Transitioning from equal-time to light-front quantization in $\phi_2^4$ theory

We implement the limiting procedure of Hornbostel for the quantization of two-dimensional $\phi^4$ theory in a sequence of coordinate systems that interpolate between equal-time and light-front coordinates. This allows computation of the vacuum state in the odd and even sectors of the theory and computation of massive states built on these vacua. Results are compared with those of the equal-time calculations of Rychkov and Vitale and those of standard light-front calculations.


I. INTRODUCTION
Recently, there has been a resurgence of interest in the spectrum of two-dimensional φ 4 theory [1-6, 8, 9], 1 partly because of what appeared to be an inconsistency between results from equal-time quantization and light-front quantization. Although the apparent inconsistency has been resolved, as a difference in mass renormalizations [8,10,11], there remain various issues related to the structure of the vacuum. In light-front quantization [12][13][14], the vacuum is famously trivial, 2 but in equal-time quantization, it is as complex as any of the other eigenstates.
In order to see more clearly what may be happening for the light-front vacuum, we apply the interpolation procedure championed by Hornbostel [16] 3 and emphasized by Ji [24], in which the coordinates are chosen to be 4 with x + chosen as the time coordinate. The parameter c ranges from 0 to 1, with 0 being the light-front limit [26], 5 with x ± = (t ± z)/ √ 2, and 1 the equal-time limit, with x + = t and x − = −z. The minus sign for the equal-time spatial coordinate may seem incongruous, but it is a permissible choice that simplifies the notation.
The conjugate energy and momentum are Dot products of the momentum and spatial two-vectors are then given by p·x = p + x + +p − x − . The mass-shell condition becomes The positive root for p + yields For the c = 1 and c = 0 limits, this expression becomes Clearly, the zero modes (p − = 0) and negative p − states have infinite light-front energy and are removed from the spectrum, as c → 0.
However, these modes can contribute to light-front computations and, in particular, to vacuum expectation values [16,24]. A standard illustration of this is in the spectrum and VEV of a free scalar field that has been shifted by a constant. The shift introduces to the Lagrangian a term that is linear in the field; on the light-front, this can only contribute via zero modes. This can be seen quite cleanly in the c → 0 limit, where the Hamiltonian eigenvalue problem in an x − box has an analytic solution for any c > 0. A numerical solution in a truncated Fock space works just as well. This is discussed in Sec. II.
A more interesting case is that of φ 4 theory, where, as already mentioned, the vacuum needs to be better understood. In Sec. III, we will explore what the solutions with c = 0 and the c → 0 limit can tell us. The calculations are done numerically, in a Fock basis of discrete momentum states in an x − box. A brief summary is given in Sec. IV.

