---
abstract: 'We use the N–quantum approach (NQA) to quantum field theory to construct a solution of the two flavor effective instanton induced ‘t Hooft interaction model valid for any temperature ($T$) and chemical potential ($\mu$) beyond the mean field theory.  The model contains only the $\bar{q}q$ channels.  In constructing this solution we calculate the masses, widths and coupling constants of bound $\sigma$ and resonant color $\bar{3}$ scalar diquark states.  We find that the chemical potential induced by the interaction cancel exactly for all values of $T$ in contrast to the Nambu–Jona–Lasinio model.  Our method can also be extended to include the $qq$ channels in the model Lagrangian and we discuss how the NQA can be used to study the properties of bound diquark states at finite $T$ and $\mu$. PACS numbers: 03.70.+k, 12.40.$-$y 0.25cm Keywords: Quantum Field Theory at Finite $T$ and $\mu$; Haag Expansion; ‘t Hooft Interaction 0.25cm Submitted to Physics Letters B'
address: |
    Instituto de Física Corpuscular – C.S.I.C.\
    Departamento de Física Teòrica, Universitat de València\
    E–46100 Burjassot, València, Spain
author:
- Yasuo Umino and Vicente Vento
title: |
    The ’t Hooft interaction at finite $T$ and $\mu$:\
    Beyond the mean field theory
---

Introduction {#intro}
============

In this letter we apply the N–quantum approach (NQA) to quantum field theory [@gre94a] to the effective instanton induced ‘t Hooft interaction Lagrangian [@tho96] and construct a non–perturbative solution to the field theory beyond the mean field approximation for any temperature ($T$) and chemical potential ($\mu$).  Developed by Greenberg [@gre65], NQA uses an expansion in on–shell asymptotic fields, known as the Haag expansion [@haa55], to solve the operator equations of motion.  The method is particularly well suited for bound state problems since it describes bound states with the same number of kinematical variables as in non–relativistic theories while maintaining explicit Lorentz invariance (for theories in free space).  A summary of applications of NQA can be found in [@gre94a]. 

Recently we have extended the NQA to finite $T$ and $\mu$ [@umi00].  Our work was based on the thermal field dynamics formalism of Umezawa and collaborators [@umezawa], and was motivated by an application of NQA to the BCS theory of superconductivity [@gre94b].  Temperature and chemical potential were introduced simultaneously by applying a thermal Bogoliubov transformation to each of the fermionic or bosonic field operators in the Haag expansion.  At the same time we defined a thermal vacuum state which is annihilated by each of the thermal annihilation operators.  This takes care of the doubling of the Hilbert space which characterizes the formalism of thermal field dynamics.  Furthermore, in the NQA one works only in the space of asymptotic field operators and therefore there is no need to specify the structure of this vacuum state. 

We have successfully tested our formalism on the Nambu–Jona–Lasinio (NJL) model [@nam61] where the mean field (Hartree–Fock) results at finite $T$ and $\mu$ are well established [@reviews].  These results were recovered using only the first term in the Haag expansion.  In this work we take the second order terms in the Haag expansion into account to solve the ‘t Hooft model beyond the mean field theory.  Although non–renormalizable and non–confining, the ‘t Hooft interaction incorporates the symmetries of QCD and naturally provides a mechanism for chiral symmetry breaking and resolves the $U_A(1)$ problem [@sch98].  As is well known it also supports the formation of various kinds of condensates.  Among them are the color singlet chiral condensate ($\langle \bar{q}q \rangle$) which breaks chiral symmetry dynamically and generates a mass to the fermions and the color anti–triplet scalar diquark condensate ($\langle qq \rangle$) which triggers color superconductivity at finite density [@diquark]. 

In this work we only include the $\bar{q}q$ channels in the model Lagrangian.  Our solution is constructed by calculating the masses, widths and coupling constants of the $\sigma$ and [*resonant*]{} diquark states for any $T$ and $\mu$ both in the Nambu–Goldstone and in the Wigner–Weyl phases of the model field theory.  In the concluding section we discuss how our method can be used to study the properties of the [*bound*]{} diquark states in hot and dense matter. 

Model Lagrangian and the NQA Ansatz {#model}
===================================

We start by defining the model Lagrangian which is the SU(2)$_{\rm f}$ $\otimes$ SU(3)$_{\rm c}$ version of the ‘t Hooft interaction [@tho96] with massless quarks and a chemical potential term.  $$\begin{aligned}
%
{\cal L} 
& = & 
i\bar{\Psi} \left( \partial\!\!\!/  -i \mu_0 \gamma_0 \right) \Psi + 
g_0 \Bigl[ (\bar{\Psi} \Psi )^2 + (\bar{\Psi} i \vec{\tau}\gamma_5 \Psi)^2 
- (\bar{\Psi} i\gamma_5 \Psi)^2 - (\bar{\Psi}\vec{\tau} \Psi)^2 \Bigr]
%
\label{eq:tHooft} \\
& = &
{\cal L}_{\rm NJL} - g_0 \Bigl[ (\bar{\Psi} i\gamma_5 \Psi)^2 
+ (\bar{\Psi}\vec{\tau} \Psi)^2 \Bigr]
\label{eq:tHooft2} 
%\end{aligned}$$ In the above Lagrangian $\bar{\Psi} \Psi$, $\bar{\Psi} i \vec{\tau}\gamma_5 \Psi$, $\bar{\Psi}i\gamma_5 \Psi$ and $\bar{\Psi}\vec{\tau} \Psi$ terms correspond to the $\bar{q}q$ channels with quantum numbers of the $\sigma$, $\pi$, $\eta'$ and $a_0$ mesons, respectively.  The coupling constant $g_0$ has the dimension of $[{\rm mass}]^{-2}$. In Eq.  (\[eq:tHooft2\]) the ‘t Hooft Lagrangian is expressed as a sum of the NJL lagrangian and interactions in the $\eta'$ and $a_0$ channels.  Note that $\mu_0$ is in general not the total chemical potential of the interacting many body system since the interaction may induce a contribution to the total chemical potential. 

