On Spontaneous Symmetry Breaking in Discretized Light Cone Field Theory

The problem of spontaneous symmetry breaking in scalar field theories quantized on the light cone is considered. Within the framework of "discretized" light-cone field theory, a constrained zero mode of the scalar field, which is necessary for obtaining a consistent dynamics, is responsible for supporting nonzero vacuum expectation values classically. This basic structure is shown to carry over to the quantum theory as well, and the consistency of the formalism is checked in an explicit perturbative calculation in (1+1)-dimensional ${\ensuremath{\varphi}}^{4}$ theory.


I. INTRODUCTION
There has recently been a renewal of interest in developing a practical Hamiltonian approach to relativistic field theory, based on light-cone quantization [1]. Many of the technical difficulties that plagued the original scheme of Tamm [2] and Dancoff [3] seem to disappear, or are at least rendered more tractable, when a relativistic system is quantized at equal "light-cone time" x+=x +x . The most striking aspect of quantum field theories formulated in this way is surely that in some cases the bare Fock vacuum is the full physical vacuum state of the theory. There is a simple (that is, naive) kinematical argument for why this is the case: the lightcone Hamiltonian conserves light-cone momentum p+, so that the bare vacuum can only mix with other states of @+=0. But in many theories, in particular those with only massive excitations, there are no such states; p+ is strictly positive for states containing quanta. ' Thus the Fock vacuum is an eigenstate of the full interacting Hamiltonian.
The occurrence of nontrivial vacuum structure is therefore somewhat mysterious in the light-cone framework [7]. At present phenomena such as spontaneous symmetry breaking (SSB) and the formation of condensates are poorly understood. It is clearly important to study how e8'ects such as these might be incorporated into the formalism, since the QCD vacuum, for example, is believed to be quite complex.
Hornbostel has recently given an illuminating analysis of these issues [6], using a quantization surface that inter-polates between t =0 and x =0. In many cases he finds that the physical vacuum state does indeed approach the bare Fock vacuum as the quantization surface approaches the light cone. Nonezero vacuum expectation values can be supported in the limit, however, by singularities in the field operators near p+ =0.
In this paper I consider the problem of SSB in scalar field theories quantized ab initio on the light cone. So that everything is well defined, I shall work within the framework of "discretized" light-cone (DLC) quantization, a popular method of setting up actual numerical calculations. In the present context, the nice feature of DLCQ is that it provides a systematic regularization of the small-p singularities that are endemic to light-cone field theory. The aim is to see whether, e.g. , vacuum expectation values and multiple vacua can arise in this formalism.
The basic formulation of DLC field theory has recently been reexamined by several authors [8 -10], who have pointed out the generic need for a constrained p+ =0 mode in theories with bosonic fields. In light of the kinematical argument described above, if SSB can be accommodated in DLC field theory, it may be expected to be closely connected to the properties of this zero mode.
This connection was worked out for classical scalar field theory in Ref. [9]; in the present paper I shall verify that the same basic structure obtains in the quantum theory, and check the consistency of the formalism in perturbative calculations.
Other work extending the results of Ref. [9] to the quantum theory of self-interacting scalar fields has recently been described in Refs. [11,12]. Particles with p = -~d o carry vanishing p+; however, these constitute a set of measure zero in the space of states and so can usually be neglected.
2This argument obviously fails if there are massless particles in the theory; for example, gauge particles. In this case there are states with p+ =0 that can mix with the bare vacuum, so that nontrivial vacuum structure can in principle be supported. Experience with theoretical models suggests, however, that even in these cases the vacuum structure on the light cone is far simpler than in the equal-time representation [4][5][6].

