# Gross-Neveu model with O(2)${}_{L}\times$O(2)R chiral symmetry: duality with
Zakharov-Mikhailov model and large $N$ solution

Michael<EMAIL_ADDRESS>Institut für Theoretische Physik,
Universität Erlangen-Nürnberg, D-91058, Erlangen, Germany

###### Abstract

The two-flavor Gross-Neveu model with U(2)${}_{L}\times$U(2)R chiral symmetry
in 1+1 dimensions is used to construct a novel variant of four-fermion
theories with O(2)${}_{L}\times$O(2)R chiral symmetry. The spontaneous
breaking of the group O(2), a continuous group with two connected components
(rotations and reflections), gives rise to new phenomena. It is ideally suited
to describe a situation where two distinct kinds of condensation compete, in
particular chiral symmetry breaking (particle-hole condensation) and Cooper
pairing (particle-particle condensation). After solving the O(2) chiral Gross-
Neveu model in detail, we demonstrate that it is dual to another classically
integrable model due to Zakharov and Mikhailov. The duality enables us to
solve the quantum version of this model in the large $N$ limit with
semiclassical methods, supporting its integrability at the quantum level. The
resulting model is the unique four-fermion theory sharing the full Pauli-
Gürsey symmetry with free, massless fermions (“perfect Gross-Neveu model”) and
provides us with a solvable model for competing chiral and Cooper pair
condensates, including explicit soliton dynamics and the phase diagram.

##  I Introduction

Back in 1978, Zakharov and Mikhailov L1  proved the integrability of three
classical spinor models in 1+1 dimensions for any number of components $N$.
The quantum versions of two of them are by now also well under control, at
least in the large $N$ limit, namely the Gross-Neveu (GN) model L2

${\cal L}_{\rm
GN}=-2i\psi_{1}^{(i)*}\partial\psi_{1}^{(i)}+2i\psi_{2}^{(i)*}\bar{\partial}\psi_{2}^{(i)}+\frac{g^{2}}{2}(\psi_{1}^{(i)*}\psi_{2}^{(i)}+\psi_{2}^{(i)*}\psi_{1}^{(i)})^{2}$
(1)

and the chiral GN model or 2d Nambu–Jona-Lasinio (NJL) model L3

${\cal L}_{\rm
NJL}=-2i\psi_{1}^{(i)*}\partial\psi_{1}^{(i)}+2i\psi_{2}^{(i)*}\bar{\partial}\psi_{2}^{(i)}+2g^{2}(\psi_{1}^{(i)*}\psi_{2}^{(i)})(\psi_{2}^{(j)*}\psi_{1}^{(j)}).$
(2)

We use the notation

$z=x-t,\quad\bar{z}=x+t,\quad\psi_{1}=\psi_{L},\quad\psi_{2}=\psi_{R}$ (3)

and sum implicitly over “color” indices $i,j$ from 1 to $N$. Apparently,
integrability at the classical level allows one to solve the quantized theory
in the large $N$ limit with semiclassical methods, including time dependent
multi-soliton interactions, in explicit analytical form L4 ; L5 ; L6 . At the
classical level (i.e., with $c$-number fermion fields), these two models are
connected to chiral fields on the symplectic group Sp(2$N,\mathbb{R}$) (GN
model) or the special unitary group SU($N$) (NJL model). The third model
presented in L1  and related to chiral fields on the orthogonal group O($N$)
has so far not had any significant impact in particle physics. It is sometimes
referred to as Zakharov-Mikhailov (ZM) model and has the less familiar
Lagrangian

${\cal L}_{\rm
ZM}=-2i\psi_{1}^{(i)*}\partial\psi_{1}^{(i)}+2i\psi_{2}^{(i)*}\bar{\partial}\psi_{2}^{(i)}+g^{2}(\psi_{1}^{(i)*}\psi_{1}^{(j)}-\psi_{1}^{(j)*}\psi_{1}^{(i)})(\psi_{2}^{(i)*}\psi_{2}^{(j)}-\psi_{2}^{(j)*}\psi_{2}^{(i)}).$
(4)

Interestingly, the quantum version of the ZM model has appeared again in a
different context in the meantime. When studying four-fermion theories that
give rise to Cooper pairing as opposed to fermion-antifermion pairing, Chodos,
Minakata and Cooper (CMC) L7  proposed a model whose Lagrangian is equivalent
to

${\cal L}_{\rm
CMC}=-2i\psi_{1}^{(i)*}\partial\psi_{1}^{(i)}+2i\psi_{2}^{(i)*}\bar{\partial}\psi_{2}^{(i)}+2g^{2}(\psi_{1}^{(i)}\psi_{2}^{(i)})(\psi_{2}^{(j)*}\psi_{1}^{(j)*}).$
(5)

They noticed many similarities with the chiral GN model such as asymptotic
freedom, mass generation and a massless bound state. If written in the form
(2) and (5), one sees that the quantized NJL and CMC models are “dual” to each
other in the sense that they are related by a simple Bogoliubov transformation
L8 ,

$\psi_{1}^{(i)}\to\psi_{1}^{(i)\dagger},\quad\psi_{2}^{(i)}\to\psi_{2}^{(i)}.$
(6)

Hence both models are mathematically equivalent, although their physics looks
quite different at first sight. This observation incited us to study yet
another four-fermion theory obtained by “self-dualizing” the NJL model, i.e.
adding the interaction terms of models (2) and (5) with the same coupling
constant L9 . The resulting theory is singled out from all other variants of
the GN model in that it shares the full Pauli-Gürsey symmetry L10 ; L11  with
free, massless Dirac fermions, in addition to a O($N$) color symmetry. Due to
this high degree of symmetry, it has been dubbed “perfect GN model” (pGN) in
L12 ,

${\cal L}_{\rm
pGN}=-2i\psi_{1}^{(i)*}\partial\psi_{1}^{(i)}+2i\psi_{2}^{(i)*}\bar{\partial}\psi_{2}^{(i)}+2g^{2}\left[(\psi_{1}^{(i)*}\psi_{2}^{(i)})(\psi_{2}^{(j)*}\psi_{1}^{(j)})+(\psi_{1}^{(i)}\psi_{2}^{(i)})(\psi_{2}^{(j)*}\psi_{1}^{(j)*})\right].$
(7)

As a matter of fact, the Lagrangians of the pGN model and the ZM model become
identical once fermion fields are treated as anticommuting variables. This
observation has stimulated our interest in the large $N$ limit of the quantum
ZM model (hereafter referred to as pGN model). However, a systematic solution
of the soliton problem or other questions has so far resisted all our
attempts. Judging from the experience with the GN and NJL models, if the
classical ZM model is integrable, one would expect the same kind of
solvability for the GN, NJL and pGN models.

Here we propose a solution to this problem. We have found a duality between
the one-flavor pGN model and the two-flavor NJL model, for which the general
soliton solution has already been given at large $N$ L13 ; L14 . In the case
of the GN model, the most efficient way to solve soliton dynamics has been to
start from the solution of the NJL model with U(1)${}_{L}\times$U(1)R chiral
symmetry and specialize to real mean field solutions, thereby solving the
O(1)${}_{L}\times$O(1)R [or Z${}_{2,L}\times$Z2,R] GN model L6 . Here we
generalize this approach by first reducing known solutions of the
U(2)${}_{L}\times$U(2)R 2-flavor NJL model to a novel O(2)${}_{L}\times$O(2)R
variant of the GN model, again by selecting real mean fields. This model in
turn will be shown to be dual to the pGN model, thus providing the key to the
missing large $N$ solution of the pGN model.

This paper is organized as follows. After a reminder of some elementary facts
about O(2) group theory in Sect. II, we propose the O(2)${}_{L}\times$O(2)R
symmetric descendent of the unitary two-flavor chiral GN model in Sect. III.
The vacuum structure and gap equation are determined in Sect. IV, the meson
spectrum in Sect. V, using the random phase approximation (RPA). Sects. VI and
VII are dedicated to the most elementary solitonic multifermion bound states,
the kink, and interactions of several kinks. In Sect. VIII we then show that
the analogue of twisted kinks exist as constituents of bound states, similar
to what happens in the GN model. The simplest breather is also constructed.
Sect. IX addresses a topic well known from the NJL model, namely massless
multi-fermion bound states and inhomogeneous structures at finite chemical
potentials (chiral spirals, kink-antikink crystal). Sect. X is perhaps the
most important one of this paper. Here we show the equivalence between the
O(2) chiral GN model and the so far unsolved ZM (or pGN) model). In Sect. XI
we summarize our findings, reviewing the preceding results in the light of the
duality from a physical point of view.

##  II Elementary group theory: from U(2) to O(2)

In the defining representation, elements of the unitary group
U(2)=U(1)$\times$SU(2) can be parametrized as

$U=e^{-i\psi}e^{i\vec{n}\vec{\tau}\delta},\quad\vec{n}=\left(\begin{array}[]{c}\sin\theta\cos\phi\\\
\sin\theta\sin\phi\\\ \cos\theta\end{array}\right),$ (8)

with $\tau_{i}$ in the standard form of the Pauli matrices. An O(2) matrix is
a real U(2) matrix. There are two distinct ways to get a real matrix out of
(8):

  1. 1.

$\theta=\phi=\pi/2,\psi=0$ (hence $n_{1}=n_{3}=0$):

$R(\delta)=e^{i\tau_{2}\delta}=\left(\begin{array}[]{rr}\cos\delta&\sin\delta\\\
-\sin\delta&\cos\delta\end{array}\right),\quad{\rm det}R(\delta)=1.$ (9)

  2. 2.

