Isoscalar $\pi \pi$ Scattering and the $\sigma$ Meson Resonance from QCD

We present for the first time a determination of the energy dependence of the isoscalar $\pi \pi$ elastic scattering phase shift within a first-principles numerical lattice approach to QCD. Hadronic correlation functions are computed including all required quark propagation diagrams, and from these the discrete spectrum of states in the finite volume defined by the lattice boundary is extracted. From the volume dependence of the spectrum, we obtain the $S$-wave phase shift up to the $K\overline{K}$ threshold. Calculations are performed at two values of the $u$, $d$ quark mass corresponding to $m_{\pi }=236,391\mathrm{MeV}$, and the resulting amplitudes are described in terms of a $\sigma$ meson which evolves from a bound state below the $\pi \pi$ threshold at the heavier quark mass to a broad resonance at the lighter quark mass.

Figure 1

Schematic quark propagation diagrams which contribute to the isoscalar correlation functions required in this Letter.

Figure 2

Contributions of various diagrams [falling into the categories (a)–(d) presented in Fig. 1—two different variants each of (a) and (c) appear] to the correlation function having an operator ${\pi }_{[000]}{\pi }_{[110]}$ at both $t^{\prime }=0$ and $t$. The time dependence is weighted by $e^{E_{0}t}$ with $E_0$ the energy of the lightest state with $\overrightarrow{P}=[110]$. The complete correlation function, which corresponds to the sum of the pieces shown, is shown by the black squares. Computation on a $32^3\times 256$ lattice with $m_{\pi }=236\mathrm{MeV}$.

Figure 3

Center-of-momentum frame energies (black and gray points) extracted from variational analysis of correlation matrices at five values of $\overrightarrow{P}$ plotted versus $L$. Upper panel shows spectra for $m_{\pi }=391\mathrm{MeV}$ and the lower panel for $m_{\pi }=236\mathrm{MeV}$. The vacuum contribution to [000] correlation functions, which is time independent, and thermal contributions for other $\overrightarrow{P}$ are removed using the technique described in Ref. [18]. Red curves indicate noninteracting $\pi \pi$ energies. Small dashed red and green horizontal lines show the $\pi \pi$ and $K\overline{K}$ thresholds. Orange points show the finite-volume spectrum obtained from the $K$-matrix pole-plus-constant parametrization mentioned in the text.

Figure 4

Upper panel: $S$-wave $\pi \pi$ elastic scattering phase shift ${\delta }_0$ plotted against the scattering momentum $p^2=(E_{\mathrm{cm}}/2{)}^2-m_{\pi }^2$. Lower panel: Same data presented as $p\cot {\delta }_0$ (some points with large uncertainty have not been plotted). The colored curves are the result of a $K$-matrix pole-plus-constant with Chew-Mandelstam phase-space parametrization. The gray points show experimental data [1], and the gray curve shows the constrained dispersive description of these data presented in Ref. [3].

Figure 5

Upper panel: $t$-matrix pole positions for a variety of parametrizations: $K$-matrix pole-plus-polynomial forms (1a)–(1c) with and (2) without Chew-Mandelstam phase space and/or (3a),(3b) Adler zero [29], (4) relativistic Breit-Wigner, and (5) effective range expansion. See the Supplemental Material for a full description [28]. Green points: bound-state pole (physical sheet) at $m_{\pi }=391\mathrm{MeV}$. Red points: resonant pole (unphysical sheet) at $m_{\pi }=236\mathrm{MeV}$. Black point: Resonant pole from dispersive analysis of experimental data (conservative average presented in Ref. [5]). Lower panel: Coupling $g_{\sigma \pi \pi }$ from $t$-matrix residue at the pole—points from various parametrizations are shifted horizontally for clarity.