AND THE CONSTRAINED ZERO MODE
The formulation of DLC field theory has been described in many places (see, for example, Ref. [ [9]. In this approach, it follows from requiring that the vanishing of the momentum conjugate to $0 be preserved in This is just a general expansion in periodic functions, with the zero-mode piece explicitly separated out. The a are Fourier coeKcients which, in the quantum theory, satisfy the standard commutation relations: [a, at]=5 (2.3) The Fock space is generated by acting with a~o n the bare vacuum state~0). Note that the integer q =2, 4, . . . is associated with a (conserved) light-cone momentum p+ =qnIL. It the.refore follows quite generally that~0) is an eigenstate of the full interacting DLC Hamiltonian, since it is the only state in the theory with p+ =0. Now, the conjugate momentum of a scalar field is with U a c-number constant. We can regard the last two terms as (the negative of) a potential with minimum at P =v, so that after solving the theory we expect to find (niacin)= u- In this case the relation defining the constrained zero mode is trivial; integration in x from -I. to  If P is assumed to be periodic in x for all x+, then integration of Eq. (2.5) in x from L to L results in- 3I define light-cone coordinates x -=x +x', and corresponding derivatives B~=-3/Bx+. As is conventional, x+ will here play the role of time.
Thus $0 is a constrained (operator) functional of the dynamical fields in the theory. In general, this constraint is very complicated. For example, $0 actually appears in the current Pg as well, so that Eq. (2.6) really defines (to implicitly [10]. Further examples of these constraint re-where I have made use of f~d x y=o. From Eq. (3.5) we see that the Fock vacuum is in fact the physical vacu- Note that the energy of this state is just the value of the classical potential-energy density at the minimum times the volume of space. Furthermore, the expectation value of P in the physical vacuum is (n~y~n) =(0~(y, +q )~0) =u . (3.7) Thus our expectations regarding this simple system are realized. Note, however, that the language used to describe these results is somewhat unconventional. The vacuum state is simple in this theory. The constraint relation, however, forces a particular condition on the fi'eld itself, namely, that it have a c-number zero-mode piece, and this reproduces the correct vacuum expectation value (VEV) and vacuum energy.
Let us now turn to the more complicated problem of the spontaneous breaking of refiection symmetry in (p )3 theory. We take for the Lagrangian density y(2)-0 1 --gAg -/ B+O(g ) v'g 2 (3.18) -1+ -gA -g / B+O(g ), (3. (3.17) In the quantum theory the presence of complicated inverses of roots of the operators A and B renders these expressions intractable.
We can expand the Po" in powers of g, however, resulting in approximate solutions that are polynomial in A and B. It is straightforward to show that as the equation defining Po. This is obviously more complicated than our previous relation. First of all, Po is an explicit functional of y, that is, it is an operator, not a c number. Furthermore, it will not in general commute with y, so that the ordering of factors on the right-hand side of (3.10) must be prescribed. A general solution of Eq. (3.10) in the quantum theory thus seems difficult to achieve.
If y is a c-number field, however, then Eq.  The correction to (0~/~0) is therefore (3.23) The third contains no c-number piece, and furthermore, since B is trilinear in y we have (O~B~0) =0. Thus (0~/~0) =0 through O(g) for Po '. It is clear that these solutions correspond, in the usual language, to versions of the theory expanded about one of the three extrema of the classical potential. Here, however, we speak of a choice of solution of the constraint relation (3.10), rather than a choice of vacuum state, characterizing the theory.
If we let po~po, corresponding to g~g, we then obtain the A, P theory without symmetry breaking. Solutions Po ' and Po ' become imaginary in this case, and must therefore be rejected. Thus there is only one physically acceptable solution to the constraint, namely, Po( ' with g~g, and hence only one phase of the theory. The solution Po( ' is slightly pathological in the brokensymmetry case, however. It turns out that, although the Fock vacuum is an eigenstate of the full P (for the general reasons discussed in Sec II), it is not the state with the lowest eigenvalue of P . Thus the true vacuum state, that with the lowest light-cone energy, is complicated. This is in accordance with the standard equal-time analysis: were we foolish enough to write the Lagrangian in terms of the unshifted field we would encounter spurious instabilities and "tachyonic" modes. We anticipate that P is in fact bounded from below, based on our experience at equal times, but to see this would require going beyond perturbation theory. For the remainder of this paper I shall focus on the better-behaved solution (&)

4'o
In order to check that this formalism makes sense, let us now calculate the lowest-order correction to (0~/~0), due to the A term in Eq. (3.18 can simply imagine that the sum is regulated with a cutoff. Notice that this correction is completely independent of L, and so survives in the L~00 limit. %'e must also renorrnalize the boson mass at this order. The bare mass po is connected to the renormalized mass p via Po=P +6P (3.24) where 5)u /(M =O(g). In terms of p, then, the c-number background field part of $0(" is Note that I have written everything in normal order, discarding leftover c-number constants but nothing else. Thus I explicitly retain the self-induced inertia terms in Eq. (3.32). Note also that the free Hamiltonian Po has come out with the correct sign so that the zeroth-order eigenvalues of P for states containing particles are strictly positive.
Since perturbative corrections to these are by assumption small, we conclude that, at least in perturbation theory, the Pock vacuum~0& will be the eigenstate of lowest p and hence the physical vacuum state of the theory. (We would obtain the same result were we to use the solution Po( ' to define the theory. ) That the vacuum is simple in the light-cone representation is therefore seen to be true even in the presence of symmetry breaking. In the equal-time representation this state is of course very complicated, containing an infinite number of bare quanta.
We can now calculate the O(A. ) correction to the energy of the one-boson state at~0&. As we are interested in the mass counterterm, we retain only on the divergent part of this quantity. The contribution which proceeds through a two-boson intermediate state via the trilinear coupling turns out to be finite. From the self-induced inertia term in Eq. ( This is reminiscent of the results of Hornbostel [6], in that VEV's are associated with properties of the field operators themselves. In the framework of his interpolating quantization surface these were singularities near p+ =0, which pick out the leading corrections to the bare vacuum state as the light cone is approached. In DLCQ the p+ =0 singularity is regulated by the boundary conditions, but the information it contains survives, in a sense, encoded into the constrained zero mode. It is straightforward to extend these results to higher dimensions and continuous symmetries [9,10]. An important feature is worth noting, however: the constrained zero mode will have a nontrivial dependence on any transverse coordinates x in generalit is only completely independent of x . See Ref. [10] for an explicit example of this in a (3+ 1)-dimensional Yukawa theory.
It is perhaps not surprising that a part of the price paid for a simple vacuum structure is the appearance in the theory of other complexities; in this case, complicated operator constraints. It will be of great importance to the DLCQ program to find ways of going beyond the perturbative type of solution of the constraint relation describe here. A "mean-field" ansatz has been proposed in Ref. [11], and used to study the phase transition in the ((t) )2 model [15]. (The transition is of course invisible to a perturbative analysis. ) This leads to a correct prediction of the second-order nature of the transition, along with a reasonable estimate of the critical coupling g, . In the context of a Tamm [10,16,17]. This is certainly true when massless fields are present [16]. Work on understanding the role of these "boundary" degrees of freedom in the massive case is currently in progress [18].