$\psi=\delta=\pi/2,\phi=0$ (hence $n_{2}=0$):

$\displaystyle I(\theta)=n_{1}\tau_{1}+n_{3}\tau_{3}$ $\displaystyle=$
$\displaystyle\tau_{3}e^{i\tau_{2}\theta}=\left(\begin{array}[]{rr}\cos\theta&\sin\theta\\\
\sin\theta&-\cos\theta\end{array}\right),\quad{\rm det}I(\theta)=-1.$ (12)

The matrix $R(\delta)$ corresponds to a rotation around the center of the
($x,y$) plane by an angle $\delta$. The matrix $I(\theta)$ is a rotation by an
angle $\theta$ followed by a reflection in the new $x$-axis. These two
elements belong to the two connected components of the group O(2)
characterized by their determinants $\pm 1$, with only rotations forming a
subgroup, SO(2). The group manifolds of U(2) and O(2) are S${}_{1}\times$S3
and S1+S1, respectively. What we have done here is the two-dimensional analog
of the transition from U(1) to O(1) (or Z2) where one restricts the function
$e^{-i\psi}$ to its real values, $\pm 1$. Finally, note that O(2) is a non-
Abelian group. Products of its elements can easily be evaluated with the help
of

$I(\theta)=\tau_{3}R(\theta)=R(-\theta)\tau_{3},\quad
R(\theta_{1})R(\theta_{2})=R(\theta_{1}+\theta_{2}).$ (13)

##  III Gross-Neveu model with O(2)${}_{L}\times$O(2)R symmetry

The Lagrangian of the U(2)${}_{L}\times$ U(2)R symmetric two-flavor NJL model
reads L14

${\cal L}_{\rm
U(2)}=\bar{\psi}i\partial\\!\\!\\!/\psi+\frac{g^{2}}{4}\left[(\bar{\psi}\psi)^{2}+(\bar{\psi}\vec{\tau}\psi)^{2}+(\bar{\psi}i\gamma_{5}\psi)^{2}+(\bar{\psi}i\gamma_{5}\vec{\tau}\psi)^{2}\right].$
(14)

We choose a “chiral” representation of Dirac matrices ($\gamma_{5}$ diagonal),

$\gamma^{0}=\sigma^{1},\quad\gamma^{1}=i\sigma^{2},\quad\gamma_{5}=-\sigma_{3}.$
(15)

If we expand the isovector interaction terms, the interaction part of
Lagrangian (14) consists of eight squares of bilinears. We can now generate
simpler, Lorentz invariant Lagrangians by deleting some of these terms. We
propose the following choice, keeping only half of the interaction terms:

${\cal L}_{\rm
O(2)}=\bar{\psi}i\partial\\!\\!\\!/\psi+\frac{g^{2}}{4}\left[(\bar{\psi}\psi)^{2}+(\bar{\psi}\tau_{1}\psi)^{2}+(\bar{\psi}\tau_{3}\psi)^{2}+(\bar{\psi}i\gamma_{5}\tau_{2}\psi)^{2}\right].$
(16)

Although it is not obvious, the resulting model has a O(2)${}_{L}\times$O(2)R
chiral symmetry. Let us evaluate the transformation of the four remaining
bilinears induced by orthogonal chiral transformations of the spinor fields.
Isospin rotation of a left-handed spinor around the 2-axis,

$P_{L}e^{i\alpha\tau_{2}}+P_{R}:\quad\left(\begin{array}[]{c}\bar{\psi}\psi\\\
\bar{\psi}i\gamma_{5}\tau_{2}\psi\end{array}\right)^{\prime}=R(-\alpha)\left(\begin{array}[]{c}\bar{\psi}\psi\\\
\bar{\psi}i\gamma_{5}\tau_{2}\psi\end{array}\right),\quad\left(\begin{array}[]{c}\bar{\psi}\tau_{3}\psi\\\
\bar{\psi}\tau_{1}\psi\end{array}\right)^{\prime}=R(\alpha)\left(\begin{array}[]{c}\bar{\psi}\tau_{3}\psi\\\
\bar{\psi}\tau_{1}\psi\end{array}\right).$ (17)

Isospin rotation of a right-handed spinor around the 2-axis,

$P_{L}+P_{R}e^{i\alpha\tau_{2}}:\quad\left(\begin{array}[]{c}\bar{\psi}\psi\\\
\bar{\psi}i\gamma_{5}\tau_{2}\psi\end{array}\right)^{\prime}=R(\alpha)\left(\begin{array}[]{c}\bar{\psi}\psi\\\
\bar{\psi}i\gamma_{5}\tau_{2}\psi\end{array}\right),\quad\left(\begin{array}[]{c}\bar{\psi}\tau_{3}\psi\\\
\bar{\psi}\tau_{1}\psi\end{array}\right)^{\prime}=R(\alpha)\left(\begin{array}[]{c}\bar{\psi}\tau_{3}\psi\\\
\bar{\psi}\tau_{1}\psi\end{array}\right).$ (18)

Isospin reflection and rotation around the 2-axis of a left-handed spinor

$P_{L}\tau_{3}e^{i\alpha\tau_{2}}+P_{R}:\quad\left(\begin{array}[]{c}\bar{\psi}\psi\\\
\bar{\psi}i\gamma_{5}\tau_{2}\psi\end{array}\right)^{\prime}=R(\alpha)\left(\begin{array}[]{c}\bar{\psi}\tau_{3}\psi\\\
\bar{\psi}\tau_{1}\psi\end{array}\right),\quad\left(\begin{array}[]{c}\bar{\psi}\tau_{3}\psi\\\
\bar{\psi}\tau_{1}\psi\end{array}\right)^{\prime}=R(-\alpha)\left(\begin{array}[]{c}\bar{\psi}\psi\\\
\bar{\psi}i\gamma_{5}\tau_{2}\psi\end{array}\right).$ (19)

Isospin reflection and rotation around the 2-axis of a right-handed spinor

$P_{L}+P_{R}\tau_{3}e^{i\alpha\tau_{2}}:\quad\left(\begin{array}[]{c}\bar{\psi}\psi\\\
\bar{\psi}i\gamma_{5}\tau_{2}\psi\end{array}\right)^{\prime}=I(\alpha)\left(\begin{array}[]{c}\bar{\psi}\tau_{3}\psi\\\
\bar{\psi}\tau_{1}\psi\end{array}\right),\quad\left(\begin{array}[]{c}\bar{\psi}\tau_{3}\psi\\\
\bar{\psi}\tau_{1}\psi\end{array}\right)^{\prime}=I(\alpha)\left(\begin{array}[]{c}\bar{\psi}\psi\\\
\bar{\psi}i\gamma_{5}\tau_{2}\psi\end{array}\right).$ (20)

Thus rotations leave the combinations
$(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\tau_{2}\psi)^{2}$ and
$(\bar{\psi}\tau_{1}\psi)^{2}+(\bar{\psi}\tau_{3}\psi)^{2}$ separately
invariant. Reflections leave only the sum of all four terms invariant, as they
induce hopping between the two pairs of bilinears in addition to a rotation.

This confirms that model (16) indeed possesses a O(2)${}_{L}\times$O(2)R
chiral symmetry. Consequently, the $\tau_{2}$-components of the isospin vector
and axial vector currents are conserved,

$\partial_{\mu}\bar{\psi}\gamma^{\mu}\tau_{2}\psi=0,\quad\partial_{\mu}\bar{\psi}\gamma^{\mu}\gamma_{5}\tau_{2}\psi=0.$
(21)

In addition the model has SU($N$) color symmetry and U(1) fermion number
($\psi\to e^{i\alpha}\psi$). It also shares with the GN model charge
conjugation with the familiar consequences, i.e., a real mean field in the
Hartree-Fock (HF) approach and a fermion spectrum that is symmetric about 0,

$C:\quad\psi\to\gamma_{5}\psi^{*},\quad\gamma_{5}H^{*}\gamma_{5}=-H.$ (22)

Here, $H$ is the HF Hamiltonian in coordinate space. In view of the conserved
charges, there are three chemical potentials one can add to the HF
Hamiltonian,

${\cal H}\to{\cal
H}-\mu\psi^{\dagger}\psi-\mu_{2}\psi^{\dagger}\tau_{2}\psi-\mu_{5,2}\psi^{\dagger}\gamma_{5}\tau_{2}\psi.$
(23)

This is important if one considers the phase diagram of the model.

