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Abstract
Correctness of the results obtained in the recent analysis of the ππ interactions using new dispersion relations with imposed crossing symmetry condition is checked. The performed proof is based on purely mathematical relations and properties of analytic functions and concerns position of the scalar-isoscalar ƒ0(500) (former σ) pole. Mere analysis of amplitudes expressed by the trigonometric functions and their derivatives clearly defines the area in which mass of the σ and its width must be located.
Highlights
Its shape is completely given by crossing symmetry condition
Real parts of the "Old" S 0 input and output amplitude in the GKPY equations are presented on the Fig. 2
Comparison of the gradients of the imaginary and real parts seen on this figure shows that just the imaginary part changes faster than the real one. Taking into account those facts and the shape of the dominant part of the GKPY equations shown on Fig. 1, defined almost entirely by crossing symmetry condition, one can conclude that mentioned ae-mail: robert.kaminski@ifj.edu.pl
Summary
Results on the σ pole position presented in [1] have been obtained using dispersive analysis without any model assumptions about specific energy dependence of the ππ amplitudes in the S (S 0) and P waves [2]. To prove uniqueness of the new position of the σ pole one can use results of analysis presented in [3].
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