For our purpose the equation of motion corresponding to Eq.  (\[eq:tHooft\]) is most conveniently written in terms of colored quark fields in momentum space.  For example, the charge conjugation symmetric form of the equation of motion for the up quark field may be written as $$\begin{aligned}
%
(q\!\!\!/ + \mu_0 \gamma_0)_{\alpha\beta}(u_a)_\beta(q) 
& = &
-2 g_0 \int d^4p_1 d^4p_2 d^4p_3\; \delta^4(q - p_1 - p_2 - p_3) \nonumber \\
&   &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
\otimes \;
\Biggl\{ \left[(\bar{d}_b)_\beta(p_1), (d_b)_\beta(p_2) \right]_- (u_a)_\alpha(p_3)
+ \left[(\bar{d}_b)_\beta(p_1), (d_a)_\alpha(p_2) \right]_- (u_b)_\beta(p_3) \nonumber \\
&   &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
+ \;\; 
(\gamma_5)_{\alpha\beta}(\gamma_5)_{\epsilon\rho} \Biggl[
\left[(\bar{d}_b)_\epsilon(p_1), (d_b)_\rho(p_2) \right]_- (u_a)_\beta(p_3)
+ \left[(\bar{d}_b)_\epsilon(p_1), (d_a)_\beta(p_2) \right]_- (u_b)_\rho(p_3)
\Biggr] \Biggr\}
%
\label{eq:motion}\end{aligned}$$ where $[A, B]_- = AB - BA$. $u$ and $d$ are up and down quark fields with color and Dirac indices indicated in Roman and Greek letters, respectively.  Summation convention for repeated indices is implied. 

We proceed to construct an ansatz for the ‘t Hooft model by Haag expanding each of the quark fields in Eq.  (\[eq:motion\]) to second order following [@gre86].  For example, the second order Haag expansion for the up quark field with color index 1 may be written as $$\begin{aligned}
%
(u_1)_\alpha(q)
& = &
(u_1^0)_\alpha(q) + \int d^4k d^4b\; \delta^4(q + k -b)
\Biggl\{ f_{\alpha\beta}(k, b):(u_1^0)_\beta(-k)\sigma^0(b): \nonumber \\
&   &
+ d_{\alpha\beta}(k, b) (C^{-1})_{\beta\epsilon}
:\Bigl[ (\bar{d}_2^0)_\epsilon(-k) D_2^0(b) + (\bar{d}_3^0)_\epsilon(-k)
D_5^0(b)\Bigr]:\Biggr\} 
\label{eqn:haagu1} 
%\end{aligned}$$ where the symbol $:\hspace{0.5cm}:$ denotes normal ordering.  The asymptotic fields in the Haag expansion are choosen to be in–coming fields which implies that the corresponding propagators obey the retarded boundary conditions.  These fields are defined as $u_a^0(k) = u_a^{\rm IN}(k)\delta(k^2 - m^2)$, $d_a^0(k) = d_a^{\rm IN}(k)\delta(k^2 - m^2)$, $\sigma^0(b) = \sigma^{\rm IN}(b)\delta(b^2 - m_\sigma^2)$ and $D^0_A(b) = D_A^{\rm IN}(b)\delta(b^2 - m_D^2)$. In Eq.  (\[eqn:haagu1\]) $\sigma$ is a color singlet scalar field which transforms as $\bar{\Psi}\Psi$ while $D_A$ is a color anti–triplet scalar field transforming as $\Psi^T C^{-1} \gamma_5 \tau_2 \lambda_A \Psi$ with $A = 2, 5, 7$ and $C^{-1} = -C = -i \gamma_2\gamma_0$ being the charge conjugation operator. 

In this work we only include the $\sigma$ and $D_A$ fields in the second order Haag expansion.  Other composite fields may be included in the second order terms as shown, for example, in [@gre86].  We emphasize that $\sigma$ and $D_A$ fields need [*not*]{} be interpreted as bound states.  As we shall see below, in medium they can be either bound or resonant states.  Thus, in order to avoid confusion with the bound diquark state [@diquark] we shall henceforth refer the color $\bar{3}$ $qq$ state simply as the $D_A$ state. 

The asymptotic quark mass $m$ will shortly be identified as the chiral gap while the $\sigma$ and $D_A$ masses $m_\sigma$ and $m_D$ will be determined below as functions of $T$ and $\mu$. The Haag amplitudes $f(k, b)$ and $d(k, b)$ are interpreted as amplitudes for the creation of the $\sigma$ and $D_A$ states, respectively.  Corresponding expansion for $(\bar{u}_1)_\alpha(q)$ can be found by using the relation $(\bar{u}_1)(q) = u_1^\dagger(-q)\gamma^0$. Second order Haag expansions for the remaining quark fields in Eq.  (\[eq:motion\]) can be constructed in a similar manner. 