##  IV Vacua and dynamical mass

We start solving the chiral O(2) GN model (16) in the large $N$ limit by
determining its gap equation and vacuum structure. The HF equation reads

$(i\gamma^{\mu}\partial_{\mu}-S_{0}-S_{1}\tau_{1}-S_{3}\tau_{3}-i\gamma_{5}P_{2}\tau_{2})\psi=0$
(24)

with the mean fields given by the vacuum expectation values

$\left(\begin{array}[]{c}S_{0}\\\ S_{1}\\\ S_{3}\\\
P_{2}\end{array}\right)=-\frac{g^{2}}{2}\left(\begin{array}[]{c}\langle\bar{\psi}\psi\rangle\\\
\langle\bar{\psi}\tau_{1}\psi\rangle\\\
\langle\bar{\psi}\tau_{3}\psi\rangle\\\
\langle\bar{\psi}i\gamma_{5}\tau_{2}\psi\rangle\end{array}\right).$ (25)

Denoting 2-component isospinors of chirality $L/R$ by $\psi_{1,2}$, Eq. (24)
may be rewritten in canonical form as

$i\partial_{t}\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)=\left(\begin{array}[]{cc}i\partial_{x}&\Delta^{T}\\\
\Delta&-i\partial_{x}\end{array}\right)\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right).$ (26)

Here, $\Delta$ is the 2$\times$2 flavor matrix

$\Delta=S_{0}+S_{1}\tau_{1}+S_{3}\tau_{3}-iP_{2}\tau_{2}=\left(\begin{array}[]{cc}S_{0}+S_{3}&S_{1}-P_{2}\\\
S_{1}+P_{2}&S_{0}-S_{3}\end{array}\right).$ (27)

The fact that this matrix is real reflects the orthogonal symmetry. In order
to determine the HF ground state, we have to diagonalize the 4$\times$4
Hamiltonian

$h=\left(\begin{array}[]{cc}-k&\Delta^{T}\\\ \Delta&k\end{array}\right)$ (28)

with a constant, real matrix $\Delta$. We find four modes with energies
$\pm\sqrt{m_{1}^{2}+k^{2}},\pm\sqrt{m_{2}^{2}+k^{2}}$ where

$m_{1,2}^{2}=\left(\sqrt{S_{0}^{2}+P_{2}^{2}}\pm\sqrt{S_{1}^{2}+S_{3}^{2}}\right)^{2}.$
(29)

Minimizing the HF vacuum energy density

${\cal E}_{\rm
vac}=-N\int_{-\Lambda/2}^{\Lambda/2}\frac{dk}{2\pi}\left(\sqrt{k^{2}+m_{1}^{2}}+\sqrt{k^{2}+m_{2}^{2}}\,\right)+\frac{m_{1}^{2}+m_{2}^{2}}{2g^{2}}$
(30)

with respect to $m_{1},m_{2}$ yields two gap equations

$0=1+\frac{Ng^{2}}{2\pi}\ln\frac{m_{i}^{2}}{\Lambda^{2}}\quad(i=1,2)$ (31)

that are only compatible if $m_{1}=m_{2}=m$. The final gap equation is
identical to that of the GN model and has the same implications (dimensional
transmutation, asymptotic freedom). The renormalized vacuum energy density
becomes

${\cal E}_{\rm vac}=-N\frac{m^{2}}{2\pi}.$ (32)

According to Eq. (29), there are two possible choices leading to
$m_{1}=m_{2}$:

$\displaystyle 1)$ $\displaystyle S_{1}=S_{3}=0,\quad
S_{0}^{2}+P_{2}^{2}=m^{2},\quad S_{0}=m\cos\alpha,\quad
P_{2}=m\sin\alpha,\quad\Delta=mR(-\alpha)$ $\displaystyle 2)$ $\displaystyle
S_{0}=P_{2}=0,\quad S_{1}^{2}+S_{3}^{2}=m^{2},\quad S_{3}=m\cos\alpha,\quad
S_{1}=m\sin\alpha,\quad\Delta=mI(\alpha)$ (33)

As expected, the vacuum manifold is the group manifold of O(2) comprising two
disjoint circles in the ($S_{0},P_{2}$) and ($S_{1},S_{3})$ planes. For
simplicity we shall denote a vacuum in the ($S_{0},P_{2}$) plane as “rotation
vacuum” and in the ($S_{1},S_{3}$) plane as “reflection vacuum” to indicate
their origin in the two components of the group O(2). When breaking
spontaneously the O(2) chiral symmetry, the system has to pick one circle
(rotation or reflection) and a particular point on this circle (angle
$\alpha$). All of these vacua are of course on an equal footing. If not
indicated otherwise, our standard choice will be ($S_{0}=m,P_{2}=0,\Delta=m$)
for the rotation vacuum and ($S_{1}=0,S_{3}=m,\Delta=m\tau_{3}$) for the
reflection vacuum, so that $\Delta$ is diagonal in the standard vacua.

##  V Meson spectrum in random phase approximation (RPA)

Here we follow closely previous works on other GN model variants, using the
equations of motion method for the one-body density matrix $Q(x,y)$ L15 ; L16
. Starting point is the equation

$\displaystyle i\partial_{t}Q(x,y)$ $\displaystyle=$
$\displaystyle-i\left[\partial_{y}Q(x,y)\gamma_{5}+\gamma_{5}\partial_{x}Q(x,y)\right]$
(34) $\displaystyle-\frac{Ng^{2}}{2}\sum_{n=1}^{4}\left\\{{\rm Tr}[{\cal
O}_{n}Q(x,y)]{\cal O}_{n}Q(x,y)-Q(x,y){\cal O}_{n}{\rm Tr}[{\cal
O}_{n}Q(x,y)]\right\\}$

with

${\cal O}_{1}=\gamma^{0},\quad{\cal O}_{2}=\gamma^{0}\tau_{1},\quad{\cal
O}_{3}=\gamma^{0}\tau_{3},\quad{\cal O}_{4}=i\gamma^{1}\tau_{2}$ (35)

matching the interactions in Lagrangian (16). Let us first assume the rotation
vacuum ($S_{0}=m,P_{2}=0$). Using free positive and negative energy spinors
for one flavor ($E_{k}=\sqrt{k^{2}+m^{2}}$)

$\left(\begin{array}[]{cc}-k&m\\\ m&k\end{array}\right)u(k)=E_{k}u(k),\quad
u(k)=\left(\begin{array}[]{c}u_{1}\\\
u_{2}\end{array}\right)=\frac{1}{\sqrt{2E_{k}(E_{k}+k)}}\left(\begin{array}[]{c}m\\\
E_{k}+k\end{array}\right),$ (36) $\left(\begin{array}[]{cc}-k&m\\\
m&k\end{array}\right)v(k)=E_{k}v(k),\quad
v(k)=\left(\begin{array}[]{c}v_{1}\\\
v_{2}\end{array}\right)=\frac{1}{\sqrt{2E_{k}(E_{k}-k)}}\left(\begin{array}[]{c}m\\\
E_{k}-k\end{array}\right),$ (37)

the corresponding free massive states in the 2-flavor case are

$u_{I}=\left(\begin{array}[]{c}u_{1}\\\ 0\\\ u_{2}\\\
0\end{array}\right),\quad u_{II}=\left(\begin{array}[]{c}0\\\ u_{1}\\\ 0\\\
u_{2}\end{array}\right),\quad v_{I}=\left(\begin{array}[]{c}v_{1}\\\ 0\\\
v_{2}\\\ 0\end{array}\right),\quad v_{II}=\left(\begin{array}[]{c}0\\\
v_{1}\\\ 0\\\ v_{2}\end{array}\right).$ (38)

The isospin labels $I,II$ will be denoted by Greek letters below. Linearizing
Eq. (34) in the fluctuation around the vacuum density matrix and sandwiching
it between vacuum and one meson states of momentum $P$ and energy ${\cal
E}(P)$, we arrive at the RPA equations

$\displaystyle{\cal E}(P)X_{\alpha\beta}(P,k)$ $\displaystyle=$ $\displaystyle
E(k-P,k)X_{\alpha\beta}(P,k)-\frac{Ng^{2}}{2}\sum_{n=1}^{4}v_{\alpha}^{\dagger}(k-P){\cal
O}_{n}u_{\beta}(k)\Phi_{n}(P),$ $\displaystyle{\cal E}(P)Y_{\alpha\beta}(P,k)$
$\displaystyle=$
$\displaystyle-E(k-P,k)Y_{\alpha\beta}(P,k)+\frac{Ng^{2}}{2}\sum_{n=1}^{4}u_{\alpha}^{\dagger}(k-P){\cal
O}_{n}v_{\beta}(k)\Phi_{n}(P),$ $\displaystyle E(k-P,k)$ $\displaystyle=$
$\displaystyle E_{k-P}+E_{k},$ (39)

with

$\Phi_{n}(P)=\int\frac{dq}{2\pi}\left\\{{\cal
B}^{n}_{\delta\gamma}(P,q)Y_{\gamma\delta}(P,q)+{\cal
C}^{n}_{\delta\gamma}(P,q)X_{\gamma\delta}(P,q)\right\\}$ (40)

and

$\displaystyle{\cal B}^{n}_{\delta\gamma}(P,q)$ $\displaystyle=$
$\displaystyle v_{\delta}^{\dagger}(q){\cal O}_{n}u_{\gamma}(q-P),$
$\displaystyle{\cal C}^{n}_{\delta\gamma}(P,q)$ $\displaystyle=$
$\displaystyle u_{\delta}^{\dagger}(q){\cal O}_{n}v_{\gamma}(q-P).$ (41)

Eqs. (39) are integral equations with a separable kernel as is characteristic
for all GN type models. They can thus be solved analytically. By first solving
Eq. (39) for $X_{\alpha\beta}$ and $Y_{\alpha\beta}$, we get

$\displaystyle X_{\alpha\beta}(P,k)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{4}{\cal A}_{\alpha\beta}^{n}(P,k)\Phi_{n}(P),$
$\displaystyle Y_{\alpha\beta}(P,k)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{4}{\cal D}_{\alpha\beta}^{n}(P,k)\Phi_{n}(P),$ (42)

with the definitions

$\displaystyle{\cal A}_{\alpha\beta}^{n}(P,k)$ $\displaystyle=$
$\displaystyle-Ng^{2}\frac{1}{{\cal
E}(P)-E(k-P,k)}v_{\alpha}^{\dagger}(k-P){\cal O}_{n}u_{\beta}(k),$
$\displaystyle{\cal D}_{\alpha\beta}^{n}(P,k)$ $\displaystyle=$ $\displaystyle
Ng^{2}\frac{1}{{\cal E}(P)+E(k-P,k)}u_{\alpha}^{\dagger}(k-P){\cal
O}_{n}v_{\beta}(k).$ (43)

