[**[INFRARED REGION OF QCD AND CONFINEMENT]{}**]{}

R.Parthasarathy[[^1]]{}\
The Institute of Mathematical Sciences\
C.P.T.Campus, Taramani Post\
Chennai - 600 113 , India\

[[*[**[Abstract]{}**]{}*]{}]{}

Gauge field configurations appropriate for the infrared region of QCD is proposed. Using the usual QCD action, confinement is realized as in the London theory of Meissner effect.

[[PACS numbers: 11.15.-q; 12.38Aw]{}]{}

The mechanism of confinement of gluons and quarks is very important to understand. A suggestion of ’tHooft$^{1}$ and Mandelstam$^{2}$ is that magnetic monopoles must play a crucial role in confinement. The high energy limit of QCD has been studied by perturbation theory due to the small coupling strength of the QCD running coupling constant and experimental results confirm the use of perturbation theory in this region$^{3}$. Here the gluons are massless and the color symmetry is an exact symmetry. At low energies, the coupling is strong and non-perturbative methods are to be invoked. This region is not completely understood. Color confinement is possibly due to a dual Meissner effect$^{2,4,5}$ and this suggests that in the infrared region, some other variables, other than $A^a_{\mu}$, may be more relevant. The idea of ’tHooft$^{1}$ is to make an Abelian projection and Kondo$^{6}$ has examined this for $SU(2)$ gauge theory by integrating the non-Abelian degrees of freedom to obtain an Abelian projected effective field theory. Here, the relation between the non-Abelian gauge fields and the monopole configuration is not clear. Recently, Faddeev and Niemi$^{7}$ have proposed a set of variables for describing the infra-red limit of 4-dimensional $SU(2)$ and $SU(N)$ gauge theory. This is a continuation of the earlier works of Corrigan et.al.,$^{8}$ and Cho$^{9}$. In Ref.9, a complex scalar field $\phi$ for the monopole has been introduced and confinement is suggested by dynamically breaking (Coleman-Weinberg mechanism) the magnetic symmetry. On the other hand, Faddeev and Niemi$^{7}$ propose a non-linear sigma model action as relevant in the infra-red limit of Yang-Mills theory, motivated by Wilsonian renormalization group arguments.

In this Letter, a gauge field configuration for the infra-red limit of QCD is proposed in which the magnetic confinement of gluons and quarks will be realized. By considering the dual version, the electric confinement will as well be realized.

The proposed $SU(3)$ gauge field configurations $A^a_{\mu}$ , for the infrared region of QCD, is taken to satisfy $$\begin{aligned}
D^{ab}_{\mu}{\omega}^b\ =\ {\partial}_{\mu}{\omega}^a + g
f^{acb}A^c_{\mu}{\omega}^b &=& 0,\end{aligned}$$ where ${\omega}^a\ \in \ su(3)$ and are chosen such that $$\begin{aligned}
{\omega}^a{\omega}^a &=& 1 \nonumber \\
d^{abc}{\omega}^b{\omega}^c& = & 
\frac{1}{\sqrt{3}}\ {\omega}^a,\end{aligned}$$ with $d^{abc}$ as the symmetric Gell-Mann $SU(3)$ tensor. Now, in the infra-red region, the gauge group manifold is not the complete $su(3)$ group manifold spanned by arbitrary ${\omega}^a$’s but a [*[submanifold spanned by those ${\omega}^a$’s satisfying (2) ]{}*]{} and the relevant gauge field configurations $A^a_{\mu}$ are determined by (1). Classically, the Wu and Wu$^{10}$ ansatz for the $SU(3)$ gauge field, satisfying the Euler-Lagrange equations of pure $SU(3)$ gauge theory and the construction ${\omega}^a\ =\
d^{abc}F^b_{\mu\nu}F^c_{\mu\nu}$ satisfy (1) and (2)$^{11}$. [*[Instead, here we will choose to solve (1) for $A^a_{\mu}$]{}*]{}. The second relation in (2) is very [*[special]{}*]{} to $SU(3)$ and will be [*[crucial]{}*]{} in what follows.

