arXiv : Nonperturbative Matching Between Equal-Time and Lightcone Quantization

Fitzpatrick, A. Liam; Katz, Emanuel; Walters, Matthew T.

doi:10.1007/JHEP10(2020)092

Article
Report number	arXiv:1812.08177 ; CERN-TH-2018-277
Title	Nonperturbative Matching Between Equal-Time and Lightcone Quantization
Author(s)	Fitzpatrick, A. Liam (Boston U.) ; Katz, Emanuel (Boston U.) ; Walters, Matthew T. (CERN)
Publication	2020-10-14
Imprint	2018-12-19
Number of pages	22
Note	23+9 pages, 10 figures; references added, discussion of perturbative matching clarified
In:	JHEP 2010 (2020) 092
DOI	10.1007/JHEP10(2020)092
Subject category	hep-th ; Particle Physics - Theory
Abstract	We investigate the nonperturbative relation between lightcone (LC) and standard equal-time (ET) quantization in the context of $\lambda \phi^4$ 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.
Copyright/License	preprint: (License: arXiv nonexclusive-distrib 1.0) publication: © 2020 The Authors (License: CC-BY-4.0), sponsored by SCOAP³

${\it Left, top}: Plot of $\< \phi^2\>$ as a function of $\bar{\lambda}_{\rm ET}$. {\it Right, top}: Mass-squared gap $\bar{\mu}^2_{\rm ET}$ in ET quantization as a function of $\bar{\lambda}_{\rm ET}$. {\it Left, bottom:} $m^2_{\rm LC}$ as a function of $m^2_{\rm ET}$, setting $\lambda_{\rm ET}= \lambda_{\rm LC} =1$, according to eq. (\ref{eq:PertMatching}). {\it Right, bottom:} $\bar{\lambda}_{\rm LC}$ as a function of $\bar{\lambda}_{\rm ET}$, again according to eq. (\ref{eq:PertMatching}). The turnaround in the bottom 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 $\bar{\lambda}_{\rm LC}$.$ ${\it Left, top}: Plot of $\< \phi^2\>$ as a function of $\bar{\lambda}_{\rm ET}$. {\it Right, top}: Mass-squared gap $\bar{\mu}^2_{\rm ET}$ in ET quantization as a function of $\bar{\lambda}_{\rm ET}$. {\it 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}). {\it Right, middle:} $\bar{\lambda}_{\rm LC}$ as a function of $\bar{\lambda}_{\rm ET}$, again according to eq.~(\ref{eq:PertMatching}). {\it Bottom:} $\fr{\p\<\phi^2\>}{\p \bar{\lambda}_{\rm ET}}$ (solid black line) compared with the turnaround threshold $\fr{1}{12\bar{\lambda}_{\ET}^2}$ (dashed gray line). All five plots include results obtained using both renormalized ET Hamiltonian truncation~\cite{Rychkov:2014eea} (``HT'') and Borel resummation~\cite{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 $\bar{\lambda}_{\rm LC}$.$ Show more plots

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