ππscattering

TL;DR

The paper combines Roy equations with two-loop chiral perturbation theory to determine the ππ scattering amplitude in the low-energy region. By matching a subthreshold χPT representation to the phenomenological Roy representation, the authors predict S-wave threshold parameters a_0^0 = 0.220 ± 0.005 and a_0^2 = −0.0444 ± 0.0010, along with correlated quantities and phase shifts, while analyzing infrared singularities and quark-mass dependence. The analysis yields precise values for the effective Lagrangian couplings (e.g., ℓ_1, ℓ_2, ℓ_4, r_5, r_6) and supports the standard chiral picture with a dominant quark-condensate parameter, consistent with lattice and phenomenological inputs. They also provide accurate predictions for the ρ-meson properties and the π⁺π⁻ atom lifetime, linking low-energy theorems to experimental observables and lattice extrapolations.

Abstract

We demonstrate that, together with the available experimental information, chiral symmetry determines the low energy behaviour of the $ππ$ scattering amplitude to within very small uncertainties. In particular, the threshold parameters of the S-, P-, D- and F-waves are predicted, as well as the mass and width of the $ρ$ and of the broad bump in the S-wave. The implications for the coupling constants that occur in the effective Lagrangian beyond leading order and also show up in other processes, are discussed. Also, we analyze the dependence of various observables on the mass of the two lightest quarks in some detail, in view of the extrapolations required to reach the small physical masses on the lattice. The analysis relies on the standard hypothesis, according to which the quark condensate is the leading order parameter of the spontaneously broken symmetry. Our results provide the basis for an experimental test of this hypothesis, in particular in the framework of the ongoing DIRAC experiment: The prediction for the lifetime of the ground state of a $π^+π^-$ atom reads $τ=(2.90\pm 0.10)10^{-15} sec$.

Figures (11)

• Figure 1: Constraints imposed on the $S$--wave scattering lengths by chiral symmetry. The three full circles illustrate the convergence of the chiral perturbation series at threshold, according to eq. (12.2). The one at the left corresponds to Weinberg's leading order formulae. The error ellipse represents our final result. The other elements of the figure are specified in the text.
• Figure 2: $S$--wave scattering lengths as functions of $\ell_3$.
• Figure 3: Phase relevant for the decay $K\rightarrow \pi\pi e\nu$. The three bands correspond to the three indicated values of the $S$--wave scattering length $a_0^0$. The uncertainties are dominated by those from the experimental input used in the Roy equations. The triangles are the data points of Rosselet et al. rosselet, while the full circles represent the preliminary E865 results Truol.
• Figure 4: Values of the coupling constants $\ell_1$ and $\ell_2$. The shaded ellipse shows
• Figure 5: Result for $b_3$ and $b_4$. The errors in our result are dominated by those