Charged critical behavior and nonperturbative continuum limit of three-dimensional lattice SU($N_c$) gauge Higgs models
Claudio Bonati Andrea Pelissetto Ivan Soler Calero Ettore Vicari





 
Dipartimento di Fisica dell'Università di Pisa and 
INFN Sezione di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy


Dipartimento di Fisica dell'Università di Roma Sapienza
and INFN, Sezione di Roma I, I-00185 Roma, Italy

 
Dipartimento di Fisica dell'Università di Pisa and 
INFN Sezione di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy


 
Dipartimento di Fisica dell'Università di Pisa,
        Largo Pontecorvo 3, I-56127 Pisa, Italy





§ ABSTRACT 
  We consider the three-dimensional (3D) lattice SU(N_c) gauge Higgs
  theories with multicomponent (N_f>1) degenerate scalar fields and
  U(N_f) global symmetry, focusing on systems with N_c=2, to
  identify critical behaviors that can be effectively described by the
  corresponding 3D SU(N_c) gauge Higgs field theory.  The
  field-theoretical analysis of the RG flow allows one to identify a
  stable charged fixed point for large values of N_f, that would
  control transitions characterized by the global symmetry-breaking
  pattern U(N_f)→SU(2)⊗U(N_f-2).  Continuous transitions with the same
  symmetry-breaking pattern are observed in the SU(2) lattice gauge
  model for N_f ≥ 30.  Here we present a detailed finite-size
  scaling analysis of the Monte Carlo data for several large values of
  N_f.  The results are in substantial agreement with the
  field-theoretical predictions obtained in the large-N_f
  limit. This provides evidence that the SU(N_c) gauge Higgs field
  theories provide the correct effective description of the 3D
  large-N_f continuous transitions between the disordered and the
  Higgs phase, where the flavor symmetry breaks to
  SU(2)⊗U(N_f-2).  Therefore, at least for
  large enough N_f, the 3D SU(N_c) gauge Higgs field theories with
  multicomponent scalar fields can be nonperturbatively defined by the
  continuum limit of lattice discretizatized models with the same
  local and global symmetries.




Charged critical behavior and nonperturbative continuum
  limit
 of three-dimensional lattice SU(N_c) gauge Higgs models
    Ettore Vicari
    September 9, 2024
========================================================================================================================








§ INTRODUCTION




Local gauge symmetries play a fundamental role in the construction of
quantum and statistical field theories that describe phenomena in
various physical contexts: In high-energy physics they are used to
formulate the theories of fundamental
interactions <cit.>, in condensed-matter physics their application spans
from superconductors to systems with topologically ordered
phases <cit.>, in statistical mechanics they
are needed to describe classical and quantum critical phenomena with
(also emergent) gauge fields <cit.>.

The physical properties of lattice gauge models with scalar fields
crucially depend on the behavior of gauge and scalar
modes <cit.>.  Their interplay can give rise
to continuous phase transitions, which are associated with notrivial
continuum limits of the corresponding gauge theories. The
corresponding critical behavior depends both on the breaking pattern
of the global symmetry and on the local gauge symmetry, which
determines which scalar degrees of freedom can become
critical. Moreover, in the presence of gauge symmetries, scalar
systems show Higgs phases <cit.>, a fundamental
feature of many modern-physics systems.

In this paper we focus on a class of three-dimensional (3D)
non-Abelian Higgs (NAH) field theories, which are characterized by
SU(N_c) gauge invariance and by the presence of N_f degenerate
scalar fields transforming in the fundamental representation of the
gauge group. The fundamental fields are a complex scalar field
Φ^af(x), where a=1,...,N_c and f=1,…,N_f, and a
gauge field A_μ^c(x), where c=1,…,N_c^2-1.  The most
general renormalizable Lagrangian consistent with the local SU(N_c)
color symmetry and the global U(N_f) flavor symmetry of
the scalar sector is

    L=   1/g^2 Tr F_μν^2 +  Tr  [(D_μΦ)^† (D_μΦ)]
    
      +    r  Tr Φ^†Φ 
      +  u 4 ( Tr Φ^†Φ)^2  +  v 4  Tr (Φ^†Φ)^2  ,

where F_μν = ∂_μ A_ν -∂_ν A_μ -i[A_μ,
  A_ν] (with A_μ, ab=A_μ^c t_ab^c), and D_μ, ab =
∂_μδ_ab -i t_ab^c A_μ^c where t^c_ab are the
SU(N_c) Hermitian generators in the fundamental representation.

