Thirring Model as a Gauge Theory
T. Itoh
0
Y. Kim
0
M. Sugiura
0
K. Yamawaki
0
0
Taichi lTOH, Yoonbai KIM, Masaki SUGIURA and Koichi YAMAWAKI
We reformulate the Thirring model in D (2:<0:D<4) dimensions as a gauge theory by introducing U(l) hidden local symmetry (HLS) and study the dynamical mass generation of the fermion through the SchwingerDyson (SD) equation. By virtue of such a gauge symmetry we can greatly simplify the analysis of the SD equation by taking the most appropriate gauge ("nonlocal gauge") for the HLS. In the case of even number of (2component) fermions, we find the dynamical fermion mass generation as the second order phase transition at certain fermion number, which breaks the chiral symmetry but preserves the parity in (2+ 1) dimensions (D=3). In the infinite fourfermion coupling (massless gauge boson) limit in (2 + 1) dimensions, the result coincides with that of the (2 +I)dimensional QED, with the critical number of the 4component fermion being Ncr=l28/3n2 As to the case of odd number (2component) fermion in (2 + 1) dimensions, the regularization ambiguity on the induced ChernSimons term may be resolved by specifying the regularization so as to preserve the HLS. Our method also applies to the (1 + 1) dimensions, the result being consistent with the exact solution. The bosonization mechanism in (1 +I)dimensional Thirring model is also reproduced in the context of dualtransformed theory for the HLS.

Fermion dynamical mass generation is the central issue of the scenario of the
dynamical electroweak symmetry breaking such as the technicolor1> and the top
quark condensate.2> Special attention has recently been paid to the role of four
fermion interaction in the context of walking technicolor,3>strong ETC technicolor4>
and top quark condensate.2> These models are based on the NambuJonaLasinio
(NJL) model5> of the scalar/pseudoscalartype fourfermion interactions combined
with the gauge interactions, socalled gauged NJL model, whose phase structure has
been extensively studied through the SchwingerDyson (SD) equation (see Ref. 6)). It
has been shown6> that the phase structure of such a gauged NJL model in (3+ 1)
dimensions (D=4) is quite similar to that of the D(2<D<4) dimensional fourfermion
theory of scalar/pseudoscalartype (without gauge interactions)/> often called Gross
Neveu model, which is renormalizable in 1/N expansion.8>
What about the fourfermion interaction of vector/axialvectortype? Does it
also give rise to the fermion mass generation? Of course it can be transformed into
the scalar /pseudoscalartype interaction through the Fierz transformation, but they
are independent of each other in the usual framework of 1/N expansion (see, e.g.,
Ref. 9)). This type of fourfermion interaction in fact has been studied in combina
tion with the scalar/pseudoscalartype ("generalized NJL model"), which is now a
popular model as a low energy effective theory of QCD. It is wellknown that at the
usual1/N leading order in the "generalized NJL model" only the scalar/pseudoscalar
fourfermion interaction contributes to the gap equation or to the fermion mass
generation, while the vector/ axialvectortype fourfermion interaction does not.
Thus we may address the following question: If the scalar/pseudoscalartype
fourfermion interaction does not exist at all and the formaliiN leading order is missing
in the above gap equation, is the fermion dynamical mass still generated by the
vectorI axialvectortype alone? If it is the case, such a dynamics would be interesting
for the model building beyond the standard model. It would also be interesting if
there arises a situation similar to the case of scalarlpseudoscalartype: Namely, the
phase structure of the gauged NJL model of vectorI axialvectortype might have some
resemblance to that of the Thirring model 10> (without gauge interactions) in D(2<D
<4) dimensions which is known to be renormalizable in IIN expansion.8>.n>
Thus we wish to study the fermion dynamical mass generation in the D(2~D<4)
dimensional Thirring model including the D=2 case. The Thirring model has been
extensively studied in (I+ I) dimensions since it is explicitly solvable.10> However, it
is only recent that the fermion dynamical mass generation in (2 +I)dimensional
Thirring model has been studied by several authors. 12H 6> The results of these
papers, however, are different from each other and rather confusing partly due to lack
of the discipline of the analysis. In Refs. I2) and I3), for example, they introduced
vector auxiliary field and pretended it as a gauge field despite the absence of manifest
gauge symmetry.
In this paper we reformulate the Thirring model as a gauge theory by introducing
the hidden local symmetry (HLS)9>(see also 17>.">). When we fix the gauge of HLS to
the unitary gauge, we get back to the original Thirring model written in terms of the
vector auxiliary field. However, the unitary gauge is notorious for making the actual
analysis difficult, while the existence of such a gauge symmetry has some virtues to
make the analysis consistent and systematic.
In the case of odd number of 2component fermions, a peculiarity arises in (2 +I)
dimensions, namely, the possibility of induced ChernSimons (CS) term.14> We may
take advantage of existence of the gauge symmetry to resolve the problem of regular
ization ambiguity concerning the induced CS term; the regularization must be chosen
in such a way as to keep the gauge symmetry (PauliVillars regularization) as in the
(2+ I)dimensional QED (QEDa).18> The parity violating CS term will arise from the
PauliVillars regulator even in the symmetric phase where the fermion mass is not
dynamically generated.
As to the case of even number of 2component fermions, on the other hand, we
expect that parity violation in (2 +I) dimensions will not be induced, since the above
induced CS term of each fermion species can be arranged in pair of opposite sign to
cancel each other. We shall demonstrate that an appropriate HLS gauge fixing
(nonlocal gauge19>) other than the unitary gauge actually leads to the simple and
consistent analysis of the SD equation for the D(2~D<4) dimensional Thirring
model. The nonlocal gauge is the gauge having no wave function renormalization
for the fermion and is the only way to make the bare vertex (ladder) approximation
to be consistent with the WardTakahashi (WT) identity for the current conservation.
