The Continuity of the Gauge Fixing Condition $n\cdot\partial n\cdot A=0$ for $SU(2)$ Gauge Theory

0.14 MB·

by  Gao-Liang Zhou

The Continuity of the Gauge Fixing Condition n ∂n A = 0 for SU(2) Gauge Theory
· ·
Gao-Liang Zhou,∗ Zheng-Xin Yan, and Xin Zhang
College of Science, Xi’an University of Science and Technology, Xi’an 710054, People’s Republic of China
Thecontinuityofthegaugefixingconditionn·∂n·A=0forSU(2)gaugetheoryonthemanifold
RNS1NS1NS1 isstudiedhere,wherenµ standsfordirectional vectoralongxi-axis(i=1,2,3).
Itisprovedthatthegaugefixingconditioniscontinuousgiventhatgaugepotentialsaredifferentiable
with continuousderivativeson themanifold RNS1NS1NS1 which is compact.
7 PACSnumbers: 11.15.-q 12.38.-t, 12.38.Aw
1 Keywords: Faddeev-Popov quantization, Gribovambiguity,continuous gauge
0
2

n
a I. INTRODUCTION.
J

9
It is well known that the Faddeev-Popov quantization[1] of non-Abelian gauge theory is hampered by the Gribov
1
ambiguity [2, 3]. Especially, as proved in [3], there is no continuous gauge condition that is free from the Gribov
] ambiguity for non-Abelian gauge theories on 3-sphere(S3) and 4 sphere(S4) once the gauge group is compact. The
h −
conventional Faddeev-Popov quantization procedure[1] of gauge theory is based on the equation:
p
- δ(G(A))
n [ α(x)]det( )δ(G(A)) =1, (1)
e D δα
Z
g
whereα(x)istheparameterofGaugetransformation,G(A)representsthegaugefixingfunction. Accordingtoresults
.
s in [3], it is impossible to choose suitable continuous gauge fixing function G(A) so that the equation (1) holds for
c
i non-Abelian gauge theory on 3-sphere(S3) and 4 sphere(S4) given that the gauge group is compact. In addition,
s −
degeneracy of the gauge fixing condition relies on configurations of gauge potentials and may affect calculations of
y
h physical quantities.
p While concerning infinitesimal gauge transformations, the Gribov ambiguity originates from zero eigenvalues(with
[ nontrivialeigenvectors)oftheFaddeev-Popovoperator[2,4–6]. ThusonecanworkintheregioninwhichtheFaddeev-

Popov operator is positive definite to eliminate infinitesimal Gribov copies. Such region is termed as Gribov region
2
in literatures[2, 6]. Studies on the Gribov regionare interesting and fruitful(see, e.g. Refs.[7–17]). The Gribov region
v
6 method is extended to linear covariant gauges in [18] through the field dependent BRST transformation[19–21]. It
9 can also be extended to other gauge conditions(see, e.g. Refs. [22–26]). We emphasize that the Gribov ambiguity
6 does not vanish even for one works in the Gribov region, although expectation values of gauge invariant quantise are
4 not affected by such ambiguity while working in the Gribov region[27]. Other works on the Gribov ambiguity can
0
been seen in [28–32] and references therein.
.
1 In[33],Wepresentanewgaugeconditionn ∂n A=0fornon-AbeliangaugetheoryonthemanifoldR S1 S1 S1,
0 where nµ is the directional vector along x -ax·is(i·=1,2,3). We have proved that n ∂n A=0 is a con⊗tinuo⊗us g⊗auge
i
7 for non-Abelian gauge theory on R S1 S1 S1 given that generators of Wilson·line·s along nµ are continuous on
1 ⊗ ⊗ ⊗
the manifold R4, where the generator of a unitary matrix U(x) = exp(iφ(x)) means the Hermitian matrix φ(x) not
:
v infinitesimal generators of Lie group in this paper. In addition, we have proved that the gauge condition is free from
i the Gribov ambiguity except for configurations with zero measure.
