Stable, finite energy density solutions in the effective theory of non-abelian gauge fields

TL;DR

This paper derives static solutions in an effective non-Abelian gauge theory that have finite energy density and suggest a natural mass scale related to the gluon condensate, providing insights into the low-energy phase.

Contribution

It introduces a new class of static solutions in the effective theory of non-Abelian gauge fields with finite energy density and a natural mass scale.

Findings

Solutions are Gaussian in the z component of momentum.

Solutions are proportional to delta functions of other momentum components.

Parameters relate to the fourth root of the gluon condensate.

Abstract

We consider the gauge fixed partition function of pure $SU(N_c)$ gauge theory in axial gauge following the Halpern's field strength formalism. We integrate over $3 (N_c^2-1)$ field strengths using the Bianchi identities and obtain an effective action of the remaining $3 (N_c^2-1)$ field strengths in momentum space. We obtain the static solutions of the equations of motion (EOM) of the effective theory. The solutions exhibit Gaussian nature in the $z$ component of momentum and are proportional to the delta functions of the remaining components of momentum. The solutions render a finite energy density of the system and the parameters are found to be proportional to fourth root of the gluon condensate. It indicates that the solutions offer a natural mass scale in the low energy phase of the theory.

Tables

Table 1. Table 1: divergent parameters

$d_0$	$1.38161\times {10}^{-16}$	$m_0$	4.00046
$d_1$	$8.45785\times {10}^{-8}$	$m_1$	0.99998
$d_2$	$5.15311\times {10}^{-16}$	$m_2$	2.999873
$d_3$	$2.56904\times {10}^{-6}$	$m_3$	1.00214
$d_4$	$5.53533\times {10}^{-16}$	$m_4$	4.00008
$d_5$	$3.54694\times {10}^{-8}$	$m_5$	1.0009
$d_6$	$2.13056\times {10}^{-17}$	$m_6$	3.00199

Table 2. Table 2: finite parameters

$h_0$	$3.3454\times {10}^{-12}$	$f_1$	$6.73501\times {10}^{-7}$	$n_1$	2
$h_1$	$39.4784$	$f_2$	$1.23925\times {10}^{-14}$	$n_2$	3.9974
$h_2$	$9.2534\times {10}^{-12}$	$f_3$	$9.9540\times {10}^{-6}$	$n_3$	2.99996
		$f_4$	$1.84434\times {10}^{-14}$	$n_4$	4.99795
		$f_5$	$1.81293\times {10}^{-4}$	$n_5$	3.0001
		$f_6$	$1.85485\times {10}^{-4}$	$n_6$	5.99762

Equations

$$\mathcal{L}=-\frac{1}{4}{G}^{a\mu \nu }{G}_{\mu \nu }^a$$

$$Z=\int\mathcal{D}G\delta[I[G]]\exp\Big{[}-\frac{i}{4}\int d^{4}xG^{a}_{\mu\nu}G^{a\mu\nu}\Big{]}.$$

$$\delta [I[G]]=\prod _{\mu ,a,x}\delta (I_{\mu }^a(x)),$$

$$I^{a\mu }(x)$$

$$\mathrm{and}{\tilde{G}}^{a\mu \nu }$$

$$G_{\mu \nu }^a(x)$$

$$\mathrm{and}{A}_{\mu }^a(x)$$

$$A_0^a(k)$$

$$A_2^a(k)$$

$$Z=\int \mathcal{D}G\delta [I[G]]e^{iS[G]},$$

$$S[G]$$

$$\mathrm{and}\delta [I[G]]$$

$$I^{a0}(k)$$

$${\mathbf{I}}^a(k)$$

$$I^{a0}(k)$$

$$I_x^a(k)$$

$$\mathrm{and}{I}_y^a(k)$$

$$f_1^a(k)$$

$$f_2^a(k)$$

$$f_3^a(k)$$

$$X_1^a=B_x^a,X_2^a=B_y^a\mathrm{and}{X}_3^a=E_z^a.$$

$$\delta [I(G)]$$

$$Z=\mathcal{N}\int \mathcal{D}X{e}^{iS[X]},$$

$$S[X]=S^{(0)}[X]+S^{(1)}[X],$$

$$S^{(0)}[X]=\frac{1}{2}\int \frac{d^{4}p}{(2\pi {)}^4}{X}_j^a(p)(M^{-1}{)}_{jk}(p)X_k^a(-p)$$