We use the method developed in [@umi00] to introduce temperature and chemical potential into Eq.  (\[eqn:haagu1\]).  The on–shell fermionic field $\Psi^0$ is extended to finite $T$ and $\mu$ as follows.[^1] $$\begin{aligned}
%
\Psi^0(q)
& = &
\Psi(q)\delta(q^2 - m^2) \nonumber \\
& = &
\frac{1}{(2\pi)^3}\frac{m}{\omega_q} \sum_{s=\pm} 
\biggl[ b(\vec{q},s) u(\vec{q},s) \delta(q^0-\omega_q) + d^{\dagger}(-\vec{q},s) v(-\vec{q},s) 
\delta(q^0+\omega_q) \biggr]  \nonumber \\
& \rightarrow &
 \frac{1}{(2\pi)^3}\frac{m}{\omega_q} \sum_{s=\pm} \Biggl\{
\biggl[ \alpha_q B(\vec{q},s) - s \beta_q \tilde{B}^{\dagger}(-\vec{q},s)\biggr] 
u(\vec{q},s) \delta(q^0-\omega_q)  \nonumber \\
&    &
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; +
\biggl[ \gamma_q D^{\dagger}(-\vec{q},s) - s \delta_q 
\tilde{D}(\vec{q},s) \biggr] v(-\vec{q},s) \delta(q^0+\omega_q) \Biggr\}
%
\label{eq:FERMITMU}\end{aligned}$$ Here $B$ and $\tilde{B}^{\dagger}$ are quasi–particle annihilation and quasi–hole creation operators, respectively, while $\tilde{D}$ and $D^{\dagger}$ are the annihilation operator for a quasi–anti–hole and creation operator for a quasi–anti–particle, respectively.  These thermal operators satisfy the fermion anti–commutation relations and annihilate the interacting thermal vacuum state $|{\cal G}(T,\mu)\rangle$. $$%
B(\vec{q},s) |{\cal G}(T,\mu)\rangle 
=\tilde{B}(\vec{q},s) |{\cal G}(T,\mu)\rangle 
= D(\vec{q},s) |{\cal G}(T,\mu)\rangle 
=\tilde{D}(\vec{q},s) |{\cal G}(T,\mu)\rangle 
= 0
%
\label{eq:AVAC1}$$ The $T$ and $\mu$ dependent coefficients of the thermal Bogoliubov transformation are given by $\alpha_q = \sqrt{1-n_q^-}$, $\beta_q = \sqrt{n_q^-}$, $\gamma_q = \sqrt{1-n_q^+}$ and $\delta_q = \sqrt{n_q^+}$ where $n_q^{\pm}= [e^{(\omega_q \pm \mu)/(k_B T)}+1]^{-1}$ are the Fermi distribution functions. 

The extension to finite $T$ and $\mu$ for the on–shell bosonic field $\phi^0$ proceeds through an analogous path.  $$\begin{aligned}
%
\phi^0(q)
& = &
\phi(q)\delta(q^2 - m^2) \nonumber \\
& = &
\frac{1}{2 \omega_q} \biggl[ a(\vec{q}\:) \delta(q^0-\omega_q) 
+ a^{\dagger}(-\vec{q}\:) \delta(q^0+\omega_q) \biggr]  \nonumber \\
& \rightarrow &
\frac{1}{2 \omega_q} \Biggl\{
\biggl[ c_q A(\vec{q}\:) + d_q \tilde{A}^{\dagger}(\vec{q}\:) \biggr] 
\delta(q^0-\omega_q) \nonumber \\
&    &
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + 
\biggl[ d_{q} \tilde{A}(-\vec{q}\:) + c_{q} A^{\dagger}(-\vec{q}\:) \biggr] 
\delta(q^0+\omega_q) \Biggr\}
%
\label{eq:BOSETMU}\end{aligned}$$ The coefficients of the bosonic thermal Bogoliubov transformation are given by $c_q = \sqrt{1 + d_q^2}$ and $d_q = \sqrt{n_q^B}$ where $n_q^B = [e^{(\omega_q- \mu)/(k_B T)}-1]^{-1}$ is the Bose distribution function.  The thermal operators now obey the boson commutation relations and annihilate the thermal vaccum as in Eq.  (\[eq:AVAC1\]), $$%
A(\vec{q}\:) |{\cal G}(T,\mu)\rangle
=\tilde{A}(\vec{q}\:) |{\cal G}(T,\mu)\rangle 
= 0
%
\label{eq:AVAC2}$$ Thus our ansatz for the model Lagrangian Eq.  (\[eq:tHooft\]) at finite $T$ and $\mu$ is constructed in three steps.  First, each of the colored quark fields in the equations of motion are Haag expanded to second order as in Eq.  (\[eqn:haagu1\]).  Second, each of the asymptotic quark fields in the Haag expansion are subjected to a thermal Bogoliubov transformation of type shown in Eq.  (\[eq:FERMITMU\]).  Finally, each of the asymptotic bosonic fields $\sigma^0$ and $D_A^0$ are subjected to a thermal Bogoliubov transformation of type shown in Eq.  (\[eq:BOSETMU\]).  The basic idea of NQA is to use the operator equation of motion Eq.  (\[eq:motion\]) to solve for the Haag amplitudes $f(k, b)$ and $d(k, b)$ and thus construct a solution to the field theory defined by Eq.  (\[eq:tHooft\]). 

The Gap and Self–consistency Equations {#equations}
======================================

We now derive the gap equation and the self–consistency equations for the Haag amplitudes using the operator equation of motion in the one–loop approximation.  This approximation corresponds to keeping only those terms involving a single contraction when renormal ordering the field operators with respect to the interacting vacuum $|{\cal G}(T,\mu)\rangle$. We begin by inserting the first order terms in the Haag expansion of the quark fields in Eq.  (\[eq:motion\]) which is equivalent to invoking the mean field approximation.  After renormal ordering and keeping only the linear terms in the field operators we obtain the following equation $$%
m = 
(24 + 4) g_0 \int d^3\bar{p}\; \frac{m}{\omega_p} 
\left( 1- \beta^2_p - \delta^2_p\right)
%
\label{eqn:m}$$ where $d^3\!\bar{p} \equiv d^3\!p/(2\pi)^3$. The factors of 24 and 4 represent contributions from the Hartree and Fock terms, respectively.  In our approach both of these contributions arise naturally from the equation of motion.  This gap equation determines the asymptotic quark mass $m$ which is interpreted as the chiral gap.  As in the NJL model when $m(T, \mu) = 0$ the model is in the Wigner–Weyl phase while if $m(T,\mu) \neq 0$ then it is in the Nambu–Goldstone phase. 

