Phase of directCPviolation from threshold pion production

A recent series of experiments on pion production near threshold provides accurate values of the $s$-wave $\pi -\pi$ isospin scattering lengths. Combined with dispersion relations we find the $s$-wave phase shift difference at the $K$ mass is ${\delta }_0-{\delta }_2=42\mathrm{}\pm 4°$ and the phase of direct $\mathrm{CP}$ violation is $\arg {\epsilon }^{\prime }=\frac{\pi }{2}-({\delta }_0-{\delta }_2)=48\mathrm{}\pm 4°$. We also evaluate the $p$-wave scattering length to be $a_1=0.035\mathrm{}\pm 0.001m_{\pi }^{-3\mathrm{}}$.