We find dynamical chiral symmetry breaking which is parityconserving in (2+ I)
dimensions in accord with the VafaWitten theorem20> and establish a "second order"
(continuous) phase transition at a certain number of 4component fermions N for each
given value of the dimensionless fourfermion coupling g, namely, the critical line on
the (N, g) plane. This is somewhat analogous to the existence of the critical N in the
QED3,21H 3> which has been confirmed by the lattice Monte Carlo simulation.24>
Actually, when the fourfermion coupling constant g goes to infinity (massless gauge
boson limit) for D=3, our critical N is explicitly evaluated as Ncr=128/3J'(2 in perfect
agreement with that of the QED3 in the nonlocal gauge.22>'23> Note that this massless
vector limit is smooth thanks to the HLS in contrast to the original Thirring model
(unitary gauge) where this limit is singular and illdefined. For D=2 we explicitly
solve the gap equation, which turns out to be consistent with the exact solution.
Although at tree level the HLS gauge boson is merely the auxiliary field, it is a
rather common phenomenon that the HLS gauge boson acquires kinetic term by the
quantum corrections and hence becomes dynamical.9> We shall show that the HLS
gauge boson also becomes dynamical in the Thirring model once the fermion mass is
dynamically generated. In the infinite fourfermion coupling limit this dynamical
gauge boson becomes massless and the HLS becomes a spontaneously unbroken
gauge symmetry. Were it not for the HLS, on the other hand, this limit becomes
illdefined due to lack of the manifest gauge symmetry.9>
Another advantage of the HLS is that we can use the dual transformation in the
Thirring model. Dual transformation is one which manifests the propagating
degrees of freedom and maps the theory with strong gauge coupling to that with weak
coupling constant.25> In (1 +1) dimensions the HLS together with the dual transfor
mation offers us a straightforward method for the bosonization of the Thirring model
in the context of path integral.
The rest of this paper is organized as follows. In 2 we introduce the model in
D(2sD<4) dimensions and reformulate it by use of the hidden local symmetry. The
nonlocal gauge is introduced at the Lagrangian level rather than in the Schwinger
Dyson equation, which makes the BRS invariance transparent. In 3 we study the
SD equations under the nonlocal Re gauge fixing for HLS and establish the existence
of chiral symmetry breaking dynamical mass generation and of the associated critical
line on the (N, g) plane in the case of even number of 2component fermions. In (2
+1) dimensions this mass keeps the parity in accord with the VafaWitten theorem.
In (1 +1) dimensions the fermion mass generation always takes place as far as g is
positive. In 4 we demonstrate that the gauge boson of the HLS develops a pole due
to quantum correction (fermion loop) in the broken phase where the fermion mass is
dynamically generated. Section 5 is devoted to the dual transformation of the
Thirring model as a new feature of the HLS and to the study of various aspects of the
dual transformed theory, particularly the bosonization of the Thirring model in (1 + 1)
dimensions. We conclude in 6 with some discussion. In the Appendix a proof is
given of the BRS invariance of the Thirring model Lagrangian with the HLS in the
nonlocal gauge.
2. Hidden local symmetry
In this section we introduce the HLS9> into the massless Thirring model in D (2
sD<4) dimensions. The Lagrangian density of the Thirring model is given by
where 1lfa is a 4component Dirac spinor (although formal in D dimensions) and a, b
are summed over from 1 toN. Let us rewrite the theory by introducing an auxiliary
vector field A":
where D"=o"(i/ /N)A". Note that D" is not the covariant derivative in spite of its
formal similarity, since the field A" is just a vector field which depicts the fermionic
current and does not transform as a gauge field. Actually, the Lagrangian (22) has
no gauge symmetry. It is easy to see that when we solve away the auxiliary field A"
through the equation of motion for A", Eq. (22) is reduced back to the original
Thirring model (21).
Based on the "U(1) /U(1)" nonlinear sigma model, we now show that Eq. (2 2) is
gauge equivalent to another model possessing a symmetry U(1)giobai X U(1)Iocai, with
the U(1)Iocai being HLS:9>'17),ll)
and D" =a" (if /N)A" is the covariant derivative for A" which is a gauge field in
contrast to A" in Eq. (22). It is obvious that Eq. (23) possesses a U(1) gauge
symmetry and is invariant under the transformation:
Actually, is the fictitious NambuGoldstone (NG) boson field which is to be absorbed
into the longitudinal component of AI' If we fix the gauge by the gauge transforma
tion into the unitary gauge '=0(a= )
then the Lagrangian (2 3) precisely coincides with Eq. (2 2). Thus the original
Thirring model is nothing but the gaugefixed (unitary gauge) form of our HLS model.
The mass of the vector boson A~' is now regarded as that of the gauge boson A~'
generated through the Higgs mechanism.9> This U(1) case is actually identical with
what is known as the Sttickelberg formalism for the massive vector boson.
There are several virtues of the existence of such a gauge symmetry: First, the
gauge symmetry enables us to prove straightforwardly the Smatrix unitarity through
the BRS symmetry (see Ref. 26)). Secondly, actual calculations, particularly loop
calculations, are generally hopeless in the unitary gauge, while the HLS provides us
with the privilege to take the most appropriate gauge for our particular purpose. Let
us consider a general gauge F[A] = 0 and introduce the gauge fixing term into
Eq. (23):
..[M=~ ?Jair"Dpif;a+ 2~ (Ap~/N ap>)2  ~ F[AJ( ,;(~2) F[A])'
where, by introducing the momentum (derivative) dependence of the gauge fixing
parameter .;, we have formulated at Lagrangian level the socalled nonlocal gauge19>
which has been discussed only at the SD equation level. The covariant gauge is given
by F[A] = aA" in which the fictitious NG boson > is not decoupled (except the
Landau gauge .;=O).
More interesting gauge is the Re gauge,27l
which can again be a nonlocal gauge through the dependence of.; on the derivative.
The gauge fixing term in the nonlocal Re gauge is given by
..[,A=~ iair"D"a+ 2~ (A")2 ~ aA"( ,;(~2) avA11 ),
.."'= ~ a",pa"> 2~ (,;(a2)),
where we have rescaled the > as IN/G >~>if;. Thus the fictitious NG boson if; is
completely decoupled independently of ~(a2) in theRe gauge, whether.; is nonlocal or
not. It is straightforward to prove that the above Lagrangian (even in the nonlocal
gauge) possesses the BRS symmetry (see the Appendix). Equation (2 12) might
appear as if we added the "covariant gauge fixing term" to Eq. (22), although there
is no gauge symmetry and .1L is not a gauge field in Eq. (22). Such a confusion was
actually made by several authors12>'13> who happened to arrive at the Lagrangian
having the same form as Eq. (2 2) in the case of constant .;. This Lagrangian (2 12),
whether the gauge parameter is nonlocal or not, can only be justified through the HLS
in the Re gauge. If we take the covariant gauge, on the other hand, the field if; does
not decouple except for the Landau gauge as we have already mentioned.
