Form factors of the $ρ$ meson from effective field theory and the lattice

Abstract

The calculation of resonance form factors in effective field theory as well as on the lattice is a highly challenging task. In a recent paper, we proposed a novel method based on the introduction of a background field and the Feynman-Hellmann theorem to address the problem, and applied it to a toy model. In the present work we use this method for the electromagnetic form factors of the $ρ$-meson. By matching the results to Chiral Perturbation Theory, we provide a first, crude estimate of all three form factors of the $ρ$-meson within the effective field theory. Contact contributions to these form factors turn out to be substantial. A procedure for lattice calculations is outlined, paving the way for an ab initio approach to the problem.

Figures (4)

• Figure 1: Contributions to the rho-meson form factor in NREFT: (a) The so-called triangle diagram, where the external photon is hooked on to the charged pion, and (b) the external photon emitted from the local five-point vertex. Both types of diagrams are dressed by an infinite number of two-pion bubbles describing the initial- and the final-state interactions.
• Figure 2: Invariant form factors $G_1(k^2),G_2(k^2),G_3(k^2)$ up to $-k^2\le (1\,\hbox{GeV})^2$. Two different phenomenological parameterizations of the $\pi\pi$ amplitude are used. The shaded areas correspond to the crude estimate of the low-energy couplings $g_1,g_2,g_3$ at the next-to-leading order in ChPT. For better visibility of the bands, the inset on the third plot shows an enlarged region for $0.1\,\hbox{GeV}^2\leq -k^2\leq 1\,\hbox{GeV}^2$.
• Figure 3: The real and imaginary parts of the function $\epsilon_i(k^2)$ (a relative error resulting from the use of two different phenomenological parameterizations), which is given in Eq. (\ref{['eq:epsilon']}). As seen, in a rather large interval, the magnitude of this function amounts up to a few percent.
• Figure 4: The P-wave $\pi\pi$ phase shift in the channel with total isospin $I=1$, the parameterizations from Ref. Heuser:2024biq and Oller:1998hw. Experimental data are taken from Refs. Protopopescu:1973shEstabrooks:1974vu.