1. L.V. Fil’kov, V.L. Kashevarov Lebedev Physical Institute Dipole and quadrupole polarizabilities of the pion NSTAR 2007

2. 1. Introduction 2.     3.  p    n 4.     5.     A   A 6. Discussion 7. Summary NSTAR 2007

3. The dipole (  ,   ) and quadrupole (  ,   ) pion polarizabilities are defined through the expansion of the non-Born helicity amplitudes of the Compton scattering on the pion over t at s=   s=(q 1 +k 1 ) 2, u=(q 1 –k 2 )2, t=(k 2 –k 1 ) 2 M ++ (s=μ 2,t  2(α 1 - β 1 ) + 1/6(α 2 - β 2 )t ] + O(t 2 ) M +- (s=μ 2,t  2(α 1 + β 1 ) + 1/6(α 2 +β 2 )t] + O(t 2 ) (α 1, β 1 and α 2, β 2 in units 10 -4 fm 3 and 10 -4 fm 5, respectively)

4.  →  0  0 L. Fil’kov, V. Kashevarov, Eur. Phys. J. A5, 285 (1999); Phys. Rev. C72, 035211 (2005)

5. s-channel: ρ(770), ω(782), φ(1020); t-channel: σ, f 0 (980), f 0 (1370), f 2 (1270), f 2 (1525) Free parameters: m σ, Γ σ, Γ σ→ , (α 1 -β 1 ), (α 1 +β 1 ), (α 2 -β 2 ), (α 2 +β 2 ) The σ-meson parameters were determined from the fit to the experimental data on the total cross section in the energy region 270 - 825 MeV. As a result we have found: m σ =(547± 45) MeV, Γ σ =(1204±362) MeV, Γ σ→  =(0.62±0.19) keV  0 meson polarizabilities have been determined in the energy region 270 - 2250 MeV. A repeated iteration procedure was used to obtain stable results.

6. The total cross section of the reaction  →  0  0 H.Marsiske et al., Phys.Rev.D 41, 3324 (1990) J.K.Bienlein, 9-th Intern. Workshop on Photon-Photon Collisions, La Jolla (1992) our best fit

7.  0 meson polarizabilities [1] L.Fil’kov, V. Kashevarov, Eur.Phys.J. A 5, 285 (1999) [2] L. Fil’kov, V. Kashevarov, Phys.Rev. C 72, 035211 (2005) [3] J. Gasser et al., Nucl.Phys. B728, 31 (2005) [4] A. Kaloshin et al., Z.Phys. C 64, 689 (1994) [5] A. Kaloshin et al., Phys.Atom.Nucl. 57, 2207 (1994) Two-loop ChPT calculations predict a positive value of (α 2 +β    , in contrast to experimental result. One expects substantial correction to it from three-loop calculations.

8.  + p →  +  + + n (MAMI) J. Ahrens et al., Eur. Phys. J. A 23, 113 (2005)

9. where t = (p p –p n ) 2 = -2m p T n The cross section of  p→   + n has been calculated in the framework of two different models: I.Contribution of all pion and nucleon pole diagrams. II.Contribution of pion and nucleon pole diagrams and  (1232), P 11 (1440), D 13 (1520), S 11 (1535) resonances, and σ-meson.

10. To decrease the model dependence we limited ourselves to kinematical regions where the difference between model-1 and model-2 does not exceed 3% when (α 1 – β 1    =0. I. The kinematical region where the contribution of (α 1 – β 1 )  + is small: 1.5  2 < s 1 < 5  2 Model-1 Model-2 Fit of the experimental data The small difference between the theoretical curves and the experimental data was used for a normalization of the experimental data.

