[Paper Review] Spin Polarizabilities on the Lattice

Frank Lee, Andrei Alexandru|arXiv (Cornell University)|Nov 18, 2011

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper proposes a lattice QCD framework to isolate spin polarizabilities of spin-1/2 hadrons using an effective non-relativistic quantum mechanical action in Euclidean space. By engineering specific time- and space-dependent electromagnetic fields and analyzing double ratios of spin- and field-asymmetric correlators, the authors demonstrate a method to disentangle all eight polarizability parameters—including the challenging spin-dependent ones like $\gamma_{E1}, \gamma_{M1}, \gamma_{E2}, \gamma_{M2}$—and validate the approach via numerical tests and a preliminary lattice QCD study showing feasibility despite statistical noise challenges.

ABSTRACT

Spin polarizabilities provide information on the internal structure of hadrons in the presence of weak external electromagnetic fields, and are actively studied by Compton scattering experiments. They provide finer detail than the regular polarizabilities since they require space and time-varying fields. Using an effective action in the weak field limit, we have identified methods to isolate each of the physical quantities ($μ, α, β, γ_{E1}, γ_{M1}, γ_{E2}, γ_{M2}$) for spin-1/2 hadrons, both neutral and charged. We also perform a lattice QCD simulation to investigate the feasibility of the effective action approach.

Motivation & Objective

• To develop a systematic method for isolating all spin polarizabilities ($\mu, \alpha, \beta, \gamma_{E1}, \gamma_{M1}, \gamma_{E2}, \gamma_{M2}$) for spin-1/2 hadrons in lattice QCD.
• To address the challenge of extracting spin-dependent polarizabilities, which require time- and space-varying fields and are not accessible via standard polarizability measurements.
• To test the feasibility of the effective action approach using lattice simulations with controlled boundary conditions and field configurations.
• To enable precise determination of hadron structure in weak electromagnetic fields by matching effective theory correlators with lattice QCD data.

Proposed method

• Use an effective non-relativistic QM action in Euclidean space with bilinear fermion terms coupled to external electromagnetic fields.
• Discretize the effective action on a lattice with finite spacing $a$, using Dirichlet and periodic boundary conditions to control state spectra.
• Compute the two-point correlation function $G_{ss'}(t, \vec{p}, A_\mu)$ as the inverse of the matrix $K$ in the path integral, using a BiCGSTAB solver with $10^{-15}$ convergence.
• Engineer vector potentials $\vec{A}$ to isolate individual polarizabilities: e.g., $\vec{A} = (e_1 t^2/a, e_2 t, 0)$ for $\gamma_{E1}$, with field-dependent terms in the Lagrangian.
• Extract polarizabilities via double ratios of correlators (e.g., $R_{\gamma_{E2}} = [G_{11}(e_1,e_2)G_{11}(-e_1,-e_2)]/[G_{22}(e_1,e_2)G_{22}(-e_1,-e_2)]$) that isolate specific spin and field dependencies.
• Match effective theory correlators with lattice QCD data under identical boundary conditions and field configurations to extract physical values.

Experimental results

Research questions

• RQ1 Can all spin polarizabilities of spin-1/2 hadrons be isolated using a controlled effective field theory approach on the lattice?
• RQ2 How can the field and spin dependence in the effective action be exploited to disentangle $\gamma_{E1}, \gamma_{M1}, \gamma_{E2}, \gamma_{M2}$ from each other and from regular polarizabilities?
• RQ3 What role do boundary conditions and field configurations play in isolating specific polarizabilities in lattice simulations?
• RQ4 Is the effective action approach feasible for lattice QCD when Monte Carlo noise is present in the correlators?
• RQ5 Can the double-ratio method reliably extract spin polarizabilities from noisy lattice data, as demonstrated in a preliminary QCD simulation?

Key findings

• The effective QM correlator successfully isolates all eight polarizability parameters ($\mu, \alpha, \beta, \gamma_{E1}, \gamma_{M1}, \gamma_{E2}, \gamma_{M2}$) through tailored field configurations and correlator ratios.
• The double ratio $R_{\gamma_{E2}} \sim \exp(1/2 \gamma_{E2} e_1 e_2 t / a)$ isolates $\gamma_{E2}$ by exploiting symmetry under $ (e_1,e_2) \to (-e_1,-e_2) $, applicable to both charged and neutral hadrons.
• For $\gamma_{M2}$, a similar double ratio with reversed fields yields $\sim \exp(-1/2 \gamma_{M2} e_1 e_2 t / a)$, confirming the method's generality.
• The combination $2\gamma_{M1} - \gamma_{E2}$ is isolated via a field configuration, and $\gamma_{M1}$ can be extracted once $\gamma_{E2}$ is known from prior analysis.
• A preliminary lattice QCD study with $24^3 \times 48$ at $a = 0.093$ fm shows that $\gamma_{E1}$ can be probed via $\sin[\gamma_{E1} e_1 e_2 t / a]$ in the ratio $\text{Im}[G_{11}/G_{22}]$, though statistical errors are large.
• The study concludes that noise reduction by at least a factor of 100 is required to extract $\gamma_{E1}$ reliably, highlighting the main challenge for future lattice QCD applications.