#  Implications of the topological Chern-Simons mass in the Gap Equation

Caroline P. Felixa , Chung Wen Kaoa ,
a Department of Physics and Center for High Energy Physics,
Chung-Yuan Christian University, Chung-Li 32023, Taiwan
felix<EMAIL_ADDRESS>
###### Abstract

In this paper, we analyze the Yang-Mills-Chern-Simon theory in the scope of
Gribov formalism considering the first order in $M^{2}$ (the topological
Chern-Simons mass) of the gap equation. By doing this, we show that the
character of physical excitaions, i.e. real massive poles in the gluon
propagator, can be trick, because there are three possible solutions for the
Gribov parameter.

##  1 Introduction

It is well know that the non-perturbative non-Abelian gauge field theory is
plagued by gauge copies, the famous Gribov copies [1]. Gribov realized that
the standard gauge fixing procedure by Faddeev-Popov is not sufficient to
remove all equivalent gauge field configurations in non-perturbative non-
Abelian gauge field theory. To solve this problem, Gribov found a region that
is free from this kind of copies, which is called Gribov Region. In 44 years,
a big range of research has been done based in the Gribov problem, for
curiosity the reader can see [2, 3, 4, 5, 6, 7, 8]. The pioneer work was done
by Zwanziger, who figured out how to implement the Gribov region in the
action, hence developing a new formalism called Gribov-Zwanziger formalism [9,
10, 11], what some authors have used to understand and explain
confinement/deconfinement phase transition, see some example here [12, 13, 14,
15].

Another very interesting method to study confinement/deconfinement is through
$3d$ Euclidean Yang-Mills-Chern-Simons theory [16, 17], because the
topological Chern-Simons (CS) mass term gives the gluon field an extra mass
generating a confinement/deconfinement transition phase [12, 13] in $3d$
Euclidean Yang-Mills.

Now, it sounds much more interesting if we join Gribov-Zwanziger formalism and
Yang-Mills-Chern-Simons theory and analyze what happens with the gap equation
(the equation that defines the Gribov horizon [1, 9, 10, 11]) and with the
gluon propagator poles in the presence of a topological mass and the Gribov
parameter $\gamma^{4}$. The gluon propagator poles in Yang-Milss-Chern-Simons
theory in the presence of Gribov ambiguity has already been analyzed in [12].
However, in [12], the authors considered only the zero order of the gap
equation expansion in CS mass term. Despite, the zero order is the most
dominant term, we will see that the first order in the CS mass has notable
physical implications.

Our starting point is the Gribov-Zwanziger action in linear covariant gauge in
$3d$ dimensions Enclidean space,

$\begin{split}S~{}=~{}&\int
d^{3}x\frac{1}{4}\left(F_{\mu\nu}^{a}\right)^{2}+\int
d^{3}x\left(\frac{\alpha}{2}b^{a}b^{a}+\mathrm{i}b^{a}\,\partial_{\mu}A^{a}_{\mu}+\bar{c}^{a}\partial_{\mu}D^{ab}_{\mu}c^{b}\right)\\\
&+\int
d^{3}x\left[\bar{\varphi}^{ac}_{\mu}\partial_{\nu}D^{ab}_{\nu}\varphi^{bc}_{\mu}-\bar{\omega}^{ac}_{\mu}\partial_{\nu}(D^{ab}_{\nu}\omega^{bc}_{\mu})-g(\partial_{\nu}\bar{\omega}^{an}_{\mu})f^{abc}D^{bm}_{\nu}c^{m}\varphi^{cn}_{\mu}\right]\\\
&-\gamma^{2}g\int
d^{3}x\left[\vphantom{\frac{1}{2}}f^{abc}A^{a}_{\mu}\varphi^{bc}_{\mu}+f^{abc}A_{\mu}^{a}\bar{\varphi}_{\mu}^{bc}+\frac{3}{g}(N_{c}^{2}-1)\gamma^{2}\vphantom{\frac{1}{2}}\right],\end{split}$
(1.1)

where $\gamma$111$\gamma^{4}=\frac{2\beta Ng^{2}}{3V(N^{2}-1)}$ is the Gribov
parameter; $F^{a}_{\mu\nu}$ is the field strength tensor,

$F^{a}_{\mu\nu}~{}=~{}\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+gf^{abc}A^{b}_{\mu}A^{c}_{\nu}\;;$
(1.2)

$(\phi,\bar{\phi})$ are pair of complex-conjugate bosonic fields;
$(\omega,\bar{\omega})$ anti-commuting complex-conjugate fields; the fields
$({\bar{c}}^{a},c^{a})$ are the Faddeev-Popov ghosts; $\alpha$ is the gauge
parameter, which is zero for the Landau gauge, $\partial_{\mu}A_{\mu}=0$;
$b^{a}$ accounts for the Lagrange multiplier implementing the gauge condition;
and $D^{ab}_{\mu}=(\delta^{ab}\partial_{\mu}+gf^{acb}A^{c}_{\mu})$ is the
covariant derivative in the adjoint representation of $SU(N)$.

Now, by coupling the Chern-Simons action,

$S_{\text{CS}}~{}=~{}-\frac{\mathrm{i}M}{2}\epsilon^{\mu\nu\lambda}\int
d^{3}x\left(A_{\mu}^{a}\partial_{\lambda}A_{\nu}^{a}-\frac{2}{3!}gf^{abc}A_{\mu}^{a}A_{\nu}^{b}A_{\lambda}^{c}\right)\,,$
(1.3)

to (1.1), since the Chern-Simons action gives a topological mass, $M$, to the
gauge field creating a deconfined massive excitation. Therefore, the
action(1.1) becomes

