Nonperturbative Matching Between Equal-Time and Lightcone Quantization

TL;DR

This work establishes a nonperturbative bridge between equal-time and lightcone quantization for two-dimensional lambda phi^4 theory by extracting a map between ET and LC bare couplings via Borel resummation of the mass gap. The approach bypasses the breakdown of naive perturbative matching, using the fully resummed mass gap computed from ET data to define LC parameters and then testing the resulting map with Hamiltonian truncation data in both quantizations. The results show strong agreement for the mass gap and the Z residue across ET and LC under the mapping, and reveal the distinct critical-point scaling in LC, which the mapping captures. These findings support the view that ET and LC quantizations describe the same theory nonperturbatively, and they provide a practical procedure to relate observables across quantizations in nonperturbative regimes.

Abstract

We investigate the nonperturbative relation between lightcone (LC) and standard equal-time (ET) quantization in the context of $λφ^4$ theory in $d=2$. We discuss the perturbative matching between bare parameters and the failure of its naive nonperturbative extension. We argue that they are nevertheless the same theory nonperturbatively, and that furthermore the nonperturbative map between bare parameters can be extracted from ET perturbation theory via Borel resummation of the mass gap. We test this map by using it to compare physical quantities computed using numerical Hamiltonian truncation methods in ET and LC.

Figures (10)

• Figure 1: General structure of "plant" diagrams.
• Figure 2: Left, top: Plot of $\langle \phi^2\rangle$ as a function of $\bar{\lambda}_{\rm ET}$. Right, top: Mass-squared gap $\bar{\mu}^2_{\rm ET}$ in ET quantization as a function of $\bar{\lambda}_{\rm ET}$. Left, middle:$m^2_{\rm LC}$ as a function of $m^2_{\rm ET}$, setting $\lambda_{\rm ET}= \lambda_{\rm LC} =1$, according to eq. (\ref{['eq:PertMatching']}). Right, middle:$\bar{\lambda}_{\rm LC}$ as a function of $\bar{\lambda}_{\rm ET}$, again according to eq. (\ref{['eq:PertMatching']}). Bottom:$\frac{\partial\langle\phi^2\rangle}{\partial \bar{\lambda}_{\rm ET}}$ (solid black line) compared with the turnaround threshold $\frac{1}{12\bar{\lambda}_{\textrm{ET}}^2}$ (dashed gray line). All five plots include results obtained using both renormalized ET Hamiltonian truncation Rychkov:2014eea ("HT") and Borel resummation Serone ("Borel"). The close agreement between the HT and Borel methods is evidence of their accuracy. The turnaround in the middle two plots indicate that the literal interpretation of (\ref{['eq:PertMatching']}) would incorrectly imply that the map from $\bar{\lambda}_{\rm ET}$ to $\bar{\lambda}_{\rm LC}$ is not invertible; two different values of $\bar{\lambda}_{\rm ET}$ would correspond to the same $\bar{\lambda}_{\rm LC}$.
• Figure 3: Outline of the procedure for extracting the map between bare couplings in LC and ET from the ET perturbation series.
• Figure 4: Plots of estimates of the gap from 6th, 7th, and 8th order, for (blue, dot-dashed), (red, dotted), and (black, solid), respectively. The upper and lower lines are the upper and lower values from moving $b,s$ away from their "best-fit" values as described in the text. Additional errors due to the change from one order in perturbation theory to the next can be read off by comparing the different lines. We also show in purple, dashed, a plot of the Taylor series truncated at $\lambda^8$.
• Figure 5: Left: Gap $\bar{\mu}^2$ as a function of $\bar{\lambda}_{\rm ET}$, from Borel resumming its perturbation series at eighth order. Center: Gap $\bar{\mu}^2$ as a function of $\bar{\lambda}_{\rm LC}$ from Borel resumming its perturbation series, also at eighth. Right: Inferred map $\bar{\lambda}_{\rm LC}(\bar{\lambda}_{\rm ET})$ from imposing $\bar{\mu}^2_{\rm LC}(\bar{\lambda}_{\rm LC})/\bar{\lambda}_{\rm LC}=\bar{\mu}^2_{\rm ET}(\bar{\lambda}_{\rm ET})/\bar{\lambda}_{\rm ET}$. In the left and center plot, errors (barely visible) are calculated as in Fig. \ref{['fig:BorelGap']}.