The Lagrangian (<ref>) has been written in the standard
continuum form, in which perturbative computations are usually carried
out (after gauge fixing). An important issue is whether it is possible
to give a definition of the model that goes beyond perturbation
theory.  To investigate this issue, one may proceed as it is usually
done in quantum chromodynamics (QCD), where the question is studied by
considering the lattice QCD formulation <cit.>.  In
this setting a nonperturbative continuum limit exists if the lattice
regularized model undergoes a continuous transition with a divergent
length scale, in which all fields become critical.

Thus, the crucial point is the identification of critical transitions
in 3D lattice NAH models.  In the field-theoretical setting this is
equivalent to the existence of a stable fixed point (FP) of the
renormalization-group (RG) flow of the 3D NAH field theory
(<ref>).  Its existence allows us to define a continuum limit
and therefore it would provide a nonperturbative definition of the
model, as it occurs in the case of QCD <cit.>.

This program has been carried out in 3D Abelian Higgs (AH) theories
(scalar electrodynamics).  Noncompact lattice formulations of the U(1)
gauge fields <cit.>, and compact formulations with
higher-charge scalar fields <cit.> undergo continuous
transitions, where scalar and gauge modes become critical, allowing us
to define a corresponding scalar-gauge statistical field theory.  Note
that the identification of the correct nonperturbative continuum limit
is not trivial, since 3D lattice AH models also undergo continuous
transitions that are not related with the gauge field theory. Indeed,
there are transitions where gauge modes play no role and that have an
effective Landau-Ginzburg-Wilson (LGW) description with no local gauge
symmetry <cit.>, and topological transitions only driven
by the gauge fields, where scalar fields play no role <cit.>.
None of these transitions, even if continuous, allows one to define
the continuum limit of the gauge Higgs field theory, which requires
both gauge and scalar modes to be critical.

For this reason, in order to correctly identify the continuous
transitions that provide the continuum limit for the corresponding
field theory, it is crucial to compare the lattice results with an
independent calculation. In the case of the lattice AH models, the
identification was supported by the comparison of the numerical
lattice results with nonperturbative field-theoretical computations in
the limit of a large number of components of the scalar
field <cit.>.


In this paper, we wish to pursue the same program for the NAH field
theory (<ref>). The RG flow in the space of the Lagrangian
couplings has been analyzed to one-loop order <cit.>, close
to to four dimensions, in the ε≡ 4-d
expansion <cit.>. It has a stable infrared FP, with positive
quartic coupling v, for any N_c and sufficiently large N_f
 <cit.>.  We qualify this FP as charged, because the
gauge coupling assumes a nonzero positive value, thus implying
nontrivial critical correlations of the gauge field. These one-loop
ε-expansion results only indicate that a continuum limit
can be defined for large N_f but do not provide a quantitative
characterization of the behavior in three dimensions and thus, they do
not provide quantitative results that can be compared with numerical
estimates obtained in the corresponding three-dimensional lattice
model. For this purpose the nonperturbatice large-N_f expansion at
fixed N_c is more useful: O(1/N_f) estimates of critical exponents
 <cit.> can be used to verify the correspondence of lattice
results and field-theory estimates.

In this work we mostly focus on lattice NAH models with SU(2) gauge
symmetry. Their phase diagram was investigated in
Ref. <cit.>, identifying different transition lines.  In
this paper we present an accurate numerical study of some of these
transitions, with the purpose of verifying if the observed critical
behavior is consistent with the predictions of the NAH field theory.
We perform Monte Carlo (MC) simulations for sufficiently large N_f
and perform a finite-size scaling (FSS) analysis of the MC results to
estimate the universal features of the transitions.  The numerical
estimates of the N_f-dependent critical exponents are then compared
with the results obtained by using the 1/N_f field-theoretical
expansion  <cit.>.  The numerical results for the
length-scale exponent ν that we present here nicely agree with the
1/N_f prediction, providing a robust evidence of the fact that the
lattice NAH models develop critical behaviors that can be associated
with the stable charged FP of the RG flow of the NAH field theory.