X
Inthispaper,westudySU(2)gaugetheoryonR S1 S1 S1 toinvestigatethecontinuityofthegaugecondition
r ⊗ ⊗ ⊗
a n ∂n A = 0. Although the theory is quite simple, studies on SU(2) gauge theory are helpful for understanding
· ·
properties of the Gribov ambiguity of non-Abelian gauge theories. The work in this paper is also meaningful for
the study on effects of the Gribov ambiguity on the weak interaction. We will prove that the gauge fixing condition
n ∂n A = 0 is continuous for the theory considered here once gauge potentials are differentiable with continuous
· ·
derivatives. To simplify the proof, we consider the theory on the manifold R S1 S1 S1 with finite lengths along
⊗ ⊗ ⊗
every direction(including the time direction) in this paper. That is to say, the manifold considered here is compact.
For the case that the length along the time direction tends to , the conclusion is not affected given that gauge
∞
potentials satisfy suitable boundary conditions.
The paper is organized as follows. In Sec.II, we describe the gauge theory on R S1 S1 S1 briefly. In SecIII,
we consider SU(2) gaugetheory on R S1 S1 S1 and presentthe proofthat th⊗e gau⊗ge con⊗dition n ∂n A=0 is
⊗ ⊗ ⊗ · ·
∗ [email protected]
2
continuousgiventhatgaugepotentialsaredifferentiablewithcontinuousderivativesonthemanifold. Ourconclusions
and some discussions are presented in Sec.IV.
II. GAUGE THEORIES ON R⊗S1⊗S1⊗S1
Inthis section,wedescribethe gaugetheoryonthefinite 3+1dimensionalsurfaceR S1 S1 S1. Themanifold
considered here can be obtained from the manifold R4 through the identification ⊗ ⊗ ⊗
(t,x ,x ,x ) (t,x +L ,x ,x ) (t,x ,x +L ,x ) (t,x ,x ,x +L ), (2)
1 2 3 1 1 2 3 1 2 2 3 1 2 3 3
∼ ∼ ∼
where L (i = 1,2,3) are large constants. In addition, we require that the surface is finite along the time direction.
i
That is,
T t T, (3)
− ≤ ≤
whereT isalarge(positive)constant. Itisclearthatthemanifoldconsideredhereiscompact. Weconsidercontinuous
gauge potentials on the surface R S1 S1 S1 in this paper. We have,
⊗ ⊗ ⊗
Aµ(t,x +L ,x ,x )=Aµ(t,x ,x ,x )
1 1 2 3 1 2 3
Aµ(t,x ,x +L ,x )=Aµ(t,x ,x ,x )
1 2 2 3 1 2 3
Aµ(t,x ,x ,x +L )=Aµ(t,x ,x ,x ). (4)
1 2 3 3 1 2 3
Gauge potentials are continuous functions on the compact manifold. As a result, we have,
Aµa(x) < , (5)
| | ∞
where a representsthe colorindex. For simplicity, we considerSU(2) gaugetheory in this paper and havea=1,2,3.
Effects of center vortexes like those shown in[34] are not considered here. As a result, we require that
U(t,x +L ,x ,x )=U(t,x ,x ,x )
1 1 2 3 1 2 3
U(t,x ,x +L ,x )=U(t,x ,x ,x )
1 2 2 3 1 2 3
U(t,x ,x ,x +L )=U(t,x ,x ,x ), (6)
1 2 3 3 1 2 3
for continuous gauge transformation on the manifold considered here.
