Relativistic three-particle quantization condition for nondegenerate scalars

Three-particle finite-volume formalism for $π^+π^+ K^+$ and related systems

We consider three-particle systems consisting of two identical particles and a third that is different, with all being spinless. Examples include $π^+π^+ K^+$ and $K^+K^+π^+$. We derive the formalism necessary to extract two- and three-particle infinite-volume scattering amplitudes from the spectrum of such systems in finite volume. We use a relativistic formalism based on an all-orders diagrammatic analysis in generic effective field theory, adopting the methodology used recently to study the case of three nondegenerate particles. We present both a direct derivation, and also a cross-check ba…

Progress on the three-particle quantization condition

We report progress on extending the relativistic model-independent quantization condition for three particles, derived previously by two of us, to a broader class of theories, as well as progress on checking the formalism. In particular, we discuss the extension to include the possibility of 2->3 and 3->2 transitions and the calculation of the finite-volume energy shift of an Efimov-like three-particle bound state. The latter agrees with the results obtained previously using non-relativistic quantum mechanics.

A relativistic, model-independent, three-particle quantization condition

We present a generalization of Luescher's relation between the finite-volume spectrum and scattering amplitudes to the case of three particles. We consider a relativistic scalar field theory in which the couplings are arbitrary aside from a Z2 symmetry that removes vertices with an odd number of particles. The theory is assumed to have two-particle phase shifts that are bounded by π/2 in the regime of elastic scattering. We determine the spectrum of the finite-volume theory from the poles in the odd-particle-number finite-volume correlator, which we analyze to all orders in perturbation theory…

Can stochastic quantization evade the sign problem? -- the relativistic Bose gas at finite chemical potential

A nonperturbative study of field theories with a complex action, such as QCD at finite baryon density, is difficult due to the sign problem. We show that the relativistic Bose gas at finite chemical potential has a sign and `Silver Blaze' problem, similar to QCD. We then apply stochastic quantization and complex Langevin dynamics to study this theory with nonperturbative lattice simulations. Independence of chemical potential at small and a transition to a condensed phase at large chemical potential are found. Lattices of size N^4, with N=4,6,8,10, are used. We show that the sign problem is se…

Three-body scattering and quantization conditions from $S$ matrix unitarity

Two methodologies have been presented in the literature which connect relativistic three-particle scattering amplitudes with lattice QCD spectra -- the ``relativistic effective field theory'' approach and the ``finite-volume unitarity'' method. While both methods have been shown to be equivalent in various works, it has not been shown how to arrive at the relativistic effective field theory results directly from $S$ matrix unitarity. In this work, we provide a simple proof of the relativistic effective field theory form of the scattering equations directly from unitarity. Motivated by the fini…

Numerical study of the relativistic three-body quantization condition in the isotropic approximation

We present numerical results showing how our recently proposed relativistic three-particle quantization condition can be used in practice. Using the isotropic (generalized $s$-wave) approximation, and keeping only the leading terms in the effective range expansion, we show how the quantization condition can be solved numerically in a straightforward manner. In addition, we show how the integral equations that relate the intermediate three-particle infinite-volume scattering quantity, $\mathcal K_{\text{df},3}$, to the physical scattering amplitude can be solved at and below threshold. We test …

Relating the finite-volume spectrum and the two-and-three-particle $S$ matrix for relativistic systems of identical scalar particles

Working in relativistic quantum field theory, we derive the quantization condition satisfied by coupled two- and three-particle systems of identical scalar particles confined to a cubic spatial volume with periodicity $L$. This gives the relation between the finite-volume spectrum and the infinite-volume $\textbf 2 \to \textbf 2$, $\textbf 2 \to \textbf 3$ and $\textbf 3 \to \textbf 3$ scattering amplitudes for such theories. The result holds for relativistic systems composed of scalar particles with nonzero mass $m$, whose center of mass energy lies below the four-particle threshold, and for …

Generalizing the relativistic quantization condition to include all three-pion isospin channels

We present a generalization of the relativistic, finite-volume, three-particle quantization condition for non-identical pions in isosymmetric QCD. The resulting formalism allows one to use discrete finite-volume energies, determined using lattice QCD, to constrain scattering amplitudes for all possible values of two- and three-pion isospin. As for the case of identical pions considered previously, the result splits into two steps: The first defines a non-perturbative function with roots equal to the allowed energies, $E_n(L)$, in a given cubic volume with side-length $L$. This function depends…

Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states

In this work, we use an extension of the quantization condition, given in Ref. [1], to numerically explore the finite-volume spectrum of three relativistic particles, in the case that two-particle subsets are either resonant or bound. The original form of the relativistic three-particle quantization condition was derived under a technical assumption on the two-particle K matrix that required the absence of two-particle bound states or narrow two-particle resonances. Here we describe how this restriction can be lifted in a simple way using the freedom in the definition of the K-matrix-like quan…

标度仿射 φ₄⁴ 量子欧几里得协变相对论场论中格林函数的连续极限

Continuum limit of the Green function in scaled affine $φ^4_4$ quantum Euclidean covariant relativistic field theory

本文通过路径积分蒙特卡罗实验证明，标度仿射量子化方法使 φ₄⁴ 欧几里得协变相对论标量场论成为有效量子场论，其单点和两点函数具有良好定义的连续极限，并揭示了仅出现在量子层面的 ħ²/φ² 项。

Relativistic, model-independent, three-particle quantization condition

This is a combined write-up for two talks which were given consecutively and which described different aspects of the same topic. We present a generalization of Lüscher's relation between the finite-volume spectrum and S-matrix to three particles. Specifically, we consider a scalar field theory, which has a $\mathbb{Z}_2$ symmetry that prevents even/odd coupling. The theory is assumed to have no two-particle bound states and to have two-particle phase shifts that are bounded by $π/2$ in the regime of elastic scattering. Considering center of mass energies between one and five particle masses, …

Implementing the three-particle quantization condition including higher partial waves

We present an implementation of the relativistic three-particle quantization condition including both $s$- and $d$-wave two-particle channels. For this, we develop a systematic expansion about threshold of the three-particle divergence-free K matrix, $\mathcal{K}_{\mathrm{df,3}}$, which is a generalization of the effective range expansion of the two-particle K matrix, $\mathcal{K}_2$. Relativistic invariance plays an important role in this expansion. We find that $d$-wave two-particle channels enter first at quadratic order. We explain how to implement the resulting multichannel quantization c…