It is worth emphasizing that the existence of these new universality
classes – characterized by the presence of a non-Abelian gauge
symmetry – not only establish the nonperturbative existence of a new
class of 3D quantum field theories, but also allow us to extend the
phenomenology of continuous transitions of 3+1 dimensional lattice
gauge theories at finite temperature, see, e.g., Refs. <cit.>.

The paper is organized as follows. In Sec. <ref> we collect the
known results on the RG flow of the NAH field theory (<ref>),
based on ε expansion, and the large-N_f nonperturbative
predictions.  In Sec. <ref> we define the lattice NAH models,
essentially obtained by discretizing the NAH field theory, and discuss
some general features of their phase diagram.  In Sec. <ref> we
present the FSS analyses of the numerical MC data obtained for N_c=2
and N_f =30,40,60.  Finally, we draw our conclusions in
Sec. <ref>.





§ NAH FIELD THEORY





 §.§ RG flow and large-N_f predictions



The RG flow of the field theory (<ref>) was determined close to
four dimensions in the framework of the ε≡ 4-d
expansion <cit.>. The RG functions were computed by using
dimensional regularization and the minimal-subtraction (MS)
renormalization scheme, see, e.g., Ref. <cit.>.  The RG
flow is determined by the β functions associated with the
Lagrangian couplings u, v, and α = g^2.  At one-loop order
they are given by  <cit.>

    β_α   =    -
      εα + (N_f-22N_c) α^2,
    
    β_u    =    -ε u + (N_f N_c + 4) u^2   
    + 2 (N_f+N_c) u v + 3 v^2  
           - 18  (N_c^2 -1) N_c  u α 
    + 27 (N_c^2 + 2) N_c^2 α^2,
    
    β_v    =    - ε v + (N_f+N_c)v^2 + 6uv
    - 18  (N_c^2 -1) N_c  v α
          + 27 (N_c^2 - 4) N_c α^2.

Some numerical factors, which can be easily inferred from the above
expressions, have been reabsorbed in the normalizations of the
renormalized couplings to simplify the expressions.

The analysis of the common zeroes of the β
functions <cit.> shows that the RG flow close to four
dimensions has a stable charged FP with a nonvanishing α if
N_f > N_f^*, where N_f^* depends on N_c and on the space
dimension.  Close to four dimensions, we have N_f^*=
375.4+O(ε) for N_c=2, and N_f^* = 638.9+O(ε)
for N_c=3. The stable charged FP lies in the region v>0 for any
N_c.  The number of components N_f^* necessary to have a stable
charged FP is quite large in four dimensions. However, we expect
N_f^* to significantly decrease in three dimensions, as it happens
in the AH theories <cit.>, where it
varies from N_f^*≈ 183 in four dimensions <cit.> to a
number in the range 4<N_f^*<10 in three dimensions <cit.>
(see also Refs. <cit.>).

As we already mentioned in the introduction, the one-loop
ε expansion provides only qualitative informations for
three dimensional systems.  A more quantitative approch is the 1/N_f
expansion at fixed N_c  <cit.>. Assuming the existence of
a charged critical behavior for finite N_f, this approach provides
exact predictions of critical quantities for large values of
N_f. The length-scale critical exponent ν for was computed to
O(N_f^-1) <cit.>, obtaining

    ν =  1 - 48 N_cπ^2 N_f + O(N_f^-2),

for tree-dimensional systems.  In particular, ν≈ 1 - 9.727/N_f for
N_c=2.




 §.§ Relevance of the field-theoretical results




The studies of the continuous transitions and critical behaviors of
lattice Abelian and non-Abelian gauge theories with scalar matter,
see, e.g.,
Refs. <cit.>, have shown
the emergence of several qualitatively different types of transitions.

