Physics From Breit-Frame Regularization Of a Lattice Hamiltonian

TL;DR

This paper introduces a Hamiltonian lattice formulation using Breit-frame regularization, enabling efficient computation of physical observables in quantum field theories like scalar $ ext{phi}^4$ and QCD, with promising results for mass spectra and parton distributions.

Contribution

It proposes a novel Breit-frame based Hamiltonian regularization scheme that reduces degrees of freedom and facilitates calculations of spectra, structure functions, and thermodynamics in lattice quantum field theories.

Findings

Observed scaling behavior in the scalar $ ext{phi}^4$ theory mass spectrum.

Calculated parton distribution functions with QCD-like behavior.

Compared results with renormalization group predictions.

Abstract

We suggest a Hamiltonian formulation on a momentum lattice using a physically motivated regularization using the Breit-frame which links the maximal parton number to the lattice size. This scheme restricts parton momenta to positive values in each spatial direction. This leads to a drastic reduction of degrees of freedom compared to a regularization in the rest frame (center at zero momentum). We discuss the computation of physical observables like (i) mass spectrum in the critical region, (ii) structure and distribution functions, (iii) $S$-matrix, (iv) finite temperature and finite density thermodynamics in the Breit-frame regularization. For the scalar $\phi^{4}_{3+1}$ theory we present numerical results for the mass spectrum in the critical region. We observe scaling behavior for the mass of the ground state and for some higher lying states. We compare our results with renormalization group results by L\"uscher and Weisz. Using the Breit-frame, we calculate for $QCD$ the relation between the $W^{\mu \nu}$ tensor, structure functions (polarized and unpolarized) and quark distribution functions. We use the improved parton-model with a scale dependence and take into account a non-zero parton mass. In the Bjorken limes we find the standard relations between $F_{1}$, $F_{2}$, $g_{1}$ and the quark distribution functions. We discuss the r\^ole of helicity. We present numerical results for parton distribution functions in the scalar model. For the $\phi^{4}$-model we find no bound state with internal parton structure. For the $\phi^{3}$-model we find a distribution function with parton structure similar to Altarelli-Parisi behavior of $QCD$.