We find that the chemical potentials induced by the scalar and pseudoscalar Fock terms in the ‘t Hooft interaction cancel exactly.[^2] Therefore the total chemical potential, which appears in the Fermi and Bose distribution functions, is simply the bare chemical potential $ \mu_0$. This situation is in contrast with the NJL model where in the mean field approximation the Fock term induces a chemical potential proportional to the quark number density making it necessary to solve the gap equation self–consistently with the total chemical potential [@asa89].  As a result, it becomes impossible to clearly define critical points in the $T-\mu$ plane [@asa89].  Eq (\[eq:tHooft2\]) shows that the ‘t Hooft Lagrangian can be written as a sum of the NJL Lagrangian and interactions in the $a_0$ and the $\eta'$ channels.  Thus the addition of these channels leads to the cancellation of the chemical potential induced by the $\sigma$ and the $\pi$ channels.  This is one important difference between the NJL and the ‘t Hooft model at finite $\mu$ in the mean field approximation. 

To go beyond the mean field theory we add the second order terms in the Haag expansion to our ansatz for the quark fields.  Then, in addition to Eq.  (\[eqn:m\]), we obtain the following self–consistency equations for the Haag amplitudes from the equation of motion in the one–loop approximation $$\begin{aligned}
%
\left( b\!\!\!/ - k\!\!\!/ - m \right) f(k, b) 
& = &
\nonumber \\
&   &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
2 g_0 \int d^3\bar{p}\; \frac{m}{\omega_p}
\Biggl\{ \left( 1- 2\beta^2_p \right)
\Biggl[ f(p^-, b) + \bar{f}(p^+, b) + \gamma_5 f(p^-, b)\gamma_5 
+ \gamma_5\bar{f}(p^+, b)\gamma_5 
\nonumber \\
&   &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
+ \;\; 3 {\rm Tr}\Bigl( f(p^-, b) + \bar{f}(p^+, b) \Bigr) 
+ 3 {\rm Tr}\Bigl( \gamma_5 f(p^-, b) + \gamma_5\bar{f}(p^+, b) \Bigr)\gamma_5 \Biggr]
\nonumber \\
&   &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
+ \;\left( 1- 2\delta^2_p \right) 
\Biggl[ f(p^+, b) + \bar{f}(p^-, b) + \gamma_5 f(p^+, b)\gamma_5 
+ \gamma_5\bar{f}(p^-, b)\gamma_5 
\nonumber \\
&   &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
+  \;\; 3 {\rm Tr}\Bigl( f(p^+, b) + \bar{f}(p^-, b) \Bigr) 
+ 3 {\rm Tr}\Bigl( \gamma_5 f(p^+, b) + \gamma_5\bar{f}(p^-, b) \Bigr)\gamma_5 
\Biggr]\Biggr\}
%
\label{eqn:f} 
%\end{aligned}$$ and $$\begin{aligned}
\left( b\!\!\!/ - k\!\!\!/ - m \right) d(k, b) 
& = &
\nonumber \\
&   &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
-4 g_0 \int d^3\bar{p}\; \frac{m}{\omega_p}
\Biggl\{ \left( 1- 2\beta^2_p \right)
\Biggl[d(p^+, b) + \gamma_5 d(p^+, b)\gamma_5 
\nonumber \\
&   & 
+ \;\;C^{-1T} d^{T}(p^+, b) C^{T} 
+ \gamma_5 C^{-1T} d^{T}(p^+, b) C^{T} \gamma_5\Biggr]
\nonumber \\
&   &
+ \;\left( 1- 2\delta^2_p \right)
\Biggl[d(p^-, b) + \gamma_5 d(p^-, b)\gamma_5 
\nonumber \\
&   & 
+ \;\; C^{-1T} d^{T}(p^-, b) C^{T} 
+ \gamma_5 C^{-1T} d^{T}(p^-, b) C^{T} \gamma_5 \Biggr]\Biggr\}
%
\label{eqn:d}
%\end{aligned}$$ In the above expressions $p^\pm = (\pm\omega_p, \vec{p\:})$, $\bar{f}(p, b) = \gamma_0 f^\dagger(-p, -b) \gamma_0$, $b^2 = m^2_{\sigma}$ for Eq.  (\[eqn:f\]) and $b^2 = m^2_D$ for Eq.  (\[eqn:d\]).  These three dimensional linear integral equations for $f(k, b)$ and $d(k, b)$ can be solved exactly. 

The $\sigma$ Amplitude
======================

In the broken symmetry phase where $m \neq 0$, the discrete $C$, $P$ and $T$ transformations constrain the structure of the $\sigma$ amplitude $f(k, b)$ to be $$%
f^{\rm NG}(k, b) = \frac{1}{b\!\!\!/ - k\!\!\!/ - m} \left( A + Bb\!\!\!/ \right) 
\frac{(-k\!\!\!/ + m)}{2m}
%
\label{eq:fsoln}$$ where $A$ and $B$ are real and constant diagonal matrices.  The superscript NG indicates that we are in the Nambu–Goldstone phase of the model.  The first factor in Eq.  (\[eq:fsoln\]) is the quark propagator obeying the retarded boundary condition.  At finite $\mu$ it is necessary to choose a reference frame since Lorentz invariance is lost at finite density.  We shall work in the rest frame of the bound states which is the appropriate frame to calculate their masses, widths and coupling constants as will be shown shortly. 