$$S^{(1)}[X]$$

$$M^{-1}(p)=\left(\begin{array}[]{ccc}\frac{p^{2}+p_{y}^{2}}{p_{z}^{2}}&-\frac{p_{x}p_{y}}{p_{z}^{2}}&-\frac{p_{0}p_{y}}{p_{z}^{2}}\\ -\frac{p_{x}p_{y}}{p_{z}^{2}}&\frac{p^{2}+p_{x}^{2}}{p_{z}^{2}}&\frac{p_{0}p_{x}}{p_{z}^{2}}\\ -\frac{p_{0}p_{y}}{p_{z}^{2}}&\frac{p_{0}p_{x}}{p_{z}^{2}}&\frac{|{\bf{p}}|^{2}}{p_{z}^{2}}\\ \end{array}\right),$$

$$b_{jkl}^{(3)}(p)$$

$$b_{jklm}^{(4)}$$

$$\Big{(}\frac{\delta S[X]}{\delta X^{a}_{j}(p)}\Big{)}_{X=\bar{X}}=0$$

$$(M^{-1}{)}_{kl}(p){\bar{X}}_l^a(-p)$$

$$-g{C}^{abc}\frac{1}{p_z}\int \frac{d^4{q}^{\prime }}{(2\pi {)}^4}\frac{{\tilde{b}}_{klm}^{(3)}(p,q^{\prime })}{q_z^{\prime }(p_z+q_z^{\prime })}{\bar{X}}_l^b(-p-q^{\prime }){\bar{X}}_m^c(q^{\prime })$$

$$-\frac{g^2}{2}{C}^{abe}{C}^{ecd}\frac{1}{p_z}\int \frac{d^4{q}^{\prime }}{(2\pi {)}^4}\frac{d^4{q}^{\prime \prime }}{(2\pi {)}^4}\frac{{\tilde{b}}_{klmn}^{(4)}}{q_z^{\prime }{q}_z^{\prime \prime }(p_z+q_z^{\prime }+q_z^{\prime \prime })}$$

$$\times {\bar{X}}_l^b(q^{\prime }){\bar{X}}_m^c(-q^{\prime }-q^{\prime \prime }-p){\bar{X}}_n^d(q^{\prime \prime })=0,$$

$${\tilde{b}}_{klm}^{(3)}(p,q)$$

$${\tilde{b}}_{klmn}^{(4)}$$

$${\bar{X}}_j^a(p)=(2\pi {)}^4{\delta }^{(3)}(p_{\perp }){\xi }_j^a(p_z)$$

Full text

††preprint: APS/123-QED

I Introduction

It is well known that non abelian gauge theory (NAGT) has asymptotic freedom Gross1973 which indicates that the theory is almost free at very high energy and at very small distance scale. It also indicates that at very low energy it exhibits the confinement of gauge degrees of freedom and results in the non-zero values of gluon condensates at this low energy phase of the theoryShifman1979 . Owing to non linear nature of NAGT its exact quantization is still lacking and as a consequence of which the dynamics of this theory at low energy remains least understood till today. It is believed that stable classical solutions of EOM of the theory may be of use to understand the behaviour of the system at low energy and it may also shed some light on the issue of quantization of the theory at this energyJackiw1977 .

The endeavour for obtaining the classical solutions of EOM of NAGT has a long history since its inception. Attempt has been madeMaciejko1978 to caste those equations in the form of Maxwell equations of electrodynamics. The exact periodic solutions of the $SU(2)$ gauge theory have been constructedOh1979 in Minkowski space-time. There also exists solutions which are the non-abelian analogues of electromagnetic plane wavesColeman1977 . A relationship has been establishedCorrigan1977 between the solutions of a $\phi^{4}$ scalar field theory and a class of solutions of EOM of $SU(2)$ gauge theory. The most general self-dual, non-abelian, plane wave solutions have been obtainedLo1980 in NAGT. A comprehensive discussion on the solutions of EOM in Minkowski space-time are given in Ref.Actor1979 ; Malec1987

It has been proposed long beforeHalpern1979 that a non-abelian gauge theory can be formulated in terms of field strengths in axial gauge. The unique inversion of gauge potentials to field strength requires Bianchi identities to be satisfied in the quantized theory. We use Bianchi identities to integrate over the $3(N_{c}^{2}-1)$ degrees of freedom and obtain an effective action in terms of the remaining $3(N_{c}^{2}-1)$ field strengths in the momentum space. Then we obtain a solution of the equations of motion of the effective theory. Solutions indicate that the non abelian magnetic fields have Gaussian nature in their $z$ component of the momentum and non abelian electric fields are zero. We expand the effective action around this solution to quadratic order in fluctuation and obtain the energy density of the system by integrating out the fluctuations in the partition function. The energy density has been found to have a minimum with respect to the parameter of the solution and it thus indicates the stability of the obtained solution.