In the next section we shall demonstrate that the analysis of the fermion dynami
cal mass generation in the ladder SD equation can be greatly simplified by taking the
nonlocal gauge19> of the Re gauge for HLS. The nonlocal gauge is the gauge in which
the fermion gets no wave function renormalization, i.e., A( p2)=1 for the fermion
propagator iS1(P)=A( P2)jj B( P2). This gauge is necessary for the ladder
(bare vertex) approximation to be consistent with the WT identity for the U(1) gauge
symmetry (or the current conservation of the global U(1) symmetry in the original
Thirring model), which actually requires no wave function renormalization. In the
usual gauge (.;=constant) including the Landau gauge (.;=0), on the other hand, we
cannot arrange A( P2)= 1 without modifying the bare vertex into a complicated one
consistent with the WT identity in a rather arbitrary way.
Thirdly, the massless vector boson limit (limit of infinite fourfermion coupling
constant) can be taken smoothly in the gauges other than the unitary gauge (original
Thirring model) in the HLS formalism, so that our result in (2+ 1) dimensions can be
compared with QED3, which is impossible in the unitary gauge. Such a massless limit
is also interesting in the composite models of gauge bosons.9>17>28>
Fourthly, the HLS can also be used to settle the regularization ambiguity on the
induced CS term in (2 +I)dimensional Thirring model. Without gauge symmetry,
any regularization could be equally allowed, which then leads to contradictory result
on whether or not the CS term is induced by the fermion loop. Once the HLS is
explicit, regularization must be such as to preserve the HLS (PauliVillars regulariza
tion), which then concludes that the CS term is actually induced in the same way as
in QED3. 18> Then there exists CS term for the odd number of 2component fermions
even in the symmetric phase where the fermion mass is not generated, while for the
even number of 2component fermions it can be arranged to cancel each other within
the pairs of the regulators.
At this point one might still suspect that the HLS is just a redundant degree of
freedom and plays no significant role on physics, since there is no kinetic term for AP
at tree level. However, as is well known in the D(2:::;;:D<4)dimensional nonlinear
sigma model like CPN! model, the HLS gauge boson acquires kinetic term through
loop effects.9> Moreover, there exist realistic examples of such dynamical gauge
bosons of HLS realized in Nature: The vector mesons (p, co,) are successfully
described as the dynamical gauge bosons of the HLS in the nonlinear chiral Lagran
gian.29>9> We shall demonstrate in the next section that in the case at hand this
phenomenon actually takes place, once the fermion gets dynamical mass from the
nonperturbative loop effects in the SD equation. In passing, the limit of infinite
fourfermion coupling can be taken only through the HLS, namely, the massless gauge
boson can be generated dynamically only when the manifest gauge symmetry does
exist.9>
Our HLS Lagrangian can easily be extended to the nonAbelian case by use of the
"U(n)/U(n)" nonlinear sigma model which is gauge equivalent to U(n)giobai X U(n)Iocai
model:9>'17>
where AP=Apaya, and u=e;1, ='aTa, with ya being the U(n) generators.
Actually, Eq. (2 14) is gauge equivalent to the Thirring model having the interaction
In contrast to the U(1) case, however, the fictitious NG bosons in the nonAbelian
case are not decoupled even in the Rt gauge, which would make the SD equation
analysis rather complicated.
3. SchwingerDyson equation
In this section we study the fermion dynamical mass generation in the D(2S.D
<4)dimensional massless Thirring model through the SD equation. First of all, we
must clarify what symmetry is to be dynamically broken by the dynamical generation
of fermion mass. In this respect D=3 is rather special. Since we wish to study the
fermion mass generation in D(2S.D<4) dimensions which contain D=3, we here
identify the types of fermion mass and the symmetries to be broken in (2+ 1) dimen
sions.30>
3.1. Chiral and parity symmetries in (2 + 1) dimensions
In (2+ 1) dimensions the simplest representation of r matrices is the one with
respect to the 2 X 2 matrices, or the Pauli matrices,
The 2component fermions are denoted by Xa with the flavor index a=1, , N. The
parity transformation is defined by
for x'=(t, x, y).
Note that the mass terms for 2component fermion, mxaxa, are odd under the parity
symmetry. We are restricting our interest to even number of fermion species and
addressing ourselves to the question whether the symmetries of the classical Lagran
gian are preserved at quantum level or not. In this case it is convenient to write the
theory in terms of 4component spinors <Pa=(x~:J with flavor index a=1, , N as we
did in 2. The three 4 X 4 r matrices can be taken to be
and then there are three more 4 X 4 matrices
which, together with the identity, constitute generators of the global U(2) symmetry,
i.e., ~ 0 =1, ~ 1 =iy3 , ~ 2 =y5 , ~ 3 =r. Then the (2+1)dimensional massless Thirring
model is invariant under the socalled "chiral" transformation
where Ta denote the generators of U(N), so that the full chiral symmetry of the
theory is U(2N) as expected. The parity transformation for 4component fermions
is composed of that of 2component fermions and the exchange between the upper and
the lower 2component fermions, specifically
and the corresponding operation on the gauge field becomes
Now we identify the peculiarity of D=3 dimensions; the full global symmetries of
the Lagrangian (21) (or equivalently Eq. (23)) are the parity and the global U(2N)
chiral symmetry. Accordingly, in (2 +I)dimensional case, order parameter of the
chiral symmetry breaking U(2N)> U(N) X U(N) is the parityinvariant mass defined
by
Though at this stage we do not yet know whether the dynamical symmetry
breaking really occurs or not, we know what the breaking pattern should be once it
happens, thanks to the VafaWitten theorem.20> Namely, since the treelevel gauge
action corresponding to Eq. (23) is real and positive semidefinite in Euclidean space,
energetically favorable is a parity conserving configuration consisting of half the
2component fermions acquiring equal positive masses and the other half equal
negative masses. Such a parityconserving mass is indeed generated, as we shall
show through the SD equation. It was also confirmed in QED3 where the classical
action shares the same structure as ours except for the kinetic term of the gauge field,
both satisfying condition of the real positivity in Euclidean space.30> Moreover, the
parity violating pieces including the induced CS term18> do not appear in the gauge
sector whenever the number of 2component fermions is even. According to the
above arguments, the pattern of symmetry breaking we shall consider is not for the
parity but for the chiral symmetry. Thus we investigate the dynamical mass of the
type mf in the SD equation. In D( ,:3) dimensions, on the other hand, such a mass
breaks the chiral U(N) X U(N) symmetry of Eq. (22) (also Eq. (23)) down to a
diagonal U(N) symmetry. Incidentally, the U(l) subgroup of this diagonal U(N)
was actually enlarged into the U(l)8Iobai X U(l)Jocai by the HLS in 2. (See Ref. 9).)