$\begin{split}S_{YMGZCS}~{}=~{}&\int
d^{3}x\left[\frac{1}{4}\left(F_{\mu\nu}^{a}\right)^{2}-\frac{iM}{2}\epsilon_{\mu\nu\lambda}\left(A_{\nu}^{a}\partial_{\lambda}A_{\mu}^{a}-\frac{2}{3!}gf^{abc}A_{\mu}^{a}A_{\nu}^{b}A_{\lambda}^{c}\right)\right]\\\
&+\int
d^{3}x\left(\frac{\alpha}{2}b^{a}b^{a}+\mathrm{i}b^{a}\,\partial_{\mu}A^{a}_{\mu}+\bar{c}^{a}\partial_{\mu}D^{ab}_{\mu}c^{b}\right)\\\
&+\int
d^{3}x\left[\bar{\varphi}^{ac}_{\mu}\partial_{\nu}D^{ab}_{\nu}\varphi^{bc}_{\mu}-\bar{\omega}^{ac}_{\mu}\partial_{\nu}(D^{ab}_{\nu}\omega^{bc}_{\mu})-g(\partial_{\nu}\bar{\omega}^{an}_{\mu})f^{abc}D^{bm}_{\nu}c^{m}\varphi^{cn}_{\mu}\right]\\\
&-\gamma^{2}g\int
d^{3}x\left[\vphantom{\frac{1}{2}}f^{abc}A^{a}_{\mu}\varphi^{bc}_{\mu}+f^{abc}A_{\mu}^{a}\bar{\varphi}_{\mu}^{bc}+\frac{3}{g}(N_{c}^{2}-1)\gamma^{2}\vphantom{\frac{1}{2}}\right].\end{split}$
(1.4)

##  2 The gap equation

In this section, we analyze the contribution of the Chern-Simon mass term to
the gap equation. The gap equation is a self consistent condition obtained
through the saddle-point approximation, which becomes exact in the
thermodynamic limit, see [1, 9, 10, 11]. In other words, the gap equation can
be obtained by taking the first derivative of the vacuum energy density,
(A.25), with respect to $\beta$, computed at the specific value $\beta^{*}$
that minimizes $\mathcal{E}_{v}$, see Appendix A for more details about the
calculus of the vacuum energy density. In this work, the gap equation reads,

$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\;\frac{k^{4}+\gamma_{*}^{4}}{\left(k^{4}+\gamma_{*}^{4}\right)^{2}+k^{6}M^{2}}~{}=~{}\frac{3}{2Ng^{2}}\,.$
(2.1)

The integral on the left side of this equation can be solved by residues
theorem. However, the solutions for the gap equation (2.1) is only given by
graphic analysis. Let us to analyze this equation, by comparing its left side,
that is shown in Figure 1, with its right side. In the non-perturbative
regime, i.e in the infrared, $g^{2}$ is strong, the right side of (2.1) is
small. In the Figure 1, we can see that the left side from (2.1) is also very
small for $\gamma^{4}$ large and $M$ small. In the perturbative regime, right
side from (2.1) is big. The left side of the same equation, the big values are
reached when $M$ is big or $\gamma^{4}$ is small, what is consistent with
theory, since $\gamma\propto\exp(-1/g^{2})$. By Figure 1, we also notice that
the left side of the gap equation (2.1) is not defined by all values of the
Chern-Simons mass $M$, for $M>3$ in the plot, the function is not defined.

Figure 1: The right side from (2.1).

As we are working with Gribov formalism, we are only interested in the
infrared (IR) regime, where Gribov copies show up. By the graphic analysis
above, we see that the IR regime is given by small $M$. Then, we can expand
the left side of (2.1) in $M^{2}$ by using Taylor series. Hence, the gap
equation can be written as follows

$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{n=0}^{\infty}\frac{(-i)^{2n}\left(k^{6n}M^{2n}\right)}{\left(\gamma^{4}+k^{4}\right)^{2n+1}}~{}=~{}\frac{3}{2Ng^{2}}$
(2.2) $\displaystyle\sum_{n=0}^{\infty}(-i)^{2n}\frac{4\pi}{(2\pi)^{3}}\int
dk\frac{k^{6n}M^{2n}}{\left(\gamma^{4}+k^{4}\right)^{2n+1}}k^{2}~{}=~{}\frac{3}{2Ng^{2}}$
(2.3)

In this paper, we only consider the two first terms of (2.3). This equation
has been solved in zero order in [12] as it was already said before.

To solve (2.3), we used the residue theorem:

$\int_{0}^{\infty}f(x)dx=\frac{1}{2}\int_{-\infty}^{\infty}f(x)dx=\pi\mathrm{i}\textnormal{Res}\left[f(z)\right]_{z=z_{0}}=-\pi\mathrm{i}\textnormal{Res}\left[f(z)\right]_{z=-z_{0}}\\\
$ (2.4)

where $z_{0}$ is the pole of the function $f(z)$. In spherical coordinates,
the zero order term from (2.3) is

$\begin{split}\frac{1}{2\pi^{2}}\int_{0}^{\infty}\frac{k^{2}}{k^{4}+\gamma^{4}}dk=\frac{1}{4\pi\sqrt{2}\gamma},\end{split}$
(2.5)

that is the result from [12]. The first order term in $M^{2}$ of (2.3) is

$\begin{split}-\frac{4\pi}{(2\pi)^{3}}\int_{0}^{\infty}\frac{k^{8}M^{2}}{\left(k^{2}-i\gamma^{2}\right)^{3}\left(k^{2}+i\gamma^{2}\right)^{3}}dk=-\frac{5M^{2}}{128\sqrt{2}\pi\gamma^{3}}.\end{split}$
(2.6)

By replacing (2.5) and (2.6) in (2.3), we obtain

$\gamma^{3}-\frac{Ng^{2}}{6\sqrt{2}\pi}\gamma^{2}+\frac{5Ng^{2}}{192\sqrt{2}\pi}M^{2}=0,$
(2.7)

which is a cubic equation. Hence, there are three solutions for this equation
and the value of its discriminant says if these roots are complex or real.
Hence, if the discriminant of (2.7) is positive,

$\begin{split}\Delta=&-\left(1215\pi^{2}M^{2}-16g^{4}N^{2}\right)>0\\\
&0<M<\frac{4g^{2}N}{9\sqrt{15}\pi},\end{split}$ (2.8)

there are three real roots, it means three real values for $\gamma$. For
values of $M$ bigger than in (2.8), $\gamma$ assumes complex values. On that
account, $M$ cannot be all possible values, it is limited in an interval. By
using François Viète’s formula, we find three real solutions for the cubic
equation:

$\gamma_{t}=\frac{g^{2}N}{18\sqrt{2}\pi}+\frac{g^{2}N}{9\sqrt{2}\pi}\cos\left[\frac{2\pi
t}{3}-\frac{1}{3}\arccos\left(1-\frac{1215\pi^{2}M^{2}}{8g^{4}N^{2}}\right)\right],$
(2.9)

with $t=0,~{}1,~{}2$, the solutions plots are in Figures 2(a), 2(b) and 2(c).
Notice, if we take $M=0$, we recover the result from [12].