II Gauge fixed effective action

We start with the pure $SU(N_{c})$ gauge theory Lagrangian

[TABLE]

where $G^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+gC^{abc}A^{b}_{\mu}A^{c}_{\nu}$ ($a\,,b\,,c=1,\,\cdots,\,N_{c}^{2}-1$) is the field strength of the non-abelian gauge field $A^{a}_{\mu}$. Here the axial gauge $A_{3}^{a}(x)=0$ is chosen. The unique inversion $A(G)$ makes it possible to change variables to field strengths and the gauge fixed partition function takes the formHalpern1979

[TABLE]

where

[TABLE]

[TABLE]

We use the Fourier transform of

[TABLE]

and write the solutions of HalpernHalpern1979 for the gauge potentials in momentum space as

[TABLE]

where ${\mathbf{E}}^{a}(k)$ and ${\mathbf{B}}^{b}(k)$ are the non abelian electric and magnetic field vectors in momentum space respectively. We use eq.(6) and (7) to write the partition function in terms of the fields in momentum space as

[TABLE]

where

[TABLE]

$I^{a\mu}(k)$ in momentum space in terms of non abelian electric and magnetic fields read

[TABLE]

where $\hat{x}$, $\hat{y}$, $\hat{z}$ are the unit vectors along $x$, $y$ and $z$ axes respectively. We write

[TABLE]

$f^{a}_{j}(k)$ in terms of the fields $X^{a}_{j}(k)$ ($j=1,2,3$) are given as

[TABLE]

where

[TABLE]

We write using eq.(13)

[TABLE]

and then integrating over the fields $E^{a}_{x}$, $E^{a}_{y}$ and $B^{a}_{z}$ obtain the partition function as

[TABLE]

where the factor ${\cal N}=\delta(0)\prod_{k_{z}}\frac{i}{k_{z}^{3}}$. The factor of $\delta(0)$ comes from $\delta(I^{a}_{z}(G))$ when $I^{a}_{z}(G)=0$, after performing the integration over the remaining delta functions. The effective action

[TABLE]

where the quadratic part

[TABLE]

and the interaction part

[TABLE]

Here $3\times 3$ matrix $M^{-1}(p)$ reads

[TABLE]

[TABLE]

III Solutions of EOM

Equations of motion

[TABLE]

take the form

[TABLE]

where

[TABLE]

We assume a solution of eq.(24) as

[TABLE]

where $p_{\perp}=(p_{0},p_{x},p_{y})$ and

[TABLE]

Upon substitution of this solution into eq.(24), it takes the form

[TABLE]

Using the property in eq.(28) we evaluate the integral

[TABLE]

With the use of this result, eq.(29) simplifies to

[TABLE]

We then make the following ansatz:

[TABLE]

that are consistent with the property in eq.(28). Here $\Delta$ is a scale of mass dimension one and $y_{j}^{a}(n)$($j=1,2$) satisfy the following equations:

[TABLE]

We assume that in case of $SU(2)$ gauge theory

[TABLE]

Then $y^{a}_{j}(n)$ ($j=1,2$) for few different $n$ ($n\geq 1$) are as follows:

[TABLE]

IV Stability of the solution

To address the issue of the stability of the obtained solution we compute the vacuum energy density $\epsilon(\phi)$ of the system. The solutions will be stable if $\epsilon(\phi)$ has a minimum with respect to $\phi$ and it remains bounded even for limited fluctuation of $\phi$ around the minimum. From trace anomalyPeskin we know that the trace of the energy momentum tensor

[TABLE]

where $\beta(g)$ ($=\mu\frac{\partial g}{\partial\mu}$) is negative for pure NAGT. Lorentz invariance requires that $\langle 0\mid\Theta_{\mu\nu}\mid 0\rangle=\epsilon g_{\mu\nu}$ which is also consistent with the fact that $\Theta_{00}=\epsilon$. We therefore obtain $\langle 0\mid\Theta_{\mu\nu}\mid 0\rangle=4\epsilon$ and then the use of eq.(38) gives

[TABLE]