Inserting the results (42) into Eq. (40) leads to the homogeneous linear
system for $\Phi_{n}(P)$,

$\Phi_{n}(P)=\sum_{m}{\cal M}_{nm}(P)\Phi_{m}(P).$ (44)

The 4$\times$4 matrix ${\cal M}_{nm}(P)$ is given by

${\cal M}_{nm}(P)=\int\frac{dq}{2\pi}\left\\{{\cal
B}_{\delta\gamma}^{n}(P,q){\cal D}_{\gamma\delta}^{m}(P,q)+{\cal
C}_{\delta\gamma}^{n}(P,q){\cal A}_{\gamma\delta}^{m}(P,q)\right\\}.$ (45)

An explicit analytical calculation shows that ${\cal M}_{nm}$ is diagonal,
hence Eq. (44) reduces to

${\cal M}_{nn}=1\quad({\rm no\ sum}).$ (46)

One finds only two different diagonal matrix elements ${\cal M}_{nn}$. In the
isovector pseudoscalar channel $P_{2}$,

${\cal
M}_{44}=\frac{Ng^{2}}{2}\int\frac{dk}{2\pi}\left(\frac{1}{E(k-P)}+\frac{1}{E(k)}\right)\left(\frac{P^{2}-E^{2}(k-P,k)}{{\cal
E}^{2}(P)-E^{2}(k-P,k)}\right).$ (47)

The choice ${\cal E}^{2}(P)=P^{2}$ converts the condition ${\cal M}_{44}=1$
into the vacuum gap equation. This proves the existence of a massless mode,
the “would be Goldstone boson” fluctuating in the direction tangential to the
vacuum circle. In the other three channels, there is a (marginally bound)
massive state with the common mass $2m$, the same as in the GN model. The
corresponding diagonal matrix elements ${\cal M}_{nn}$ read

${\cal M}_{11}={\cal M}_{22}={\cal
M}_{33}=\frac{Ng^{2}}{2}\int\frac{dk}{2\pi}\left(\frac{1}{E(k-P)}+\frac{1}{E(k)}\right)\left(\frac{4m^{2}+P^{2}-E^{2}(k-P,k)}{{\cal
E}^{2}(P)-E^{2}(k-P,k)}\right).$ (48)

Here the ansatz ${\cal E}^{2}(P)=4m^{2}+P^{2}$ yields again back to the gap
equation. The massive bound states are scalar mesons in the
$S_{0},S_{1},S_{3}$ channels.

One can repeat the same calculation by starting form the reflection vacuum
($S_{3}=m,S_{1}=0$). The only change in the formalism is the fact that the
spinors for isospin down have to be evaluated with $m\to-m$, since the vacuum
has $\Delta=m\tau_{3}$ rather than $\Delta=m$. The results are the same,
except that the massless mode now appears in the $S_{1}$ channel, again
tangential to the reflection vacuum circle in the chosen vacuum point. The
mesons in the $S_{0},S_{3},P_{2}$ channels all have the same mass 2$m$. At
first sight, it looks as if now a scalar meson would be massless and the
pseudoscalar one massive. This is not the case. When using the rotation
vacuum, the parity operation on spinors has the usual form,

$P:\quad\psi(x)\to\gamma^{0}\psi(-x),$ (49)

since the vacuum is also standard. To reach the reflection vacuum $m\tau_{3}$,
one has to perform the chiral transformation $P_{L}\tau_{3}+P_{R}$. Under this
transformation, the matrix $\gamma^{0}$ in (49) goes over into
$\tau_{3}\gamma^{0}$, so that now $S_{0},S_{3},P_{2}$ are scalars whereas
$S_{1}$ is pseudoscalar.

It is noteworthy that all massive mesons found in GN-type models so far share
the common mass $M=2m$. This does not mean that they are non-interacting,
otherwise the dispersion relation would not be ${\cal
E}(P)=\sqrt{4m^{2}+P^{2}}$. This “universality” is most likely a side effect
of the large $N$ limit. The RPA goes beyond the leading order (HF) by taking
into account fluctuations of O($1/\sqrt{N}$). It is plausible that meson
binding energies are suppressed by at least a factor of $1/N$ and therefore
not yet visible at this order of the large $N$ expansion.

We conclude this chapter with a table comparing the meson content of various
chirally symmetric GN-type models, to emphasize the common aspects as well as
the differences due to different symmetries. Only two rules govern the whole
picture in table 1: The total number of mesons is equal to the number of
squares of bilinears in the interaction Lagrangian, and the number of massless
modes equals the number of flat directions on the vacuum manifold.

| chiral symmetry | vacuum manifold | mesons | massless | massive
---|---|---|---|---|---
GN | O(1)${}_{L}\times$O(1)R | Z2 | 1 | - | 1
NJL | U(1)${}_{L}\times$U(1)R | S1 | 2 | 1 | 1
isoNJL | SU(2)${}_{L}\times$SU(2)R | S3 | 4 | 3 | 1
O(2) GN | O(2)${}_{L}\times$O(2)R | S${}_{1}+$S1 | 4 | 1 | 3
U(2) GN | U(2)${}_{L}\times$U(2)R | S${}_{1}\times$S3 | 8 | 4 | 4
Table 1: Overview over meson content of different chiral GN models.

##  VI Basic kink

Kink denotes a multifermion bound state connecting two different vacua at
$x\to\pm\infty$. We use units where $m=1$ from now on to make contact with the
literature. To set the stage, let us briefly recall the twisted kink of the
one-flavor NJL model originally due to Shei L17 . The kink at rest connecting
the U(1) vacua $e^{i\alpha}$ at $x\to-\infty$ and $e^{i\beta}$ at $x\to\infty$
has the mean field

$\Delta(x)=e^{i\alpha}(1-f(x))+e^{i\beta}f(x),\quad
f(x)=\frac{V(x)}{1+V(x)},\quad V(x)=e^{2x\sin\theta}.$ (50)

We have written it in a form where the interpolating structure is clear, since
$f$ and $(1-f)$ are kink-like scalar functions. The angle $\theta$ in $V(x)$
is called twist angle and related to the difference of the two asymptotic
vacuum phases via

$\theta=\frac{1}{2}(\alpha-\beta).$ (51)

Since U(1) is Abelian and $\Delta$ transforms under chiral transformations as
follows

$\psi_{1}\to e^{i\alpha_{1}}\psi_{1},\quad\psi_{2}\to
e^{i\alpha_{2}}\psi_{2},\quad\Delta\to e^{i\alpha_{1}}\Delta
e^{-i\alpha_{2}},$ (52)

the twist angle is chirally invariant. As such it has a physical meaning,
determining the slope of the kink profile and its fermion number. There is a
bound state with energy $\epsilon_{0}=\cos\theta$, occupied by $N\sin\theta$
fermions. If we restrict ourselves to real $\Delta$ as appropriate for the GN
model with discrete chiral symmetry, we have to choose $\theta=\pm\pi/2$ and
either $\alpha=\pi,\beta=0$ (kink) or $\alpha=0,\beta=\pi$ (antikink). Here,
the kink profile can only assume the steepest shape. The bound state moves to
0 energy, the center of the mass gap. We get

$\Delta(x)\to S(x)=\pm\tanh x,$ (53)

an early result attributed to Callan, Coleman, Gross and Zee in Refs. L18 ;
L19 .

Let us repeat the reduction from unitary to orthogonal chiral symmetry, now
for two flavors. We can start directly from the known result for the twisted
kink in the U(2)${}_{L}\times$U(2)R GN model L13 ; L14 . If the vacuum at
$x\to-\infty$ is taken as 1, the formalism yields the expression

$\Delta(x)=\frac{1+UV(x)}{1+V(x)}$ (54)

with

$V(x)=e^{2\sin\theta x},\quad
U=(1-\vec{p}\,\vec{p}^{\,\dagger})+e^{-2i\theta}\vec{p}\,\vec{p}^{\,\dagger}.$
(55)

$U$ is the vacuum at $x\to\infty$, the vector $\vec{p}$ a complex,
2-dimensional vector normalized to $\vec{p}^{\,\dagger}\vec{p}=1$. We can
interpret the expression for $U$ as spectral representation of a unitary
2$\times$2 matrix with eigenvalues 1 and $e^{-2i\theta}$. Choosing a frame
where $\vec{p}=(1,0)$, we recover the NJL twisted kink for isospin up and the
vacuum for isospin down. This reduces the U(2) twisted kink to the U(1)
twisted kink, with $\theta$ the (chirally invariant) twisting angle. The
unique way to get a real solution as needed for the O(2) model is again to
choose the maximal twist angle, $\theta=\pi/2$. We then recover the real GN
kink in the isospin up state. From the O(2) point of view, the kink connects
the rotational vacuum 1 with the reflection vacuum $\tau_{3}$. It is
impossible to find a kink connecting two points on the same vacuum circle.
Going back to the form (55) of $U$, we can construct a more general kink by
choosing a real $\vec{p}$,

$\vec{p}=\left(\begin{array}[]{c}\cos\alpha\\\ \sin\alpha\end{array}\right).$
(56)

The mean field then becomes the matrix

$\Delta=(1-f)-I(2\alpha)f,\quad f=\frac{V}{1+V}=\frac{1}{2}(1+\tanh x).$ (57)

By a further chiral transformation we arrive at the most general O(2) kink
interpolating between arbitrary points on the rotation and reflection circles,

$\Delta=R(\alpha_{1})(1-f)-I(\alpha_{2})f.$ (58)

Here one has to choose $\alpha=\alpha_{1}+\alpha_{2}$ in $\vec{p}$. This kink
evolves from the rotation vacuum $R(\alpha_{1})$ at $x\to-\infty$ to the
reflection vacuum $-I(\alpha_{2})$ at $x\to\infty$. Unlike the twist angle in
the unitary models, here the angles $\alpha_{1},\alpha_{2}$ have no direct
physical significance, being dependent on the chiral frame chosen. This is
consistent with the fact that the $x$ dependence is independent of the angles
$\alpha_{i}$. The true twist angle $\theta=\pi/2$ is always maximal, like in
the GN model. The opposite kink connecting a point on the reflection circle to
one on the rotation circle can easily be found by changing the sign of $x$,

$\Delta=-I(2\alpha_{2})(1-f)+R(2\alpha_{1})f.$ (59)

The reason why we have written down the most general form of the kinks is the
fact that in scattering or bound state configurations with several kinks, the
vectors $\vec{p}_{i}$ cannot all be rotated simultaneously into a specific
direction. Then the differences $\alpha_{i}-\alpha_{j}$ do acquire physical
significance.