We study the continuity of the gauge condition n ∂n A=0 in this paper, where nµ represents directionalvectors
along x -axis(i=1,2,3). Without loss of generality,·we·take nµ as
i
nµ =(0,0,0,1). (7)
It is proved in [33] that the gauge condition n ∂n A=0 is continuous on R S1 S1 S1 given that generator of
· · ⊗ ⊗ ⊗
the Wilson line
x3
W(x ,x ,x ,x ) exp(ig dzn A(x ,x ,x ,z)) (8)
0 1 2 3 0 1 2
≡P ·
Z0
is acontinuousonthe manifoldR4, wherethe generatorofa unitarymatrixU(x)=exp(iφ(x)) shouldbe understood
as the Hermitian matrix φ(x) not infinitesimal generators of Lie group in this paper. For Abelian gauge theory,
generatoroftheWilsonline(8)isacontinuousfunctiononR4 giventhatn A(x)iscontinuousR4. Infact,theWilson
·
line (8) can be written as
x3
exp(ig dzn A(x ,x ,x ,z)) (9)
0 1 2
·
Z0
for Abelian gauge theory. Generator of the Wilson line (9) reads
x3
g dzn A(x ,x ,x ,z), (10)
0 1 2
·
Z0
which is continuous on R4. Thus the transformation
x L3 x3
V(x) exp(ig 3 dzn A(x ,x ,x ,z))exp( ig dzn A(x ,x ,x ,z)) (11)
≡ L · 0 1 2 − · 0 1 2
3 Z0 Z0
3
is continuous on R S1 S1 S1. One can verity that
⊗ ⊗ ⊗
n ∂n AV =0 (12)
· ·
for Abelian gauge theory, where
i
n AV(x)=n A(x)+ Vn ∂V†(x). (13)
· · g ·
For non-Abelian gauge theory, the situation is more subtle. For example, we consider the following Wilson line
exp(iπcos(2πx3)σ ) (x 0)
W(x) L3 1 3 ≤ , (14)
≡(cid:26)exp(iπcos(2πLx33)σ3) (x3 ≥0)
f
where σ (i=1,2,3) are Pauli matrixes. The gauge potential corresponding to such Wilson line reads,
i
2π2σ1 sin(2πx3)exp(iπcos(2πx3)σ1) (x 0)
n A(x)= − gL3 L3 L3 2 3 ≤ , (15)
· (−2gπL2σ33 sin(2Lπx33)exp(iπcos(2Lπx33)σ23) (x3 ≥0)
which is continuous on R S1 S1 S1. The generator of the Wilson line reads,
⊗ ⊗ ⊗
(2N (x)+cos(2πx3))σ (x 0)
φ(x) 1 L3 1 3 ≤ , (16)
≡(cid:26)(2N2(x)+cos(2Lπx33))σ1 (x3 ≥0)
e
whereN (x) andN (x) arearbitraryintegerswhichmayvaryfrompointto point. Itis clearthatonecannotchoose
1 2
suitable N (x)(i=1,2) so that φ(x) is continuous on R4.
i
Inthispaper,weconsidertheSU(2)non-Abeliangaugetheoryforsimplicity. Westudythecontinuityofthe gauge
condition n ∂n A=0 in followeing texts.
· ·
III. CONTINUITY OF THE GAUGE CONDITION n·∂n·A=0
In this section we consider the continuity of generator of the Wilson line
x3
W(x ,x ,x ,x ) exp(ig dzn A(x ,x ,x ,z)). (17)
0 1 2 3 0 1 2
≡P ·
Z0
Fora unitary matrixU(x)=exp(iφ(x)), wecallthe Hermitianmatrix φ(x) as the generatorofU(x). Oneshouldnot
confuse it with infinitesimal generators of Lie group.
W(x) is an element of SU(2) group and can be written as
W(x)=eiθ~l·~σ(x) =cos(θ)(x)I +i~σ ~lsin(θ)(x), (18)
·
where θ representsanarbitraryrealnumber, I representsthe unit matrix in colorspace,~σ represents Paulimatrixes,
~l representsanarbitraryunit vector. We emphasizethat~l maynotbe welldefinedfor sin(θ)=0. GeneratorofW(x)
can be written as θ~l ~σ(x), which may be singular for sin(θ)=0,θ =0.
· 6
The Wilson line W(x) is differentiable on R4 given that the gauge potential n A(x) is differentiable on R4.
