[Paper Review] Pion mass difference from vacuum polarization

JLQCD Collaboration, Eigo Shintani+6 more|ArXiv.org|Oct 3, 2007

Quantum Chromodynamics and Particle Interactions4 references4 citations

TL;DR

This paper presents the first lattice QCD calculation of the pion mass difference Δm²_π using the Das-Guralnik-Mathur-Low-Young (DGMLY) sum rule, employing two-flavor dynamical overlap fermions on a 16³×32 lattice at a ≈ 0.12 fm. The exact chiral symmetry of overlap fermions enables precise control of the VV−AA vacuum polarization, yielding Δm²_π = 1024(100) MeV², in reasonable agreement with the experimental value of 1261.2 MeV².

ABSTRACT

We calculate the electromagnetic contribution to the pion mass difference, $Δm^2_π=m^2_{π^+}-m^2_{π^0}$, in the chiral limit through the $VV-AA$ type vacuum polarization using Das-Guralnik-Mathur-Low-Young (DGMLY) sum rule. The calculation is made with two-flavors of dynamical overlap fermions on a $16^3 imes 32$ lattice at $a\sim$0.12 fm. The exact chiral symmetry of the overlap fermion is essential to control the systematic error in the difference $VV-AA$. We obtain $Δm_π^2 = 1024(100) { m MeV^2}$ combining the lattice data with the perturbative contribution in the high momentum region evaluated by the operator product expansion. By analyzing the momentum dependence of the vacuum polarization, we also obtain pion decay constant $f_π$ and the low-energy constants $L_{10}^r$ in the chiral limit.

Motivation & Objective

• To calculate the electromagnetic contribution to the pion mass difference Δm²_π = m²_π⁺ − m²_π⁰ in the chiral limit using the DGMLY sum rule.
• To overcome systematic errors in VV−AA vacuum polarization by employing overlap fermions with exact chiral symmetry.
• To extract the pion decay constant f_π and low-energy constant L₁₀ʳ from the momentum dependence of the vacuum polarization.
• To combine non-perturbative lattice data with perturbative OPE results in the high-momentum region for a complete evaluation.

Proposed method

• Apply the DGMLY sum rule to express Δm²_π as a momentum integral of Q²Π_V−A(Q²), the difference of vector and axial-vector vacuum polarization functions.
• Use two-flavor dynamical overlap fermions on a 16³×32 lattice with a ≈ 0.12 fm spacing to preserve exact chiral symmetry and control VV−AA matrix elements.
• Perform chiral extrapolation of lattice data for Q²Π_V−A(Q²) to the chiral limit using a fit function incorporating chiral perturbation theory and OPE behavior.
• Split the momentum integral at Λ² = 2.0604 (aQ)², using the fit function below and OPE at one-loop order (including ⟨αsO₈⟩) above.
• Use the perturbative OPE expression for Π_V−A(Q²) at high Q² with μ₀ = 2 GeV and ⟨¯ψψ⟩ = −(251 MeV)³ to model the UV contribution.
• Extract f_π and L₁₀ʳ by fitting the Q² dependence of Q²Π_V−A(Q²) to CHPT predictions in the chiral limit.

Experimental results

Research questions

• RQ1 Can the pion mass difference Δm²_π be accurately computed in the chiral limit using the DGMLY sum rule with lattice QCD?
• RQ2 How does the use of overlap fermions improve the control of systematic errors in the VV−AA vacuum polarization compared to Wilson or domain-wall fermions?
• RQ3 What is the value of f_π and L₁₀ʳ extracted from the momentum dependence of the vacuum polarization in the chiral limit?
• RQ4 How well does the combined lattice+OPE approach reproduce the experimental value of Δm²_π?
• RQ5 What is the impact of finite-volume and topological sector effects on the final result?

Key findings

• The calculation yields Δm²_π = 1024(100) MeV², combining statistical error and OPE uncertainty, which is consistent with the experimental value of 1261.2 MeV².
• The pion decay constant is extracted as f_π = 107(15) MeV, consistent with previous hadron spectrum studies using the same configurations.
• The low-energy constant L₁₀ʳ is determined from the fit to the Q² dependence of Q²Π_V−A(Q²), with the result being compatible with QCD sum rule estimates.
• The fit function Q²Π_V−A^fit(Q²) successfully describes the lattice data up to (aQ)² = 2.0604, incorporating chiral behavior, resonance poles, and logarithmic terms.
• The OPE contribution at high momentum is evaluated using the one-loop expression with μ₀ = 2 GeV and ⟨¯ψψ⟩ = −(251 MeV)³, contributing +48 MeV² to the final result.
• The result suggests feasibility for extracting non-perturbative matrix elements like ⟨O₈⟩ from lattice data, as the OPE behavior is well reproduced.