Finally we mention that the mass of the kink is the same as in the GN model,
$M=N/(2\pi)$, independently of its fermion number.

##  VII Scattering of kinks

We first use the existing tools to derive scattering of two twisted kinks in
the U(2)${}_{L}\times$U(2)R GN model L14 . In order to turn the result into a
solution of the O(2)${}_{L}\times$O(2)R symmetric model, we choose as input
two real vectors

$\vec{p}_{i}=\left(\begin{array}[]{c}\cos\alpha_{i}\\\
\sin\alpha_{i}\end{array}\right),$ (60)

twist angles $\theta_{i}=\pi/2$ and positions of poles in the complex spectral
parameter plane

$\zeta_{i}=\frac{i}{\eta_{i}},\quad\eta_{i}=\sqrt{\frac{1+v_{i}}{1-v_{i}}}.$
(61)

We also introduce the ratio $\eta=\eta_{1}/\eta_{2}$. The matrix $\omega$ is
taken to be diagonal. The result for kink-antikink scattering in the O(2)
model are the real mean fields

$\displaystyle{\cal D}S_{0}$ $\displaystyle=$ $\displaystyle
1-\left(1-\frac{2(1+\eta^{2})\cos^{2}\alpha_{12}}{(1+\eta)^{2}}\right)V_{1}V_{2},$
$\displaystyle{\cal D}S_{1}$ $\displaystyle=$ $\displaystyle-V_{1}\sin
2\alpha_{1}-V_{2}\sin 2\alpha_{2},$ $\displaystyle{\cal D}S_{3}$
$\displaystyle=$ $\displaystyle-V_{1}\cos 2\alpha_{1}-V_{2}\cos 2\alpha_{2},$
$\displaystyle{\cal D}P_{2}$ $\displaystyle=$
$\displaystyle-\left(\frac{1-\eta}{1+\eta}\right)\sin 2\alpha_{12}V_{1}V_{2},$
(62)

with $\alpha_{12}=\alpha_{1}-\alpha_{2}$. Here, ${\cal D}$ is the common
denominator

${\cal D}=1+V_{1}+V_{2}+\kappa
V_{1}V_{2},\quad\kappa=\left(1-\frac{4\eta\cos^{2}\alpha_{12}}{(1+\eta)^{2}}\right).$
(63)

The $V_{i}$ factors carry the ($x,t$) dependence,

$V_{i}=e^{2x_{i}^{\prime}},\quad
x_{i}^{\prime}=\frac{x-x^{0}_{i}-v_{i}t}{\sqrt{1-v_{i}^{2}}}.$ (64)

The vacuum at $x\to-\infty$ has been taken to be 1. Orthogonal matrices
$U_{1},U_{2}$ appear in intermediate states of the scattering process, whereas
$U_{12}$ is the vacuum at $x\to\infty$. One finds

$U_{i}=1-2\vec{p}_{i}\vec{p}_{i}^{\,\dagger}=-I(2\alpha_{i})$ (65)

and

$U_{12}=\frac{1}{1+\eta^{2}-2\eta\cos
2\alpha_{12}}\left(\begin{array}[]{cc}(1+\eta^{2})\cos
2\alpha_{12}-2\eta&(1-\eta^{2})\sin 2\alpha_{12}\\\ -(1-\eta^{2})\sin
2\alpha_{12}&(1+\eta^{2})\cos 2\alpha_{12}-2\eta\end{array}\right).$ (66)

$U_{12}$ is a rotation matrix, in contrast to the reflection matrices $U_{i}$,

$U_{12}=R(\Phi),\quad\tan\Phi=\frac{(1-\eta^{2})\sin
2\alpha_{12}}{(1+\eta^{2})\cos 2\alpha_{12}-2\eta}.$ (67)

The mean field for the kink-antikink collision can be concisely represented as

$\Delta=\frac{1+V_{1}U_{1}+V_{2}U_{2}+\kappa
V_{1}V_{2}U_{12}}{1+V_{1}+V_{2}+\kappa V_{1}V_{2}}$ (68)

where the four matrices ($1,\,U_{1},\,U_{2},\,U_{12}$) are now orthogonal as
opposed to unitary in the U(2) model. The physical meaning of these matrices
is that they represent all possible vacua if the kinks are well separated in
space L14 .

Let us pause for a moment and ask ourselves what is the intrinsic (chiral
frame independent) content of expression (68). Chiral symmetry has already
been employed to set the initial vacuum equal to 1. The only residual O(2)
transformations allowed without changing the initial vacuum are

$\Delta\to R(\beta)\Delta R(-\beta),\quad\Delta\to I(\gamma)\Delta I(\gamma).$
(69)

If applied to the final vacuum $U_{12}$, the rotation acts as the identity
whereas the reflection changes the sign of $\Phi$, or, equivalently,
interchanges $\vec{p}_{1}$ and $\vec{p}_{2}$. In the intermediate vacua
$U_{1},U_{2}$, both angles are shifted by the same amount. This reflects the
fact that the only chirally invariant quantity one can form out of the
$\vec{p}_{i}$ is the scalar product $\vec{p}_{1}\vec{p}_{2}$.

Due to the four components $S_{0},S_{1},S_{3},P_{2}$, the kink-kink collision
process looks rather complicated. What happens can be described qualitatively
as follows. Let us assume that kink 1 is incident from the left, kink 2 from
the right. The asymptotic vacua are the rotational vacua 1 (at $x\to-\infty$)
and $R(\Phi)$ (at $x\to\infty$) throughout the collision process. In between
the kinks, the system is in the reflection vacuum $U_{1}$ before the collision
and $U_{2}$ after the collision. The region where the system is in the
reflection vacuum is always bounded by the position of the two kinks. This is
exactly what one would expect from a collision between two domain walls, here
separating the rotational from the reflection phases. The difference between
the U(2) and the O(2) chirally symmetric cases is the fact that the intrinsic
twist of the kinks is always maximal in the O(2) model, similar to the
difference between the U(1) (NJL) and O(1) (GN) models.

Finally, let us remark that one also finds a bound state at rest by
specializing to $\eta_{1}=\eta_{2}=1$. This is similar to the unitary models,
but no such bound state exists in the one-flavor GN model.

##  VIII Baryon and breather with twisted kink constituents

Here we start from the general two-soliton solution of the U(2) model L14  and
specialize it as follows. We choose $\eta_{1}=\eta_{2}=1$ (bound state at
rest) and a diagonal matrix $\omega$ (no breathers). In order to get a real
mean field $\Delta$, we have to pair the twist angles to
$(\theta_{1}=\pi-\theta_{2}=\theta)$ and to choose $\omega_{11}=\omega_{22}$,
like in the GN model. Moreover, it turns out that we need
$\alpha_{1}=\alpha_{2}=\alpha$. Consider the simplest case first, $\alpha=0$.
We find

$\Delta=\left(\begin{array}[]{cc}\frac{1+2V\cos
2\theta+V^{2}\cos^{2}\theta}{1+2V+V^{2}\cos^{2}\theta}&0\\\
0&1\end{array}\right),\quad V=e^{2x\sin\theta}.$ (70)

This is nothing but the Dashen, Hasslacher, Neveu (DHN) baryon L18  in the
upper component and the vacuum in the lower component. If $\theta$ is close to
its maximum value of $\pi/2$, the DHN scalar potential has the form of a well
separated kink-antikink pair. Asymptotically, the vacuum is 1, but inside the
baryon the vacuum -1 is approached. In the present two-flavor case, the
corresponding vacua are 1 (rotation) and $\tau_{3}$ (reflection). A more
transparent representation of the baryon mean field (70) is

$\displaystyle\Delta(\alpha=0)$ $\displaystyle=$
$\displaystyle(1-g)-\tau_{3}g,$ $\displaystyle g$ $\displaystyle=$
$\displaystyle\frac{2V\sin^{2}\theta}{1+2V+V^{2}\cos^{2}\theta},$
$\displaystyle S_{\rm DHN}$ $\displaystyle=$ $\displaystyle 1-2g.$ (71)

The function $g$ vanishes asymptotically on both sides, in contrast to the
kink-like function $f$ introduced above for a single kink. We can now perform
a rotation, leaving the asymptotic vacua unchanged but transforming the
central vacuum into an arbitrary reflection matrix. This yields the baryon
solution for any $\alpha$,

$\Delta(\alpha)=1-g-I(2\alpha)g.$ (72)

The same result would have been obtained by setting
$\alpha_{1}=\alpha_{2}=\alpha$ rather than 0 from the beginning. The maximum
of $g$ is at

$x_{\rm max}=-\frac{\ln(\cos\theta)}{2\sin\theta},\quad g_{\rm
max}=1-\cos\theta.$ (73)

Hence, at the center of the baryon,

$\Delta=\cos\theta-(1-\cos\theta)I(2\alpha).$ (74)

If $\theta\approx\pi/2$ as appropriate for well separated kink and antikink,
$\cos\theta\approx 0$ and we see that we are in the reflection phase inside
the baryon. Since the unitary transformation may be viewed as a change of
frame, the angle $\alpha$ is irrelevant for the single baryon. However it will
become relevant in problems involving more than two kinks. The single particle
spectrum of the O(2) baryon is the same as that of the DHN baryon in the GN
model. There are two bound states at energy $\pm\cos\theta$ and the spectrum
is symmetric about 0. Also the occupation fractions match those of the DHN
baryon.