(a) $\gamma_{0}(g,M)$

(b) $\gamma_{1}(g,M)$

(c) $\gamma_{2}(g,M)$

Figure 2: Plot of (2.9), $t=0$, $t=1$ and $t=2$ with $N=3$.

When $\Delta=0$, we obtain

$M=\frac{4g^{2}N}{9\sqrt{15}\pi}\;,$ (2.10)

and the solutions of the gap equation are

$\gamma_{t}=\frac{g^{2}N}{18\sqrt{2}\pi}+\frac{g^{2}N}{9\sqrt{2}\pi}\sin\left(\frac{2\pi
t}{3}+\frac{\pi}{6}\right),$ (2.11)

there are three roots, but two of them are similar $\gamma_{0}=\gamma_{1}$. We
see that (2.10) is the maximum allowed value of $M$ to obtain real roots for
the gap equation. In [12], the weak coupling constant regime is given by the
condition without restriction to Gribov horizon, that is

$M>\frac{Ng^{2}}{6\pi},$

eq. (8) in their paper, what is bigger than the maximum value of $M$ given by
(2.10). Therefore, in the case of real roots, the regime is always in the
strong coupling regime, since, $M$ is always smaller than $Ng^{2}/6\pi$.

When $\Delta<0$,

$M>\frac{4g^{2}N}{9\sqrt{15}\pi},$

there are one real root $\gamma_{c1}(=\gamma_{2})$ and two complex conjugated
roots, $\gamma_{c2}$ and $\gamma_{c3}$. We do not show their expression in
this paper, because they are complicated expressions in function of $g$ and
$M$. Instead, we only plot their imaginary part, see Figure 3. Of course, the
imaginary part of $\gamma_{c1}$ is null.

(a) Im[$\gamma_{c1}(g,M)$]

(b) Im[$\gamma_{c2}(g,M)$]

(c) Im[$\gamma_{c3}(g,M)$]

Figure 3: Plot of imaginary part of $\gamma$ when $\Delta<0$, $N=3$.

There are two very interesting results from our work: $\gamma^{4}$ has three
solutions, what implies that there are three Gribov horizons, i.e three Gribov
regions; two of them intercepts each other when $M$ is given by (2.10), which
can be visualized in Figure 4, it is the line where both function meet each
other.

Figure 4: $\gamma_{0}(g,M)$ (blue) and $\gamma_{1}(g,M)$ (yellow) in the same
plot, $N=3$.

##  3 The Yang-Mills-Chern-Simons gluon propagator

To calculate the gluon propagator from (1.4), it is necessary to take only the
quadratic part in the gauge field of action and integrating it out. Following
this steps, one should end up with

$S~{}=~{}\int\frac{d^{3}k}{(2\pi)^{3}}\left(-\frac{1}{2}\tilde{A}_{\mu}^{a}(k)Q_{\mu\nu}^{ab}\tilde{A}_{\nu}^{b}(-k)\right)$
(3.1)

where

$Q_{\mu\nu}^{ab}~{}=~{}\delta^{ab}\left(\frac{k^{4}+\gamma^{4}}{k^{2}}\delta_{\mu\nu}+\left(\frac{1}{\alpha}-1\right)k_{\mu}k_{\nu}+M\epsilon_{\mu\nu\lambda}k_{\lambda}\right)$
(3.2)

and $\gamma^{4}=\frac{2\beta Ng^{2}}{3V(N^{2}-1)}$ is the Gribov parameter. In
order to obtain the propagator we have to compute the inverse of (3.2), which
can be obtained through the following expression,

$Q_{\mu\nu}^{ab}(Q_{\nu\delta}^{bc})^{-1}~{}=~{}\delta^{ac}\delta_{\mu\delta}\,.$
(3.3)

The ansatz for the inverse of (3.2) reads

$(Q_{\nu\delta}^{bc})^{-1}~{}=~{}\delta^{bc}\left(F(k)\delta_{\nu\delta}+B(k)\frac{k_{\nu}k_{\delta}}{k^{2}}+C(k)M\,\epsilon_{\delta\nu\alpha}\frac{k_{\alpha}}{k^{2}}\right)\,,$
(3.4)

where the coefficients are dimensionless.

The Landau gauge is recovered in the limit $\alpha\to 0$, and the propagator
reads

$\displaystyle\langle A^{a}_{\mu}(k)A^{b}_{\nu}(-k)\rangle$ $\displaystyle=$
$\displaystyle\delta^{ab}{F(k)}\left[\left(\delta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)-\frac{k^{4}}{(k^{4}+\gamma^{4})}M\,\epsilon_{\mu\nu\alpha}\frac{k_{\alpha}}{k^{2}}\right]\,.$
(3.5)

The overall factor $F(k)$ can be written as

$F(k)=\frac{(k^{4}+\gamma^{4})k^{2}}{(k^{4}+\gamma^{4})^{2}+k^{6}M^{2}}\,.$
(3.6)

In order to find out the poles, the authors of [12] pointed out that the roots
of the polynomial

$\displaystyle P(k^{2})$ $\displaystyle=$
$\displaystyle(k^{4}+\gamma^{4})^{2}+k^{6}M^{2}$ (3.7) $\displaystyle=$
$\displaystyle(k^{2}+m_{1}^{2})(k^{2}+m_{2}^{2})(k^{2}+m_{3}^{2})(k^{2}+m_{4}^{2})$

where $m_{i}$ stands for the solutions of the polynomial $P(k^{2})$.