Upon inserting Eq.  (\[eq:fsoln\]) into the self–consistency equation for $f(k, b)$ we find the following relation between $A$ and $B$ $$%
\left(A + Bb\!\!\!/\right) \left(-k\!\!\!/ + m \right) = \left(I_f A + 2m B \right)
\left(-k\!\!\!/ + m \right)
%
\label{eq:fsoln0}$$ where $$%
I_f= 56 g_0  \int d^4\bar{p}\; \delta(p^2 - m^2) 
\frac{2m^2 - b\cdot p}{(b - p + i\epsilon^0)^2 - m^2}
\left( 1- \beta^2_p - \delta^2_p \right)
%$$ with $b_\mu = (\pm m_\sigma, \vec{0}\;)$. Following [@gre86] we trace Eq.  (\[eq:fsoln0\]) after multiplying by 1 and $b\!\!\!/$ to obtain two homogeneous equations for $A$ and $B$. The solution for this set of equations exists when $I_f = 1$ which implies $$%
f^{\rm NG}(k, b) = \frac{\left( A + 2mB \right)}{b\!\!\!/ - k\!\!\!/ - m}  
\frac{(-k\!\!\!/ + m)}{2m}
%
\label{eq:fsoln1}$$ It remains to determine the $T$ and $\mu$ dependent normalization constant $A + 2mB$.

From the structure of the second order Haag expansion Eq.  (\[eqn:haagu1\]) we see that this normalization constant corresponds to the $\sigma qq$ coupling constant $g_{\sigma qq}$. This coupling constant, as well as the $\sigma$ mass and its width, can be determined by rewriting the condition $I_f = 1$ as $$%
0 = 1 - 2g_0 \Biggl[ 28\int d^4\bar{p}\; \delta(p^2 - m^2) 
\frac{2m^2 - b\cdot p}{(b - p + i\epsilon^0)^2 - m^2}
\left( 1- \beta^2_p - \delta^2_p \right) \Biggr]
%
\label{eq:sigpf}$$ In the standard treatment of the NJL model the $\sigma$ mass is determined by the lowest pole of the $\sigma$ propagator in the random phase approximation (RPA) at zero three momentum.  This pole is a solution of the equation $$%
0 = 1 - 2G \Pi_\sigma(m_\sigma, \vec{0}\;)
%$$ where $G$ is the coupling constant in the NJL Lagrangian and $\Pi_\sigma$ is the $\sigma$ polarization function [@reviews].  Since we are working in the rest frame of the $\sigma$ we can identify the term in square brackets in Eq.  (\[eq:sigpf\]) as the $\sigma$ polarization function in the broken symmetry phase $$%
\Pi^{\rm NG}_\sigma(b) = 28\int d^4\bar{p}\; \delta(p^2 - m^2) 
\frac{2m^2 - b\cdot p}{(b - p + i\epsilon^0)^2 - m^2}
\left( 1- \beta^2_p - \delta^2_p \right)
%
\label{eq:spol}$$ For $b_\mu = (b_0, \vec{0}\;)$ and $b_0^2 \geq 4m^2$, the real and imaginary parts of Eq.  (\[eq:spol\]) are given by[^3] $$\begin{aligned}
%
{\rm Re}\; \Pi^{\rm NG}_\sigma(b_0) 
& = &
\frac{7}{\pi^2} {\rm P}\; \int^\infty_m dx\; \frac{(x^2 - m^2)^{3/2}}{x^2 -
\frac{1}{4}b_0^2} \Biggl\{ 1 - \left[ e^{(x-\mu)/(k_B T)}\right]^{-1}
- \left[ e^{(x+\mu)/(k_B T)}\right]^{-1} \Biggr\}
\label{eq:respol} \\
%
{\rm Im}\; \Pi^{\rm NG}_\sigma(b_0) 
& = &
\frac{7}{\pi}\frac{1}{b_0}\left[\frac{1}{4}b_0^2-m^2\right]^{3/2}
\Biggl\{ 1 - \left[ e^{(\frac{|b_0|}{2}-\mu)/(k_B T)}\right]^{-1}
- \left[ e^{(\frac{|b_0|}{2}+\mu)/(k_B T)}\right]^{-1} \Biggr\} 
\label{eq:imspol} 
%\end{aligned}$$ The real part of $\Pi^{\rm NG}_\sigma$ is equivalent to the one obtained by Asakawa and Yazaki in Eq.  (30) of [@asa89] except for the factor in front of the principal value integral.  They obtain a factor of $\frac{13}{2}$ whereas our factor is 7.[^4] Note that ${\rm Re}\;\Pi^{\rm NG}_\sigma$ is a function of $b_0^2$ and is independent of the choice of the $b^\pm \equiv (\pm b_0, \vec{0}\:)$ frames. 

The $\sigma$ mass can be obtained directly from Eq.  (\[eq:sigpf\]) by noting that the gap equation Eq.  (\[eqn:m\]) can be rewritten as $$%
1 = 28g_0 \int d^4\bar{p}\; \delta(p^2 - m^2) \frac{m_\sigma^2 - 2 b\cdot p}{(b-p)^2 - m^2}
\left( 1- \beta^2_p - \delta^2_p \right) 
%$$ Using this expression in Eq.  (\[eq:sigpf\]) implies $m_\sigma = 2m$ for all $T$ and $\mu$. Moreover, since the width of the $\sigma$ state is given by $$%
\Gamma_\sigma = \frac{1}{m_\sigma}|{\rm Im}\;\Pi_\sigma|
%
\label{eq:widtheq}$$ it follows immediately from Eq.  (\[eq:imspol\]) that the $\sigma$ has vanishing width in the broken symmetry phase. 

Finally, in the RPA the $\sigma qq$ coupling constant is given by [@reviews] $$%
g^2_{\sigma qq} = \left[ \frac{\partial {\rm Re}\; \Pi_\sigma}{\partial
b^2}\right]^{-1}_{b^2 = m_\sigma^2}
%
\label{eq:coupling}$$ which fixes the normalization of the $\sigma$ amplitude in the broken symmetry phase.  Thus, we have solved for the Haag amplitude of the $\sigma$ state which can be written as $$%
f^{\rm NG}(k, b) = \frac{g_\sigma qq}{b\!\!\!/ - k\!\!\!/ - m}  
\frac{(-k\!\!\!/ + m)}{2m}
%
\label{eq:fsoln3}$$ and obtained its mass, width and coupling constant.  In the broken symmetry phase the $\sigma$ is a stable bound state with zero width and a mass of exactly twice the chiral gap for all values of $T$ and $\mu$. This conclusion is in agreement with the one obtained by Zhuang, Hüfner and Klevansky in the NJL model [@zhu94]. 