3.2. SchwingerDyson equation in the nonlocal Re gauge
Taking the above arguments into account, we now study the SD equation to
confirm whether the chiral symmetry is spontaneously broken or not in the D(2~D
<4) dimensional Thirring model. We write the full fermion propagator as S(p)
= i[A( p2)ji B( P2)] 1, with B being the order parameter of the chiral symmetry
which preserves the parity in (2 + 1) dimensions. Then the SD equation for Eq. (2 12)
is written as follows:
(A( P2)1)Ji B( p2)
 _l_f dDq A( q2)4 +B( q2) 0 I'll
 N i(2n)n /P. Az( qz)qz Bz( qz) Tv(P, q)zD (p q),
where Tv(P, q) and Dp.v(pq) denote the full vertex function and the full gauge boson
propagator, respectively. We should apply some appropriate approximations to this
equation so as to reduce it to the solvable integral equation for the mass function
M( pz)=B( pz)/A( pz).
Following the spirit of the analysis of QEDa,30l we here adopt an approximation
based on the large N arguments, in which Tv(p, q) and Dp.v(pq) are those at the 1/N
leading order, namely, the bare vertex and the oneloop vacuum polarization of
massless fermion loop, respectively:
Tv(P, q)= lv,
 iD~'ll(k)= d( k2)( gl'll_ 7J( k2) k;r),
where we have adopted a nonlocal R, gauge, Eq. (210), with the momentum
dependent gauge parameter ~( k2), and II( k2) is the oneloop vacuum polarization
of massless fermions:
which is readily calculated to be
Then the SD equation (3 10) is reduced to the following coupled equations for
A( P2) and B( p2 ):
2  1 f dDq A(q2)
A( P )1 Npz i(2n)n Az( qz)qz Bz( _ qz)
x d( k2)[{7]( k2)+2 D}(p q) 2(k P2}k q) 7J( k 2)J,
B(P2)=ivfi&:)n Az(q~~~!~z(qz)d(k2)[D7J(k2)],
where kp.=p,q,. These are our basic equations.
At first sight Eqs. (316) and (317) might seem to be trivial in 1/N expansion,
since L.H.S. is formally of 0(1/N) whereas R.H.S. is of 0(1), which then would imply
a trivial solution, A( P2) = 1 and B(  P2) =0, at the 1/N leading order. However, as
was realized in QED3,30> these equations are selfconsistent nonlinear equations
through which A( P2) and B( P2) may arrange themselves to balance the N
dependence of L.H.S. and that of R.H.S. in a nontrivial manner. In fact, N
dependence of the solution might be nonanalytic inN as was the case in QED3.21l We
wish to find such a nonperturbative nontrivial solution by just examining Eqs. (316)
and (3 17) for finite N.
A technical issue to solve Eqs. (3 16) and (3 17) is how to handle the coupled SD
equations Eqs. (3 16) and (3 17). Here we follow the nonlocal gauge proposed by
Georgi et al. 19>'23> which reduces the coupled SD equations into a single equation for
B( P2) by requiring A( P2) = 1 in Eq. (3 16) by use of the freedom of gauge choice.
This is actually the gauge in which the bare vertex approximation can be consistent
with the WT identity for HLS (or the current conservation), i.e., A(O)=l. This is in
sharp contrast to the ordinary (momentumindependent) covariant gauge or even the
Landau gauge with 7J(k2)=1 (~(k2)=0), in which the bare vertex approximation
is not consistent with the WT identity. In the nonlocal gauge B( P2) itself is a mass
function, i.e., M( p2)=B( p2).
Requiring A( P2) = 1 in the nonlocal gauge, we perform the angular integration
of Eqs. (316) and (317) in the Euclidean space (hereafter in this section we use the
Euclidean notation):
*> For D=2 we should separately consider Eqs. (316) and (317). Then we obtain '7(k2)=0 and the gap
equation for B(P2)=constant (see Eq. (341)).
with k2 =P2 +q2 2Pqcos8. It is easily seen that Eq. (318) is used to determine the
gauge fixing function 7J(k2):
which takes the same form as that in the Ddimensional QED.23>*> Substituting d(k2)
in Eq. (3 13) into the above relation, we determine 7J(k2):
with zFt(a, b, c; z) being the hypergeometric function. Equation (322) actually
forces the gauge fixing parameter~ to be a function of k 2 Once we determined r;(k2)
and hence the kernel Eq. (320), our task is now reduced to solving a single SD
equation for the mass function B(P2), Eq. (3 19), which is much more tractable.
The kernel K(p, q; G) in Eq. (320) is a positive function for positive arguments
p, q and G, since Dr;(k2) is positive for positive k 2 Moreover, the kernel depends
on the arguments p and q only through k 2=P2+q22pqcos8, so that it has the
symmetry under the exchange of P and q. These kinematical properties of the kernel
K(p, q; G) are essential to proving the existence of a nontrivial solution of the SD
equation (3 19) in the next subsection.
3.3. Existence of nontrivial solution and critical line
Now we investigate existence of the nontrivial solution for the SD equation (319)
for z::;;:D<4, based on the method of Refs. 31) and 32). In D=2 dimensions the SD
equation reduces to the gap equation for the constant dynamical fermion mass as in
the Gross Neveu model. We can explicitly solve this full gap equation. For 2< D<4
we shall use the bifurcation method32> to solve the SD equation.