As it is pointed in [12], the poles of the gauge field propagator is well
defined by two parameters, namely $(g,M)$. Such dependence of the poles of the
gluon propagator on the coupling constant, $g$, happens by means of the gap
equation of the Gribov parameter.

In [12], it is also mentioned is the possibility of split up the propagator in
two parts concerning the breaking of the parity symmetry, namely

$\langle
A^{a}_{\mu}(k)A^{b}_{\nu}(-k)\rangle~{}=~{}\mathcal{G}^{ab}_{\mu\nu}\big{|}_{par}+\mathcal{G}^{ab}_{\mu\nu}\big{|}_{par-
viol}\,,$ (3.8)

with

$\displaystyle\mathcal{G}^{ab}_{\mu\nu}\big{|}_{par}$ $\displaystyle=$
$\displaystyle\delta^{ab}F(k)\left(\delta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)$
(3.9)

and

$\displaystyle\mathcal{G}^{ab}_{\mu\nu}\big{|}_{par-viol}$ $\displaystyle=$
$\displaystyle\delta^{ab}\frac{k^{6}M}{(k^{4}+\gamma^{4})^{2}+k^{6}M^{2}}\epsilon_{\nu\mu\alpha}\frac{k_{\alpha}}{k^{2}}\,.$
(3.10)

Using partial fractional decomposition for each part, i.e, non-parity
violation and parity violation parts, we have

$\begin{split}&\mathcal{G}^{ab}_{\mu\nu}\big{|}_{par}=\delta^{ab}\left(\frac{\alpha_{1}}{k^{2}+m_{1}^{2}}+\frac{\alpha_{2}}{k^{2}+m_{2}^{2}}+\frac{\alpha_{3}}{k^{2}+m_{3}^{2}}+\frac{\alpha_{4}}{k^{2}+m_{4}^{2}}\right)\left(\delta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\\\
&\mathcal{G}^{ab}_{\mu\nu}\big{|}_{par-
viol}=\delta^{ab}\left(\frac{\beta_{1}}{k^{2}+m_{1}^{2}}+\frac{\beta_{2}}{k^{2}+m_{2}^{2}}+\frac{\beta_{3}}{k^{2}+m_{3}^{2}}+\frac{\beta_{4}}{k^{2}+m_{4}^{2}}\right)M\epsilon_{\nu\mu\alpha}k_{\alpha}\end{split}$
(3.11)

where $m_{i}^{2}$ are the roots of the polynomial $P(k^{2})$ in (3.7) and the
residues are given by

$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle-\frac{m_{1}^{4}}{(m_{1}^{2}-m_{2}^{2})(m_{1}^{2}-m_{3}^{2})(m_{1}^{2}-m_{4}^{2})},$
$\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle\frac{m_{2}^{4}}{(m_{1}^{2}-m_{2}^{2})(m_{2}^{2}-m_{3}^{2})(m_{2}^{2}-m_{4}^{2})},$
$\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle\frac{m_{3}^{4}}{(m_{1}^{2}-m_{3}^{2})(-m_{2}^{2}+m_{3}^{2})(m_{3}^{2}-m_{4}^{2})},$
$\displaystyle\alpha_{4}$ $\displaystyle=$
$\displaystyle\frac{m_{4}^{4}}{(m_{1}^{2}-m_{4}^{2})(-m_{2}^{2}+m_{4}^{2})(-m_{3}^{2}+m_{4}^{2})},$
$\displaystyle\beta_{1}$ $\displaystyle=$
$\displaystyle-\frac{m_{1}^{2}(\gamma^{4}+m_{1}^{4})}{(m_{1}^{2}-m_{2}^{2})(m_{1}^{2}-m_{3}^{2})(m_{1}^{2}-m_{4}^{2})},$
$\displaystyle\beta_{2}$ $\displaystyle=$
$\displaystyle\frac{m_{2}^{2}(\gamma^{4}+m_{2}^{4})}{(m_{1}^{2}-m_{2}^{2})(m_{2}^{2}-m_{3}^{2})(m_{2}^{2}-m_{4}^{2})},$
$\displaystyle\beta_{3}$ $\displaystyle=$
$\displaystyle\frac{m_{3}^{2}(\gamma^{4}+m_{3}^{4})}{(m_{1}^{2}-m_{3}^{2})(-m_{2}^{2}+m_{3}^{2})(m_{3}^{2}-m_{4}^{2})},$
$\displaystyle\beta_{4}$ $\displaystyle=$
$\displaystyle\frac{m_{3}^{2}(\gamma^{4}+m_{3}^{4})}{(m_{1}^{2}-m_{4}^{2})(-m_{2}^{2}+m_{4}^{2})(-m_{3}^{2}+m_{4}^{2})}.$
(3.12)

In this point, the analysis is not diferent from [12]. Then, the discriminant
of $P(k^{2})$ is

$\Delta_{p}=256M^{4}\gamma^{20}-27M^{8}\gamma^{16}.$

So, there are four complex masses, if $\Delta_{p}>0$ or
$\gamma>\sqrt[4]{27}M/4$, and there are two complex and two real masses, if
$\Delta_{p}<0$ or $\gamma<\sqrt[4]{27}M/4$. However, we need to be more
careful than [12] about the physical excitation in the gluon propagator, since
$\gamma$ does not only depend on $g$, it is also depends on $M$ and we need to
check all solutions for the gap equation222The following conditions have been
obtained via Reduce method in Wolfram Mathematica software..