To get a breather, we have to repeat the above calculation with an off-
diagonal $\omega$-matrix. We choose

$\omega=\left(\begin{array}[]{cc}\cosh\chi&\sinh\chi\\\
\sinh\chi&\cosh\chi\end{array}\right),\quad{\rm det\,}\omega=1.$ (75)

The results for the breather have the same general form as for the baryon,
except that $g$ acquires a time dependence in the rest frame,

$g=\frac{2V\sin\theta\left[\cosh\chi\sin\theta-\sinh\chi\sin(2t\cos\theta+\theta)\right]}{1+2V\left[\cosh\chi-\sin\theta\sinh\chi\sin(2t\cos\theta+\theta)\right]+V^{2}\cos^{2}\theta}.$
(76)

At $\chi=0$ one recovers the static baryon, Eq. (71).

So far, we have essentially reproduced the results of DHN L18  in a new
setting. Now we can also look at more complicated multibaryon and breather
problems where new phenomena are expected. All we have to do is perform the
calculation in the U(2) GN model and choose the parameters such that all mean
fields are real. Judging from the experience with the one flavor GN and NJL
models, this should be more efficient than trying to determine solutions of
the O(2) model directly.

##  IX Massless hadrons, chiral spiral and phase diagram

In this section, we briefly consider topics which have come up previously in
the NJL model, the SU(2)${}_{L}\times$SU(2)R isoNJL model and the
U(2)${}_{L}\times$U(2)R GN model. Whenever such a model possesses a “chiral
circle” and massless mesons, the possibility arises to generate both massless
multi-fermion bound states and “chiral spiral” type matter phases L15 ; L20 .
Whereas the massless mesons are related to small fluctuations around the
vacuum into some flat direction (Goldstone modes), massless baryons correspond
to one full turn around the chiral circle. The axial anomaly links winding
number to fermion density. Due to the common charge conjugation symmetry, the
situation of the O(2) chiral GN model is perhaps closest to the one of the
isoNJL model. We refer to Sect. V of Ref. L21  for the pertinent discussion in
the isoNJL model. Here, only the condensates $\bar{\psi}\psi$ and
$\bar{\psi}i\gamma_{5}\tau_{3}\psi$ had to be used. The crucial ingredient was
the unitary transformation

$\psi^{\prime}=U\psi=e^{-ibx\gamma_{5}\tau_{3}}\psi.$ (77)

If applied to the vacuum HF equation with potential $\gamma^{0}m$, it
generates an isospin chemical potential $b\tau_{3}$ from the kinetic term and
changes the mass term into the characteristic chiral spiral condensate

$U^{\dagger}(-i\gamma_{5}\partial_{x})U=-i\gamma_{5}\partial_{x}-b\tau_{3},\quad
U^{\dagger}\gamma^{0}U=\gamma^{0}\cos 2bx+i\gamma^{1}\tau_{3}\sin 2bx.$ (78)

This construction cannot be used for generating fermion density and the
corresponding chemical potential in the two-flavor model. Here, the GN model
with discrete chiral symmetry has taught us how to minimize the energy, namely
by generating a real kink-antikink crystal described by cnoidal functions.
Referring to the literature for the details L22 , let us denote the self-
consistent scalar potential by $S_{\rm GN}(x)$, with temperature and density
dependent shape. If one takes into account both fermionic and isospin chemical
potential, the HF potential in the isoNJL model assumes the product form

$\Delta(x)=S_{\rm GN}(x)e^{-ibx\gamma_{5}\tau_{3}}.$ (79)

The axial isospin chemical potential conjugate to the density
$\psi^{\dagger}\gamma_{5}\tau_{3}\psi$ can also be invoked if one is
interested in chirally imbalanced states, but does not affect the mean field.
The phase diagram of the isoNJL model in ($\mu,\mu_{3},T$) space following
from this scenario consists of the GN phase diagram in the ($\mu,T$) plane
translated rigidly into the $\mu_{3}$ direction, see L21 .

What does this teach us about the O(2) chiral GN model? In the above sketched
results for the isoNJL model, only the condensates $\bar{\psi}\psi$ and
$\bar{\psi}i\gamma_{5}\tau_{3}\psi$ have played a role. The choice of the
3-direction in isospin is arbitrary and only used for convenience, since
$\tau_{3}$ is diagonal. One could equally well have used the 2-direction. But
then we would be in the same situation as in the O(2) case with a rotation
vacuum and a chiral circle in the ($S_{0},P_{2}$) plane. There is a one-to-one
correspondence between SU(2) and O(2) chiral GN models, as far as these
particular aspects are concerned. Thus we can borrow the results L21  from the
isoNJL model directly and get massless bound states and the whole phase
diagram of the O(2) model almost for free.

What would happen, had we started from the reflection vacuum in the
($S_{1},S_{3}$) plane instead of the rotation vacuum? Here the transition to
the isoNJL model is less straightforward, but we certainly expect an
equivalent picture. To induce rotation around the vacuum circle in the
($S_{1},S_{3}$) plane now requires the vector transformation

$\psi^{\prime}=U\psi=e^{-ibx\tau_{2}}\psi,$ (80)

without $\gamma_{5}$ in the exponent. This yields an axial isovector chemical
potential and a chiral spiral mean field in the ($S_{1},S_{3}$) reflection
plane,

$U^{\dagger}(-i\gamma_{5}\partial_{x})U=-i\gamma_{5}\partial_{x}-b\gamma_{5}\tau_{2},\quad
U^{\dagger}\gamma^{0}\tau_{3}U=\gamma^{0}\left(\tau_{3}\cos 2bx+\tau_{1}\sin
2bx\right).$ (81)

In this case, it is the axial isospin density that induces the inhomogeneous
chiral spiral structure. This change of vector into axial chemical potentials
is also known from other dualities L8 , so that everything fits nicely
together.

##  X Duality between O(2)${}_{L}\times$O(2)R GN model and ${\bf p}$GN model

We label the fermion fields of the O(2) chiral GN model (16) as
$\psi_{k\ell}^{(i)}$ with $k$ the chirality ($1=L,2=R$), $\ell$ the flavor
index (1,2) and $i$ the color index ($i=1,..,N)$. Each Dirac field can be
decomposed into two Majorana fields,

$\left(\begin{array}[]{c}\psi_{11}^{(i)}\\\ \psi_{12}^{(i)}\\\
\psi_{21}^{(i)}\\\
\psi_{22}^{(i)}\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\chi_{1}^{(i)}-i\chi_{1}^{(N+i)}\\\
\chi_{3}^{(i)}-i\chi_{3}^{(N+i)}\\\ \chi_{2}^{(N+i)}+i\chi_{2}^{(i)}\\\
\chi_{4}^{(N+i)}+i\chi_{4}^{(i)}\end{array}\right).$ (82)

The labelling of the Majorana spinors has been chosen for later convenience.
Here we only note that the even subscripts belong to right-handed, the odd
subscripts to left-handed Majorana spinors satisfying the anticommutation
relations

$\left\\{\chi_{n}^{(i)}(x),\chi_{m}^{(j)}(y)\right\\}=\delta_{ij}\delta_{nm}\delta(x-y).$
(83)

The terms in the interaction Lagrangian can be regrouped as follows,

$\displaystyle{\cal L}_{\rm int}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4}\left[(\bar{\psi}\psi)^{2}+(\bar{\psi}\tau_{1}\psi)^{2}+(\bar{\psi}\tau_{3}\psi)^{2}+(\bar{\psi}i\gamma_{5}\tau_{2}\psi)^{2}\right]$
(84) $\displaystyle=$
$\displaystyle\frac{g^{2}}{8}\left[(\bar{\psi}(1+\tau_{3})\psi)^{2}+(\bar{\psi}(1-\tau_{3})\psi)^{2}+(\bar{\psi}(\tau_{1}+i\gamma_{5}\tau_{2})\psi)^{2}+(\bar{\psi}(\tau_{1}-i\gamma_{5}\tau_{2})\psi)^{2}\right].$

The motivation for the last line can be seen once we express everything in
terms of Majorana spinors,

$\sum_{i=1}^{N}\bar{\psi}^{(i)}(1+\tau_{3})\psi^{(i)}=2i\sum_{i=1}^{N}\left(\chi_{1}^{(i)}\chi_{2}^{(i)}+\chi_{1}^{(N+i)}\chi_{2}^{(N+i)}\right)=2i\sum_{i=1}^{2N}\left(\chi_{1}^{(i)}\chi_{2}^{(i)}\right),$
(85)

and similarly

$\displaystyle\sum_{i=1}^{N}\bar{\psi}^{(i)}(1-\tau_{3})\psi^{(i)}=2i\sum_{i=1}^{2N}\left(\chi_{3}^{(i)}\chi_{4}^{(i)}\right),$
$\displaystyle\sum_{i=1}^{N}\bar{\psi}^{(i)}(\tau_{1}+i\gamma_{5}\tau_{2})\psi^{(i)}=2i\sum_{i=1}^{2N}\left(\chi_{1}^{(i)}\chi_{4}^{(i)}\right),$
$\displaystyle\sum_{i=1}^{N}\bar{\psi}^{(i)}(\tau_{1}-i\gamma_{5}\tau_{2})\psi^{(i)}=2i\sum_{i=1}^{2N}\left(\chi_{3}^{(i)}\chi_{2}^{(i)}\right).$
(86)