A solution to (1)$^{12}$ is $$\begin{aligned}
A^a_{\mu} &=& C_{\mu}\ {\omega}^a - \frac{4}{3g}f^{abc}\
{\omega}^b\ {\partial}_{\mu}{\omega}^c,\end{aligned}$$ where $C_{\mu}$ is arbitrary. In obtaining (3), we have derived a useful relation for ${\omega}^a$’s in (2), $$\begin{aligned}
\frac{4}{3}f^{abc}f^{edc}{\omega}^b{\omega}^e{\partial}_{\mu}
{\omega}^d &=& - {\partial}_{\mu}{\omega}^a,\end{aligned}$$ which will be used subsequently. The gauge field configuration (3) and the submanifold determined by (2) are proposed to describe the infra-red region of QCD. Relations (1), (2) and (3) are self-consistent. It is to be noted here that ${\omega}^a\ A^a_{\mu}\ =\ C_{\mu}\ \neq \ 0$, in contrast to the classical configuration described in Ref.11.

There are two important consequences that follow from (1). Usually a mass term for the gauge fields, $m^2
A^a_{\mu}A^a_{\mu}$ cannot be present in the Lagrangian density due to its gauge non-invariance. Since under a gauge transformation, the field changes as $\delta A^a_{\mu}\ =\
D^{ab}_{\mu}{\omega}^b$, in view of (1), if we consider gauge transformations within the submanifold, then such a mass term is allowed in it. This is similar to defining the state vector in the Heisenberg picture in quantum mechanics$^{13}$. ${\psi}_H\ =\ exp(-i\frac{Ht}{\hbar}){\psi}_S(t)$, i.e., as ${\psi}_S(t)$ evolves in time, it is brought back to ${\psi}_H\ =\ {\psi}_S(0)$ at every instant. Second, if we choose a Lorentz covariant gauge ${\partial}_{\mu}A^a_{\mu}\
=\ 0$, for $A^a_{\mu}$’s in (3), then there will be no Gribov ambiguity$^{14}$ in fixing the gauge in view of (1), as long as we are within the submanifold.