In some cases only gauge-invariant scalar-matter correlations become
critical at the transition, while the gauge variables do not display
long-range correlations.  At these transitions, gauge fields prevent
non-gauge invariant scalar correlators from acquiring nonvanishing
vacuum expectation values and from developing long-range order.  In
other words, the gauge symmetry hinders some scalar degrees of
freedom—those that are not gauge invariant—from becoming critical.
In this case the critical behavior or continuum limit is driven by the
condensation of a scalar order parameter.  This operator plays the
role of fundamental field in the LGW theory which provides an
effective description of the critical behavior.  The effective model
depends only on the scalar order-parameter field, and is only
characterized by the global symmetry of the model.  Gauge invariance
is only relevant in determining the gauge-invariant scalar order
parameter.  Examples of such continuous transitions are found in
lattice AH models <cit.>, and lattice NAH
models <cit.>. A more complex example is
the finite-temperature chiral transitions in QCD.  Ref. <cit.>
(see also Refs. <cit.>) assumed this transition to be
only driven by the fermionic related modes, proposing an effective LGW
theory in terms of a scalar gauge-invariant composite operator
bilinear in the fermionic fields, without gauge fields.

There are also examples of phase transitions in lattice gauge models
where scalar-matter and gauge-field correlations are both critical. In
this case the critical behavior is expected to be controlled by a
charged FP in the RG flow of the corresponding continuum gauge field
theory. This occurs, for instance, in the 3D lattice AH model with
noncompact gauge fields <cit.>, and in the compact model
with scalar fields with higher charge Q≥ 2 <cit.>, for a
sufficiently large number of scalar components. Indeed, the critical
behavior along one of their transition lines is associated with the
stable FP of the AH field
theory <cit.>,
characterized by a nonvanishing gauge coupling.

As already mentioned in the introduction, at present, there is no
conclusive evidence that 3D NAH lattice models undergo continuous
transitions with both scalar and gauge critical correlations, which
can be associated with the stable charged FP of their RG flow
discussed in Sec. <ref>.  A preliminary study was reported in
Ref. <cit.>.  In this paper we return to this issue,
comparing more accurate, numerical analyses with the results obtained
in the field-theoretical 1/N_f expansion. In particular, we
investigate whether, along some specific transition lines, the
critical behavior is characterized by a critical exponent ν that
is consistent, for large values of N_f, with the nonperturbative
1/N_f result (<ref>).



§ LATTICE SU(N_C) GAUGE MODELS WITH MULTIFLAVOR SCALAR FIELDS





 §.§ The lattice model



As in lattice QCD <cit.>, we consider lattice SU(N_c)
gauge models which are lattice discretizations of the NAH field theory
(<ref>). They are defined on a cubic lattice of linear size L
with periodic boundary conditions.  The scalar fields are complex
matrices Φ^af_ x (with a=1,...,N_c and f=1,...,N_f),
satisfying the unit-length constraint Tr Φ_ x^†Φ_ x^† = 1, defined on the lattice sites,
while the gauge variables are SU(N_c) matrices U_
    x,μ <cit.> defined on the lattice links.  The
lattice Hamiltonian reads <cit.>

    H =   - J N_f∑_ x,μ Re  Tr Φ_
        x^†  U_ x,μ Φ_
          x+μ̂^† + v 4∑_ x
        Tr (Φ_ x^†Φ_ x)^2 
       -
      γ N_c∑_ x,μ>ν Re  Tr 
      [U_ x,μ U_ x+μ̂,ν U_
            x+ν̂,μ^† U_ x,ν^†].

In the following we set J=1, so that energies are measured in units
of J, and write the partition function as Z = ∑_{Φ,U}exp(-β H) where β=1/T.

The Hamiltonian H is invariant under local SU(N_c) and global
U(N_f) transformations. Note that U(N_f) is not a simple group and
thus we may separately consider SU(N_f) and U(1) transformations,
that correspond to Φ^af→∑_g V^fgΦ^ag, V∈
SU(N_f), and Φ^af→ e^iαΦ^ag, α∈
[0,2π), respectively.  Since the diagonal matrix with entries
  e^2π i/N_c is an SU(N_c) matrix, α can be restricted
  to [0,2π/N_c) and the global symmetry group is more precisely
    U(N_f)/ℤ_N_c when N_f≥ N_c (if N_f<N_c a global
    U(1)/ℤ_N_c transformation can be reabsorbed by a
    SU(N_c) gauge transformation, see Ref. <cit.>).

