Dispersive analysis of the $S$-, $P$-, $D$-, and $F$-wave $\pi\pi$   amplitudes
  A reanalysis of $\pi\pi$ amplitudes for all important partial-waves below about 2 GeV is presented. A set of once subtracted dispersion relations with imposed crossing symmetry condition is used to modify unitary multi-channel amplitudes in the $S$, $P$, $D$, and $F$ waves. So far, these specific amplitudes constructed in our works and many other analyzes have been fitted only to experimental data and therefore do not fulfill the crossing symmetry condition. In the present analysis, the self consistent, i.e. unitary and fulfilling the crossing symmetry, amplitudes for the $S$, $P$, $D$, and $F$ waves are formed. The proposed very effective and simple method of modification of the $\pi\pi$ amplitudes does not change their previous-original mathematical structure and the method can be easily applied in various other analyzes. 
P. Byd\v{z}ovsk\'y, R. Kami\'nski, and V. Nazari
hep-ph nucl-th