Let us first consider 2<D<4. The integral equation (319) always has a trivial
solution B(p2)=0. We are interested in the vicinity of the phase transition point
where the nontrivial solution also starts to exist without gap (we call this continuous
phase transition a "second order" phase transition in distinction from the first order
one). Such a bifurcation point is identified by the existence of an infinitesimal
solution oB(p2) around the trivial solution B(P2)=0.32> Then we obtain the linearized
equation for oB(P2):
oB(p2)= 11.:~. d(qD2)K(p, q; G)oB(q2),
K(x, y; g)=K(y, x; g)>O
This is the most important property for the existence proof of the nontrivial
soluwhere we introduced the IR cutoff m. It is enough for us to show the existence of a
nontrivial solution of the bifurcation equation (323).31) Particularly, we can obtain
the exact phase transition point where the bifurcation takes place. Since we normal
ize the solution as m= oB(m2), m is nothing but the dynamically generated fermion
mass.
Rescaling p=Ax 11<D2>and oB(p2)==AJ:(x), we rewrite Eq. (323) as follows:
where we introduced <Jm=(m/A)D2 (0< <Jm::;;:1), the dimensionless fourfermion cou
pling constant g=G/A2D and
K(x, y; g)=K(xti<D2>, yti<D2>; g).
Let us consider the linear integral equation:
~1 An2(g, am)=1:1:dxdy[K(x, y; g))2< co.
whose eigenvalues and eigenfunctions are denoted by An(g, am) (IAni2:IAn+II; n=1, 2, )
and rl>n(x), respectively. The kernel K(x, y; g) is a symmetric one and hence satisfies
the following property:
The R.H.S. of Eq. (328) gives the upper bound for each eigenvalue An(g, am).
Furthermore, using the positivity of the symmetric kernel (see Eq. (326)), we can
prove that the maximal eigenvalue AI(g, am) is always positive and the corresponding
eigenfunction r/>I(x) has a definite sign (nodeless solution).
In the bifurcation equation (3 24) this implies the following: If N is equal to the
maximal eigenvalue of the kernel Az(g, am); N=Az(g, am), then there exists a nontrivial
nodeless solution I:(x)=r/>z(x) besides a trivial one. N=AI(g, am) determines a line on
(N, g) plane which is specified by am. Hence the above statement means each line
with the parameter am corresponds to the dynamically generated mass m=
A(am)11<D2>. Now we introduce Ncr(g) defined by
Ncr(g)=ih(g, am~O).
As am approaches zero, the corresponding line also approaches the critical line on (N,
g) plane; N=Ncr(g). Moreover, since AI(g, am) is the maximal eigenvalue of the
kernel, there is no nonzero solution for N larger than AI(g, am). Through these
consideration we can conclude that ~f the inequality N <Ncr(g) is satisfied, then the
fermion mass is dynamically generated. Existence of the critical line, N=Ncr(g) or g
= gcr(N), in the twoparameter space is somewhat analogous to that in the gauged
NJL model.33)
Although it is difficult to obtain the explicit form of the critical line N = Ncr(g) in
the general case, we can do it in the limit of infinite fourfermi coupling constant, g
~co Let us discuss the (2+ 1) dimensions for definiteness, in which case the kernel
reads
K(x ' y' g~co)=33J2r2m.m{1X' Y1} .
plus boundary conditions
I:'(am)=O, (IR B.C.)
sin(fa) (;J"'sin{ ~ [10:, +a])'
w=/Ncr/N 1, 6'=2w1arctanco,
where 6m is given by the UV boundary condition (333):
~[In ;m +26' ]=nJr, n=1, 2, ....
KQEo(P, q; a)= 4~21"desin8 dQEo(k2)[37J(k2)],
The solution with n=1 is the nodeless (ground state) solution whose scaling behavior
is read from Eq. (335):
The critical number Ncr=128/3n2 is equivalent to the one in QED3 with the nonlocal
gauge fixing. 22>'23>
Here we discuss the reason why our bifurcation equation at g ~ oo in (2 + 1)
dimensions is the same as the one in QED3. In the nonlocal gauge the SD equation for
QED3 reads23'
[x.l''(x)+ .l'(x)]x=I =0. (UV B.C.)
Equations (331)~(333) are the same as those in QEDg.30>'22>'23> When N>Ncr
=128/3.7r2, there is no nontrivial solution of Eq. (331) satisfying the boundary condi
tions, while for N <Ncr the following bifurcation solutions exist:
where the scale a is defined by a=Ne2 with the gauge coupling e and the nonlocal
gauge function 7J(k2) is given by Eq. (321) with d(k2) replaced by adQEo(k2). It is
known that contribution to the kernel comes mainly from the momentum region k
< a.30> Noting that II(k2) ~ k from Eq. (3 15), we can expand adQEo(k2) in k/a:
In spite of the big difference in the general form, adQEo(k2) at k/a4:..1 and d(k2) at g 1
4:..1 are both dominated by the same vacuum polarization II(k2), which yields the same
7J(k2) and hence the same kernel. This is the reason for the coincidence of the value
of Ncr with that of the QED3.
However, we should note an essential difference of our case from the QEDs.
Since the asymptotic behavior of the HLS gauge boson propagator is ~ 1/k, the loop
integration appearing on R.H.S. of the SD equation (310) is logarithmically diver
gent. This is due to lack of the kinetic term of the HLS gauge boson at tree level, in
contrast to the photon in QEDs whose asymptotic behavior is ~ 1/k2 Hence, in order
to keep the integral in Eq. (310) to be finite, we must introduce the cutoff A, whereas
in QEDs the gauge coupling constant a provides an intrinsic mass scale which plays
a role of the natural cutoff.30> Therefore, different from QEDs, A should be removed
by taking the limit A~ oo in such a way as to keep the physical quantity like the
fermion dynamical mass to be finite. This procedure corresponds to the renormaliza
tion a Ia Miransky proposed in the strong coupling QED4.34> This renormalization
defines the continuum theory at the UV fixed point located at the critical line N
=Ncr(g) or g=gcr(N) in much the same way as the gauged NJL model.6>
Next we discuss the D=2 case. By direct inspection of Eq. (316) we obtain r;(k2)
=0 and thereby the kernel does not depend on p. Then the SD equation (3 17) is
written as
Note that G=g for D=2. The R.H.S. of Eq. (341) has no p dependence, so that the
fermion mass function is just a constant mass B(P2) = m. This salient feature can
only be realized under the nonlocal gauge ~(k2) =Fconst. we have chosen. Therefore,
as in the GrossNeveu model, the above equation gives the gap equation:
which is solvable in its full nonlinear form and has a nontrivial solution m=FO for
arbitrary N when G >O or G< ;r:
ln( 1+ ~: ) = 2N(1+ ~) .