In the following we first expand the action around the solutions to quadratic order in fluctuations and then proceed to compute $\langle 0\mid G^{a}_{\alpha\beta}G^{a\alpha\beta}\mid 0\rangle$ using this expanded action around the solutions. Solutions in eq.(32) are dependent on a mass scale $\Delta$ which is dynamic in nature. It will be shown later using a rough estimate that it falls inversely as $g$ when $g\gg 1$ and goes to zero when $g\rightarrow 0$. Since, $\Delta$ turns out to be small in the non-perturbative ($g\gg 1$) region, we take $\Delta\ll 1$ and expand eq.(32) in the following in powers of $\Delta$:

[TABLE]

We introduce $3(N_{c}^{2}-1)$ sources and write the partition function of eq.(8) as

[TABLE]

where

[TABLE]

Consider an operator $O({\mathbf{E}}^{a},{\mathbf{B}}^{a})$ which is a functional of non abelian electric and magnetic fields. The vacuum expectation value

[TABLE]

Then after performing the integration over $E^{a}_{x}$, $E^{a}_{y}$ and $B^{a}_{z}$ as discussed in section II we obtain

[TABLE]

where the action of eq.(18) is modified as

[TABLE]

Moreover, according to eq.(14) $f^{a}_{j}$ are functions of $X^{a}_{j}$ ($j=1,2,3$), so we use

[TABLE]

To compute the vacuum average of the operator we adopt the stationary phase approximation in $Z[J]$:

[TABLE]

We expand the action about the classical solution and keep terms to order $(X-\bar{X})^{2}$:

[TABLE]

where $\Phi_{j}^{a}(p)=X^{a}_{j}(p)-\bar{X}^{a}_{j}(p)$. $\Phi_{j}^{a}(p)$ is assumed as a very slowly varying function of $p_{z}$ over the momentum scale $\Delta$ ($\Delta\ll 1$). Then

[TABLE]

Then we change the integration variable in eq.(47) from $X$ to $\Phi$ and integrate over $\Phi$ to obtain the final result as

[TABLE]

where,

[TABLE]

and the convention $J^{a}_{j}(p)=J^{a}_{j}(p_{\perp},p_{z})$ has been adopted. We take the operator as

[TABLE]

and use eq.(44) to obtain its average as

[TABLE]

where the expressions for $O_{2}$, $O_{3}$ and $O_{4}$ are as follows:

[TABLE]

and

[TABLE]

We evaluate the integrals in $O_{2}$, $O_{3}$ and $O_{4}$ numerically for $SU(2)$ gauge group. First we change the momentum integrals from Minkowski to Euclidean space: $p_{4}=ip_{0}$ and $d^{4}p=-id^{4}p_{E}$ where the Euclidean four momenta $p_{E}=(p_{4},{\mathbf{p}})$. We then use the system of coordinates in a manner such that the triad ($p_{x}$, $p_{y}$, $p_{4}$) form a spherical system of coordinates and write $p_{4}=r\cos\theta$, $p_{x}=r\sin\theta\cos\theta_{1}$ and $p_{y}=r\sin\theta\sin\theta_{1}$. The integrals are evaluated over the following ranges: $0\leq r\leq\Lambda$, $0\leq\theta\leq\pi$ and $0\leq\theta_{1}\leq 2\pi$, where $\Lambda$ is an ultraviolet cut off used in the calculation. Since the integrands are found to be even function of $p_{z}$ we fold the region of integration from $-\Lambda\leq p_{z}\leq\Lambda$ to $s\Lambda\leq p_{z}\leq\Lambda$, where $s$ ($0<s<1$) is an infrared scale parameter used in the calculation. Next we use the dimensionless variables as $x=r/\Lambda$, $y=p_{z}/\Lambda$, $\phi_{1}=\frac{\sqrt{\pi}g\phi}{\Lambda}$ and $\Delta_{1}=\frac{\Delta}{\Lambda}$. As a result the integrals in eq.(54), (55) and (56) when expressed in terms of those new variables will receive a factor of $\Lambda^{4}$ outside the integrals. Therefore, we Taylor expand the integrands to quartic order of $\phi_{1}$ and $\Delta_{1}$ in the limit when $\Lambda\rightarrow\infty$ for finite $\phi$ and $\Delta$ and compute the coefficients for different values of $s$. The analytic expressions of the coefficients as a function of $s$ is obtained by fitting the numerically obtained data points for small values of $s$ ($\approx 10^{-4}$).

We numerically compute the momentum integrals in $O_{2}$, $O_{3}$ and $O_{4}$ for $SU(2)$ theory and obtain the vacuum average as

[TABLE]

where the values of the parameters are given in the following TABLE -I and TABLE-II.

