\rm YM}+\mathcal{L}_{\rm
m}+\mathcal{P})\right]$ (73)

where $dU_{\mu}$ is now the Haar measure. After the change of variable (39),
omitting an unimportant factor, it can be rewritten in the form

$T_{\rm G}=\int_{-\gamma}^{\infty}ds_{\mu}\int
dU_{\mu}\exp\left(-\sum_{x}a^{4}\mathcal{L}_{\rm G}\right)$ (74)

where $\mathcal{L}_{\rm G}$ is the total gauge fields Lagrangian density in
polar coordinates

$\mathcal{L}_{\rm G}=\mathcal{L}_{\rm G}^{\rm W}+\mathcal{L}_{\rm
int}+\mathcal{L}_{\rm m}+\mathcal{L}_{\rho}+\mathcal{L}_{l}+\mathcal{P}\,,$
(75)

$\mathcal{L}_{l}$ being the same as in Abelian case, Eq. (26).

###  IV.2 Modified compact SU(2) Lagrangian

At variance with the Abelian case, the couplings $\beta,\gamma$ are not
independent from each other. Due to asymptotic freedom perturbation theory is
justified, and to one loop it gives [Becc2, ]

$\gamma=\gamma_{1}\,\beta+\gamma_{2}$ (76)

where $\gamma_{1},\gamma_{2}$ are independent of $\beta$, $\gamma_{1}$ is
arbitrary and $\gamma_{2}$ can be, but it has not been evaluated. This result
has been confirmed by a calculation in the Hamiltonian formalism [Bora, ] (see
Appendix B for details and comparison with numerical simulations). In
conclusion, we cannot vary $\gamma$ arbitrarily, but in the scaling region it
is certainly large so that the field $s_{\mu}$ is coupled linearly to the
Wilson variables as in the Abelian case, and it can be integrated out giving
the modified gauge fields Lagrangian

$\mathcal{L}_{\rm G}=\mathcal{L}_{\rm G}^{\rm
W}-\frac{\beta^{2}\,a^{4}}{8\gamma^{2}}\,\sum_{\mu}\left[{\rm
Tr}\sum_{\nu}U_{P\mu\nu}(x)\right]^{2}\,.$ (77)

Also the evaluation of matter fields Lagrangians proceeds in the same way as
in the Abelian case. The covariant derivative of a scalar with flavour quantum
number $f$ and colour quantum number $c$ in the fundamental representation for
small lattice spacing reads

$\displaystyle\left({\mathcal{D}}_{\mu}\phi^{f}\right)^{c}(x)$
$\displaystyle=a\rho_{\mu}(x)\left({\mathcal{D}}^{\rm
W}_{\mu}\phi^{f}(x)\right)^{c}$
$\displaystyle+\frac{a}{2}\,\nabla_{\mu}^{-}\rho_{\mu}(x)\,\sum_{d}(U^{\dagger})^{cd}_{\mu}(x-\mu)\phi^{fd}(x-\mu)$

where

$\displaystyle\left({\mathcal{D}}^{\rm W}_{\mu}\phi^{f}\right)^{c}(x)$
$\displaystyle=\frac{1}{2a}\,\sum_{d}\left[U^{cd}_{\mu}(x)\phi^{fd}(x+\mu)\right.$
$\displaystyle\left.\qquad\qquad-(U^{\dagger})^{cd}_{\mu}(x-\mu)\phi^{fd}(x-\mu)\right]$
(79)

is the Wilson covariant derivative. The modified Lagrangian including the
interaction with scalars is therefore

$\mathcal{L}^{\rm mod}=\mathcal{L}_{\rm G}^{\rm W}+\mathcal{L}_{\phi}^{\rm
W}-\frac{\beta^{2}\,a^{4}}{\gamma^{2}}\,\sum_{\mu}\left(\frac{2}{\beta}\,\delta\mathcal{L}^{\phi}_{\mu}+\sum_{\nu}{\mbox{T}r}\,U_{P\mu\nu}\right)^{2}$
(80)

while as far as fermions are concerned there is nothing new with respect to
the Abelian case.

##  V Three-dimensional case

In four space-time dimensions the gauge fields have energy dimension 1 and the
coupling constant is dimensionless, so that with or without it all the terms
in the covariant derivative have the same dimension. In three dimensions
instead both gauge fields and gauge coupling constant have energy dimension
1/2, but again all the terms of the Wilson variables have the same dimension.
The same is true for the angular fields of the Wilson variables

$U_{\mu}=\exp(i\,a\,g\,\alpha_{\mu})\,.$ (81)

Scalar and fermion fields have dimension 1/2, 1 respectively so that their
Lagrangians remain unchanged as well.

The auxiliary fields $\rho_{\mu}$ and $s_{\mu}$ remain of dimension 1, 0
respectively, but the coupling $\gamma$ has dimension -1/2, so that the change
of variables (33) becomes

$\rho^{2}_{\mu}=\frac{1}{a^{2}}\,\left(1+\frac{\sqrt{a}}{\gamma}\,s_{\mu}\right)\,,$
(82)

and

$\mathcal{L}_{\rm m}=\frac{1}{2}\,\sum_{\mu}\frac{1}{a^{3}}s^{2}_{\mu}\,.$
(83)

For U(1) the modified Lagrangian

$\mathcal{L}^{\rm mod}=\mathcal{L}_{\rm G}^{\rm W}+\mathcal{L}_{\phi}^{\rm
W}-\frac{a^{4}}{2\gamma^{2}}\,\sum_{\mu}\left(\delta\mathcal{L}^{\phi}_{\mu}+\beta\,\sum_{\nu}U_{P\mu\nu}\right)^{2}\,,$
(84)

identical to the corresponding expression in four dimensions (44) (the
dimensions match because $\gamma^{2},\beta$ have dimensions -1). All the
changes in the other expressions are obvious.