To obtain the $\sigma$ amplitude in the chirally symmetric phase we repeat the same exercise elucidated above but with $m = 0$. The structure of the Haag amplitude now becomes $$%
f^{\rm BW}(k, b) = -\frac{g_\sigma qq}{b\!\!\!/ - k\!\!\!/} k\!\!\!/ 
%
\label{eq:fsoln4}$$ while the real and imaginary parts of the extracted $\sigma$ polarization function are found to be $$\begin{aligned}
%
{\rm Re}\;\Pi_\sigma^{\rm BW}(b)
& = & 
\frac{7}{\pi^2} {\rm P}\; \int^\infty_0 dx\; 
\frac{x^3}{x^2 - \frac{1}{4}m_\sigma^2} 
\Biggl\{ 1 - \left[ e^{(x-P_F)/(k_B T)}\right]^{-1}
- \left[ e^{(x+P_F)/(k_B T)}\right]^{-1} \Biggr\}
\\
{\rm Im}\;\Pi_\sigma^{\rm BW}(b)
& = &
\frac{7}{8\pi} 
\Biggl\{ 1 - \left[ e^{(\frac{1}{2}m_\sigma-P_F)/(k_B T)}\right]^{-1}
- \left[ e^{(\frac{1}{2}m_\sigma+P_F)/(k_B T)}\right]^{-1} \Biggr\}
%
%\end{aligned}$$ where $P_F$ is the Fermi momentum.  In this case the $\sigma$ mass must be determined by solving the equation $$%
0 = 1 - 2g_0 {\rm Re}\Pi_\sigma^{\rm BW}(b)
%$$ for each $T$ and $P_F$. This mass determines the width and coupling constant through Eqs.  (\[eq:widtheq\]) and (\[eq:coupling\]). 

The $\sigma$ state in the Wigner–Weyl phase is in general not a stable bound state.  For $T \neq 0$ it is a resonant state with a finite width which can decay into a quark–anti–quark pair.  However, when $T = 0$ and $P_F \neq 0$ the decay width vanishes for $m_\sigma < 2P_F$ and the $\sigma$ becomes a stable bound state.  This is a consequence of Pauli blocking which allows the $\sigma$ to decay only when $m_\sigma \geq 2P_F$ [@zhu94]. 

The $D_A$ Amplitude
===================

Determination of the $D_A$ amplitude, mass, widths and coupling constant is analogous to that of the $\sigma$ case.  In the Nambu–Goldstone phase the $D_A$ amplitude in the broken symmetry phase is given by $$%
d^{\rm NG}(k, b) = \frac{g_{Dqq}^{\rm NG}}{b\!\!\!/ - k\!\!\!/ - m} i\gamma_5 
\frac{(-k\!\!\!/ + m)}{2m}
%
\label{eq:dsoln3}$$ where $g_{Dqq}$ is the $D_A$ coupling constant.  The extracted polarization function is $$%
\Pi^{\rm NG}_D(b) = \frac{1}{\pi^3} \int d^4\bar{p}\; \delta(p^2 - m^2) 
\frac{-b\cdot p}{(b - p + i\epsilon^0)^2 - m^2} 
\left[ 1 - 2\beta_p^2 \theta(p_0) -  2\delta_p^2 \theta(-p_0) \right]
%
\label{eq:dpol1}$$ Similarly, the $D_A$ amplitude and the polarization function in the Wigner–Weyl phase are $$%
d^{\rm BW}(k, b) = -\frac{g_{Dqq}^{\rm BW}}{b\!\!\!/ - k\!\!\!/} 
i\gamma_5 k\!\!\!/
%
\label{eq:dsoln4}$$ and $$\begin{aligned}
%
\Pi^{\rm BW}_D(b) 
& = &
\frac{1}{\pi^3} \int d^4\bar{p}\; \delta(p^2) 
\frac{-b\cdot p}{(b - p + i\epsilon^0)^2} 
\nonumber\\
&   &
\;\;\;\;\;\;\;
\otimes
\left\{ 1 - 2\left[e^{(p-\mu)/(k_B T)}+1\right]^{-1} \theta(p_0) 
-  2\left[e^{(p+\mu)/(k_B T)}+1\right]^{-1} \theta(-p_0) \right\}
%
\label{eq:dpol2}\end{aligned}$$ Note that the $D_A$ polarization functions Eq.  (\[eq:dpol1\]) and (\[eq:dpol2\]) are [*not*]{} symmetric under the interchange of particle and anti–particle Fermi distribution functions due to the presence of the Heaviside functions $\theta(p_0)$ and $\theta(-p_0)$. As a consequence there are two possible branches of the $D_A$ state.  For example the real part of $\Pi^{\rm NG}_D$ on the positive mass shell $b^+ = (+m_D, \vec{0}\,)$ is $$\begin{aligned}
%
{\rm Re}\Pi^{\rm NG}_D(b^+)
& = &
\frac{2}{\pi^2} \Biggl\{ {\rm P}\int_m^\infty dx\; 
\frac{x^2\sqrt{x^2-m^2}}{x^2 - \frac{1}{4}m_D^2}
+2 {\rm P}\int_m^\infty dx\; 
\frac{x\sqrt{x^2-m^2}}{m_D - 2x} \left[e^{(x-\mu)/(k_B T)}+1\right]^{-1}
\nonumber \\
&   &
\;\;\;\;\;\;\;\;\;\;\;\;
-2 \int_m^\infty dx\; 
\frac{x\sqrt{x^2-m^2}}{m_D + 2x}\left[e^{(x+\mu)/(k_B T)}+1\right]^{-1}
\Biggr\} 
%\end{aligned}$$ while on the negative mass shell $b^- = (-m_D, \vec{0}\,)$ the corresponding result is $$\begin{aligned}
%
{\rm Re}\Pi^{\rm NG}_D(b^-)
& = &
\frac{2}{\pi^2} \Biggl\{ {\rm P}\int_m^\infty dx\; 
\frac{x^2\sqrt{x^2-m^2}}{x^2 - \frac{1}{4}m_D^2}
-2\int_m^\infty dx\; 
\frac{x\sqrt{x^2-m^2}}{m_D + 2x} \left[e^{(x-\mu)/(k_B T)}+1\right]^{-1}
\nonumber \\
&   &
\;\;\;\;\;\;\;\;\;\;\;\;
+ 2 {\rm P}\int_m^\infty dx\; 
\frac{x\sqrt{x^2-m^2}}{m_D - 2x}\left[e^{(x+\mu)/(k_B T)}+1\right]^{-1}
\Biggr\} 
%\end{aligned}$$ The origin of the two branches is due to the unequal treatment of particles and anti–particles at finite density.  When $\mu = 0$, the particles and anti–particles are described by the same Fermi distribution function and thus the $D_A$ polarization functions become invariant under the $b^+$ and $b^-$ interchange leading to a single branch. 