Note that the parameter v in the lattice Hamiltonian corresponds to
the Lagrangian parameter v in Eq. (<ref>).  Therefore, if the
lattice model (<ref>) develops a critical behavior described
by the charged FP of the NAH field theory, then this is expected to
occur for positive values of v.

    


 §.§ The phase diagrams for N_f>N_c



A thorough discussion of the phase diagram of the lattice NAH models
(<ref>) was reported in Ref. <cit.>. In this
section we recall the main features that are relevant for the present
study. For N_f=1 the phase diagram is trivial, as only one phase is
present  <cit.>.  For N_f>1, the lattice model
has different low-temperature Higgs phases, which are essentially
determined by the minima of the scalar potential v 
  Tr(Φ^†Φ)^2 with the unit-length constraint Tr Φ^†Φ=1.  Their properties crucially depend on the
sign of the parameter v, the number N_c of colors, and the number
N_f of flavors. Substantially different behaviors are found for
N_f>N_c, N_f=N_c, and N_f< N_c. Also N_c is relevant and one
should distinguish systems with N_c=2 from those with N_c≥ 3.
Since we are interested in phase transitions that can be described by
the stable charged FP of the NAH field theory, and we want to compare
their features with the large-N_f predictions at fixed N_c, we
focus on the case N_f>N_c.







Sketches of the phase diagrams for N_c=2 and N_c≥ 3 when
N_f>N_c are shown in Figs. <ref> and
<ref>, respectively. They are qualitatively similar,
with two different Higgs phases and a single high-temperature
phase. The only difference is the shape of the line that separates the
two Higgs phases.  For N_c=2, the model with v=0 is invariant
under a larger global symmetry group, the Sp(N_f)/ℤ_2 group <cit.>. In this case, the line v=0, that is a
first-order line for N_f≥ 3, separates the Higgs phases. For N_c
> 2 instead, there is no additional symmetry and the boundary of the
two Higgs phases is a generic curve that lies in the positive v
region, see Fig. <ref>.

In the following we focus on the SU(2)-gauge NAH theory
(<ref>), which should be already fully representative for the
problem we address in this paper.  We recall that the analysis of the
RG flow of the NAH field theory, see Sec. <ref>, indicates that
the attraction domain of the stable charged FP must be located in the
region v>0.  Therefore, we should focus on the continuous
transitions occurring in the domain v>0, where the symmetry-breaking
pattern is <cit.>

    U(N_f)→SU(2)⊗U(N_f-2).





§ NUMERICAL ANALYSES OF THE MULTIFLAVOR LATTICE SU(2) NAH MODELS



The numerical results reported in Ref. <cit.> for the SU(2)
lattice gauge model provided good evidence of continuous transitions
for v=1, γ=1, and N_f=40.  First-order transitions were
instead observed for N_f=20, for several values of γ and
v. Therefore, a natural hypothesis is that for v=1 and γ=1
(more generally, for generic positive v and sufficiently large
values of γ) the transitions are continuous for N_f>N_f^*,
with N_f^* in the interval 20<N_f^*<40.

To understand whether these transitions are associated with the
charged FP of the NAH field theory, we need accurate numerical results
that can be compared with predictions obtained from the 3D NAH field
theory. We will focus on the critical exponent ν, comparing the
numerical estimates with the large-N_f result, Eq. (<ref>).
For this purpose, we have performed numerical simulations for v=1,
γ=1, and N_f=30, N_f=40, and N_f=60, varying β
across the transition line. Simulations have been performed on cubic
lattices with periodic boundary conditions.  Some technical details on
the MC simulations have been already reported in
Ref. <cit.>, to which we refer for more details.



 §.§ Observables and finite-size scaling



To study the breaking of the global SU(N_f) symmetry, we monitor
correlation functions of the gauge-invariant bilinear operator

    Q_ x^fg = ∑_a Φ̅_ x^afΦ_ x^ag
      - 1 N_fδ^fg.

