[Paper Review] A new moving frame to extract scattering phases in lattice QCD

Xu Feng, Karl Jansen+1 more|arXiv (Cornell University)|Apr 1, 2011

Cold Atom Physics and Bose-Einstein Condensates4 references3 citations

TL;DR

This paper derives finite-size formulae for a new moving frame in lattice QCD with total momentum $\mathbf{P} = (2\pi/L)(\mathbf{e}_1 + \mathbf{e}_2)$, enabling precise extraction of S-wave and P-wave scattering phases at multiple energies on a single lattice size. The method extends Lüscher's formalism to this novel frame, allowing improved resonance parameter determination with reduced computational cost.

ABSTRACT

We present a derivation of the finite-size formulae in a moving frame with total momentum P=(2π/L)(e1+e2). These formulae allow us to calculate the S-wave and P-wave scattering phases at more energies with a fixed lattice size and thus help us to determine the resonance parameters precisely.

Motivation & Objective

• To extend the moving frame formalism in lattice QCD to a new frame with total momentum $\mathbf{P} = (2\pi/L)(\mathbf{e}_1 + \mathbf{e}_2)$.
• To derive finite-size formulae that relate energy eigenvalues in this frame to scattering phases in infinite volume.
• To enable higher-resolution determination of scattering phases in the resonance region without increasing lattice volume.
• To support precise extraction of resonance parameters such as mass and width from lattice data.

Proposed method

• Derives the singular $\mathbf{d}$-periodic solutions of the Helmholtz equation for the new moving frame with $\mathbf{d} = \mathbf{e}_1 + \mathbf{e}_2$.
• Uses Lorentz transformation to relate the wave function in the moving frame to the center-of-mass frame.
• Expands the wave function in terms of spherical harmonics and spherical Bessel functions, incorporating symmetry-adapted basis functions.
• Constructs the $\mathcal{M}^{\mathbf{d}}_{lm,l'm'}(p)$ matrix elements using the modified zeta function $\mathcal{Z}^{\mathbf{d}}_{lm}(q^2)$.
• Derives the finite-size formulae by equating two expansions of the wave function and enforcing consistency via determinant conditions.
• Expresses the scattering phase via $\tan^{-1}\delta_l(p) = M^{\mathbf{d}}_{ln,l'n'}(\Gamma)$ for irreducible representations $A^+$, $B_1^-$, $B_2^-$, $B_3^-$.

Experimental results

Research questions

• RQ1 Can finite-size formulae be derived for a new moving frame with total momentum $\mathbf{P} = (2\pi/L)(\mathbf{e}_1 + \mathbf{e}_2)$ in lattice QCD?
• RQ2 How do the scattering phases for S-wave and P-wave channels depend on the energy eigenvalues in this new frame?
• RQ3 What is the structure of the $\mathcal{M}^{\mathbf{d}}_{lm,l'm'}(p)$ matrix for this frame, and which matrix elements are non-zero?
• RQ4 Can this formalism improve the resolution of scattering phase determination in the resonance region?

Key findings

• The finite-size formulae for the new moving frame with $\mathbf{P} = (2\pi/L)(\mathbf{e}_1 + \mathbf{e}_2)$ are derived, enabling extraction of scattering phases from lattice energy eigenvalues.
• The $\mathcal{M}^{\mathbf{d}}_{lm,l'm'}(p)$ matrix elements are explicitly computed for $l,l' = 0,1$, with non-zero entries listed in Table 1.
• For the S-wave channel, $\tan^{-1}\delta_0(p) = M^{\mathbf{d}}_{01,01}(A^+)$, with $M^{\mathbf{d}}_{01,01}(A^+) = (\gamma\pi^{3/2}q)^{-1}\mathcal{Z}^{\mathbf{d}}_{00}$.
• For the P-wave channels, $\tan^{-1}\delta_1(p) = M^{\mathbf{d}}_{11,11}(\Gamma)$, with explicit expressions involving $\mathcal{Z}^{\mathbf{d}}_{00}$, $\mathcal{Z}^{\mathbf{d}}_{20}$, and $\mathcal{Z}^{\mathbf{d}}_{22} \pm \mathcal{Z}^{\mathbf{d}}_{2\bar{2}}$ for $\Gamma = B_1^-, B_2^-, B_3^-$.
• The derived formulae allow determination of scattering phases at multiple energies on a single lattice size, improving resonance parameter resolution.