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José Antonio Oller Universidad de Murcia. Spain.

Scalar radius of the pion. Introduction. Dispersion relation. Omnès representation. Watson’s final state theorem. Ynduráin’s approach. Extended method. Results. Conclusions I. . Introduction. Dispersive Approach. . Conclusions II.

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José Antonio Oller Universidad de Murcia. Spain.

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  1. Scalar radius of the pion. Introduction. • Dispersion relation. Omnès representation. Watson’s final state theorem. • Ynduráin’s approach. • Extended method. • Results. Conclusions I. • . Introduction. • Dispersive Approach. • . • Conclusions II. José Antonio Oller Universidad de Murcia. Spain. In collaboration with L. Roca (Murcia), C. Schat (Murcia; CONICET and U.Buenos Aires, Argentina) TexPoint fonts used in EMF: AAAAAAAAAAAAAAAAAAAA

  2. 1. Introducion The non-strange I=0 pion scalar form factor:

  3. ² E l a s t i c O m n µ e s r e p r e s e n t a t i o n a n d n u m e r i c a l - C H P T t o t w o l o o p s G a s s e r , M e i s s n e r N P B 3 5 7 , 9 0 ( 1 9 9 1 ) ² T w o - l o o p C H P T , B i j n e n s , C o l a n g e l o a n d T a l a v e r a J H E P 9 8 0 5 , 0 1 4 ( 1 9 9 8 ) ² O n e l o o p U C H P T , M e i ¼ n e r , J A O , N P A 6 7 9 , 6 7 1 ( 2 0 0 1 ) ; M e i ¼ n e r , L a h d e , P R D 7 4 , 0 3 4 0 2 1 ( 2 0 0 6 ) ² Y n d u r ¶ a i n ' s a p p r o a c h b a s e d o n t h e O m n µ e s r e p r e s e n t a t i o n o f F ( t ) Y n d u r a ¶ i n , P L B 5 7 8 , 9 9 ( 2 0 0 4 ) ; ( E ) - i b i d B 5 8 6 , 4 3 9 ( 2 0 0 4 ) ( Y 1 ) Y n d u r ¶ a i n , P L B 6 1 2 , 2 4 5 ( 2 0 0 5 ) ( Y 2 ) Y n d u r ¶ a i n , a r X i v : h e p - p h / 0 5 1 0 3 1 7 ( Y 3 ) 2 ¼ h i r = 0 s Yall 2 2 ¼ 2 ± h i ± : 7 5 0 : 0 7 f m \ r o b u s t " l o w e r b o u n d : r = 0 : 7 0 0 : 0 6 f m s Consequences in the scattering lengths of CGL

  4. 2. Disperson Relations The pion non-strange scalar form factor

  5. π t Hard gluon 1/t π For the scalar form factor F(z) vanishes as 1/z because of QCD. (Brodsky-Farrar counting rules).

  6. One must first remove the zeroes (also the poles for the general case, not in the present one) of F(t) and consider the function

  7. Corolary:

  8. 3. Ynduráin’s method Weak point of the argument Not always compatible

  9. From F.J.Ynduráin, PLB578,99(2004)

  10. This approach was critized by Ananthanarayan, Caprini, Leutwyler, IJMP A21,954 (2006) (ACGL).

  11. 4. Extended Y’s method L. Roca and J.A.O. Phys. Lett. B651,139(2007) arXiv:0704.0039 [hep-ph]

  12. 360 degrees 180

  13. 5. Numerical Analysis

  14. Explodes at around 1 GeV

  15. No axial vector exchanges

  16. 6. Multipion states Extremely small

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