II. SHIFTED FREE SCALAR
A free scalar field that is shifted by a constant provides an interesting example of the impact of zero modes on a light-front calculation. This can be seen explicitly in the c → 0 limit, where a nonzero contribution is found for the vacuum energy and the VEV of the field. These analytic results [16,24] can be replicated in a numerical calculation using a Fock basis of zero modes. We illustrate this here.
The Lagrangian of a free scalar field of mass µ is In terms of the interpolating coordinates (1.1) with arbitrary c in two dimensions, this becomes [16], The (free) Hamiltonian is with π = c∂ + φ + s∂ − φ. The mode expansion for the field is The normal-ordered free Hamiltonian can then be written as Similarly, the momentum operator is Discretization consistent with discrete light-cone quantization (DLCQ) [27] is invoked by placing the system in a box −L < x − < L with periodic boundary conditions. The momentum is then discrete, p − = nπ/L, as set by the integer n; however, unlike DLCQ, n ranges over all integers, not just the positive ones. As shown in (1.5), negative p − is removed from the spectrum only for c = 0. An energy cutoff is then required for a finite basis. We do still define a positive integer K as the resolution [27], so that in the c → 0 light-front limit, the total momentum is P − = Kπ/L. The index n for individual momentum then ranges from 1 to K in the light-front limit, and momentum fractions p − /P − are just n/K.
The discrete mode expansion for arbitrary c is with w n ≡ n 2 + cL 2 ,L ≡ µL/π, and [a n , a † m ] = δ nm . The free Hamiltonian becomes We now shift the field: and the Hamiltonian, having dropped a constant, is with the interaction part In the light-front limit c → 0 this interaction term completely disappears, as it does in a native light-front calculation where zero modes are neglected. Without this term, the shift in the field and the shift in the energy cannot be recovered. However, a calculation for arbitrary c > 0 succeeds, and the light-front limit can then be taken. This was discussed by Hornbostel [16], and we repeat the argument here. The vacuum eigenstate is a coherent state of zero modes This works because the coherent state is, as always, an eigenstate of the annihilation operator a 0 |vac = −α|vac (2.14) and, therefore, Given w 0 =L √ c, we only need α = v L π √ c to eliminate the a † 0 terms and make this coherent state indeed an eigenstate of P + , with an eigenenergy of −µ v √L π c 1/4 α = − 1 2 µ 2 v 2 (2L). This restores the constant originally dropped from the Hamiltonian. In the light-front limit c → 0, α also becomes zero, and this state becomes the empty state |0 , but the energy is independent of c. All massive states are decoupled and remain as free.
The VEV of the field is given by This, of course, reflects the original shift in the field. Obviously, this is independent of the value of c. A non-zero result is obtained because the vanishing coefficients of zero mode contributions are compensated by the 1/c 1/4 divergence in the zero-mode part of the field. We need not rely on having an analytic solution to see this result for the vacuum state. A numerical solution in a finite basis of zero modes (a † 0 ) n |0 , truncated to n = 10, yields the spectrum shown in Fig. 1 as a  This nontrivial light-front limit provides a connection with the known results for equaltime quantization. In the equal-time approach, the linear interaction term is not lost but makes a direct contribution to the Hamiltonian. The solution for the vacuum state then includes the consequences of the shift in the field, as can be seen here for c = 1. The light-front limit c → 0 reproduces the results obtained from equal-time quantization.
To explore these connections further, we consider two-dimensional φ 4 theory, where it is known that equal-time and light-front quantizations differ in the vacuum contributions to mass renormalization [10]. Thus, the remainder of the paper is an analysis of φ 4 theory in terms of the interpolating coordinates (1.1). The discrete form of the theory is constructed in the next subsection, and the results of the numerical solution are discussed in Sec. III B.

A. Analysis
The Lagrangian for φ 4 theory is We construct the (discrete) interaction Hamiltonian from the φ 4 term as Substitution of the discrete mode expansion (2.8), with L =Lπ/µ, and evaluation of the now-trivial integrals, yields, P I + = µ gL 4 n 1 ...n 4 1 √ w n 1 · · · w n 4 1 12 (a n 1 · · · a n 4 + a † n 1 · · · a † n 4 )δ n 1 +···+n 4 ,0 (3.3) + 1 3 (a † n 1 a n 2 a n 3 a n 4 + a † n 2 a † n 3 a † n 4 a n 1 )δ n 1 ,n 2 +n 3 +n 4 (3.4) + 1 2 a † n 1 a † n 2 a n 3 a n 4 δ n 1 +n 2 ,n 3 +n 4 , (3.5) with g ≡ λ/(4πµ 2 ) the dimensionless coupling. The Hamiltonian eigenstates are constructed as Fock-state expansions To take into account the symmetrization of states with k identical bosons, we rewrite this sum as where N n i is the number of bosons with momentum index n i and the wave functions are related by The normalization is The probability P k for the Fock sector with k bosons is then given by The eigenstates must satisfy (P 0 + +P I + )|ψ = E|ψ . For simplicity, we look for eigenstates at rest, with total P − = 0, and either an odd or even number of constituents; the Hamiltonian changes particle number by only even amounts and therefore does not mix odd and even Fock states. The sums over the number of constituents k are then limited to even or odd values. In particular, we have expansions in the form For the purpose of having a finite numerical matrix calculation, the infinite Fock basis is truncated both in the sum over constituents and in energy. First, the number of constituents is limited to a maximum of K, so that the sum over k in |ψ is finite. Second, the total energy of each Fock state, as specified by the free Hamiltonian (2.9), is limited to be no more than a fixed energy, E max .
The total energy of a Fock state is given by μ Thus, for n ≤ 0, the contributions diverge and Fock states with such constituent momenta will be removed by the energy cutoff as c goes to zero. For eigenstates with total P − = 0, where the integers n must sum to zero, there must be at least one constituent with n ≤ 0. For such a state, a sufficiently small value of c will cause the energy cutoff to remove all the Fock states, except the trivial empty state |0 . However, this would be inconsistent with the analysis of the shifted free scalar, where the addition of a c-dependent energy cutoff would have removed the (infinite set of) Fock states needed to construct the coherent state for the vacuum eigenstate. So, we instead keep the Fock basis unchanged as c changes by imposing the energy cutoff at c = 1 and then leaving the basis fixed when c is decreasing. Therefore, for all c values, the energy limit on Fock states is given by (3.14) We do not study the dependence on these truncations, nor on the box size, in a systematic way, our purpose being a qualitative understanding of the light-front limit as the parameter c goes to zero. In equal-time quantization there has been considerable work by Rychkov and collaborators [3] on the renormalization necessary to reduce the cutoff dependence and facilitate very accurate calculations with minimal basis sizes. Attempting this for arbitrary c is certainly of some interest but is beyond the scope of the present work.