which shows that G=O is indeed a trivial UV fixed point while G= ;r is a nontrivial
IR fixed point. The region 2 allows only the trivial solution m=O. In the next
section we shall derive the bosonization in (1 +1) dimensions through the dual trans
formation for HLS and show that the theory with  ;r < G < 0 lies not in the symmetric
Now the (N, G) plane is divided into three regions with N>O; 1. G>O, 2. ;r
< G<O, and 3. G< ;r. When we take the continuum limit m/A ~o as in the case
of 2<D<4, we reach the critical line G=O and N>O which may be interpreted as a
trivial UV fixed line. On the other hand, G= ;r is only reached by A/m~o (maybe
a nontrivial "IR fixed line") and has nothing to do with the continuum limit.
Actually, the /3function in the broken phase (G>O, G< ;r) is calculated from
Eq. (343) as
phase but has no ground state so that the trivial solution m=O depicts an unstable
extremum. It means that the theory only has a broken phase with G > 0 or G <  1r.
Therefore the results based on the SD equation perfectly coincide with the exact
results obtained by operator methods10>and those in 5.*> This agreement is quite
encouraging for the reliability of the SD equation in D(2<D<4) dimensions as well
as D=2.
4. Dynamical gauge boson
In this section we discuss the dynamical pole generation of the HLS gauge boson
which is merely the auxiliary field at tree level. It is obvious that in the chiral
symmetric phase where the fermions remain massless, there is no chance for the HLS
gauge boson to develop a pole due to fermion loop effect, since a massive vector bound
state, if it is formed, should decay into massless fermion pair immediately. In fact
the gauge boson propagator in Eqs. (312) and (3 13) with the contribution from the
vacuum polarization of massless fermions in Eq. (314) has no pole in the timelike
momentum region. However, once the fermion acquires the mass, the HLS gauge
boson propagator can have a pole structure due to the massive fermion loop effect.9>
We here discuss the vacuum polarization tensor of massive fermion loop in D(2::::;: D
< 4) dimensions.
At this point one might suspect that the fermion mass effect on the vacuum
polarization tensor may affect the analysis of the SD equation in 3 where the vacuum
polarization tensor (3 14) was calculated by the massless fermion loop. However,
the fermion mass effect on the SD equation through the vacuum polarization tensor
enters the kernel only as a linear or higher terms in 8B( P2) in the bifurcation form
of Eq. (3 23). There exists a linear term of 8B( P2) already in the integral of
Eq. (3 23), so that the mass effect on the vacuum polarization tensor yields only
higher order in 8B( P2) and can be neglected in the bifurcation equation. Thus our
analysis in 3 is totally unaffected by inclusion of the dynamically generated fermion
mass in the vacuum polarization tensor.
Suppose that the fermion acquires the dynamical mass m= oB(m2). Disregard
ing the momentumdependence of the mass function for simplicity,**> we can calculate
the oneloop vacuum polarization tensor for the HLS gauge boson as
*> Our result of the physical region G >0 and G <  ;r coincides with that of Coleman35J which corre
sponds to Schwinger's definition of current described by Klaiber in Ref. 10). If we use Johnson's definition
with the coupling constant G', then the coupling constant is related to ours as G'=2;rG/(2;r+ G). Hence the
above allowed region reads 2;r< G' <2;r.
**> There is subtlety in the analytic continuation from the spacelike region to the timelike one for the
momentumdependent mass function (D=F2). Our conclusion here will not be drastically changed as far as
the constant mass is a reasonable approximation.
T. ltoh, Y. Kim, M. Sugiura and K. Yamawaki
Since the function d(k2) in Eq. (312) is defined by d(k2)=[G 1 17(k2)]1,
the pole mass Mv of the dynamical gauge boson, if it exists, is defined by the following
equation:
c 1=17( Mi), OsMi<4m2
First of all we observe that Mv+0 as G+oo for arbitrary D. This limit is well
defined only through the introduction of HLS which becomes a spontaneously unbro
ken gauge symmetry. On the other hand, the original Thirring model corresponding
to the unitary gauge of HLS becomes illdefined in this limit.
It is easily found that when k 2 approaches 4m2, 17( k2) diverges in 2sDs3.
Thus the solution of Eq. (43) exists in 2sDs3 for any magnitude of coupling
constant G (although it should be stronger than the critical value over which the
dynamical fermion mass is generated). Once the fermion acquires the mass m
dynamically, the HLS gauge boson always develops a pole at Mv< 2m. Specifically
in (1 + 1) dimensions the mass function is really momentumindependent, B( p2) =m,
and the above calculation becomes exact. Then the HLS gauge boson has a pole in
the broken phase with G >0:
71r [ Mv j 44mm22Mv2 t an~hm4J2 !__ M;v2 1]c1
In (2 + 1) dimensions the HLS gauge boson pole is given by
In the case 3<D<4, on the other hand, the R.H.S. of Eq. (43) remains finite even
when k2 +4m2, then there exists a lower bound of Gunder which the HLS gauge boson
propagator has no pole. This lower bound of G is determined by
Gv1= 17(_ 4m2) 4tr/3T(4(72r)DI2D/2) 2F1( 2, 2 _ D2, 22_., 1) x mv2 .
The intriguing feature of our case is that both Gv and m are related to a single
coupling constant G. This is in contrast to the massive Thirring model where the
fermion mass is given by hand, and also to the mixed model of GrossNeveu and
Thirring model where the fermion dynamical mass and the HLS gauge boson mass
are separately determined by the GrossNeveu coupling and the Thirring coupling,
respectively.