The field strength $F^a_{\mu\nu}\ =\
{\partial}_{\mu}A^a_{\nu}-{\partial}_{\nu}A^a_{\mu}+gf^{abc}
A^b_{\mu}A^c_{\nu}$, for $A^a_{\mu}$ in (3) is calculated, using the relations (2), (4) and the relations among the $f$ and $d$ tensors$^{15}$, as $$\begin{aligned}
F^a_{\mu\nu} &=&
({\partial}_{\mu}C_{\nu}-{\partial}_{\nu}C_{\mu})\ {\omega}^a
 -
\frac{4}{3g}\ f^{abc}{\partial}_{\mu}{\omega}^b{\partial}_
{\nu}{\omega}^c.\end{aligned}$$ It is found that $F^a_{\mu\nu}$ is “$SU(3)$ orthogonal” to ${\omega}^a$ $^{16}$, i.e., $$\begin{aligned}
f^{abc}\ {\omega}^b\ F^c_{\mu\nu} &=& 0,\end{aligned}$$ consistent with (1). Unlike the case of $SU(2)$ $^{8,9}$, this does not mean that $F^a_{\mu\nu}$ is along ${\omega}^a$. Although the first term in (5) is along ${\omega}^a$, the second term is not. If a tensor field strength $G_{\mu\nu}$ is constructed as $G_{\mu\nu}\ =\ {\omega}^a F^a_{\mu\nu}$, then $G_{\mu\nu}\
=\ f_{\mu\nu} -
\frac{4}{3g}f^{abc}{\omega}^a{\partial}_{\mu}{\omega}^b
{\partial}_{\nu}{\omega}^c$, where $f_{\mu\nu}\ =\
{\partial}_{\mu}C_{\nu}-{\partial}_{\nu}C_{\mu}$. If an action $-\frac{1}{4}\int G^2_{\mu\nu}\ d^4x$ is considered, then the action becomes, $$\begin{aligned}
S &=& -\frac{1}{4}\int \{
f^2_{\mu\nu}-\frac{8}{3g}f_{\mu\nu}X_{\mu\nu} + \frac{16}
{9g^2}X_{\mu\nu}X_{\mu\nu}\} d^4x, \nonumber\end{aligned}$$ where $X_{\mu\nu}\ =\
f^{abc}{\omega}^a{\partial}_{\mu}{\omega}^b{\partial}_{\nu}
{\omega}^c$. This action coincides with the structure of the effective action for Abelian projected QCD of Kondo$^6$. Instead of this procedure, we will consider the [*[usual]{}*]{} action for pure QCD. $$\begin{aligned}
I &=& -\frac{1}{4}\int (F^a_{\mu\nu})^2\ d^4x, \nonumber \\
 &=& -\frac{1}{4}\int \{ f^2_{\mu\nu} -
\frac{8}{3g}f_{\mu\nu}(f^{abc}{\omega}^a{\partial}_{\mu}
{\omega}^b{\partial}_{\nu}{\omega}^c) \nonumber \\
&+& \frac{16}{9g^2}f^{abc}f^{aed}{\partial}_{\mu}
{\omega}^b{\partial}_{\nu}{\omega}^c{\partial}_{\mu}
{\omega}^e{\partial}_{\nu}{\omega}^d\ \} d^4x.\end{aligned}$$ This action has $U(1)$ symmetry, $C_{\mu}\ \rightarrow \
C_{\mu} + {\partial}_{\mu}\Lambda$. It will be convenient to rescale the ${\omega}^a$’s as $g^{\frac{1}{3}}{\omega}^a$ and then in the strong coupling limit (The last term in (7) is omitted hereafter in the strong coupling limit. It can be included at the end and this will not change the conclusions), the above action becomes $$\begin{aligned}
I &\simeq & -\frac{1}{4}\int \{ f^2_{\mu\nu} -
\frac{8}{3}f_{\mu\nu}(f^{abc}{\omega}^a{\partial}_{\mu}
{\omega}^b{\partial}_{\nu}{\omega}^c)\} d^4x,\end{aligned}$$ showing the Abelian dominance for the infra-red limit of QCD. Here the field strength $f_{\mu\nu}$ is coupled to $X_{\mu\nu}\ =\
f^{abc}{\omega}^a{\partial}_{\mu}{\omega}^b{\partial}_{\nu}
{\omega}^c$. First of all it is seen that ${\partial}_{\mu}X_{\mu\nu}\ =\ f^{abc}{\omega}^a {\partial}
_{\mu}({\partial}_{\mu}{\omega}^b{\partial}_{\nu}{\omega}^c)
\ \neq \ 0$ and the dual strength $\tilde{X}_{\mu\nu}\ =\ 
\frac{1}{2}{\epsilon}_{\mu\nu\alpha\beta}X_{\alpha\beta}$ violates the Bianchi identity, $$\begin{aligned}
{\partial}_{\mu}\tilde{X}_{\mu\nu}&=& -\frac{2}{3}{\epsilon}
_{\mu\nu\alpha\beta}f^{abc}{\partial}_{\mu}{\omega}^a
{\partial}_{\alpha}{\omega}^b{\partial}_{\beta}   
{\omega}^c \ \neq \ 0.\end{aligned}$$ This along with $$\begin{aligned}
-\frac{2}{3}\oint_{S}{\epsilon}_{\mu\nu\alpha\beta}f^{abc}
{\omega}^a{\partial}_{\alpha}{\omega}^b{\partial}_{\beta}
{\omega}^c dx^{\mu}\wedge dx^{\nu}\ =\ -\frac{4}{3}\oint_
{S}\tilde{X}_{\mu\nu}dx^{\mu}\wedge dx^{\nu} \ =\ 4\pi M,\end{aligned}$$ a topological invariant, imply that magnetic monopoles are present in (8) and coupled to $f_{\mu\nu}$. Second, as in the infra-red region, we proposed the gauge field configuration (3) in the submanifold determined by (2), a mass term $m^2
A^a_{\mu}A^a_{\mu}$ can be introduced. From (3) it follows that $$\begin{aligned}
A^a_{\mu}A^a_{\mu}&=&
C_{\mu}C_{\mu}+\frac{4}{3g^2}{\partial}_{\mu}{\omega}^a
{\partial}_{\mu}{\omega}^a.\end{aligned}$$ After rescaling ${\omega}^a$ by $g^{\frac{1}{3}}$ as used in (8) and taking the strong coupling limit, $A^a_{\mu}A^a_{\mu}\ \simeq \ C_{\mu}C_{\mu}$. Thus a mass term for $A^a_{\mu}$ induces a mass term for $C_{\mu}$. This breaks the $U(1)$ invariance. A mass term for $C_{\mu}$ can also be induced by Coleman-Weinberg mechanism by introducing complex scalars $\phi$ minimally coupled to $C_{\mu}$ as in the works of Mandelstam$^2$, ’tHooft$^1$ and Cho$^{9}$. Then the action (8) becomes $$\begin{aligned}
I&=&-\frac{1}{4}\int \{
f^2_{\mu\nu}-\frac{8}{3}f_{\mu\nu}X_{\mu\nu}+m^2C_{\mu}
C_{\mu}\} d^4x.\end{aligned}$$ The partition function for the above is $$\begin{aligned}
Z&=& \int [dC_{\mu}]exp(-S),\end{aligned}$$ and by integrating over the $C_{\mu}$ field (we suppress the gauge fixing and ghost terms here as we have only Abelian fields), we find $$\begin{aligned}
Z&=& det^{-1}(\Box  - \frac{m^2}{2})\
exp(\int \frac{8}{9}{\partial}_{\mu}X_{\mu\nu}(\Box 
  -
 \frac{m^2}{2})^{-1}{\partial}_{\rho}X_{\rho\nu} d^4x ).\end{aligned}$$ The effective Lagrangian from this is, apart from the constant divergent factor, $$\begin{aligned}
L_{eff}&\simeq &-\frac{8}{9} {\partial}_{\mu}X_{\mu\nu} (\Box 
  - \frac{m^2}{2})^{-1}{\partial}_{\rho}X_{\rho
\nu},\end{aligned}$$ and as ${\partial}_{\mu}X_{\mu\nu}$ is non-vanishing, this form is identical to the London case of magnetic confinement. This provides a magnetic confinement of gluons in the submanifold providing a scenario suggested by Nambu$^{5}$ that color confinement could occur similar to the magnetic confinement in an ordinary superconductor due to the Meissner effect.