$\sigma$ and $D_A$ Masses, Widths and Coupling Constants {#results}
========================================================

We present results for the $\sigma$ and $D_A$ masses, widths and coupling constants for zero temperature and finite density.  Results for other combinations of $T$ and $\mu$ will be given elsewhere together with relevant technical details of this work.  We use the non–covariant regularization and simply cut off the divergent integrals using a cut off parameter $\Lambda$. The other input parameter is the coupling constant $g_0$. We arbitrary choose $\Lambda$ = 0.61 GeV and $g_0$ = 5.01 GeV$^{-2}$ which yields a dynamical quark mass of $m$ = 0.305 GeV $\approx$ $\frac{1}{3}M_N$ at $(T, \mu) = (0, 0)$, where $M_N$ is the nucleon mass.[^5]

In Figure 1a we plot the $\sigma$ mass as a function of Fermi momentum.  In the broken symmetry phase of the model $m_\sigma = 2m$ where $m$ is dynamical quark mass.  We see that $m_\sigma$ decreases continuously with increasing $P_F$ and vanishes at the critical Fermi momentum $(P_F)_c$ of about 0.3 GeV. The phase transition is second order like in the NJL model.  Beyond $(P_F)_c$ the $\sigma$ mass starts to increase monotonically and by $P_F$ = 0.44 GeV its value is equal to the value in free space.  For all values of $P_F$ we find that $m_\sigma < 2 P_F$ and therefore it has vanishing width.  Hence we find the $\sigma$ to be a stable bound state at finite density and zero temperature in this model. 

The behaviour of the density dependent $\sigma$ mass is reflected in the $\sigma qq$ coupling constant shown in Figure 1b as a function of $P_F$. When $P_F = 0$ the value of $g^2_{\sigma qq}/4\pi$ is about 0.342 and it continuously increases with density until the chiral symmetry restoration point where it assumes its maximum value of 0.636.  Thus as the $\sigma$ mass decreases its coupling constant increases indicating the strengthening of the $\bar{q}q$ binding.  Above $(P_F)_c$, the coupling constant starts to decrease with the increasing $\sigma$ mass.  However, when the $\sigma$ mass becomes comparable to its value in free space the coupling constant is about 50% larger than the free space value.  Hence the $\sigma$ mass is much more sensitive to changes in its coupling constant in the Wigner–Weyl phase than in the Nambu–Goldstone phase. 

We find that there are no solutions to the equation determining the $D_A$ mass in the $b^-$ frame with our choice of input parameters.  The reason is the rather small value of the regulator $\Lambda$. We can find solutions for the $D_A$ mass in this branch if we cut off the divergent integrals in the $D_A$ polarization function independently of $\Lambda$. However, we choose to work with only one regulator as in previous works of the NJL model at finite $T$ and $\mu$ [@reviews].  With a single regulator we find solutions for the diquark mass in the $b^+$ frame but only for $P_F$ $>$ 0.06 GeV. Thus there are no $D_A$ states in free space and for $T=0$ its formation occurs only above a critical density the value of which depends on the input parameters.  This implies that below this critical density there is no contribution to the solution of the ‘t Hooft model from the $D_A$ term in the Haag expansion.  We also find solutions for the $D_A$ width indicating that this is a resonant state in contrast to the $\sigma$ state.  The $D_A$ mass and its width are shown in Figure 2a. 

In the broken symmetry mode the $D_A$ mass peaks around $P_F$ = 0.2 GeV and decreases monotonically until the phase transition point.  Beyond this point it increases almost linearly as in the $\sigma$ case.  We find that $m_D > m_\sigma$ for all values of density in which there is a resonant $D_A$ state.  However, the $D_A$ coupling constant is one order of magnitude smaller than that of the $\sigma$ state as shown in Figure 2b.  In fact, $g_{Dqq}$ does not become appreciable until $P_F$ $\approx$ 0.2 GeV and continuously increases with increasing density.  Since $g_{Dqq}$ normalizes the $D_A$ amplitude this means that the contribution from the $D_A$ term in the Haag expansion becomes important with increasing density but it will always be a correction to the dominant $\sigma$ term. 

Conclusion {#concl}
==========

In this work we have constructed a solution to the ‘t Hooft interaction Lagrangian valid for finite temperature and chemical potential beyond the mean field approximation.  To our knowledge this is the first time that such a solution has been presented in the literature.  In order to construct this solution we have used the NQA to quantum field theory involving the second order Haag expansion in the one–loop approximation.  We have only included the color 1 $\sigma$ and color $\bar{3}$ $D_A$ composite fields in our ansatz to the ‘t Hooft model.  An improved solution may be obtained by adding all possible channels, both color singlet and non–singlet, allowed by the model Lagrangian. 