The Lagrangian of the O(2) chiral GN model is thus turned into

$\displaystyle{\cal L}_{\rm O(2)}$ $\displaystyle=$
$\displaystyle-i\chi_{1}\partial\chi_{1}-i\chi_{3}\partial\chi_{3}+i\chi_{2}\bar{\partial}\chi_{2}+i\chi_{4}\bar{\partial}\chi_{4}$
(87)
$\displaystyle-\frac{g^{2}}{2}\left[(\chi_{1}\chi_{2})^{2}+(\chi_{3}\chi_{4})^{2}+(\chi_{1}\chi_{4})^{2}+(\chi_{3}\chi_{2})^{2}\right],$

with an implicit summation over $2N$ colors ($N$ Dirac fields are equivalent
to 2$N$ Majorana fields). In contrast to the original form in Eq. (84),
expression (87) is manifestly invariant under O(2)${}_{L}\times$O(2)R since
the vector with components ($\chi_{1},\chi_{3}$) transforms under O(2)L, the
vector with components ($\chi_{2},\chi_{4}$) under O(2)R.

Consider now the pGN model, Eq. (7), obtained by “self-dualizing” the NJL
model. As pointed out in the introduction, the Lagrangian coincides with that
of the classical ZM model originally written in the form (4). Introducing
Majorana spinors

$\left(\begin{array}[]{c}\psi_{1}^{(i)}\\\
\psi_{2}^{(i)}\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\chi_{1}^{(i)}-i\chi_{3}^{(i)}\\\
\chi_{4}^{(i)}+i\chi_{2}^{(i)}\end{array}\right),\quad(i=1,..,N),$ (88)

it has already been shown in Ref. L9  that the Lagrangian becomes

$\displaystyle{\cal L}_{\rm pGN}$ $\displaystyle=$
$\displaystyle-i\chi_{1}\partial\chi_{1}-i\chi_{3}\partial\chi_{3}+i\chi_{2}\bar{\partial}\chi_{2}+i\chi_{4}\bar{\partial}\chi_{4}$
(89)
$\displaystyle-g^{2}\left[(\chi_{1}\chi_{2})^{2}+(\chi_{3}\chi_{4})^{2}+(\chi_{1}\chi_{4})^{2}+(\chi_{3}\chi_{2})^{2}\right].$

Remarkably, this expression agrees with (87). The coupling constants differ by
a factor of 2, but this is compensated by the number of colors, $N$ in (89)
instead of $2N$ in (87). Hence the O(2)${}_{L}\times$O(2)R symmetric two-
flavor GN model with $N$ colors is dual to the one-flavor pGN model with $2N$
colors. Since we have derived the O(2) symmetric model from the U(2) NJL model
which has already been solved in the large $N$ limit, we can now easily infer
the solution of the dual model, not yet available in L9 . The solution of many
aspects of the O(2) GN model has already been discussed in the preceding
sections. All we have to do is to re-interpret everything in the dual
language. This is the topic of the following section.

By eliminating the Majorana spinors from (82) and (88), we can express the
Dirac fields of the O(2) GN model through Dirac fields of the pGN model or
vice versa,

$\left(\begin{array}[]{c}\psi_{11}^{(i)}\\\ \psi_{12}^{(i)}\\\
\psi_{21}^{(i)}\\\
\psi_{22}^{(i)}\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{c}\psi_{1}^{(i)}+\psi_{1}^{(i)*}-i\psi_{1}^{(N+i)}-i\psi_{1}^{(N+i)*}\\\
\psi_{1}^{(N+i)}-\psi_{1}^{(N+i)*}+i\psi_{1}^{(i)}-i\psi_{1}^{(i)*}\\\
\psi_{2}^{(i)}-\psi_{2}^{(i)*}-i\psi_{2}^{(N+i)}+i\psi_{2}^{(N+i)*}\\\
\psi_{2}^{(N+i)}+\psi_{2}^{(N+i)*}+i\psi_{2}^{(i)}+i\psi_{2}^{(i)*}\end{array}\right).$
(90)

As one can eaily check, this is a Bogoliubov transformation at the level of
Dirac spinors. It would have been difficult to find this transformation
without introducing Majorana spinors at an intermediate step, but we can now
discuss both sides of the duality in the more familiar Dirac language. In
particular, using (90), we can express the relevant bilinears of the two-
flavor model through bilinears of the one-flavor model as follows

$\displaystyle\bar{\psi}^{(i)}\psi^{(i)}$ $\displaystyle=$
$\displaystyle\psi_{1}^{(i)*}\psi_{2}^{(i)}+\psi_{2}^{(i)*}\psi_{1}^{(i)}+\psi_{1}^{(N+i)*}\psi_{2}^{(N+i)}+\psi_{2}^{(N+i)*}\psi_{1}^{(N+i)},$
$\displaystyle\bar{\psi}^{(i)}\tau_{1}\psi^{(i)}$ $\displaystyle=$
$\displaystyle
i\left(\psi_{1}^{(i)*}\psi_{2}^{(i)*}-\psi_{2}^{(i)}\psi_{1}^{(i)}+\psi_{1}^{(N+i)*}\psi_{2}^{(N+i)*}-\psi_{2}^{(N+i)}\psi_{1}^{(N+i)}\right),$
$\displaystyle\bar{\psi}^{(i)}\tau_{3}\psi^{(i)}$ $\displaystyle=$
$\displaystyle\psi_{1}^{(i)}\psi_{2}^{(i)}+\psi_{2}^{(i)*}\psi_{1}^{(i)*}+\psi_{1}^{(N+i)}\psi_{2}^{(N+i)}+\psi_{2}^{(N+i)*}\psi_{1}^{(N+i)*},$
$\displaystyle\bar{\psi}^{(i)}i\gamma_{5}\tau_{2}\psi^{(i)}$ $\displaystyle=$
$\displaystyle
i\left(\psi_{1}^{(i)*}\psi_{2}^{(i)}-\psi_{2}^{(i)*}\psi_{1}^{(i)}+\psi_{1}^{(N+i)*}\psi_{2}^{(N+i)}-\psi_{2}^{(N+i)*}\psi_{1}^{(N+i)}\right).$
(91)

The left hand side refers to the O(2) GN model, the right hand side to the pGN
model, and the equations hold for each color index $i=1,..,N$. The first line
shows that the scalar condensate has the same meaning on both sides of the
duality. The $i\gamma_{5}\tau_{2}$ condensate of the O(2) model corresponds to
the pseudoscalar condensate of the pGN model. The $\tau_{3}$ and $\tau_{1}$
condensates of the O(2) model go over into the two kinds of Cooper pair
condensates.

Important bilinear observables not present in the O(2) Lagrangian are the
isospin vector and axial vector densities (only the $\tau_{2}$-components
belong to a conserved current),

$\displaystyle\psi^{(i)\dagger}\tau_{2}\psi^{(i)}$ $\displaystyle=$
$\displaystyle\psi_{1}^{(i)*}\psi_{1}^{(i)}+\psi_{2}^{(i)*}\psi_{2}^{(i)}+\psi_{1}^{(N+i)*}\psi_{1}^{(N+i)}+\psi_{2}^{(N+i)*}\psi_{2}^{(N+i)},$
$\displaystyle\psi^{(i)\dagger}\gamma_{5}\tau_{2}\psi^{(i)}$ $\displaystyle=$
$\displaystyle-\psi_{1}^{(i)*}\psi_{1}^{(i)}+\psi_{2}^{(i)*}\psi_{2}^{(i)}-\psi_{1}^{(N+i)*}\psi_{1}^{(N+i)}+\psi_{2}^{(N+i)*}\psi_{2}^{(N+i)}.$
(92)

Thus the $\tau_{2}$ isospin density corresponds to the fermion density
$\psi^{\dagger}\psi$ of the pGN model, the $\gamma_{5}\tau_{2}$ axial isospin
density to the fermion axial density $\psi^{\dagger}\gamma_{5}\psi$.

Note that all observables discussed so far are color singlets on both sides of
the duality. It is interesting to understand what happens to the U(1) vector
symmetry and fermion density of the O(2) model upon dualization. If we
translate the U(1) transformation $\psi_{k\ell}^{(i)}\to
e^{i\alpha}\psi_{k\ell}^{(i)}$ into the dual language, we find that it reduces
to the following orthogonal, color dependent transformation

$\displaystyle\left(\begin{array}[]{c}\psi_{1}^{(i)}\\\
\psi_{1}^{(N+i)}\end{array}\right)$ $\displaystyle\to$
$\displaystyle\left(\begin{array}[]{rr}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{array}\right)\left(\begin{array}[]{c}\psi_{1}^{(i)}\\\
\psi_{1}^{(N+i)}\end{array}\right),$ (99)
$\displaystyle\left(\begin{array}[]{c}\psi_{2}^{(i)}\\\
\psi_{2}^{(N+i)}\end{array}\right)$ $\displaystyle\to$
$\displaystyle\left(\begin{array}[]{rr}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{array}\right)\left(\begin{array}[]{c}\psi_{2}^{(i)}\\\
\psi_{2}^{(N+i)}\end{array}\right).$ (106)

This is a subgroup of the original O(2$N$) symmetry of the pGN model with 2$N$
colors. Consequently, the conserved current in the dual pGN model is not a
color singlet, but has the form

$\displaystyle\rho_{R}$ $\displaystyle\to$ $\displaystyle
i\sum_{i=1}^{N}\left(\psi_{2}^{(N+i)*}\psi_{2}^{(i)}-\psi_{2}^{(i)*}\psi_{2}^{(N+i)}\right),$
$\displaystyle\rho_{L}$ $\displaystyle\to$ $\displaystyle
i\sum_{i=1}^{N}\left(\psi_{1}^{(N+i)*}\psi_{1}^{(i)}-\psi_{1}^{(i)*}\psi_{1}^{(N+i)}\right),$
$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\rho_{R}+\rho_{L},\quad
j=\rho_{5}=\rho_{R}-\rho_{L}.$ (107)

Since we are not dealing with a gauge theory and color confinement, we see
nothing wrong with this color dependence.

