Phase transition in an O(N) gauge model in two dimensions*

We study the phase transition properties of the nonlinear O(N) cr model in two dimensions when O(N) gauge interactions are included. With nonzero gauge coupling, this theory exhibits a first-order phase transition in the large-N limit. The broken-symmetry phase is stabilized by the Higgs mechanism and Goldstone bosons do not appear.

Recent]. y"Bardeen and Pearson have proposed a transverse-lattice formulation of quantum chromodynamics. ' In this theory, color confinement is obtained when the vacuum is invariant under the transverse gauge symmetry. A quark-gluon phase, where color is not confined, can result if this symmetry is spontaneously broken. In this paper we will study some aspects of the mechanisms which are responsible for generating such a phase transsition.
The independent gauge degrees of freedom or the transverse-lattice theory are associated with links on the transverse lattice. The longitudinal dynamics of a single transverse link consists of a nonlinear SU(3) xSU(3) o model with SU(3) x SU(3) gauge interactions. For a single transverse link the longitudinal dynamics is two-dimensional.
As the transverse gauge symmetry is a local symmetry on the lattice, the phase-transition properties of a single transverse link are relevant to the phasetransition properties of the full gauge theory.
We shall study a somewhat simpler though analogous version of the single-link problem. The model consists of a two-dimensional nonlinear O(N) o model with O(N) gauge interactions. This theory has the advantage that the model may be systematically studied in the large-N limit. The nonlinear cr model in 2 and 2+& dimensions has recently been extensively studied by Brezin and Zinn-Justin' and by Bardeen, Lee, and Shrock. ' In this theory, spontaneous symmetry breaking can occur in 2+a dimensions but only the symmetric phase can exist in 2 dimensions. The basic result of our paper is that the broken-symmetry phase can be stabilized when gauge interactions are introduced.
The existence of nontrivial phase-transition properties in two dimensions makes this theory interesting in its own right.
In two dimensions this theory is renorma]. izable with respect to the dimensionless coupling constant I/f, ', and super-renormalizable with respect to the gauge coupling constant g'. This theory may be studied directly using the methods discussed in Ref. 4. Since the nonlinear theory is renormalizable, we must be careful to preserve the symmetry structure of the theory in our calculation. Dimensional regularization is not particularly convenient in this case as we would confront the necessity of including contributions from the transverse gauge fields. Instead, we choose to regularize the theory by considering the linear 0 model in precisely two dimensions. The nonlinear theory is recovered as a limit of the linear theory. ' The linearized theory is described by the Lagran- where the constraint P =f, ' has been relaxed. The nonlinear model is obtained by taking the limit Ap with f, ' and g' fixed. Since the nonlinear theory is renormalizable, only a logarithmic dependence on Xp can occur and is absorbed by the renormalization of f, '. In the large-N limit no such logarithms appear and the limit may be taken without the renormalization involving Xp. 14 2117 The theory is most easily studied in the lightcone gauge, A "=0, A, =(1/W)(Aa+A, ). The gauge fields A", are dependent and may be eliminated in favor of a "Coulomb" interaction. In this gauge, the Lagrangian of Eq. , 'g'(rr, a-rr))( . 8') '-(rr,.a rr, . ). t The vacuum expectation value of the o-field equation of motion may be used to determine the value of the Lagrange multiplier, J', such that (o), =f.
We obtain the expression The various vacuum expectation values in Eqs. (7), (8), and (9) may be evaluated in leading order N by using the full propagators for the 0 and z fields. We obtain the following results: (9) The large-N limit of this theory is obtained by letting N-~while holding XaN, g'N, f, '/N, and f'/N fixed. In this limit, a Hartree calculation of the o and g propagators becomes exact, with the z mass being determined self-consistently.
The propagators may be computed using the Lagrangian of Eq. (6): The divergent self-energy integrals have been cornputed using a Wick rotation and a cutoff, A'.
Using the results of Eq. (10), we may compute all the large-N contributions to Eqs. (7), (8), The Hartree calculation sums consistently, in leading order N, all tadpole graphs [ Fig. 1(a)], all cactus graphs [ Fig. 1(b)], and all rainbow graphs [ Fig. 1(c)]. All other graphs are nonleading in the large-N limit.
The m mass, m"', must be determined self-consistently from Eq. (11) and Eq. (12). In the case J = 0, there are two possible solutions corresponding to the symmetric phase, f = 0, and a spontaneously broken phase, f t 0. In the symmetric phase where Vo can depend on A. o, g', and f, ' but not on f.  Since we are using the linear O(N) model to regularize the nonlinear O(N) model, we will not discuss the linear theory but proceed to a discussion of the nonlinear theory. We note that all expresssions we have have used are unrenormalized.
The only divergent renormalization necessary in the large-N limit is a logarithmic divergence in f, .
The nonlinear O(N) gauge theory is obtained by taking the limit A. , -~w ith all other parameters held fixed. One might worry that such a limit might reorder the large-N expansion. However, we have noted that in two dimensions the nonlinear theory is renormalizable with only logarithmic divergences. Hence, the large-N limit cannot be modified by powers of N and no reordering can occur. sufficiently small. Since the o-model coupling constant is directly related to the mass in the symmetric phase, we consider the phase properties of the theory expressed in terms of this mass. We denote the w masses in the symmetric and broken phases by m, ' and m~', respectively. As mentioned above we use m, ' to parametrize theory in both phases. In Fig. 2 we plot the m masses in each phase for different values of the fine-structure constant. In Fig. 3 we plot the vacuum energy for a fixed value of the fine-structure constant. In Fig.   4, w'e plot the phase-transition line in couplingconstant space. By examining Fig. 2 Fig. 3. For small m, ' (s 0.321m), the stable phase is the largermass broken-symmetry phase (III). The vacuum energy is, of course, continuous through the phase transition. However, both m ' and j' are discontinuous, which indicates that it is a first-order phase transition. In Fig. 4 we plot the phasetransition line in coupling-constant space where region A, is the symmetric phase and region B is the broken-symmetry phase.
FIG. 4. Phase diagram in terms of coupling constants for symmetric phase (A) and broken-symmetry phase (8).
In this paper we have studied the phase-transition properties of a nonlinear O(N) o model with gauge interactions. When studied in the large-N limit the theory exhibits a first-order phase transition. The symmetric phase is characterized by the generation of a bound-state o particle degenerate with the m'' s and by all physical states being O(N) singlet bound states of cr and z' s. The broken-symmetry phase is characterized by a residual O(N -1) symmetry.
The v bound states are not formed and all physical states are O(N -1) singlet bound states. The Higgs mechanism avoids the necessity of Goldstone bosons and stabilizes the broken-symmetry phase in two dimensions.
These results indicate that the longitudinal dynamics of the gluon fields in the transverse-lattice theory of Bardeen and Pearson' is rich enough to support a phase transition in the quark-gluon theory, as would be expected in 4+& dimensions or if the number of quarks were to be sufficiently large as to destabilize confinement phase. The phasetransition properties of the full gauge theory are of course much more complex than the simple model studied in this paper. We do think that the results of this paper shed some light on the mechanisms which operate in a gauge theory.
Note added. After this paper was completed, we received a report by J. S. Kang [Phys. Rev. D 14, 1587(1976], who studies the linear 0(N) gauge model. His calculations include the leading-N contribution from the meson self-interaction but are to first order in the gauge coupling. Our results represent the full leading-N calculation of the properties of the linear and nonlinear O(N) gauge theories in two dimensions.