Two-dimensional light-front ${\varphi }^4$ theory in a symmetric polynomial basis

We study the lowest-mass eigenstates of ${\varphi }_{1+1}^4$ theory with both odd and even numbers of constituents. The calculation is carried out as a diagonalization of the light-front Hamiltonian in a Fock-space representation. In each Fock sector a fully symmetric polynomial basis is used to represent the Fock wave function. Convergence is investigated with respect to the number of basis polynomials in each sector and with respect to the number of sectors. The dependence of the spectrum on the coupling strength is used to estimate the critical coupling for the positive-mass-squared case. An apparent discrepancy with equal-time calculations of the critical coupling is resolved by an appropriate mass renormalization.

Figure 1

Lowest mass eigenvalue for odd numbers of constituents. Results are shown for different Fock-space truncations to three, five, and seven constituents. Also plotted are results for the leading light-front coupled-cluster (LFCC) approximation and for a sector-dependent modification of the three-body truncation. The errors are estimated from extrapolations in polynomial basis size.

Figure 2

Same as Fig. 1 but for even numbers of constituents, with Fock-space truncations at four, six, and eight.

Figure 3

Relative Fock-sector probabilities for the lowest mass eigenstate with odd numbers of constituents.

Figure 4

The lowest masses for the odd and even cases, as used to estimate the critical coupling, including a plot of the threshold for two-particle states at 4 times the mass squared of the odd case.

Figure 5

The renormalization shift $\mathrm{\Delta }$, defined in Eq. (3.14) of the text, as a function of the square of the dimensionless coupling $g$. The points displayed are obtained as extrapolations in the polynomial basis size. The lines are linear and quadratic fits to shifts below $g=1$, extrapolated to the region of the critical coupling.

Figure 6

Lowest equal-time mass eigenvalues for odd numbers of constituents plotted versus the dimensionless light-front coupling $g$. Each mass value is obtained from the light-front masses and the computed shift $\mathrm{\Delta }$ according to Eq. (3.16) of the text. Different points at the same $g$ value correspond to different truncations of the polynomial basis size.

Figure 7

Lowest mass for the odd case as a function of the reciprocal of the highest order $K$ of the polynomial included in the five-body Fock sector, for $g=0.5$, 1.0, and 2.0. In the three-body sector, the highest order was 15, and the Fock space was truncated at five constituents. The lines are linear fits extrapolated to infinite order.

Figure 8

Same as Fig. 7, but for the mass renormalization shift $\mathrm{\Delta }$ and $g$ values of 0.5, 1.0, and 1.5.