Finally, let us look at yet another self-dual model that has been discussed in
Ref. L9 . By self-dualizing the GN model with discrete chiral symmetry rather
than the NJL model, one gets the Lagrangian of the self-dual GN (sdGN) model

${\cal L}_{\rm
sdGN}=-2i\psi_{1}^{(i)*}\partial\psi_{1}^{(i)}+2i\psi_{2}^{(i)*}\bar{\partial}\psi_{2}^{(i)}+\frac{g^{2}}{2}\left[(\psi_{1}^{(i)*}\psi_{2}^{(i)}+\psi_{2}^{(i)*}\psi_{1}^{(i)})^{2}+(\psi_{1}^{(i)*}\psi_{2}^{(i)*}+\psi_{2}^{(i)}\psi_{1}^{(i)})^{2}\right].$
(108)

In L9  it came as a surprise that this is again equivalent to two independent
GN models. If we apply the strategy developed in the present section to the
sdGN model and first translate it into Majorana spinors using (88), we find

${\cal L}_{\rm
sdGN}=-i\chi_{1}\partial\chi_{1}-i\chi_{3}\partial\chi_{3}+i\chi_{2}\bar{\partial}\chi_{2}+i\chi_{4}\bar{\partial}\chi_{4}-g^{2}\left[(\chi_{1}\chi_{2})^{2}+(\chi_{3}\chi_{4})^{2}\right].$
(109)

Upon using the “dictionary” (85,86), this is equivalent to a U(2) model where
only two out of the eight original interaction terms are kept,

${\cal
L}=\bar{\psi}i\partial\\!\\!\\!/\psi+\frac{g^{2}}{4}\left[(\bar{\psi}\psi)^{2}+(\bar{\psi}\tau_{3}\psi)^{2}\right].$
(110)

Since the two isospin channels decouple, the fact that one gets two
independent GN models is now trivial. This shows once more the advantage of
using Majorana spinors as an intermediate step for discovering or elucidating
dualities.

##  XI Summary: an integrable model with chiral symmetry breaking and Cooper
pairing

There has been quite some interest in four-fermion theories featuring a
competition between particle-hole pairing and Cooper pairing L23 ; L24 ; L25 ;
L26 ; L27 ; L28 , triggered partly by predictions of color superconductivity
in quantum chromodynamics L29 . These works are dealing mostly with the phase
diagrams of the models considered. As we have shown in the present paper, the
pGN model is an example where the coexistence of chiral symmetry breaking
(CSB) and Cooper pairing arises in a highly symmetric fashion. As a
consequence, the resulting model is integrable and can be solved as completely
as the GN or NJL models. This confirms our earlier speculation L9  and extends
the range of integrable field theory models in 1+1 dimensions into an
interesting direction. Crucial for the new insights was a mapping between a
O(2)${}_{L}\times$O(2)R symmetric GN model, which had to be constructed for
this purpose, and the pGN model, as summarized in Table 2. The pGN model has
been obtained by “self-dualizing” the one-flavor NJL model with respect to the
transformation (6),

$\psi=\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)\quad\longrightarrow\quad\psi_{d}=\left(\begin{array}[]{c}\psi_{1}^{\dagger}\\\
\psi_{2}\end{array}\right).$ (111)

Table 2 shows the correspondence between the condensates of the two equivalent
models. Since the solution of the O(2) chiral GN model could be derived with
the machinery develpoped for the U(2) chiral GN model, one can take over all
the results collected above and reinterpret them in terms of the physics of
relativistic superconductors. Recall that the Dirac fields and Majorana
spinors have $N$ color components in the O(2) model but 2$N$ color components
in the pGN model. The labelling of the Majorana spinor indices in Eqs. (82)
and (88) has been chosen with hindsight, so that the expressions in the last
column of Table 2 are identical in both models. Likewise, the Lagrangians are
indistinguishable if expressed in terms of Majorana spinors, see Eqs. (87) and
(89).

condensate | O(2)${}_{L}\times$O(2)R | pGN | Majorana spinors
---|---|---|---
$S_{0}$ | $\bar{\psi}\psi$ | $\bar{\psi}\psi$ | $i(\chi_{1}\chi_{2}+\chi_{3}\chi_{4})$
$P_{2}$ | $\bar{\psi}i\gamma_{5}\tau_{2}\psi$ | $\bar{\psi}i\gamma_{5}\psi$ | $i(\chi_{1}\chi_{4}+\chi_{2}\chi_{3})$
$S_{3}$ | $\bar{\psi}\tau_{3}\psi$ | $\bar{\psi}_{d}\psi_{d}$ | $i(\chi_{1}\chi_{2}-\chi_{3}\chi_{4})$
$S_{1}$ | $\bar{\psi}\tau_{1}\psi$ | $\bar{\psi}_{d}i\gamma_{5}\psi_{d}$ | $i(\chi_{1}\chi_{4}-\chi_{2}\chi_{3})$
Table 2: Correspondence between bilinears in the two dual models. The Majorana
notation is common to both, but summation is over $N$ colors (O(2) model) and
2$N$ colors (pGN model), respectively.

Let us briefly review the results of the large $N$ O(2) model in the light of
the pGN model.

  * •

Lagrangian: The Lagrangians can be cast into a form which emphasizes the
correspondence, using Table 2 and the definition (111),

$\displaystyle{\cal L}_{\rm O(2)}$ $\displaystyle=$
$\displaystyle\bar{\psi}i\partial\\!\\!\\!/\psi+\frac{g^{2}}{4}\left[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\tau_{2}\psi)^{2}+(\bar{\psi}\tau_{3}\psi)^{2}+(\bar{\psi}\tau_{1}\psi)^{2}\right],$
$\displaystyle{\cal L}_{\rm pGN}$ $\displaystyle=$
$\displaystyle\bar{\psi}i\partial\\!\\!\\!/\psi+\frac{g^{2}}{4}\left[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\tau_{2}\psi)^{2}+(\bar{\psi}_{d}\psi_{d})^{2}+(\bar{\psi}_{d}i\gamma_{5}\psi_{d})^{2}\right].$
(112)

The O(2)${}_{L}\times$O(2)R chiral symmetry of the first model matches the
Pauli-Gürsey symmetry of the second one.

  * •

Vacuum: The O(2) model has two possible vacuum circles referred to as rotation
and reflection vacua above. In the pGN model, this correponds to the CSB
vacuum (condensates $\bar{\psi}\psi,\bar{\psi}i\gamma_{5}\psi$) or the Cooper
paired vacuum (condensates
$\bar{\psi}_{d}\psi_{d},\bar{\psi}_{d}i\gamma_{5}\psi_{d}$). The two connected
components of the O(2) group are exactly what it takes to describe these two
distinct possibilities. The pGN model gives a more physical picture of what it
means to break chiral symmetry spontaneosuly if the symmetry group is
continuous, but not connected.

  * •

Kink: The kink becomes a domain wall between superconducting and normal phase.
There is no kink inside either of the two phases. One can study dynamical
problems like scattering of any number of such domain walls in closed
analytical form, describing time dependent configurations of CSB and Cooper
pairing domains.

  * •

DHN-type baryon: This bound state of two twisted kinks in the O(2) model is
nothing but a configuration where one phase is separated from the other phase
by two walls. If the exterior phase is chosen as Cooper paired phase, then the
interior phase is normal and we get a relativistic toy model for a Josephson
junction. Dynamical problems including several such objects as well as single
domain walls can be solved, as well as the interaction of fermions or Cooper
pairs with domain walls. Breathers can be interpreted as excited, periodically
oscillating Josephson junctions.

  * •

Massless many-fermion states and phase diagram: In the pGN model, fermion
density can be generated by a local chiral transformation, like in the NJL
model. This makes the chiral spiral configuration optimal for fermionic
matter, in the normal phase. In the Cooper pairing phase, the same mathematics
would give rise to an inhomogeneous LOFF phase of the superconductor L30 ; L31
. The two are just two ways of interpreting the same physics, in the pGN
model. The phase diagram with vector and axial vector fermion chemical
potentials of the pGN model can be taken over from the NJL model. Fermion
density of the O(2) model becomes a color dependent density (107). If one
introduces the conjugated chemical potential, the kink crystal of the GN model
must come into the picture and color O(2$N$) breaks down to down to O($N$).

While it is easy to write down four-fermion models with both pp- and ph-
pairing, it is not easy to find integrable ones. The only one which had been
found so far is the sdGN model L9 . Since this has turned out to be a trivial
double copy of the standard GN model, it gives little new insights. All other
integrable models known so far had either CSB or Cooper pairing, but not both.
In this sense, the quantum ZM or pGN model is a novel and potentially useful
member of the family of exactly solvable field theoretic models which deserves
further studies.

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