Toward extracting scattering phase shift from integrated correlation functions V: complex $φ^4$ field model in $3+1$ dimensions

TL;DR

The paper develops a 3+1 dimensional generalization of the relativistic integrated correlation function (ICF) approach to extract infinite-volume scattering phase shifts from finite-volume data, demonstrated with a complex φ^4 lattice theory. It defines A1^+ projected two-particle integrated correlators and relates their finite-volume difference to the infinite-volume S-wave phase shift through a weighted integral governed by finite-volume Green's functions, including a Euclidean form suitable for lattice computations. Monte Carlo tests in 3+1 and 2+1 dimensions validate the method: non-interacting tests reproduce analytic results, while interacting cases yield estimates of the renormalized coupling V_R(0) and the phase shift δ(E), with convergence toward the infinite-volume limit as the volume grows. The work supports the practical viability of the ICF framework for higher-dimensional relativistic field theories and points toward future applications to lattice QCD multi-hadron systems, while highlighting the need to address renormalization in more complex theories.

Abstract

In Ref.~\cite{Guo:2024zal} and associated studies, a relativistic finite-volume formalism in $1+1$ dimensions is proposed to extract infinite-volume scattering phaseshift. It is based on the difference of integrated correlation functions (ICF) rather than energy spectrum in the finite volume, and can be regarded as complementary to the well-known L\"{uscher} formalism. In the present work, the formalism is further extended into $3+1$ dimensional spacetime. The aim is to explore and demonstrate the challenges in applying the formalism to more practical settings. Specifically, Monte Carlo simulations of a complex $φ^4$ relativistic field model are carried out in both 2+1 and 3+1 dimensions on lattices of varying sizes, and phaseshifts for the contact interaction are extracted from the formalism using modest computing resources.

Figures (14)

• Figure 1: Non-interacting single-particle effective mass from Monte Carlo data of $m^{(\phi)}_{eff}$ in Eq.\ref{['emass']} at the lowest three momenta $\mathbf{ p}=2\pi \mathbf{n}/L$ with $\mathbf{n}= (0,0,0)$ (black), $(1,0,0)$ (blue), $(1,1,0)$ (purple), and $(1,1,1)$ (green). The fit results (red bands) are also indicated.
• Figure 2: Lattice dispersion relation defined in Eq.(\ref{['omegafiniteaandL']}) is compared with the simulation results measured in Fig. \ref{['onemassL10plot']}.
• Figure 3: Comparison of lattice results for non-interacting single-particle integrated correlation function against the exact expression in Eq.(\ref{['integratedCphit']}). The parameters are: $T=60$, $L=10$, $\kappa=0.0618$ and $\lambda=0$.
• Figure 4: Similar to Fig. \ref{['onefreeL10diffCtplot']}, but for the free two-particle integrated correlation function in Eq.(\ref{['integratedCtwophit']}).
• Figure 5: Interacting single-particle effective mass plot: Monte Carlo data (colored points with error bars) vs. fit results (red bands). The color coding for momentum $\mathbf{ p}=\frac{2\pi}{L } \mathbf{n}$ with $\mathbf{n}=(0,0,0)$ (black), $(1,0,0)$ (blue), $(1,1,0)$ (purple), and $(1,1,1)$ (brown). The parameters are: $T=60$, $L=28$, $\kappa=0.0666$ and $\lambda=0.03$.