  * •

If $\Delta_{p}>0$ or $\gamma>\sqrt[4]{27}M/4$ (no physical excitation or four
complex masses):

    * –

$\gamma_{0}>\sqrt[4]{27}M/4$, if

$0<M\leq\frac{4g^{2}N}{9\sqrt{15}\pi};$ (3.13)

    * –

$\gamma_{1}>\sqrt[4]{27}M/4$, if

$\frac{\left(6\sqrt{6}-5\sqrt{2}\right)}{54\sqrt[4]{3}\pi}Ng^{2}<M\leq\frac{4g^{2}N}{9\sqrt{15}\pi};$
(3.14)

    * –

$\gamma_{2}$ is never bigger than $\sqrt[4]{27}M/4$. Therefore, $\gamma_{2}$
represents a region where there is not complex masses.

    * –

There are not complex solutions for the gap equation, i.e $\gamma_{c1}$,
$\gamma_{c2}$ and $\gamma_{c3}$ do not contribute for complex poles in the
gauge propagator.

  * •

If $\Delta_{p}<0$ or $\gamma<\sqrt[4]{27}M/4$ (two complex masses and two real
masses):

    * –

$\gamma_{0}$ is never smaller than $\sqrt[4]{27}M/4$. Therefore, there is no
physical excitation inside of the region given by $\gamma_{0}$, gluons do not
have free particles character inside of the region determined by $\gamma_{0}$.

    * –

$\gamma_{1}<\sqrt[4]{27}M/4$, if

$0<M<\frac{\left(6\sqrt{6}-5\sqrt{2}\right)}{54\sqrt[4]{3}\pi}Ng^{2};$ (3.15)

    * –

$\gamma_{2}<\sqrt[4]{27}M/4$, if

$0<M\leq\frac{4g^{2}N}{9\sqrt{15}\pi};$ (3.16)

    * –

$\gamma_{c1}(=\gamma_{2})<\sqrt[4]{27}M/4$, if

$M>\frac{4g^{2}N}{9\sqrt{15}\pi},$ (3.17)

that is the only real solution of the gap equation when $\Delta<0$, $\Delta$
from (2.8).

    * –

$\gamma_{c2}$ and $\gamma_{c3}$ do not satisfy both conditions, (3.17) and
$\gamma_{c2}$ or $\gamma_{c3}<\sqrt[4]{27}M/4$. In other words, complex
solutions for $\gamma$ do not contribute to the propagator poles in any
situation, i.e neither for real poles nor for imaginary poles.

In Figures 5 and 6, we have plotted the real poles, $m_{1}^{2}$ and
$m_{2}^{2}$, at zero and first orders in $M^{2}$. In Figure 7, $m_{1}^{2}$ and
$m_{2}^{2}$ are shown using $\gamma_{2}$. $m_{3}^{2}$ and $m_{4}^{2}$ are
always complex, they never represent physical excitation, this result is
similar what was found in [12]. However, the behavior of these parameter
change completely from zero order in $M^{2}$ to first order in $M^{2}$, see
Figures 5 and 6. Differently from [12], we have three possible values for
$\gamma$. For one of these values, $\gamma_{0}$, there is no physical
excitation, in other words, all four masses are complex. Therefore, we see
that there is a region where there is no physical excitation even with the
Chern-Simons topological mass.

(a) Zero order in $M^{2}$.

(b) First order in $M^{2}$ by using $\gamma_{1}$.

Figure 5: $m_{1}^{2}$ in function of $M$ and $g$, $N=3$.

(a) Zero order in $M^{2}$.

(b) First order in $M^{2}$ by using $\gamma_{1}$.

Figure 6: $m_{2}^{2}$ in function of $M$ and $g$, $N=3$.

Figure 7: $m_{1}^{2}$ and $m_{2}^{2}$ in function of $M$ and $g$ by using
$\gamma_{2}$, $N=3$.

Figures 8, 9, 10 and 11 show the residues correspondent to the massive pole
$m_{1}^{2}$ \- $\alpha_{1},~{}\beta_{1}$ \- and the one correspondent to the
massive pole $m_{2}^{2}$ \- $\alpha_{2},~{}\beta_{2}$ \- see (3) to check
their expressions. $\alpha_{1}$ and $\beta_{1}$ are negatives, then they are
not physical excitation, even with a real pole $m_{1}^{2}$. However,
$\alpha_{2}$ and $\beta_{2}$ are positives, then they represent physical
excitation. Therefore, $m_{2}^{2}$ is the responsible for the physical
excitation in the gluon propagator, i.e it gives the free particle nature to
the gluon propagator.

(a) $\alpha_{1}(\gamma_{1})$

(b) $\alpha_{1}(\gamma_{2})$

Figure 8: Plot of $\alpha_{1}$ in function of the $(g,M)$ for $\gamma_{1}$ and
$\gamma_{2}$, $N=3$.

(a) $\alpha_{2}(\gamma_{1})$

(b) $\alpha_{2}(\gamma_{2})$

Figure 9: Plot of $\alpha_{2}$ in function of the $(g,M)$ for $\gamma_{1}$ and
$\gamma_{2}$, $N=3$.

(a) $\beta_{1}(\gamma_{1})$

(b) $\beta_{1}(\gamma_{2})$

Figure 10: Plot of $\beta_{1}$ in function of the $(g,M)$ for $\gamma_{1}$ and
$\gamma_{2}$, $N=3$.

(a) $\beta_{2}(\gamma_{1})$

(b) $\beta_{2}(\gamma_{2})$

Figure 11: Plot of $\beta_{2}$ in function of the $(g,M)$ for $\gamma_{1}$ and
$\gamma_{2}$, $N=3$.

##  4 Conclusion

In this paper, we have seen that $\gamma^{4}$ has three possible solutions
dependent of $(g,M)$, and only one of them, $\gamma_{1}$, gives complex and
real poles. $\gamma_{0}$ does not contribute to the physical excitation, i.e
to real poles in the gluon propagator. Therefore, there is a solution from the
gap equation, $\gamma_{0}$, where gluons do not manifest as free particles
even with CS mass term. In the Gribov region defined by $\gamma_{2}$, the
gluon propagator always has real poles. This last result is very interesting
and new, since the gluons behavior could be interpreted as free particles in
the strong coupling constant regime. We also see that we do not obtain a real
value for $\gamma$ with all values of CS mass. For
$M>\frac{4g^{2}N}{9\sqrt{15}\pi}$, the gap equation has complex solution,
$\gamma$ is a complex parameter. By the analysis from Section 3, these complex
value for $\gamma$ does not contribute to the poles of the gluon propagator,
i.e they do not satisfy the poles conditions in the propagator. Therefore, the
relevant solutions from the gap equation are the real roots, i.e $\gamma$ is a
real parameter.