We define its two-point correlation function (since we use periodic
boundary conditions, translation invariance holds)

    G( x- y) = ⟨ Tr  Q_ x Q_ y⟩,

the corresponding susceptibility χ, and second-moment
correlation length ξ defined as 

    χ=∑_ x G( x),  ξ^2 = 1 4 sin^2 (π/L)G( 0) - G( p_m)G( p_m),

where p_m = (2π/L,0,0) and G(
  p)=∑_ x e^i p· x G( x) is the
Fourier transform of G( x).  In our numerical study we also
consider the Binder parameter

    U = ⟨μ_2^2⟩/⟨μ_2 ⟩^2,     μ_2 = L^-6∑_ x, y Tr Q_ x Q_ y,

and the ratio

    R_ξ=ξ/L.


At a continuous phase transition, any RG invariant ratio R, such as
the Binder parameter U or the ratio R_ξ, scales as <cit.>

    R(β,L) =  R(X) +  L^-ω R_ω(X) +
      …,

where 

    X = (β-β_c)L^1/ν,

ν is the critical correlation-length exponent, ω>0 is the
leading scaling-correction exponent associated with the first
irrelevant operator, and the dots indicate further negligible
subleading contributions.  The function R(X) is universal up
to a normalization of its argument, and also R_ω(X) is
universal apart from a multiplicative factor and normalization of the
argument [the same of ℛ(X)].  In particular, R^*≡ R(0) is universal, depending only on the boundary conditions
and aspect ratio of the lattice.  Since R_ξ defined in
Eq. (<ref>) is an increasing function of β, we can
combine the RG predictions for U and R_ξ to obtain

    U(β,L) =  U(R_ξ) + O(L^-ω),

where U now depends on the universality class, boundary
conditions, and lattice shape, without any nonuniversal multiplicative
factor. Eq. (<ref>) is particularly convenient because it
allows one to test universality-class predictions without requiring a
tuning of nonuniversal parameters.

Analogously, in the FSS limit the susceptibility defined in
Eq. (<ref>) scales as

    χ≈ L^2-η_Q C(R_ξ),

where η_Q is the critical exponent, that parametrizes the
power-law divergence of the two-point function (<ref>) at
criticality, and C is a universal function apart from a
multiplicative factor.




 §.§ Numerical results




We now present the FSS analyses of the observables introduced in
Sec. <ref>, for the SU(2) gauge theory. We set v=γ=1 and
consider N_f=30, 40, 60. We report data up to L=48 for N_f=40
and N_f=60, and up to L=42 for N_f=30. As we shall see, they are
sufficient to accurately determine the critical behavior of the
lattice SU(2)-gauge NAH models (<ref>).










To begin with, we discuss the behavior for N_f=40, a case that was
already considered in Ref. <cit.>. Here we consider
significantly larger systems and obtain more accurate data.  Estimates
of R_ξ are shown in Fig. <ref> for several values of
L, up to L=48, Data have a clear crossing point for R_ξ≈
0.32, which indicates a transition at β≈ 1.186.
Accurate estimates of the critical point β_c and of the critical
exponent ν are determined by fitting R_ξ to the expected FSS
behavior (<ref>).  We perform several fits, parametrizing the
function R(X) with an order-n polynomial (stable results
are obtained for n≳ 3) an also including O(L^-ω)
corrections with ω in the range [0.5,1.0]. Note that ω
is generally expected to be smaller than one and to approach one in
the large-N_f limit, as in the 3D N-vector
models <cit.>. In any case, results are almost independent of
the value of ω. Moreover, to have an independent check of the
role of the scaling corrections, fits have been repeated,
systematically discarding the data for the smallest lattice sizes
(i.e. including only data for L≥ L_ min with L_
  min=8,12,16,20 typically). Combining all fit results we obtain the
estimates

    β_c=1.1863(1),  ν=0.745(15),   for N_f=40,

where the errors take into account how the results change when the fit
parameters are varied in reasonable ranges (these results are in
substantial agreement with results reported in Ref. <cit.>
using smaller lattice sizes, up to L=28). In Fig. <ref>
we plot R_ξ versus X=(β-β_c)L^1/ν using the above
estimates of β_c and ν.  The resulting scaling behavior when
increasing L definitely confirm the correctness of the estimates
reported in Eq. (<ref>). Some sizeable scaling
corrections are observed only for R_ξ≲ 0.12,
corresponding to X≲ -1, however the convergence of large
lattices, L≳ 30 say, is clear also in that region.  We also
mention that consistent, but less precise, results are obtained by
analyzing the Binder parameter U.