It is possible to provide electric confinement by rewriting the action (8) in its dual form. The generating functional for (8) is $$\begin{aligned}
Z&=& \int [dC_{\mu}]exp(\frac{1}{4}\int \{ f^2_{\mu\nu} -
\frac{8}{3}f_{\mu\nu}X_{\mu\nu}\}d^4x.\end{aligned}$$ Introducing the dual field strength ${\cal{G}}_{\mu\nu}$ dual to $f_{\mu\nu}$, we can rewrite the above $Z$ as $$\begin{aligned}
Z&=&\int [dC_{\mu}][d{\cal{G}}_{\mu\nu}]exp{\left(
-\frac{1}{4}{\cal{G}}^2_{\mu\nu}+\frac{1}{2}{\cal{G}}_{\mu\nu}
f_{\mu\nu}-\frac{2}{3}f_{\mu\nu}X_{\mu\nu}\right)}d^4x,\end{aligned}$$ as the functional integration over ${\cal{G}}_{\mu\nu}$ brings (15). From (16), variation with respect to $C_{\mu}$-field ($f_{\mu\nu}\ =\
{\partial}_{\mu}C_{\nu}-{\partial}_{\nu}C_{\mu}$) gives ${\partial}_{\mu}\{ {\cal{G}}_{\mu\nu}-\frac{4}{3}X_{\mu\nu}\}\
=\ 0$, which is solved for ${\cal{G}}_{\mu\nu}$ as $$\begin{aligned}
{\cal{G}}_{\mu\nu} &=&
{\epsilon}_{\mu\nu\lambda\sigma}{\partial}_{\lambda}{\tilde{A
}}_{\sigma}+\frac{4}{3}X_{\mu\nu}.\end{aligned}$$ The field $\tilde{A}_{\mu}$ serves as dual to $C_{\mu}$. Use of (17) in (16) eliminates $C_{\mu}$-field and we obtain $$\begin{aligned}
Z&=& \int
[d\tilde{A}_{\mu}]exp[\frac{1}{4}\int\{({\partial}_{\lambda}
\tilde{A}_{\sigma}-{\partial}_{\sigma}\tilde{A}_{\lambda})^2
\nonumber \\
&+&\frac{8}{3}X_{\mu\nu}{\epsilon}_{\mu\nu\lambda\sigma}
{\partial}_{\lambda}\tilde{A}_{\sigma}+\frac{16}{9}
X_{\mu\nu}X_{\mu\nu}\}d^4x].\end{aligned}$$ This action possess dual $U(1)$ invariance and coincides with the Abelian projected effective field theory for QCD based on $SU(3)$ in its [*[dual]{}*]{} form. Denoting ${\tilde{X}}_{\lambda\sigma}\ =\ \frac{1}{2}{\epsilon}_{
\lambda\sigma\mu\nu}X_{\mu\nu}$, we have $$\begin{aligned}
Z&=& \int [d\tilde{A}_{\mu}] exp[\frac{1}{4}\int \{ (
{\partial}_{\lambda}\tilde{A}_{\sigma}-{\partial}_{\sigma}
\tilde{A}_{\lambda})^2 \nonumber \\
&+& \frac{16}{3}\tilde{X}_{\lambda\sigma}{\partial}_{\lambda}
\tilde{A}_{\sigma}+\frac{16}{9}X_{\mu\nu}X_{\mu\nu}+M^2
\tilde{A}_{\mu}\tilde{A}_{\mu}\}d^4x],\end{aligned}$$ where a mass term for $\tilde{A}_{\mu}$ is introduced which can be thought of arising from Coleman-Weinberg mechanism by invoking complex scalars for example. By functionally integrating the dual field $\tilde{A}_{\mu}$, the effective Lagrangian density, apart from constant (divergent) factors not involving fields, is found to be $$\begin{aligned}
{\cal{L}}_{eff}&=&
-\frac{8}{9}{\partial}_{\lambda}\tilde{X}_{\lambda\sigma}
(\Box  - \frac{M^2}{2})^{-1} {\partial}_{\rho}
\tilde{X}_{\rho\sigma} - \frac{4}{9}\tilde{X}_{\mu\nu}
\tilde{X}_{\mu\nu},\end{aligned}$$ which gives the [*[electric]{}*]{} (dual) confinement of gluons in the submanifold.