## Acknowledgments

This work was supported by Ministry of Science and Technology, Taiwan, grant
number: MOST 109-2112-M-033 and 109-2811-M-033 -501.

##  Appendix A The vacuum energy density calculus

The Gribov parameter is determined consistently through a consistent condition
called gap equation, necessary in the thermodynamic limit [1, 9]. In order to
compute the gap equation the quantum fields must be integrated out in the path
integral within the perturbative regime. No loop correction will be considered
to the effective action, so that only quadratic in the quantum fields should
be considered, as well as constant terms. Namely, the vacuum energy can be
written as

$\displaystyle\mathrm{e}^{-V\mathcal{E}_{v}}~{}=~{}\det\left[Q^{ab}_{\mu\nu}\right]^{-1/2}\det\left[\mathcal{P}^{ab}\right]\mathrm{e}^{\beta-\ln\beta}$
(A.1)

with $Q^{ab}_{\mu\nu}$ given by (3.2) standing for the gauge field
contribution to the vacuum energy, and

$\displaystyle\mathcal{P}^{ab}~{}=~{}k^{2}\delta^{ab}\;.$ (A.2)

space. The Gribov parameter should be read off from
$\frac{2g^{2}N\beta}{3V(N^{2}-1)}~{}=~{}\gamma^{4}$.

By using the following identity,

$(\det Q_{\mu\nu}^{ab})^{-1/2}=e^{-\frac{1}{2}\det\ln
Q_{\mu\nu}^{ab}}~{}=~{}\mathrm{e}^{-\frac{1}{2}\operatorname{Tr}\ln
Q_{\mu\nu}^{ab}}$ (A.3)

one ends up with

$\displaystyle\mathrm{e}^{-V\mathcal{E}_{v}}~{}=~{}\mathrm{e}^{-\frac{1}{2}\operatorname{Tr}\ln
Q_{\mu\nu}^{ab}+\operatorname{Tr}\ln\mathcal{P}^{ab}+\beta-\ln\beta}\;.$ (A.4)

The trace is understood to be taken over each space: color indices space;
space-time indices space; and the space of momentum configurations. The
logarithm of an operator has to be understood as the following series
expansion,

$\displaystyle\ln\left(\delta+G\right)~{}=~{}G-\frac{1}{2!}G^{2}+\cdots$ (A.5)

In the above equation (A.5), $\delta$ represents the operator corresponding to
the unitary transformation, while $G$ stands for a general transformation.
Therefore, the trace over the logarithm has to me understood as the trace over
the series (A.5). Namely,

$\displaystyle\operatorname{Tr}\ln\left(\delta+G\right)~{}=~{}\operatorname{Tr}\left\\{G-\frac{1}{2!}G^{2}+\cdots\right\\}\;.$
(A.6)

With that being said, the trace of equation (A.4) must be treated separately
in each space where the operator is acting: color space (indices of the
adjoint representation of the gauge field); the space-time indices, whose
unitary element is the (Euclidean) metric $\delta_{\mu\nu}$; and the space of
possible momentum configurations.

The trace over the color indices space results in an overall factor of
$(N^{2}-1)$ for the gauge and ghost fields contribution, namely

$\displaystyle\mathrm{e}^{-V\mathcal{E}_{v}}~{}=~{}\mathrm{e}^{-(N^{2}-1)\left[\frac{1}{2}\operatorname{Tr}\ln
Q_{\mu\nu}-\operatorname{Tr}\ln\mathcal{P}\right]+\beta-\ln\beta}\;.$ (A.7)

Let us now compute the trace over the space-time indices. In order to be clear
enough in the computation, only the term corresponding to the gauge field
contribution will be considered. After all, we will comeback to the whole
expression of the vacuum energy.

Then, looking at the gauge field contribution term one has,

$\displaystyle(N^{2}-1)\operatorname{Tr}\ln Q_{\mu\nu}$ $\displaystyle=$
$\displaystyle(N^{2}-1)\operatorname{Tr}\ln\left[\left(\frac{k^{4}+\gamma^{4}}{k^{2}}\right)\delta_{\mu\nu}+\left(\frac{1}{\alpha}-1\right)k_{\mu}k_{\nu}+M\epsilon_{\mu\nu\lambda}k_{\lambda}\right]\;.$
(A.8)

To simplify things, let us define the operator
$G~{}=~{}\left(k^{2}+\gamma^{4}k^{-2}\right)$, which coincides with the
inverse of the two point Green function of the gauge field. Now, let us put
this operator as an overall factor, such as

$\displaystyle(N^{2}-1)\operatorname{Tr}\ln Q_{\mu\nu}$ $\displaystyle=$
$\displaystyle(N^{2}-1)\operatorname{Tr}\ln\left[\left(G\delta_{\mu\sigma}\right)\left[\delta_{\sigma\nu}+G^{-1}\left(\frac{1}{\alpha}-1\right)k_{\sigma}k_{\nu}+MG^{-1}\epsilon_{\sigma\nu\lambda}k_{\lambda}\right]\right]\;,$
(A.9)

which can be rewritten as

$\displaystyle(N^{2}-1)\operatorname{Tr}\ln Q_{\mu\nu}$ $\displaystyle=$
$\displaystyle(N^{2}-1)\operatorname{Tr}\ln\left[G\delta_{\mu\sigma}\right]$
(A.10) $\displaystyle+$
$\displaystyle(N^{2}-1)\operatorname{Tr}\ln\left[\delta_{\sigma\nu}+G^{-1}\left(\frac{1}{\alpha}-1\right)k_{\sigma}k_{\nu}+G^{-1}M\epsilon_{\sigma\nu\lambda}k_{\lambda}\right]\;,$

with $G^{-1}~{}=~{}(k^{2}+\gamma^{4}k^{-2})^{-1}$ the inverse of the operator
$G$. The first term will generate a numerical factor $d$ for the dimension of
the space-time. The second term shall be worked out once more by putting in
evidence the operator

