[Paper Review] Pion form factors in two-flavor QCD

JLQCD Collaboration, S. Hashimoto+13 more|ArXiv.org|Oct 18, 2005

Quantum Chromodynamics and Particle Interactions6 references3 citations

TL;DR

This lattice QCD study computes pion electromagnetic and scalar form factors in two-flavor QCD using non-perturbatively $O(a)$-improved Wilson fermions on $20^3 \times 48$ lattices at $\beta=5.2$. It performs chiral extrapolation using one-loop chiral perturbation theory, finding a pion charge radius of $0.396(10)$ fm$^2$, 12% below the experimental value, with strong chiral logarithms evident in the data.

ABSTRACT

We present a calculation of pion electromagnetic and scalar form factors in two-flavor QCD with the non-perturbatively O(a)-improved Wilson fermion. Chiral extrapolation of the corresponding charge radius is discussed based on the chiral perturbation theory.

Motivation & Objective

• To compute pion electromagnetic and scalar form factors in two-flavor QCD using lattice QCD with $O(a)$-improved Wilson fermions.
• To investigate chiral logarithms in pion form factors by performing chiral extrapolation using one-loop chiral perturbation theory.
• To test the reliability of lattice QCD in reproducing chiral dynamics near the physical pion mass.
• To extract the pion charge radius and scalar radius with controlled systematic errors from high-statistics lattice data.

Proposed method

• Three-point correlation functions are computed with smeared and local interpolating fields to extract form factors via a double ratio method.
• The double ratio isolates the form factor ratio $G_J(q^2)/G_J(0)$ by canceling overlap factors, with time separations $t_J$ in the range [4,8] for plateau extraction.
• Two fit forms are used: a 'free pole' form with adjustable pole mass and a 'measured pole' form using the lattice-measured vector meson mass.
• Chiral extrapolation employs a one-loop ChPT formula with a logarithmic term and a quadratic analytic term: $\langle r^2 \rangle_V^\pi = C_0 - \frac{1}{(4\pi f)^2} \ln\frac{m_\pi^2}{\mu^2} + C_1 m_\pi^2$.
• For the scalar form factor, a polynomial fit is used due to the lack of clear resonance dominance, followed by chiral extrapolation with a similar ChPT form.
• High statistics (1,200 gauge configurations per sea quark mass) ensures precision in form factor and radius determination.

Experimental results

Research questions

• RQ1 Does the lattice QCD calculation reproduce the expected chiral logarithms in the pion electromagnetic form factor near the chiral limit?
• RQ2 What is the pion charge radius extrapolated to the physical point, and how does it compare to the experimental value of 0.452(10) fm²?
• RQ3 How do the form factors behave with varying pion mass, and do they show the predicted enhancement due to chiral logarithms?
• RQ4 Is the scalar radius sensitive to chiral logarithms, and can it be reliably extrapolated using ChPT?
• RQ5 Can the vector meson dominance model be consistently applied in lattice QCD with $O(a)$-improved fermions, and how does it compare to the measured vector meson mass?

Key findings

• The pion charge radius is extrapolated to $\langle r^2 \rangle_V^\pi = 0.396(10)$ fm², which is 12% below the experimental value of 0.452(10) fm².
• The chiral extrapolation shows a clear upward trend toward the chiral limit, consistent with the chiral logarithm prediction in ChPT.
• The 'measured pole' fit form yields significantly reduced statistical errors compared to the 'free pole' form, indicating better consistency with analyticity and lattice vector meson masses.
• The scalar radius is estimated at $0.60(15)$ fm², with a strong chiral logarithm effect, though systematic uncertainties from fit function choice remain a concern.
• The form factor data show a plateau in the double ratio for $t_J \gtrsim 4$, supporting reliable extraction of the form factor ratio.
• The results confirm that chiral logarithms significantly affect the pion charge radius, especially near the chiral limit, and that high-statistics lattice data are essential for resolving these effects.