Derivatives of W(x) are continuous on R4 given that derivatives of n A(x) are co·ntinuous on R4. However, as
·
analysed above, generator of the Wilson line is not necessary to be continuous on R4.
We willprovethat one canalwayschoosea continuousgaugetransformationV(x) onR S1 S1 S1 sothat the
⊗ ⊗ ⊗
generator of the Wilson line
x3
WV(x ,x ,x ,x ) exp(ig dzn AV(x ,x ,x ,z)). (19)
0 1 2 3 0 1 2
≡P ·
Z0
is differentiable with continuous derivatives on R4.
4
A. Continuity of the generator of W(x) at non-zero points of n·A(x)
In this subsection, we consider differentiable gauge potentials of which derivatives are continuous functions on
R S1 S1 S1. WewillprovethatthegeneratoroftheWilsonline(17)isdifferentiablewithcontinuousderivatives
on⊗R4 e⊗xcep⊗t for zero points of n A(x).
·
We start from the differentiable matrix
W(x)=cos(θ)(x)I +i~σ ~lsin(θ)(x), (20)
·
of which derivatives are continuous R4. We notice that
1
cos(θ)(x)= tr[W(x)], sin2(θ(x))=1 cos2(θ)(x) (21)
2 −
and conclude that cos(θ)(x) and sin2(θ(x)) are differentiable on R4 and derivatives of them are continuous on R4. In
addition, we can always choose the sign of the vector~l so that
sin(θ(x))=(sin2(θ(x)))1/2. (22)
We see thatsin(θ(x)) is differentiable withcontinuous derivativesonR4 exceptfor points atwhich sin(θ(x))=0. We
define the angle θ(x) as
θ(x)=arccos(cos(θ(x))) (23)
and conclude that θ(x) is differentiable with continuous derivatives on R4 except for points at which sin(θ(x))=0.
For the vector~l(x), we notice that
~σ ~l(x)= isin−1(θ)(U cos(θ)I)(x) (24)
· − −
and conclude that the matrix~σ ~l(x) is differentiable on R4 except for zero points of sin(θ(x)). Derivatives of~σ ~l(x)
· ·
are continuous on R4 except for these points. The generator of W(x) can be written as θ~l ~σ(x). Thus the generator
of W(x) is differentiable with continuous derivatives on R4 except for points at which W(x·)=I.
We then consider the behaviour of W(x) in the neighborhood of points at which W(x) = I. Without loss of
generality, we denote one of these points as x . We have,
0
sin(θ(x ))=0,cos(θ(x ))=1. (25)
0 0
In the neighborhood of x , we have,
0
W(x +∆x)=I+ign A(x )n ∆x+..., (26)
0 0
· ·
where the ellipsis represents higher order terms about ∆x, ∆x is defined as
∆xµ =n ∆xnµ. (27)
·
We then have,
1
~l= tr[~σn A]+.... (28)
2(tr[n A2])1/2 ·
·
For the case that~l is continuous at x , we have,
0
1
vecl= tr[~σn A](x ), (29)
2(tr[n A2])1/2 · 0
·
which is well defined for n A(x ) = 0. According to (29), we see that~l ~σ is differentiable at x and derivatives of
0 0
· 6 ·
~l ~σ are continuous at x given that~l ~σ is continuous at x and n A(x )=0. We notice that
0 0 0
· · · 6
i
sin(θ)(x)= tr[W(x)~l ~σ(x)] (30)
−2 ·
5
and conclude that sinθ is differentiable at x and derivatives of sinθ are continuous at x in this case. In addition,
0 0
we notice that
cosθdcosθ dcosθ
dsin(θ)= , dθ = , cos(θ(x ))=1 (31)
0
−√1 cos2θ −√1 cos2θ
− −
and conclude that θ and θ~l ~σ are differentiable at x and derivatives of them are continuous at x given that~l ~σ is
0 0
· ·
continuous at x and n A(x )=0.
