Towards the time-like pion form factor beyond the elastic regime using domain-wall QCD

Abstract

In this work, we investigate the time-like pion form factor from lattice QCD in the isosymmetric limit, a quantity that plays an important role in understanding hadron physics with substantial phenomenological applications. This observable can be calculated in the elastic region using the finite-volume approach, up to the first (four-particle) open channel. With the goal of accessing the exclusive two-pion form factor in the inelastic region, starting from a three-point correlator involving the vector current and two (temporally-displaced) pion interpolating operators, we examine the associated underlying spectral density and calculate the form factor using a formalism based on the LSZ reduction. A preliminary analysis on one ensemble generated by the RBC/UKQCD collaboration using domain-wall fermions is presented.

Figures (4)

• Figure 1: Left: three-point function diagram. The vector current is inserted at time-slice $\tau$ and the pion operators are temporally displaced. Right: sketch of the extraction of the correlator $G_k(\tau-t; \vb{p})$ by looking at the limits $t,\tau \to \infty$, once the operator dependence on $\mathcal{O}^\dag_{\pi,2}(0)$ is removed.
• Figure 2: Two point function $G_k(\tau-t; \vb{p})$ (summed over $k$) for $\abs{\vb{p}}^2=1$ (in units of $(2 \pi/L)^2$) as a function of $\tau-t$ for different values of $\tau$. The extraction is performed according to Eq. \ref{['eq:3pt_lim_t']}, where the factor $Z_{\pi,2}^{1/2}(\vb{p})$ and the single pion energy $E_\pi(\vb{p})$ are fitted from a two-point pseudoscalar correlator. The results are blinded and the error bars are purely statistical. The points on the $x$-axis are slightly shifted for better visualization.
• Figure 3: GEVP spectrum from the 7 operators described in the main text. The calculation of the energies has been carried out by fixing $t-t_0 = 4/a$ and by projecting the correlation matrix at each time slice onto the six eigenvectors of the one at $t_0/a = 4$, as described in Ref. Bruno:2025mig. The error bars are purely statistical. Points are horizontally displaced for better readability.
• Figure 4: Reconstruction of the finite-volume spectral density $\rho_{k,\vb{p}}$ according to Eq. \ref{['eq:rho_def']} from the matrix elements $\mel{0}{V_k(0)}{n,\vb{0}}$ and energies $E_n(\vb{0})$ determined in the GEVP of Fig. \ref{['fig:GEVP']}. The remaining $\mel{n, \vb{0}}{\mathcal{O}_{\pi,1}(0)}{\pi, \vb{p}}$ have been calculated by means of a constrained fit on $G_k$. The two different colors correspond to the two different choices of $\Gamma_5$ in the operator $\mathcal{O}_{\pi,1}$, see the legend in the bottom left corner. The error bars are purely statistical. Top: reconstruction of the imaginary part. Bottom: reconstruction of the real part.