B. Results
As a check on the calculation, the even vacuum energy for equal-time quantization (c = 1) is plotted in Fig. 2 as a function of the coupling g. This is equivalent to the results of Rychkov and Vitale (RV) [2], where g = 6g RV /π andL = L RV /(2π). We also plot the subtracted spectrum for equal-time quantization in Fig. 3, where the energy E 0 of the even vacuum state is subtracted from the energy of all other states. Again, the results are equivalent to RV. In particular, the lowest odd state becomes degenerate with the even vacuum state at and beyond a critical value of the coupling.
With the equal-time results established, we next consider the variation with c, approaching the light-front limit at c = 0. Figure 4 shows how the difference between the even and odd vacuum states varies with g for various values of c. For weak coupling the difference increases as c approaches zero; however, the critical coupling, where the difference becomes zero, remains essentially the same. Thus, the 0 < c < 1 results are at least qualitatively consistent with equal-time quantization, despite that fact that light-front quantization (c = 0) is known to give a different result for the critical coupling [8,11].
To investigate the distinction between the c → 0 limit and a c = 0 computation, we plot the even vacuum state energy as a function of 1/c for various values of the coupling g in Fig. 5. As can be seen in the figure, the spectrum diverges as c → 0. This can be understood [28] by considering the simplest contribution to the vacuum energy, from the 'basketball' graph in Fig. 6. The zero-mode contribution of this graph to the vacuum energy   pure light-front calculation.

IV. SUMMARY
We have shown that, for φ 4 theory, the c = 0 light-front limit of the coordinates (1.1) is not equivalent to a native light-front calculation. The energy of the vacuum state is unbounded from below, and the value of the critical coupling remains consistent with the equal-time calculation. Thus, unlike the case of the shifted free scalar, the light-front limit of φ 4 theory does not immediately provide a nontrivial light-front vacuum expectation value. The difference is primarily due to the way zero modes contribute; their contribution has a finite limit for the shifted scalar but an infinite limit for φ 4 theory.
This divergence means that one cannot include a nontrivial vacuum by grafting a c → 0 limit onto a light-front (c = 0) calculation for P + = P − = 0. Instead, one must consider P − = 0 for a sequence of finite c approaching zero and subtract the vacuum energy while taking the limit c → 0. An alternative is to construct an effective Hamiltonian in the lightfront limit [9,13], one for which the direct correspondence with the equal-time approach is maintained. The construction is based on the evaluation of expectation values of φ n with respect to the vacuum, something that should be readily calculable in the present formalism. We intend to pursue this construction.