0 0
· 6
We then consider the case that~l ~σ is not continuous at a point x and W(x )=I. We have,
0 0
·
W(x +∆x) W(x )
0 0
lim −
∆x→0 ∆x
| |
[cos(θ(x +∆x)) cos(θ(x ))]
= lim 0 − 0 I
∆x→0 ∆x
| |
sin(θ(x +∆x))
+i lim 0 ~l ~σ(x +∆x). (32)
∆x→0 ∆x · 0
| |
For the case that~l ~σ is not continuous at x , we have,
0
·
sin(θ(x +∆x))
lim 0 =0, (33)
∆x→0 ∆x
| |
for W(x) is differentiable at x andderivativesofW(x) arecontinuousatx . That is,sinθ is differentiable atx and
0 0 0
dsinθ(x )=0. We notice that cosθ is also differentiable at x as W(x) is differentiable at x and have
0 0 0
dcosθ(x )=0, dW(x )=0. (34)
0 0
We notice that
∂W(x)
=ign A(x )W(x )=ign A(x ) (35)
∂n x |x=x0 · 0 0 · 0
·
and conclude that n A(x )=0 given that~l ~σ is not continuous at x and W(x )=I.
0 0 0
· ·
According to above proofs, we see that the generator of the Wilson line (17) is differentiable with continuous
derivatives on R4 except for those points at which n A=0.
·
B. Gauge transformation to eliminate zero points of n·A(x)
Inthissubsection,weprovethatonecanalwayschoosecontinuousgaugetransformationV(x)onR S1 S1 S1
so that n AV(x)=0 everywhere, where ⊗ ⊗ ⊗
· 6
i
n AV(x)=Vn AV†(x)+ Vn ∂V†. (36)
· · g ·
n A(x)iscontinuousonthefinitecompactmanifoldR S1 S1 S1,soareeigenvaluesofn A(x). Thusmaximum
· ⊗ ⊗ ⊗ ·
of eigenvalues of n A(x) does exist, which is denoted as λ . In addition λ is the minimum of eigenvalues of
max max
· −
n A(x) as n A(x) is traceless. We consider the following gauge transformation,
· ·
2πNn x
V(x)=exp(ig · σ ), (37)
L 3
3
where N represents an arbitrary integer with
λ
N > max. (38)
2πL
3
One canverifythatV(x) is acontinuousgaugetransformationonR S1 S1 S1. Under the transformationV(x),
⊗ ⊗ ⊗
we have,
i 2πN
n AV(x) Vn AV†(x)+ Vn ∂V† =Vn AV†(x)+ σ , (39)
· ≡ · g · · L 3
3
6
4πN
tr[σ n AV(x)]=tr[Vn AV†(x)σ ]+
3 · · 3 L
3
4πN
=tr[n A(x)σ ]+
· 3 L
3
4πN
2λ +
≥− max L
3
>0. (40)
WeseethatonecanalwayschoosethecontinuousgaugetransformationV(x)onR S1 S1 S1,sothatn AV(x)=0
⊗ ⊗ ⊗ · 6
everywhere.
It is interesting to consider the limit T , where T is the upper bound of the time t as exhibited in (3). The
proof in this subsection relies on the fact t→hat∞the manifold R S1 S1 S1 is compact. We consider the case that
⊗ ⊗ ⊗
gauge potentials are convergentin the limit T , that is,
→∞
lim Aµ(x)=Cµ(~x), lim Aµ(x)=Cµ(~x), (41)
t→−∞ 1 t→∞ 2
where Cµ(~x) and Cµ(~x) represent arbitrary depreciable matrixes with continuous derivatives on the sub-manifold
1 2
(S )3. It is reasonable to believe that the proof in this subsection works given that eigenvalues of C (~x) and C (~x)
1 1 2
are continuous and bounded function on the sub-manifold (S )3.
1
According to proofs in this and last sections, we see that one can always choose continuous gauge transformation
V(x) on R S1 S1 S1 so that the generator of the Wilson line (19) is differentiable with continuous derivatives
⊗ ⊗ ⊗
on R4 given that gauge potentials Aµ(x) are differentiable with continuous derivatives on R S1 S1 S1.