$\delta_{\sigma\nu}+G^{-1}\left(\frac{1}{\alpha}-1\right)k_{\sigma}k_{\nu}~{}=~{}\text{P}_{\sigma\nu}\;.$

Then, one ends up with

$\displaystyle-\frac{1}{2}(N^{2}-1)\operatorname{Tr}\ln Q_{\mu\nu}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}(N^{2}-1)\operatorname{Tr}\ln\left[G\delta_{\mu\sigma}\right]-\frac{1}{2}(N^{2}-1)\operatorname{Tr}\ln\left[\delta_{\sigma\theta}+G^{-1}\left(\frac{1}{\alpha}-1\right)k_{\sigma}k_{\theta}\right]$
(A.11)
$\displaystyle-\frac{1}{2}(N^{2}-1)\operatorname{Tr}\ln\left[\delta_{\theta\nu}+\text{P}^{-1}_{\theta\kappa}G^{-1}M\epsilon_{\kappa\nu\lambda}k_{\lambda}\right]\;,$

where $\text{P}^{-1}_{\theta\kappa}$ is the inverse operator of
$\text{P}_{\sigma\theta}$ and reads

$\displaystyle\text{P}^{-1}_{\theta\kappa}~{}=~{}\delta_{\theta\kappa}-\frac{G^{-1}\left(\frac{1}{\alpha}-1\right)}{G^{-1}\left(\frac{1}{\alpha}-1\right)k^{2}+1}k_{\theta}k_{\kappa}\;.$
(A.12)

The first two terms are exactly the same as the terms of the gluon
contribution to the vacuum energy in a Yang-Mills theory within the Gribov
prescription. Only in the last term of equation (A.11) the dependence on $M$
appears. Therefore, the effect of the CS term on the vacuum energy appears
only in the third term of (A.11).

The contribution of the gauge field to the vacuum energy will be split up into
two parts, one independent of $M$ and the other part depending on it. Let us
now take care of the first part, i.e. the one independent to CS massive
parameter, namely,

$\displaystyle 3(N^{2}-1)\operatorname{Tr}\ln
G+(N^{2}-1)\operatorname{Tr}\ln\left[\delta_{\sigma\theta}+\left(\frac{1}{\alpha}-1\right)G^{-1}k_{\sigma}k_{\theta}\right]$
(A.13) $\displaystyle=$ $\displaystyle 3(N^{2}-1)\operatorname{Tr}\ln G+$
$\displaystyle+$
$\displaystyle(N^{2}-1)\operatorname{Tr}\left[\left(\frac{1}{\alpha}-1\right)G^{-1}k_{\sigma}k_{\theta}-\frac{1}{2}\left(\frac{1}{\alpha}-1\right)^{2}G^{-2}k^{2}k_{\sigma}k_{\theta}+\cdots\right]\;.$

Taking the trace over the space-time indices, one gets

$\displaystyle 3(N^{2}-1)\operatorname{Tr}\ln
G+(N^{2}-1)\operatorname{Tr}\left[\left(\frac{1}{\alpha}-1\right)G^{-1}k^{2}-\frac{1}{2}\left(\frac{1}{\alpha}-1\right)^{2}G^{-2}k^{4}+\cdots\right]$
(A.14) $\displaystyle=$ $\displaystyle 2(N^{2}-1)\operatorname{Tr}\ln
G+(N^{2}-1)\operatorname{Tr}\ln\left[G\alpha
k^{-2}+\left(1-\alpha\right)\right]+(N^{2}-1)\operatorname{Tr}\ln\frac{k^{2}}{\alpha}\;.$

Notice that, in the limit $\alpha\to 0$ the term

$\operatorname{Tr}\ln\left[G\alpha
k^{-2}+\left(\alpha-1\right)\right]~{}\to~{}0\;.$

Although the term $\operatorname{Tr}\ln(k^{2}/\alpha)$diverges (in principle).
Indeed, such a $\alpha^{-1}$ dependent term diverges in the discrete momentum
configuration space. However, in the continuum 3-dimensional momentum space,
it is possible to show that such term converges to zero. In order to prove
this, let us take a look at this term separately,

$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\;\frac{k^{2}}{\alpha}~{}=~{}\alpha^{2}\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\;k^{{}^{\prime}2}\;,$
(A.15)

which goes to zero in the limit $\alpha\to 0$. The last equality of (A.15) can
be taken by changing $k^{2}\to k^{{}^{\prime}2}=k^{2}/\alpha$.

Therefore, it is safe to say that, in the Landau gauge limit, $\alpha\to 0$,
the vacuum energy reads,

$\displaystyle-V\mathcal{E}_{v}$ $\displaystyle=$
$\displaystyle\beta-\ln\beta-(N^{2}-1)\operatorname{Tr}\ln G+$ (A.16)
$\displaystyle-$
$\displaystyle\frac{1}{2}(N^{2}-1)\operatorname{Tr}\ln\left[\delta_{\theta\nu}+\text{P}^{-1}_{\theta\kappa}G^{-1}M\epsilon_{\kappa\nu\lambda}k_{\lambda}\right]+(N^{2}-1)\operatorname{Tr}\ln[k^{2}]\;.$

In the continuum momentum space it is possible to show that the ghost
contribution, $\operatorname{Tr}\ln k^{2}$, is null by dimensional
regularization.