Further evidence of FSS is achieved by the unbiased plot of the Binder
parameter U versus R_ξ, cf. Eq. (<ref>), see
Fig. <ref>. Again we observe a nice scaling behavior for
R_ξ≳ 0.2, see in particular the inset of
Fig. <ref> where data around R_ξ≈ 0.3 are shown.
We also note that sizable scaling corrections are observed around the
peak of U, corresponding to R_ξ≈ 0.12, which is also the
region where the scaling behavior of R_ξ versus X show larger
scaling corrections.  These corrections are consistent with the expected L^-ω asymptotic approach
and ω≈ 1. It is also important to note that, although
significant corrections are present in the peak region, the peak
values decrease when increasing the lattice size, excluding a
discontinuous transitions (if the transition were of first order, the
Binder parameter would diverge for L→∞
<cit.>).

We have also estimated the exponent η_Q characterizing the
behavior of the susceptibility χ.  Using the expected FSS
behavior (<ref>), η_Q was estimated by fitting logχ to (2-η_Q) log L + C(R_ξ), using a polynomial
parametrization for the function C(x). Proceeding as in the analysis
of R_ξ, we obtain η_Q = 0.87(1).  The resulting FSS plot is
shown in Fig. <ref>.

The MC data obtained for N_f=30 and N_f=60 (again for v=1 and
γ=1) have been analyzed analogously. In both cases we observe a
clear evidence of a continuous transition.  In particular, the Binder
parameter U approaches an asymptotic FSS curve when plotted versus
R_ξ, see, e.g., Fig. <ref>.  By fitting R_ξ to the
FSS ansatz (<ref>), as we did for N_f=40, we obtain the
estimates

    β_c=1.22435(10),  ν=0.64(2),   for N_f=30,

and

    β_c=1.1416(1),  ν=0.81(2),   for N_f=60,

where again the errors take into account the small variations of the
results when changing the fit parameters. A FSS plot of R_ξ for
N_f=30 is shown in Fig. <ref>.  We have also estimated
the exponent η_Q.  Performing the same analysis of the
suscelptibility as for N_f=40, we obtain the estimates
η_Q=0.79(1) for N_f=30 and η_Q=0.910(5) for N_f=60.









We now compare the above results for ν with the large-N_f
prediction, Eq. (<ref>), see Fig. <ref>. The
agreement is satisfactory, For instance, Eq. (<ref>) predicts
ν=0.757 for N_f=40 and N_c=2, to be compared with the MC
result ν=0.745(15).  Concerning the exponent η_Q, the
numerical estimates are compatible with the limiting value η_Q=1
for N_f→∞, which holds for any bilinear
operator. Finite-N_f results are consistent with a 1/N_f
correction, as expected. A fit of the data gives η_Q≈ 1 -
c/N_f with c≈ 5 for N_f≳ 40.

The nice agreement between the numerical estimates of ν and 
the field-theoretical large-N_f prediction allows us to conclude 
that, for γ > 0 and v>0 and large values of N_f, 
transitions along the line that separates the disordered from the 
Higgs phase are continuous and naturally associated
with the charged FP of the SU(2)-gauge NAH field theory
(<ref>).  
We expect this result to hold also for larger values of N_c.






§ CONCLUSIONS



We consider 3D lattice SU(N_c) gauge Higgs models with U(N_f)
global invariance with the purpose of identifying continuous
transition lines with a critical behavior associated with the stable
charged FP of the RG flow of the NAH field theory defined by the
Lagrangian (<ref>).  This would imply that the lattice models
admit a continuum limit that provides a nonperturbative definition of
the NAH field theory, as it occurs for lattice QCD <cit.>.