⊗ ⊗ ⊗
In[33],weprovedthatn ∂n AisacontinuousgaugeonR S1 S1 S1giventhatthegeneratoroftheWilsonline
(17) is continuous on R4, w·her·e nµ represents directional ve⊗ctors⊗alon⊗g x -axis(i = 1,2,3). According to such result
i
and the result in this section, we see that n ∂n A is a continuous gauge on the compact manifold R S1 S1 S1
· · ⊗ ⊗ ⊗
for SU(2) gauge theory given that gauge potentials are differentiable with continuous derivatives on the manifold.
IV. CONCLUSIONS AND DISCUSSIONS
For SU(2) gauge theory on the manifold R S1 S1 S1 with finite length along every direction, we present
⊗ ⊗ ⊗
the proof of the continuity of the gauge condition n ∂n A = 0 given that gauge potentials are differentiable with
· ·
continuousderivativesonthemanifold. Givensuitableboundaryconditionsofgaugepotentials,itisbelievedthatthe
the continuity of the gauge condition does hold even for the length of the manifold along the time direction tends to
. Compactness of the manifold considered here plays an important role in the proof of the continuity of the gauge
∞
condition n ∂n A=0.
· ·
As displayedin above sections,differentiability ofgaugepotentials is essentialfor the proofof the continuity of the
gaugeconditionn ∂n A=0. However,suchdifferentiabilitymaybedestroyedbygaugetransformationsU(x)which
satisfy the equatio·n n· ∂n AU =0. In fact, although the Wilson line (19) is differentiable, derivatives of the Wilson
· ·
line (19) are not necessary to be differentiable. Transformations of gauge potentials involving derivatives of elements
in SU(2) group. It is possible that gauge potentials are no longer differentiable after the gauge transformations
U(x). If gauge potentials are analytic on the manifold considered here, however,then it is reasonable to believe that
analyticities of gauge potentials are not affected by these gauge transformations.
For the case that the length of the manifold along the time direction tends to , suitable boundary conditions at
∞
t are necessary to guarantee the continuity of the gauge condition considered here. In fact, special boundary
→±∞
conditions at t are necessary for the vanishing of surface terms in perturbative theory. Thus we assume that
→±∞
the boundary condition (41) does hold for quantities one concerning.
ACKNOWLEDGMENTS
G. L. Zhou thanks Doctor Hua-Ze Zhu for helpful discussions and important suggestions on the manuscript. The
workofG.L.ZhouissupportedbyTheNationalNatureScienceFoundationofChinaunderGrantNo. 11647022and
The Scientific Research Foundation for the Doctoral Program of Xi’an University of Science and Technology under
Grant No. 6310116055and The Scientific Fostering Foundation of Xi’an University of Science and Technology under
Grant No. 201709. The work of Z. X. Yan is supported by The Department of Shanxi Province Natural Science
7
Foundation of China under Grant No.2015JM1027.
[1] L. D.Faddeev and V. N.Popov, Phys. Lett. B25, 29 (1967).
[2] V.N. Gribov, Nucl.Phys. B139, 1 (1978).
[3] I.M. Singer, Commun. math. Phys.B139, 7 (1978).
[4] P.van Baal, Nucl.Phys. B369, 259 (1992).
[5] R.F.Sobreiro, S.P.Sorella, D.Dudal, andH.Verschelde,Phys.Lett. B590, 265 (2004),arXiv:hep-th/0403135 [hep-th].
[6] N.Vandersickeland D.Zwanziger, Phys.Rept.520, 175 (2012), arXiv:1202.1491 [hep-th].
[7] D.Zwanziger, Nucl.Phys. B209, 336 (1982).
[8] G. DellAntonio and D. Zwanziger, Commun. Math. Phys. 138, 291 (1991).
[9] D.Zwanziger, Nucl.Phys. B321, 591 (1989).
[10] D.Zwanziger, Nucl.Phys. B323, 513 (1989).