In order to obtain a closed form for the vacuum energy, the contribution
coming from the CS term must be evaluated. Notice that this term can be
simplified to

$\displaystyle\operatorname{Tr}\ln\left[\delta_{\theta\nu}+\text{P}^{-1}_{\theta\kappa}G^{-1}M\epsilon_{\kappa\nu\lambda}k_{\lambda}\right]~{}=~{}\operatorname{Tr}\ln\left[\delta_{\theta\nu}+G^{-1}M\epsilon_{\theta\nu\lambda}k_{\lambda}\right]\;.$
(A.17)

Notice that this trace is not zero whatsoever. Taking the logarithm expansion,
it is not difficult to see that the trace of the first term is zero. However,
the trace of the second does not vanish. Namely,

$\displaystyle\operatorname{Tr}\ln\left[\delta_{\theta\nu}+G^{-1}M\epsilon_{\theta\nu\lambda}k_{\lambda}\right]$
$\displaystyle=$
$\displaystyle\operatorname{Tr}\Bigg{\\{}G^{-1}M\epsilon_{\theta\nu\lambda}k_{\lambda}-\frac{M^{2}}{2}G^{-1}G^{-1}\epsilon_{\theta\kappa\lambda}\epsilon_{\kappa\nu\lambda^{\prime}}k_{\lambda}\;k_{\lambda^{\prime}}$
$\displaystyle+\frac{2m^{3}}{3!}G^{-1}G^{-1}G^{-1}\epsilon_{\theta\kappa\lambda}\epsilon_{\kappa\omega\lambda^{\prime}}\epsilon_{\omega\nu\lambda^{\prime\prime}}k_{\lambda}\;k_{\lambda^{\prime}}\;k_{\lambda^{\prime\prime}}-\cdots\Bigg{\\}}\;.$

Let us now compute term by term of this expansion. For that, it will be useful
to have the rule of contraction of two Levi-Civita symbols:

$\displaystyle\epsilon_{\kappa\nu\lambda}\epsilon_{\kappa\mu\delta}$
$\displaystyle=$
$\displaystyle\delta_{\nu\delta}\delta_{\lambda\mu}-\delta_{\nu\mu}\delta_{\lambda\delta}\;.$
(A.19)

  * I.

The contribution to the trace linear in $\xi$:

The trace over the first term of right-hand-side of equation (A) is clearly
zero, since the Levi-Civita symbol is anti-symmetric in the $\theta,\nu$
indices. However, not all terms are zero, and some of them must be computed.
Following, the second, third and fourth terms will be computed.

  * II.

The contribution to the trace proportional to $M^{2}$:

Making use of (A.19) one gets,

$\displaystyle\epsilon_{\nu\kappa\lambda}\epsilon_{\kappa\mu\delta}k_{\lambda}k_{\delta}$
$\displaystyle=$ $\displaystyle k_{\mu}k_{\nu}-k^{2}\delta_{\nu\mu}$

$\displaystyle\operatorname{Tr}(\epsilon_{\nu\kappa\lambda}\epsilon_{\kappa\mu\delta}\partial_{\lambda}\partial_{\delta})~{}=~{}-2\sum_{k}\;k^{2}\;.$
(A.20)

Therefore, the trace over the CS contribution to the vacuum energy is not zero
anymore.

  * III.

The contribution to the trace proportional to $\xi^{3}$:

$\displaystyle\operatorname{Tr}\Big{[}\epsilon_{\theta\kappa\lambda}\epsilon_{\kappa\omega\lambda^{\prime}}\epsilon_{\omega\nu\lambda^{\prime\prime}}k_{\lambda}\;k_{\lambda^{\prime}}\;k_{\lambda^{\prime\prime}}\Big{]}=0\,.$
(A.21)

Due to the anti-symmetry feature of the Levi-Civita simbol, the contribution
proportional to $M^{3}$ is null;

  * IV.

The contribution to the trace proportional to $M^{4}$:

$\displaystyle
Tr\Big{[}\epsilon_{\theta\kappa\lambda}\epsilon_{\kappa\rho\alpha}\epsilon_{\rho\sigma\beta}\epsilon_{\sigma\nu\omega}\;\;k_{\lambda}\;k_{\lambda^{\prime}}\;k_{\lambda^{\prime\prime}}\;k_{\tilde{\lambda}}\Big{]}$
$\displaystyle=$ $\displaystyle 2\sum_{k}\;k^{4}.$

Finally, the series expansion of the contribution to the vaccum energy that
comes from (A.17) reads,

$\displaystyle\operatorname{Tr}\ln\left[\delta_{\theta\nu}+iG^{-1}\epsilon_{\theta\nu\lambda}M\,\partial_{\lambda}\right]~{}=~{}2\sum_{k}\sum_{n=1}\;\frac{1}{2n}\left(\frac{k^{2}}{k^{4}+\gamma^{4}}\right)^{2n}k^{2n}M^{2n}\,,$
(A.22)

which gives the logarithmic function. Namely,

$\displaystyle-\frac{(N^{2}-1)}{2}\operatorname{Tr}\ln\left[\delta_{\theta\nu}+iG^{-1}\epsilon_{\theta\nu\lambda}M\,\partial_{\lambda}\right]~{}=~{}-\frac{2(N^{2}-1)}{{4}}\sum_{k}\;\ln\left\\{\left(\frac{(k^{4}+\gamma^{4})^{2}+k^{6}M^{2}}{(k^{4}+\gamma^{4})^{2}}\right)\right\\}.$
(A.23)

Finally, the expression for the vacuum energy can be explicitly written as

$\displaystyle-V\mathcal{E}_{v}$ $\displaystyle=$
$\displaystyle\beta-\ln\beta-\frac{(N^{2}-1)}{2}\sum_{k}\;\ln\left[\left(\frac{(k^{4}+\gamma^{4})^{2}+k^{6}M^{2}}{k^{4}}\right)\right].$
(A.24)

Within the thermodynamic limit, when the volume goes to infinity but still
keeping finite intensive quantities, such as the Gribov parameter that can be
written as $\beta\to V\beta$ in order to make explicit its intensive feature,
the vacuum energy density reads,

$\displaystyle\mathcal{E}_{v}$ $\displaystyle=$
$\displaystyle-\beta+\frac{(N^{2}-1)}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\ln\left[\left(\frac{(k^{4}+\gamma^{4})^{2}+k^{6}M^{2}}{k^{4}}\right)\right].$
(A.25)

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