We focus on SU(2) gauge theories. We perform MC simulations for a
relatively large number of flavors, in order to be able to compare the
MC results with field-theoretical 1/N_f predictions.  The RG flow of
the SU(2)-gauge NAH field theory has a stable charged FP in the region
v>0, for N_f > N^*_f. Close to four dimensions, N_f^* is very
large, N^*_f ≈ 376, see Sec. <ref>. However, our 3D
numerical results show that continuous transitions in the relevant
parameter region occur for significantly smaller numbers of
components.  While for N_f=20 only first-order transitions (for
different values of v and γ) are observed <cit.>,
for N_f=30 a continuous transition is found for v=γ=1. These
results suggest that 20<N_f^*<30, or equivalently that N_f^*=25(4)
in three dimensions. More importantly, the numerical estimates of the
length-scale critical exponent ν for N_f=30,40,60 are in nice
agreement with the large-N_f field-theoretical result,
Eq. (<ref>).  As far as we know, this is the first evidence
of the existence of critical behaviors in 3D lattice NAH models that
can be associated with the charged FP of the 3D SU(N_c)-gauge NAH
field theory.

As we mentioned in Sec. <ref> not all transitions in gauge
systems require an effective description in terms of a gauge field theory.
There are many instances in which gauge fields have no role.  In these cases
the effective model is a scalar LGW theory in which the fundamental field is a
(coarse-grained) gauge-invarianct scalar order parameter.  This approach was
employed in Refs. <cit.> to discuss the nature of the
finite-temperature transition of QCD in the chiral limit. Indeed, it was
assumed that the transition was only due to the condensation of a
gauge-invariant operator, bilinear in the fermionic fields.  Such operator was
then taken as fundamental field in an effective 3D LGW Φ^4 theory, whose
RG flow was supposed to determine the nature of the chiral transition.  The
implicit assumption was that only gauge-invariant fermionic related modes are
relevant critical modes.

It is thus worth discussing the predictions of the LGW approach in the 
present case, to exclude that the transitions we have discussed above 
have an effective LGW description. In the LGW approach the fundamental field is 
a hermitian traceless N_f× N_f 
matrix field Ψ( x), which represents a
coarse-grained version of the gauge-invariant bilinear operator
Q_ x defined in Eq. (<ref>). The corresponding most
general LGW Lagrangian with global SU(N_f) symmetry
is <cit.>

    L_ LGW = Tr ∂_μΨ∂_μΨ +
      r TrΨ^2
    
               +  w Tr Ψ^3 + 
          u  (Tr Ψ^2)^2 + v  Tr Ψ^4.

For N_f=2 the cubic term vanishes and the two quartic terms are equivalent.
In this case a continuous transition is possible in the SU(2)/ℤ_2,
that is in the O(3) vector, universality class.  For N_f > 2 the cubic term
is present and, on the basis of the usual mean-field arguments, one expects a
first-order transition also in three dimensions (unless a tuning of the model
parameters is performed to cancel the cubic term).  Therefore, the LGW approach
does not give the correct predictions for the transitions we have investigated.
The reason  of the failure is likely related to the fact that the LGW approach
assumes that gauge fields are not relevant at criticality. In LGW transitions
their only role is that of restricting the critical modes to the
gauge-invariant sector. Instead, the relation between the critical transitions
we observed and the NAH field theory implies that gauge fields are critical and
relevant for the critical behavior in the cases we studied.

We should note that the results presented here are valid for v > 0.  For v <
0 continuous transitions are observed for N_f=2, in the O(3) universality
class <cit.>.  The NAH field theory does not provide their correct
effective description, since there are no stable FPs in the RG flow of the NAH
field theory with negative v for any N_f.  On the other hand, the LGW
theory predicts O(3) transitions for N_f=2, since the Lagrangian (<ref>)
is equivalent to the O(3) Lagrangian for this value of N_f.  We conclude
that, for v < 0 and N_f=2, gauge modes do not play any role and the
transition admits a LGW description.

This discussion shows that the critical behavior of 3D models (or 4D models at
finite temperature) with non-Abelian gauge symmetry is quite complex  and
possibily more interesting than expected.  In particular, the knowledge of the
order parameter of the transition is not enough to characterize the critical
behavior.  Informations on the behavior of the gauge fields are required to
identify the correct effective description.  




The authors acknowledge support from project PRIN 2022 “Emerging
gauge theories: critical properties and quantum dynamics”
(20227JZKWP).  Numerical simulations have been performed on the CSN4
cluster of the Scientific Computing Center at INFN-PISA.




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