11. II. The kinematical region where the (α 1 – β 1 )  + contribution is substantial:    < s 1 < 15  2, -12  2 < t < -2  2 (α 1 – β 1 )  + = (11.6 ± 1.5 st ± 3.0 sys ± 0.5 mod ) 10 -4 fm 3 ChPT (Gasser et al. (2006)) : (α 1 –β 1    (5.7±1.0) 10 -4 fm 3

12.  →+ - →+ - L.V. Fil’kov, V.L. Kashevarov, Phys. Rev. C 73, 035210 (2006) Old analyses: energy region 280 - 700 MeV (α 1 -β 1 )  ± = 4.4 - 52.6 Our analysis: energy region 280 - 2500 MeV, DRs at fixed t with one subtraction at s=  2, DRs with two subtraction for the subtraction functions, subtraction constants were defined through the pion polarizabilities. s-channel: ρ(770), b 1 (1235), a 1 (1260), a 2 (1320) t-channel: σ, f 0 (980), f 0 (1370), f 2 (1270), f 2 (1525) Free parameters: (α 1 -β 1 )  ±, (α 1 +β 1 )  ±, (α 2 -β 2 )  ±, (α 2 +β 2 )  ±

13. Charged pion polarizabilities [1] L. Fil’kov, V. Kashevarov, Phys. Rev. C 72, 035211 ( 2005). [2] J. Gasser et all., Nucl. Phys. B 745, 84 (2006).

14. Total cross section of the process  →     our best fit Born contribution calculations with α 1 and β 1 from ChPT fit with α 1 and β 1 from ChPT

15. Angular distributions of the differential cross sections Mark II – 90 CELLO - 92 ╬ VENUS - 95 Calculations using our fit |cos  *| d  /d(|cos  *|<0.6) (nb)      : Bürgi-97,     : our fit    ,      Gasser-06

16.    A  →    A t  10    (GeV/c) 2   dominance of Coulomb bremsstrahlung t  10    Coulomb and nuclear contributions are of similar size t  10  2  dominance of nuclear bremsstrahlung Serpukhov (1983): Yu.M. Antipov et al., Phys.Lett. B121, 445(1983) E 1 =40 GeV Be, C, Al, Fe, Cu, Pb |t| < 6x10  4 (GeV/c) 2      :      13.6  2.8  2.4  2 /E 1

17. Charged pion dipole polarizabilities

18. Dispersion sum rules for the pion polarizabilities

19. The DSR predictions for the charged pions polarizabilities in units 10 -4 fm 3 for dipole and 10 -4 fm 5 quadrupole polarizabilities. The DSR predictions for the    meson polarizabilities

20. Contribution of vector mesons ChPT DSR

21. Discussion 1.(α 1 - β 1 )  ± The σ meson gives a big contribution to DSR for (α 1 –β 1 ). However, it was not taken into account in the ChPT calculations. Different contributions of vector mesons to DSR and ChPT. 2. one-loop two-loops experiment (α 2 -β 2 )  ± = 11.9 16.2 [21.6] 25 +0.8-0.3 The LECs at order p 6 are not well known. The two-loop contribution is very big (~100%). 3.(α 1, 2 +β 1, 2 )  ± Calculations at order p 6 determine only the leading order term in the chiral expansion. Contributions at order p 8 could be essential.

22. Summary 1.The values of the dipole and quadrupole polarizabilities of  0 have been found from the analysis of the data on the process  →  0  0. 2.The values of (α 1 ± β 1 )  0 and (α 2 –β 2 )  0 do not conflict within the errors with the ChPT prediction. 3. Two-loop ChPT calculations have given opposite sign for (α 2 +β 2 )  0. 4. The value of (α 1 –β 1 )  ± =13.0+2.6-1.9 found from the analysis of the data on the process  →  +  - is consisted with results obtained at MAMI (2005) (  p→  + n), Serpukhov (1983)   Z →   Z), and Lebedev Phys. Inst. (1984) (  p→  + n). 5. However, all these results are at variance with the ChPT predictions. One of the reasons of such a deviation could be neglect of the σ- meson contribution in the ChPT calculations. 6. The values of the quadrupole polarizabilities (α 2 ±β 2 )  ± disagree with the present two-loop ChPT calculations. 7. All values of the polarizabilities found agree with the DSR predictions.

23.  and  contributions to  1  –  1   (  1  1 )  ± 

24.  contribution to DSR