##  VI Conclusions

In recent works U(1) models in three dimensions were compared in compact
[Bona1, ] and half-compact form [higher-charge, ] getting quite different
phase diagrams, while SU(2) models (for which a half-compact formulation does
not exist) were studied only in compact form. It seems therefore interesting
to extend the comparison to an entirely non-compact lattice formulation which
was defined for general non-Abelian theories. Such regularization showed for
SU(2) a faster convergence to the continuum, but the price to be paid is the
introduction of one auxiliary gauge invariant field. For a first investigation
one can estimate the corrections to the Wilson formulation for large mass of
the auxiliary field, can be integrated out yielding a Lagrangian expressed in
terms of compact gauge fields and matter fields only. For a first estimate one
might just evaluate the contribution of the correction at the phase
transition.

The modified Lagrangian is quartic in the matter fields, that is impractical
in the presence of fermions, in which case it may be convenient to retain the
auxiliary field, whose Lagrangian is similar to that of the gauged Nambu–Jona-
Lasinio model [Kogu2, ] after linearization. As in the latter the linear
coupling with the auxiliary field might help the inversion of the fermion
matrix.

The present result can be extended to QCD, but in this case there are colour
auxiliary fields and $D^{\dagger}_{\mu}D_{\mu}$ is no longer a gauge invariant
function of the sites but it has an expression involving the generators of the
group. We hope to be able to tackle this problem in a future work.

##  Appendix A Boundary conditions

Because pure gauge fields are not damped in the partition function, half-
compact gauge theories need a gauge fixing, but as already noticed some gauge
fixings are not compatible with some boundary conditions [Bona3, ]. The
difficulties arising from periodic boundary conditions for quantization in a
cubic box, for instance, can be avoided giving the gauge field a mass
vanishing in the continuum limit [Azco3, ], or assuming antiperiodic boundary
conditions [Bona3, ] or, at the price of nonlocality, with a complete axial
gauge fixing [Palu5, ].

Boundary conditions however cannot be chosen on the basis of mere
calculational convenience, because they may have physical effects; quite
generally different boundary conditions favour or disfavour different phases
[Camp, ] or select different physical systems as illustrated by the following
examples.

Lüscher constructed an effective action for continuous QCD in 4 dimensions
with periodic boundary conditions in terms of zero-momentum modes (gauge
fields depending on time but constant in space) integrating out all the other
modes [Lusc, ]. The mass spectrum so obtained differs from that derived [Gonz,
] assuming the twisted boundary conditions (which forbid zero-momentum modes)
firstly studied by ’t Hooft ['tHo, ]. Indeed the latter boundary conditions
force fluxes of magnetic and electric fields in the quantization box, defining
a physical system different from the one studied by Lüscher.

Further examples are provided by QED. At strong coupling in the compact
version with periodic boundary conditions electric charges are confined, while
with appropriate quasiperiodic boundary conditions one gets a Coulombic
potential [Palu3, ]. Moreover, with periodic boundary conditions zero-momentum
modes may contribute to radiative corrections [Palu4, ].

##  Appendix B Perturbative calculations versus numerical simulations

The are two sources of difficulties in a direct comparisons among perturbative
calculations and numerical simulations:

  1. a)

the choice of the partition function (73) or (74);

  2. b)

the choice of the arbitrary polynomial $\mathcal{P}$ in the gauge fields
Lagrangian (75).

Concerning point a), in both perturbative calculations [Becc2, ; Bora, ] the
partition function (74) was adopted with the approximation $\gamma\gg 1$ in
the lower bound of integration over $s_{\mu}$. Therefore to be compared with
the quoted perturbative results the numerical simulations should have
$\gamma\gg 1$ in the scaling window.

Now, in the simulation [DiCa, ] the partition function (73) was evaluated with
the integral over $s_{\mu}$ extending from $-\infty$, which implies the same
approximation $\gamma\gg 1$ as in the perturbative calculations, while in the
simulation [Poli1, ] the partition function(73) was adopted, with the integral
over $\rho_{\mu}$ extending from zero to $+\infty$, namely without any
approximation as far as this integral is concerned.

The scaling window results to be $\gamma\approx 4$ in Ref. [Poli1, ] while in
Ref. [DiCa, ] there are two scaling regions with $\gamma>8$ and $6>\gamma>10$.

Concerning point b), the arbitrary polynomial has been chosen
$\mathcal{P}=-\mathcal{L}_{\rho}$ in the perturbative calculation [Bora, ] and
in the simulation [Poli1, ], while it has been chosen
$\mathcal{P}=-\mathcal{L}_{\rho}-\mathcal{L}_{l}$ in the perturbative
calculation [Becc2, ] and in the simulation [DiCa, ].

In conclusion, a direct comparison among all these results is not
straightforward.

###### Acknowledgements.

We thank M.-P. Lombardo for most interesting conversation, correspondence and
suggestion of references.

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