5. Dual transformation and bosonization
Now that we have reformulated the Thirring model as a gauge theory, we can
further gain an insight into the theory by using a technique inherent to the gauge
theory, namely, the dual transformation.25> Let us rewrite the theory with HLS in
Eq. (2 3) by use of the dual transformation, which in (1 +1) dimensions leads to the
bosonization of the Thirring model in the context of path integral.
We first consider the path integral for the Lagrangian (23),
ZHLs= j[dAP][d][dfa][da]expi jdDx{ ~ fair~'DPa+ 2~ (Ap/Nop)2}.
Linearizing the "mass term" of gauge field by introducing an auxiliary field Cp, we
obtain a delta functional for opC~' through an integration over the scalar field as
follows:*>
= j[d][dCP]expifdDx{ ~ CPC~'+ )ccp(A~'/No~')},
= j[dCp]o(ai'C~')expi jdDx{ ~ ci'C~'+ )cc~~'},
where the auxiliary field Cl' at tree level is nothing but the conserved current. If we
pick up CP by use of dual antisymmetrictensor field HP1Po of rank D2, which
satisfies the Bianchi identity, we have the following relation:
Substituting the above relation into the path integral and integrating out the auxiliary
field Cl', then we have
Zouat= j[dHP1Po][dAP][d"a][da]expijdDx{ ~ fair~'DPa
*l The scalar phase can in fact be divided into two parts: = f9 + TJ, where f9 expressed by multivalued
function describes the topologically nontrivial sector, e.g., the creation and annihilation of topological
solitons, and r; given by singlevalued function depicts the fluctuation around a given topological sector.
Inclusion of the topological sector f9 induces a topological interaction term;> though we neglect f9 contribu
tion in this section, since we are interested in as the NG mode.
X expz.p(dnX{..~..., ';tr'.al.YP.U::.p.,'/f,'a + 2((D1)D1) HP.t"P.Dt H"'"P.o'} ,
where HP.l"'I'Dl = Op.,Hp....P.D1 aP..HP.lP.3"'Jl.Dl +... +(l)DaP.DlHP.t"P.D2 The Lagrangian
(56) describes N "free" fermions and a "free" antisymmetric tensor field of rank D
2 which are, however, constrained through the delta functional. This implies that
the dual field is actually a composite of the fermions. In (2 +1) dimensions the dual
gauge field H" has the vector structure as A" does, while in (3+ 1) dimensions it is the
secondrank antisymmetric tensor field Hp.v, i.e., H"~.~"~ p.vpcrfra.
Although the delta functional in Eq. (56) tells us that the dual field is a composite
of the fermions, it is difficult to read directly phase structure of the Thirring model in
this formalism. If we look at the tree level Lagrangian (56) in (2+ 1) dimensions, it
might seem that the dual gauge field H" is massless independently of the phase
structure. In order to understand the pole structure of the dual gauge field Hp.,
however, we have to take into account the quantum effect. For that purpose we may
ignore the contributions from the fermion oneloop diagrams except for the vacuum
polarization, since they generate only the selfinteraction terms of the gauge field A"
or equivalently those of the dual gauge field Hp.. Therefore we compute the vacuum
polarization in Eq. (55):
Zouat~ j[dH"][dAp.]expi jd3x{  ~ Hp.vH"v+ }c"v"A"ovH"
If we compare the pole structure of n" with that of A" in Eq. (43), we easily find that
the dual gauge field H" shares exactly the same pole structure with the gauge field A"
irrespectively of the phase.
In (1 +1) dimensions the relation in the delta functional in Eq. (56) implies
nothing but the bosonization of the Thirring model in the scheme of path integral, i.e.,
(1/ /G)E"vo~.~H::::;( 1/ /N) far"tf!a. Integrating the fermions in Eq. (55), we obtain an
effective theory which consists of a pseudoscalar and a vector gauge fields:
The second term of the action in Eq. (59) is the (1 +1) dimensional analogue of axion
term which is the interaction term between the scalar and the gauge fields and takes
the form Hp.vpaF"vF"a in (3 +1) dimensions. Though the computation of fermionic
determinant with regularization generates the Abelian chiral anomaly, this problem
is resolved by the constant shift of scalar field H in axion term. Since the fermionic
determinant is computed in an exact form, i.e.,
Integrating out the HLS gauge field A", we obtain an effective Lagrangian for the dual
gauge field H" without interaction terms:
_ tNln ddeettii.!W1 __217_rP(dzxA"(g,11
If G is in the region ;r<G<O, the energy per unit volume is unbounded below and
hence the (1 +I)dimensional Thirring model with coupling constant G (G >O or G
< ;r) has only the broken phase via the fermion dynamical mass generation as in the
(I+ I)dimensional Gross Neveu model.
6. Conclusion and discussion
In this paper we have studied the Thirring model in D(2S.D<4) dimensions and
proposed how to understand it as a gauge theory through the introduction of hidden
local symmetry. The advantage of manifest gauge symmetry was to let the various
nonperturbative approaches tractable and provide the consistent method to treat such
problems.
In the case of 2N 2component fermions (or equivalently N 4component Dirac
fermions) we studied the dynamical symmetry breaking in the context of 1/N expan
sion. Since we had the manifest U(I) gauge symmetry, we took a privilege to choose
a nonlocal Re gauge, which greatly simplified the analysis of the SD equation.
By using the bifurcation technique, we found a second order (continuous) phase
transition at a certain number of Nand g, thus having established the existence of the
critical line on the (N, g) plane. We also proved existence of the nontrivial solution
rigorously. The HLS gauge boson became massless in the g..r:XJ limit, where the SD
equation was solved analytically in (2+ I) dimensions, yielding Ncr=I28/37r2 in perfect
agreement with Ncr in QED3. This limit makes sense thanks to the HLS, in sharp
contrast to the original Thirring model where this limit is illdefined.
In (I+ 1) dimensions, on the other hand, fermion mass is always generated, no
matter what value N( >O) and G= g( >O) might take: The theory is in one phase
(broken phase) for G >O as in the (I+ I)dimensional Gross Neveu model. Our result
is consistent with the exact solution of the (I+ I)dimensional Thirring model. In this
case there is no regularization ambiguity, since the regularization must respect the
HLS as in the massless Schwinger model.
