Isotensor $πππ$ scattering with a $ρ$ resonant subsystem from QCD

Abstract

This work presents a lattice quantum chromodynamics (QCD) determination of $πππ$ scattering amplitudes for the isospin-2 channel with angular momentum and parity $J^{P} = 1^+$. The calculation is performed using unphysically heavy light-quark masses, corresponding to a pion mass of $m_π \approx 400$~MeV, for which the $ρ$ meson manifests as a narrow resonance. The analysis employs a previously developed formalism that non-perturbatively relates the finite-volume spectra of two- and three-pion systems to their infinite-volume scattering amplitudes. We combine earlier lattice QCD results for $ππ$ systems with isospins 1 and 2 with new determinations of the isospin-2 three-pion finite-volume spectrum, obtained for three cubic, periodic lattice volumes with box length ranging from $\approx 4/m_π$ to $6/m_π$. These combined data constrain the low-lying partial waves of the infinite-volume three-pion K matrix, from which we solve a set of coupled integral equations to extract the corresponding scattering amplitudes.

Figures (10)

• Figure 1: Energies in the isospin-2 three-pion channel plotted as a function of the box size $L$. The black points represent lattice QCD-computed energies; inner error bars indicate statistical uncertainties, while outer error bars also include systematic uncertainties as discussed in Sec. \ref{['app:ops_spec']} of the Supplemental Material. The red curves are predictions of the finite-volume quantization condition with a vanishing three-body K-matrix $\mathcal{K}_{3} = 0$, using two-pion K-matrices from a previous analysis of the two-pion spectrum, while the blue curves show results incorporating a best-fit $\mathcal{K}_{3}$ as detailed in the text. The grey lines correspond to the hypothetical spectrum of non-interacting hadrons, as explained in the text. The first excited grey curve has a two-fold multiplicity.
• Figure 2: Representative diagrams contained in the non-perturbative scattering amplitude, where the black lines represent pions. We show the two leading contributions: (a) the three-body K matrix constrained in this work, and (b) the one-pion exchange diagram responsible for additional kinematic singularities in the amplitude. Kinematic quantities and angular momenta are defined in the text.
• Figure 3: (a) The light blue bands show fit results for a given parametrization of the reduced K matrix as a function of total energy $E$ in the channels $\{{}^3S_1,{}^3D_1,{}^1P_1\} \to {}^3S_1$. Bands represent $1\sigma$ uncertainty of a given fit, and the hue of the band is directly proportional to the likelihood of the fit, as given by the AIC described in the text, with the darker bands being preferred. The red is the weighted average of the different fits, with the band including the propagated uncertainty. (b) Shown are the $\{{}^3S_1,{}^3D_1,{}^1P_1\} \to {}^3S_1$ unsymmetrized amplitudes as a function of the incoming dipion invariant mass for a fixed $E=3.4 m_\pi$ and outgoing state defined by $M_{\pi\pi}' = 2.1 m_\pi$. The red band includes the error propagation of the K matrices shown in (a). The green curve corresponds to amplitudes for $\mathcal{K}_3=0$. The cusp near $2.1m_\pi$ is due to physical OPE.
• Figure 4: Dalitz plots of the intensity distributions in Eq \ref{['eq:int']} for the $J^P=1^+$ amplitude for $E/m_\pi = \{3.3,3.4,3.5\}$. The axes represent the invariant masses corresponding to two different choices for the dipion within a given state. Initial and final state kinematics are fixed to be the same. Amplitudes are determined from the best-fitted values of the K matrix described in the text. The insets show slices of the distribution for fixed $(M_{\pi\pi},E) = (2.1m_\pi,3.4m_\pi)$ and $(2.3m_\pi,3.5m_\pi)$.
• Figure 5: The four lowest eigenvalues on the $L/a_s = 20$ volume, scaled by $e^{E_n(t-t_0)}$, where $t_0 = 10$ was used in this case. On each plot, the curves show the result of a two-exponential fit to the highlighted time region as described in the text, and the fitted energy $E_n$ and goodness of fit are also indicated.