The reason for realizing both magnetic and electric confinement is the property of $X_{\mu\nu}\ =\
f^{abc}{\omega}^a{\partial}_{\mu}{\omega}^b{\partial}_{\nu}
{\omega}^c$ that both ${\partial}_{\mu}X_{\mu\nu}$ and ${\partial}_{\mu}\tilde{X}_{\mu\nu}$ are non-vanishing. These fields $X_{\mu\nu}$ and $\tilde{X}_{\mu\nu}$ couple to $f_{\mu\nu}$ and its dual. While the non-vanishing of ${\partial}_{\mu}\tilde{X}_{\mu\nu}$ violating the Bianchi identity is attributed to the monopole topological configuration (in (9) and (10)), that of ${\partial}_{\mu}X_{\mu\nu}$ is due to the construction itself.

Now we consider the QCD action with quarks. The standard action is $$\begin{aligned}
I&=& -\frac{1}{4}\int F^a_{\mu\nu}F^a_{\mu\nu} d^4x + \int
\bar{\psi}i{\gamma}^{\mu}({\partial}_{\mu}+ig\frac{1}{2}
{\lambda}^a\ A^a_{\mu})\psi \ d^4x.\end{aligned}$$ The first term has been expressed in terms of $C_{\mu}$ and ${\omega}^a$. The Dirac current $j^a_{\mu}\ =\
g\bar{\psi}{\gamma}^{\mu}\frac{1}{2}{\lambda}^a \psi$ couples to $A^a_{\mu}$ minimally. Using the expression (3) for $A^a_{\mu}$, we find the action becomes $$\begin{aligned}
I&=& -\frac{1}{4}\int \{ f^2_{\mu\nu} - \frac{8}{3}f_{\mu\nu}
X_{\mu\nu} + m^2 C_{\mu}C_{\mu}\} d^4x  \nonumber \\
&+& \int \{\bar{\psi}{\gamma}^{\mu}i{\partial}_{\mu}\psi -   
j^a_{\mu}{\omega}^a C_{\mu} -
\frac{4}{3g}j^a_{\mu}f^{abc}{\omega}^b{\partial}_{\mu}
{\omega}^c \} d^4x.\end{aligned}$$ In the corresponding partition function, the $C_{\mu}$ field is integrated to give the effective Lagrangian density $$\begin{aligned}
L_{eff} &=& -\frac{8}{9}{\partial}_{\mu}X_{\mu\nu}(\Box -
\frac{m^2}{2})^{-1}{\partial}_{\rho}X_{\rho\nu} - \frac{1}{2} 
j^a_{\mu}
{\omega}^a (\Box - \frac{m^2}{2})^{-1}j^b_{\mu}{\omega}^b
\nonumber \\ 
&+& \frac{4}{3}{\partial}_{\mu}X_{\mu\nu}(\Box -
\frac{m^2}{2})^{-1}j^a_{\nu}{\omega}^a - \frac{4}{3g}j^a_{\mu}
f^{abc}{\omega}^b{\partial}_{\mu}{\omega}^c ,\end{aligned}$$ apart from the kinetic energy term for quarks. This is identical to the London case of magnetic confinement including the quarks. In addition the monopole current interacts with the Dirac current through $(\Box - \frac{m^2}{2})^{-1}$. A similar procedure can be effected for the dual version by coupling quarks to dual field $\bar{A}_{\mu}$.