Our solution was obtained by solving the self–consistency equations for the $\sigma$ and $D_A$ Haag amplitudes.  This has been accomplished by calculating the masses, widths and coupling constants of the $\sigma$ and the $D_A$ states.  At finite density and zero temperature we find that the $\sigma$ is a stable bound state with a vanishing width while the $D_A$ is a resonant state with a finite width.  The latter state is resonant because we have only included the $\bar{q}q$ channels in our model Lagrangian.  Thus our model provides no mechanism to create a bound $D_A$ state.  These properties of the $\sigma$ and the $D_A$ states can also be calculated using the standard Hartree–Fock–RPA approach.  We stress that the advantage of using the NQA over the Hartree–Fock–RPA approach is that one can explicitly obtain a solution to the model field theory. 

It would be interesting to apply our method to study the properties of a bound diquark state [@diquark] at finite $T$ and $\mu$. This may be done by adding a color $\bar{3}$ $qq$ channel to our model and repeat the calculation presented here using the same ansatz.  A vanishing diquark width would indicate a bound state just as in the case of the $\sigma$ shown in Figure 1a.  In addition we can determine the color superconducting gap by calculating the off–diagonal Hamiltonian and demanding that it vanishes.  We hope to report on the outcome of this investigation in the near future. 

Y. U. would like to thank G. Ripka for a very useful discussion on this work.  This work is supported in part by SEUIYD–PB97–1227.  Y. U. is also supported by the fellowship of Ministerio de Education y Ciencia de España under the auspices of the program “Estancias Temporales de Cientificos y Tecnólogos Extranjeros en España”. 

O.W. Greenberg, Virutes of the Haag Expansion in Quantum Field Theory, Proceedings of the First Arctic Workshop on Future Physics and Accelerators, Saariselka, August 1994, Los Alamos Archives [arXiv:hep-ph/9502253]{}.  G. ’t Hooft, Phys.  Rev. D4 (1976) 3432; M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl.  Phys.  B 163 (1980) 46.  O.W. Greenberg, Phys.  Rev. 139 (1965) B1038.  R. Haag, Dan.  Mat.  Fys.  Medd.  29 (1955).  Y. Umino and V. Vento, Phys.  Letts.  B472 (2000) 5.  H. Umezawa, H. Matsumoto and M. Tachiki, Thermo Field Dynamics and Condensed States, (North–Holland, Amsterdam, 1982); H. Umezawa, Advanced Field Theory, (AIP, New York, 1992).  O.W. Greenberg, Can.  J. Phys.  74 (1994) 574.  Y. Nambu and G. Jona–Lasinio, Phys.  Rev. 122 (1961) 345; 124 (1961) 246.  For example, see the following review articles and references therein: S.P. Klevansky, Rev. Mod.  Phys.  64 (1992) 649; T. Hatsuda and T. Kunihiro, Phys.  Rep. 247 (1994) 221.  T. Schaefer and E.V. Shuryak, Rev. Mod.  Phys.  70 (1998)323.  R. Rapp, T. Schaefer, E.V. Shuryak and M. Velkovsky, Phys.  Rev. Letts.  81 (1998) 53; M. Alford, K. Rajagopal and F. Wilczek, Phys.  Letts.  B422 (1998) 247.  O.W. Greenberg and P.K. Mohapatra, Phys.  Rev. D 34 (1986) 1136; O.W. Greenberg and L.H. Orr, Phys.  Rev. D 36 (1987) 1240.  C. Itzykson and J.B. Zuber, Quantum Field Theory, (McGraw–Hill, New York, 1980).  M Asakawa and K. Yazaki, Nucl.  Phys.  A 504 (1989) 668.  G. Källen, Quantum Electrodynamics, (Springer–Verlag, 1972), Chapter VI.  P. Zhuang, J. Hüfner and S.P. Klevansky, Nucl.  Phys.  A 576 (1994) 525. 

FIGURE CAPTIONS

FIGURE 1.  $\sigma$ mass and $\sigma qq$ coupling constant as functions of Fermi momentum $P_F$ for $T= 0$. Input parameters are $\Lambda$ = 0.61 GeV and $g_0$ = 5.01 GeV$^{-2}$. The critical Fermi momentum is $(P_F)_{\rm C}$ = 0.301 GeV. (a) $m_\sigma$ vs. $P_F$. The $\sigma$ is a stable bound state and has vanishing width. ( b) $g^2_{\sigma qq}/4\pi$. 0.75cm FIGURE 2.  Color $\bar{3}$ scalar diquark mass and $Dqq$ coupling constant evaluated in the frame $b_\mu = (+m_D, \vec{0}\;)$ as functions of Fermi momentum $P_F$ for $T= 0$. Input parameters are the same as in Figure 1. ( a) $m_D$ vs $P_F$. The diquark state does not exists for $P_F$ $<$ 0.06 GeV. It is a resonant state and its width $\Gamma$ is also shown in the figure bounded by the dashed line. ( b) $g^2_{Dqq}/4\pi$.

Figure 1

Figure 2

[^1]: We use the conventions and normalizations of Itzykson and Zuber [@IZ]. 

[^2]: They are identified as coefficients of the $\gamma_0$ operator. 

[^3]: The real part of $\Pi^{\rm NG}_\sigma$ is most easily determined by calculating the imaginary part and relating the result to the real part as shown in [@kallen]. 

[^4]: Note that their coupling constant $G$ is normalized to $G = \frac{4N_{\rm c} +1}{4N_{\rm c}} g$ where $g$ is the coupling constant appearing in the NJL Lagrangian. 

[^5]: These are the same parameters used in [@umi00].