Critical coupling for two-dimensional ${\ensuremath{\phi}}^{4}$ theory in discretized light-cone quantization

Open Access

Critical coupling for two-dimensional $ϕ^{4}$ theory in discretized light-cone quantization

James P. Vary, Mengyao Huang, Shreeram Jawadekar, Mamoon Sharaf, Avaroth Harindranath, and Dipankar Chakrabarti

Phys. Rev. D 105, 016020 – Published 25 January 2022

Abstract

We solve for the critical coupling in the symmetric phase of two-dimensional $ϕ^{4}$ field theory using discretized light-cone quantization. We adopt periodic boundary conditions, neglect the zero mode, and obtain a critical coupling consistent with the critical coupling reported using conformal truncation in light-front quantization. We find a 17% difference from the critical coupling reported with light-front quantization in a symmetric polynomial basis.

Received 7 October 2021
Accepted 20 December 2021

DOI:https://doi.org/10.1103/PhysRevD.105.016020

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum field theory

Particles & Fields

Authors & Affiliations

James P. Vary ¹, Mengyao Huang ¹, Shreeram Jawadekar ¹, Mamoon Sharaf¹, Avaroth Harindranath ², and Dipankar Chakrabarti ³

¹Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
²Guruvayur, Kerala 680101, India
³Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 105, Iss. 1 — 1 January 2022

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Critical couplings $λ_{c} / μ^{2}$ for vanishing mass gaps as a function of $1 / \sqrt{K}$ obtained with DLCQ using periodic boundary conditions. Solid (open) symbols represent DLCQ results for 3 states, according to the legend, that are used (not used) in the fits. The smooth curves are results for $f_{1}$ (blue), $f_{2}$ (red) and $f_{3}$ (green) resulting from procedures using polynomial fits as discussed in the text. The extrapolated value is the intercept on the vertical axis of $22.64 \pm 0.17$ . The inset provides an expanded view of the region approaching the continuum limit. A vertical black bar represents our approximate systematic uncertainty which dominates our overall uncertainty (see text). The inset also presents, for comparison, a vertical purple bar (magenta bar) on the vertical axis representing the critical coupling result of Ref. [13] ([14]) which is $26.39 \pm 0.63$ ( $23.12 \pm 0.38$ ).
Reuse & Permissions
Figure 2
Ratio of critical coupling for the vanishing mass gaps of the lowest state at $K$ of the even sector and the lowest state of the odd sector at $K / 2$ versus $1 / K$ . Symbols represent the results from DLCQ and the smooth curve is a fourth degree polynomial in $z^{2} = 1 / K$ fit to the 29 results represented by the solid symbols. The coefficients of the fit $B_{j}$ [see Eq. (14)] are presented in Table 2. The vertical dashed line signifies the boundary between regions where finite basis artifacts are strong (to the right for lower $K$ ) yet tending to cancel in this ratio as observed by the smooth behavior over all $K$ presented.
Reuse & Permissions
Figure 3
Ratio of critical coupling for the vanishing mass gaps of the first excited state at $K$ of the odd sector to the lowest state of the odd sector at $K / 3$ versus $1 / K$ . Symbols represent the results from DLCQ and the smooth curve is a fourth degree polynomial in $z^{2} = 1 / K$ fit to the 19 results represented by the solid symbols. The coefficients of the fit $C_{j}$ [see Eq. (17)] are presented in Table 2. The vertical dashed line signifies the boundary between regions where finite basis artifacts are strong (to the right for lower $K$ ) yet tending to cancel in this ratio as observed by the smooth behavior over all $K$ presented.
Reuse & Permissions
Figure 4
Calculated $M^{2} / μ^{2}$ eigenvalues (solid symbols) for the first excited state in the odd sector for couplings that span the vanishing mass gap at $K = 75$ . The solid blue squares correspond to the full calculation while the solid red diamonds correspond to results in a reduced space that omits configurations with more than 15 bosons. Results are presented for increments of 0.2 in the coupling except for one point in the full calculation at 40.0, just above the critical coupling, where 7500 Lanczos iterations were insufficient for producing a converged result. The thick blue line (thin red line) corresponds to a second degree polynomial fit used to interpolate for the vanishing mass gap. The open symbols present 1000 times the difference between the corresponding solid symbol and fit. The inset provides an expanded view of the two quadratic fits as they cross the vanishing mass gap. As expected, the coupling must be larger for the truncated calculation relative to the full calculation to produce a vanishing mass gap.
Reuse & Permissions
Figure 5
Residuals, plotted as functions of $X = (K, K / 2, K / 3)$ for $(f_{1}, f_{2}, f_{3})$ respectively, between functions based on polynomial fits and DLCQ mass-squared eigenvalues. That is, these are the residuals of the results presented in Fig. 1. The residuals are scaled, as indicated by the scaling factors in the legend, to generate visible results on a single vertical scale. Filled (open) symbols correspond to DLCQ results employed (not employed) in the fits. Connecting straight line segments are included to help guide the eye. The horizontal axis $X$ is defined for each function according to the points from $f_{1}$ employed in its evaluation (see text for details). The inset exhibits details of the residuals in the lower-X region on an expanded scale.
Reuse & Permissions
Figure 6
Residuals, plotted as functions of $(K / 2, K / 3)$ for $(R_{2 / 1}, R_{3 / 1})$ respectively, between polynomial fits and calculated ratios of mass-squared eigenvalues. The residuals are scaled, as indicated by the scaling factors in the legend. The ratio $R_{2 / 1}$ ( $R_{3 / 1}$ ) is defined in the text by Eq. (13) [Eq. (16)] and displayed in Fig. 2 (Fig. 3). Filled (open) symbols correspond to DLCQ results employed (not employed) in the fits. Connecting straight line segments are included to help guide the eye. The horizontal axis $X$ is defined for each function according to the points from $f_{1}$ employed in its evaluation (see text for details).
Reuse & Permissions

Physical Review D

covering particles, fields, gravitation, and cosmology

Critical coupling for two-dimensional $ϕ^{4}$ theory in discretized light-cone quantization

James P. Vary, Mengyao Huang, Shreeram Jawadekar, Mamoon Sharaf, Avaroth Harindranath, and Dipankar Chakrabarti

Phys. Rev. D 105, 016020 – Published 25 January 2022

Abstract

Physics Subject Headings (PhySH)

Authors & Affiliations

Article Text

References

Issue

Physical Review D

covering particles, fields, gravitation, and cosmology

Critical coupling for two-dimensional ϕ4 theory in discretized light-cone quantization

James P. Vary, Mengyao Huang, Shreeram Jawadekar, Mamoon Sharaf, Avaroth Harindranath, and Dipankar Chakrabarti

Phys. Rev. D 105, 016020 – Published 25 January 2022

Abstract

Physics Subject Headings (PhySH)

Authors & Affiliations

Article Text

References

Issue

Authorization Required

Other Options

Download & Share

Images

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Reuse & Permissions

Log In

Search

Article Lookup

Paste a citation or DOI

Enter a citation

Critical coupling for two-dimensional $ϕ^{4}$ theory in discretized light-cone quantization