Thus, by considering a submanifold for the group $SU(3)$ defined by both the relations in (2) and proposing the gauge field $A^a_{\mu}$ configuration (3) satisfying (1), we obtain a $F^a_{\mu\nu}$ “$SU(3)$ orthogonal” to ${\omega}^a$. The action for QCD (7) leads to magnetic confinement of gluons as in the London theory of Meissner effect. The dual version exists and produces the electric confinement. The mechanism has been extended to include quarks. Here, the magnetic and electric confinement of quarks is obtained in addition to an interaction between the monopole current and the Dirac current.

[**[Acknowledgements]{}**]{}

I am thankful to H.Sharatchandra, Ramesh Anishetty, R.Sridhar and G.Baskaran for very useful discussions.

[**[References]{}**]{}

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12. To show that (3) solves (1), substitute (3) in (1) and use (2). Then, $D^{ab}_{\mu}\ {\omega}^b \ =\
    {\partial}_{\mu}\ {\omega}^a \ +\ f^{abc}f^{edc}\ {\omega}^b\
    {\omega}^e\ {\partial}_{\mu}\ {\omega}^d.$ We make use of the standard relations between $f$’ and $d$’ tensors for $SU(3)$ group. See equations 2.8, 2.9 and 2.10 of A.J.Macfarlane, A.Sudberg and P.H.Weise, Commun.Math.Phys. [**11**]{}, 77 (1968).Then, $f^{abc}f^{edc}\ =\
    \frac{2}{3}({\delta}_{ae}{\delta}_{bd}-{\delta}_{ad}{\delta}_
    {be}) + d^{aec}d^{dbc}-d^{adc}d^{bec}$. Also, ${\omega}^b\
    {\partial}_{\mu}\ {\omega}^b \ =\ 0$ using the first relation in (2). These give $D^{ab}_{\mu}\ {\omega}^b \ =\ {\partial}_{\mu}
    {\omega}^a\ +\ \frac{4}{3}\ \{ -\frac{2}{3}({\partial}_{\mu}
    {\omega}^a) \ +\ d^{aec}d^{dbc} \ {\omega}^b\ {\omega}^e \ (
    {\partial}_{\mu}{\omega}^d) \ -\ d^{adc}d^{bec}\ {\omega}^b\ 
    {\omega}^e\ ({\partial}_{\mu}{\omega}^d)\}$. From the second relation in (2), it follows $d^{abc}\ ({\partial}_{\mu}
    {\omega}^b)\ {\omega}^c\ =\ \frac{1}{2\sqrt{3}}\ {\partial}_
    {\mu} {\omega}^a$. With this, we have $d^{aec}d^{dbc}\ 
    {\omega}^b \ {\omega}^e\ ({\partial}_{\mu}{\omega}^d)\ =\ 
    \frac{1}{12}({\partial}_{\mu}{\omega}^a)$ and $d^{adc}d^{bec}
    \ {\omega}^b\ {\omega}^e\ ({\partial}_{\mu}{\omega}^d)\ =\ 
    \frac{1}{6}\ ({\partial}_{\mu}{\omega}^a)$. Then, $D^{ab}_
    {\mu}\ {\omega}^a\ =\ {\partial}_{\mu}{\omega}^a\ +\ \frac
    {4}{3}\{-\frac{2}{3}+\frac{1}{12}-\frac{1}{6}\}\ ({\partial}
    _{\mu}{\omega}^a)\ =\ 0$.

13. P.Roman, [*[Advanced Quantum Theory -\
    An outline of the fundamental ideas]{}*]{}\
    Addison-Wesley Pub.Co.Inc. Reading, Massachusetts; 1965.