The dynamical symmetry breaking in (2 +I)dimensional Thirring model with
many flavors has previously been discussed in 1/N expansion.12H 6J Here we compare
our results with those of the previous authors. When the auxiliary vector field .1L
has been used, the authors in Refs. I2) and I3) pretended it as a gauge field and added
the "gauge fixing" term, though the model carries no gauge symmetry. The hidden
local symmetry we found in such a model explains that the gauge fixing they chose is
neither the "overgauge fixing" nor the Landau gauge but the Re gauge, so that the
awkward procedure of "the gauge fixing without gauge symmetry" is justified by the
discovery of hidden local U(1) symmetry. The ladder approximation for the vertex
leads to A( P2)= 1 to be consistent with the WT identity, while it is not allowed under
the Landau gauge. It can only be realized through the nonlocal Re gauge we chose.
The dynamical gauge boson is generated when fermions get mass. The result in
2S.DS.3 has a novel feature. The gauge boson pole is always developed independent
ly of the coupling G, once the fermion acquires mass at G>gcr/Av2_ For 3<D<4,
on the other hand, the gauge boson pole can be generated only for G > Gv which may
or may not be satisfied by the coupling larger than the critical coupling 9cr/A vz. It
would be interesting to see the precise relation between the critical coupling for the
dynamical gauge boson generation and that of the fermion dynamical mass genera
tion in this case.
We rewrote the Thirring model with hidden local symmetry in terms of dual field.
In (1 + 1) dimensions, we demonstrated that this dual transformation based on the
introduction of hidden local symmetry is a straightforward way to arrive at the
bosonization of the Thirring model. In (2 + 1) dimensions, it was also shown that
both the HLS gauge field AI' and dual gauge field HI' share the same mass spectrum;
they are massless in symmetric phase and they have equal mass in broken phase.
This formulation might be useful also in D(2<D<4) dimensions.
In (2+ 1) dimensions we assumed two wellknown results such that, if the number
of 2component fermions is even and the classical Lagrangian is parityeven, the
parityviolating sector is not induced both in the effective action for the gauge field 18>
and in the pattern of dynamically generated fermion masses.20> Both of them are
consequences of exact calculations and, of course, it is consistent with our results.
However, the previous papers12>' 14H 6> claimed appearance of parityviolating piece
through quantum effect, which is opposed to the results of both Refs. 18) and 20) and
ours given in 3.
The classical action in Eq. (23) for the odd number of massless fermions is
invariant under the U(1) HLS gauge transformation and the parity in (2+ 1) dimen
sions, while in the quantized theory both of them cannot be preserved simultaneously.
Since the regularization is to be specified so as to keep the U(1) HLS (PauliVillars
regularization), there is no regularization ambiguity in our case and hence the
parityviolating anomaly arises as in QED3.18> Therefore, the gauge theory described
by the effective gauge field action lies in ChernSimons Higgs phase:
_['eff(Ap) =41 lim I~Reg I c.~'vPA.uovAP + 21G (AI' Op)2 '
7r MReg+oo 1V1Reg
where MReg is the mass of the Pauli Villars regulator. Now an emphasis is in order
on an important role of the fictitious NG boson field which is composed of the
topologically nontrivial sector e and the smooth NG degree TJ for a given topological
sector e as mentioned in the previous section. If we neglect the topological sector
and consider only the smooth NG boson mode TJ, then the nonderivative gauge mass
term dominates in the long range physics and the theory remains just in the
topologically trivial sector of the ChernSimons Higgs phase which is governed by a
parityviolating helicityone photon with mass 27r/G.36> Inclusion of the topological
sector under the guiding principle of the HLS gives rise to the generation of CS
vortices and realizes an anyonic phase.37) We may recall that the addition of the CS
term to the (2+ !)dimensional QED changed the structure of phase transition to the
firstorder one38> and this theory has also been a model describing anyonic supercon
ductivity. Then the subject of dynamical symmetry breaking in massless Thirring
model for an odd number of fermions may generate intriguing results.
We would like to thank K. 1. Kondo for very helpful discussions on the nonlocal
gauge and SD equation. Thanks are also due to T. Fujita, K. Harada, Jooyoo Hong,
D. Karabali, Choonkyu Lee, P. Maris, V. P. Nair, Q. Park, A. Shibata and S. Tanimura
for valuable discussions. Y. K. is a JSPS Postdoctoral Fellow (No. 93033) and also
thanks Center for Theoretical Physics at Seoul National University for its hospitality.
This work was supported in part by a GrantinAid for Scientific Research from the
Japanese Ministry of Education, Science and Culture (No. 05640339).
In this appendix we prove that the gauge fixing term Eq. (2 12) is BRS invariant
even for the nonlocal gauge. It might be nontrivial to see that the nonlocal gauge
fixing at Lagrangian level actually works.
According to the HLS transformations Eq. (2 5), the BRS transformation for each
field is given by
~Ba(x)= )Nc(x)a(x),
where c(x) and c(x) are ghost fields, and B(x) is the socalled NakanishiLautrap
field. Moreover, it is well known that the operator ~B is nilpotent:
.GF+FP= i~B( c/[A, >, c, c, B]).
Following the text book procedure, we have the gauge fixing term plus the
FadeevPopov ghost term:
Without knowing explicit form of ![A, ], we easily see from the nilpotency,
Eq. (A 2), that .GF+FP is BRSinvariant.
Now we show that the nonlocal Re gauge fixing term (210) is obtained, if we take
![A,] as
![A, ]=a~~'+/N g<t;) +i'.;(a2)B
=F[A]+ ~ .;(OZ)B
with F[A] being the same as Eq. (29).
we find that ..GF+FP takes the form:
_['GF=B{F[A]+ ~ .;(o2)B},
_fFP=ic( o+ g~z) )c.
In fact, substituting Eq. (A4) into Eq. (A3),
As a usual result for the Abelian gauge symmetry, FP ghost decouples from the
system completely. We then translate B(x)=B'(x).; 1(0Z)F[A] in Eq. (A5) and
integrate out B'(x), arriving finally at
where we have used the following identity:
which can easily be proved in momentum space.
Equation (A 7) is nothing but the nonlocal Re gauge fixing term in Eq. (2 10).