[11] D.Zwanziger, Nucl.Phys. B399, 477 (1993).
[12] D.Dudal, S.P. Sorella, N.Vandersickel, and H.Verschelde, Phys.Rev.D77, 071501 (2008), arXiv:0711.4496 [hep-th].
[13] D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel, and H. Verschelde, Phys.Rev.D78, 065047 (2008),
arXiv:0806.4348 [hep-th].
[14] M. A. L. Capria, D. Dudalb, M. S. Guimaraesa, L. F. Palhares, and S. P. Sorella, Phys.Lett. B719, 448 (2013),
arXiv:1212.2419 [hep-th].
[15] N.Su and K. Tywoniuk, Phys.Rev.Lett. 114, 161601 (2015), arXiv:1409.3203 [hep-ph].
[16] A. Bandyopadhyay, N. Haque, M. G. Mustafa, and M. Strickland, Phys.Rev.D93, 065004 (2016),
arXiv:1508.06249 [hep-ph].
[17] M. S. Guimaraes, B. W. Mintz, and L. F. Palhares, Phys.Rev.D92, 085029 (2015), arXiv:1505.04760 [hep-ph].
[18] P.M. Lavrov and O. Lechtenfeld, Phys.Lett. B725, 386 (2013), arXiv:1305.2931 [hep-th].
[19] S.Joglekar and B. Mandal, Phys.Rev.D51, 1919 (1995).
[20] P.M. Lavrov and O. Lechtenfeld, Phys.Lett. B725, 382 (2013), arXiv:1305.0712 [hep-th].
[21] P.Yu.MoshinandA.A.Reshetnyak,in11th International Workshop on Supersymmetries and Quantum Symmetries (SQS’15) Dubna,
(2016) arXiv:1604.03027 [hep-th].
[22] M. A. L. Capri, V. E. R. Lemes, R. F. Sobreiro, S. P. Sorella, and R. Thibes, Phys.Rev.D74, 105007 (2006),
arXiv:hep-th/0609212 [hep-th].
[23] M. A. L. Capri, A. J. Gomez, M. S. Guimaraes, V. E. R. Lemes, and S. P. Sorella, J. Phys.A43, 245402 (2010),
arXiv:1002.1659 [hep-th].
[24] M. Capri, D.Fiorentini, and S. Sorella, Phys.Lett. B751, 262 (2015), arXiv:1507.05481 [hep-th].
[25] M. A. L. Capri, D. Fiorentini, A. D. Pereira, R. F. Sobreiro, S. P. Sorella, and R. C. Terin, (2016),
arXiv:1607.07912 [hep-th].
[26] M. A. L. Capri, D. Dudal, D. Fiorentini, M. S. Guimaraes, I. F. Justo, A. D. Pereira, B. W. Mintz, L. F. Palhares, R. F.
Sobreiro, and S. P. Sorella, Phys.Rev. D94, 025035 (2016), arXiv:1605.02610 [hep-th].
[27] D.Zwanziger, Phys.Rev.D69, 016002 (2004), arXiv:hep-ph/0303028 [hep-ph].
[28] J. Serreau and M. Tissier, Phys. Lett. B712, 97 (2012), arXiv:1202.3432 [hep-th].
[29] J. Serreau, M. Tissier, and A. Tresmontant, Phys.Rev.D89, 125019 (2014), arXiv:1307.6019 [hep-th].
[30] A.D. P. Jr and R.F. Sobreiro, Eur. Phys. J. C73, 2584 (2013), arXiv:1308.4159 [hep-th].
[31] A.D. P. Jr and R.F. Sobreiro, Eur. Phys. J. C74, 2984 (2014), arXiv:1402.3477 [hep-th].
[32] A.Bassetto, I.Lazzizzera, and R. Soldati, Phys. Lett.B131, 177 (1983).
[33] G.-L. Zhou,Z.-X. Yan, and X.Zhang, arXiv:1611.04337 [hep-th].
[34] G. ’t Hooft, Nucl. Phys. B138, 1 (1978).