14. V.N.Gribov, Nucl.Phys. [**B139**]{}, 1 (1978).

15. From (3), we have ${\partial}_{\mu}A^a_{\nu}-{\partial}_{\nu}A^a_{\mu}\ =\ (
    {\partial}_{\mu}C_{\nu}-{\partial}_{\nu}C_{\mu}){\omega}^a
    +C_{\nu}({\partial}_{\mu}{\omega}^a)-C_{\mu}({\partial}_{\nu}
    {\omega}^a)-\frac{8}{3g}f^{abc}{\partial}_{\mu}{\omega}^b \ 
    {\partial}_{\nu}{\omega}^c$. Using (4), the non-Abelian part is $gf^{abc}A^b_{\mu}A^c_{\nu}\ =\ C_{\mu}({\partial}_{\nu}
    {\omega}^a)-C_{\nu}({\partial}_{\mu}{\omega}^a) + \frac{16}
    {9g}f^{abc}f^{bed}f^{c\ell m}{\omega}^e{\omega}^{\ell}(
    {\partial}_{\mu}{\omega}^d)({\partial}_{\nu}{\omega}^m)$, so that $F^a_{\mu\nu}\ =\ ({\partial}_{\mu}C_{\nu}-{\partial}_
    {\nu}C_{\mu}){\omega}^a\ -\ \frac{8}{3g}f^{abc}({\partial}_
    {\mu}{\omega}^b)({\partial}_{\nu}{\omega}^c)\ +\ \frac{16}
    {9g}f^{abc}f^{bed}f^{c\ell m}{\omega}^e{\omega}^{\ell}
    {\partial}_{\mu}{\omega}^d\ {\partial}_{\nu}{\omega}^m$. Use of Jacobi identity and the relation (4) for the last term in the previous line (the one involving three $f$’s) yields, $F^a_{\mu\nu}\ =\ ({\partial}_{\mu}C_{\nu}-{\partial}_{\nu}
    C_{\mu}){\omega}^a\ -\ \frac{4}{3g}f^{abc}{\partial}_{\mu}
    {\omega}^b\ {\partial}_{\nu}{\omega}^c\ -\ \frac{16}{9g}
    f^{a\ell c}f^{bed}f^{mbc}{\omega}^e{\omega}^{\ell}
    {\partial}_{\mu}{\omega}^d\ {\partial}_{\nu}{\omega}^m$. Now using $f^{edb}f^{cmb}\ =\ \frac{2}{3}({\delta}_{ec}{\delta}
    _{dm}\ -\ {\delta}_{em}{\delta}_{cd})\ +\ d^{ecb}d^{dmb}\ -\
    d^{deb}d^{emb}$ and the relations in (2), we find the last term in the previous line vanishes.

16. To see this, the first term in (5) trivially satisfies (6). The second term in (5), when substituted in (6) gives $f^{abc}f^{ced}{\omega}^b{\partial}_{\mu}{\omega}^e\ 
    {\partial}_{\nu}{\omega}^d\ =\ \frac{2}{3}({\delta}_{ae}
    {\delta}_{bd}\ -\ {\delta}_{ad}{\delta}_{be}){\omega}^b
    {\partial}_{\mu}{\omega}^e{\partial}_{\nu}{\omega}^d\ +\ 
    (d^{aec}d^{bdc}\ -\ d^{bec}d^{adc}){\omega}^b\ {\partial}
    _{\mu}{\omega}^e\ {\partial}_{\nu}{\omega}^d$. The Kronecker $\delta$-terms vanish due to the first relation in (2). Use of the second relation in (2) implies, $d^{abc}{\omega}^c\
    {\partial}_{\mu}{\omega}^b\ =\ \frac{1}{2\sqrt{3}}{\partial}
    _{\mu}{\omega}^a$ and this simplifies the remaining terms as $\frac{1}{2\sqrt{3}}d^{aec}{\partial}_{\mu}{\omega}^e\
    {\partial}_{\nu}{\omega}^c\ -\
    \frac{1}{2\sqrt{3}}d^{adc}{\partial}_{\mu}{\omega}^c{\partial}
    _{\nu}{\omega}^d$ which cancel each other.

[^1]: e-mail address: sarathy@imsc.ernet.in