In general we consider a scalar operator $O(x)$ and the renormalized partition function with sources reads as

[TABLE]

where, $\langle 0\mid O(x)\mid 0\rangle_{ct}$ contains the required counter terms to remove the divergences from the vacuum average of $O(x)$. The renormalized average is given as

[TABLE]

Using the minimal subtraction scheme we choose $\langle 0\mid G^{a\mu\nu}(x)G^{a}_{\mu\nu}(x)\mid 0\rangle_{ct}$ appropriately and obtain the renormalized vacuum average as

[TABLE]

where

[TABLE]

Here the scale parameter must be of the form $s=\mu/\mu_{0}$, where $\mu_{0}$ is the scale of minimal subtraction and $\mu$ ($<\mu_{0}$) is the scale of interest at low energy. We then use eq.(39) to obtain the energy density for $SU(2)$ theory as

[TABLE]

It is clear that $C>0$ for any $s$. However, $A$ and $B$ simultaneously remain positive when $s\ll 1$. Then, $\epsilon(\phi)$ is maximum at $\phi=0$ and is minimum at $\phi=\pm\phi_{0}$, where

[TABLE]

Hence it ensures that the solutions are stable. We can estimate the value of $\Delta$ which is as follows: The renormalized gluon condensate at the minima $\phi=\pm\phi_{0}$ is

[TABLE]

Since, $f_{6}\gg\frac{f_{4}^{2}}{4f_{2}}$ we obtain

[TABLE]

When $g\gg 1$, we obtain using $\langle\alpha_{s}G^{2}\rangle_{0}=0.0314\,{\rm GeV}^{4}$Furnstahl1990 $\Delta=6.79s^{\frac{3}{2}}/g\,{\rm GeV}$. It indicates that even in the non-perturbative region ($g\gg 1$) $\Delta$ remains less than one. When $g\rightarrow 0$, surely $\Delta\rightarrow 0$ because the gluon condensate vanishes in the perturbative limit of NAGT. Therefore, in the non-perturbative region $\Delta$ decreases inversely as $g$, however it goes to zero in the perturbative region. So our idea of retaining terms to order $\Delta^{4}$ indicates that we have worked to obtain energy density of the system at an energy scale which lies in the non-perturbative phase of the theory. The use of $\Delta$ in eq.(63) gives

[TABLE]

Therefore, $\phi_{0}\rightarrow 5.85s/g^{2}$ when $g\gg 1$ and $\phi_{0}\rightarrow 0$ when $g\rightarrow 0$.

V Conclusion

We have taken the gauge fixed partition function of NAGT in axial gauge using the field strength formalism as given by Halpern. We have integrated out the variables $E_{x}^{a}$, $E_{y}^{a}$ and $B_{z}^{a}$ using Bianchi Identities in momentum space and have obtained an effective action of the remaining $3(N_{c}^{2}-1)$ variables $E_{z}^{a}$, $B_{x}^{a}$ and $B_{y}^{a}$. We have obtained a static solution of EOM of the effective theory in terms of parameters $\phi$ and $\Delta$. The solutions in the momentum space exhibit a Gaussian nature in the $z$ component of momentum and are proportional to Dirac delta functions of the remaining components of momentum. To check the stability of the solutions first we expand the effective action around the solutions to quadratic order in fluctuations and then compute the vacuum average $\langle 0\mid G^{a\mu\nu}(0)G^{a}_{\mu\nu}(0)\mid 0\rangle$ using this expanded action around the solutions. Then we use trace anomaly to obtain the energy density as a function of $\phi$. The energy density has minima at $\phi=\pm\phi_{0}$, where $\phi_{0}$ is proportional to the fourth root of the gluon condensate $\langle\alpha_{s}G^{2}\rangle_{0}$, which is evaluated at the minima of the energy density. Since the energy density does not change appreciably for small fluctuation around these minima, the obtained solutions are stable and make the energy density finite. It is known that the gluon condensate vanishes when computed using the method of small coupling constant perturbation. So, $\phi_{0}$ tends to zero in the perturbative limit. Since $\phi_{0}$ is non-vanishing in the non-perturbative limit, it acts as a mass scale in the low energy limit of NAGT. The parameter $\Delta$ is also found to be proportional to $\sqrt[4]{\langle\alpha_{s}G^{2}\rangle_{0}}$. Hence, $\Delta$ remains non-zero in the non-perturbative phase and vanishes in the perturbative phase of the theory.