Dipole and quadrupole polarizabilities of the pion

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

2. 1. Introduction 2. g g  p0 p0 3.g p  g p+n 4. g g  p+ p- 5. p-A  g p- A 6. Discussion 7. Summary NSTAR 2007

3. The dipole (a1, b1) and quadrupole (a2, b2) pion polarizabilities are defined through the expansion of the non-Born helicity amplitudes of the Compton scattering on the pion over t at s=m2 s=(q1+k1)2, u=(q1–k2)2, t=(k2–k1)2 M++(s=μ2,t)=pm[ 2(α1 - β1) + 1/6(α2 - β2)t ] + O(t2) M+-(s=μ2,t)=p/m[ 2(α1 + β1) + 1/6(α2+β2)t] + O(t2) (α1, β1 and α2, β2 in units 10-4 fm3 and 10-4 fm5, respectively)

4. g g→p0p0 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, f0(980), f0(1370), f2(1270), f2(1525) Free parameters: mσ, Γσ, Γσ→ gg, (α1-β1), (α1+β1), (α2-β2), (α2+β2) 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: mσ=(547± 45) MeV, Γσ=(1204±362) MeV, Γσ→ gg=(0.62±0.19) keV p0 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 gg→p0p0 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. p0 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 (α2+β2)p0, in contrast to experimental result. One expects substantial correction to it from three-loop calculations.

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

9. where t = (pp –pn )2 = -2mp Tn The cross section of g p→ gp+ n has been calculated in the framework of two different models: • Contribution of all pion and nucleon pole diagrams. • Contribution of pion and nucleon pole diagrams and • D(1232), P11(1440), D13(1520), S11(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 (α1 – β1)p+ =0. I. The kinematical region where the contribution of (α1 – β1)p+ is small: 1.5 m2 < s1 < 5 m2 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 (α1 – β1)p+ contribution is substantial: 5m2 < s1 < 15m2, -12m2 < t < -2m2 (α1 – β1)p+= (11.6 ± 1.5st ± 3.0sys ± 0.5mod) 10-4 fm3 ChPT (Gasser et al. (2006)): (α1 –β1)p+ = (5.7±1.0) 10-4 fm3

12. gg→p+p- L.V. Fil’kov, V.L. Kashevarov, Phys. Rev. C 73, 035210 (2006) Old analyses: energy region 280 - 700 MeV (α1-β1)p±= 4.4 - 52.6 Our analysis: energy region 280 - 2500 MeV, DRs at fixed t with one subtraction at s=m2, DRs with two subtraction for the subtraction functions, subtraction constants were defined through the pion polarizabilities. s-channel: ρ(770), b1(1235), a1(1260), a2(1320) t-channel: σ, f0(980), f0(1370), f2(1270), f2(1525) Free parameters: (α1-β1)p±, (α1+β1)p±, (α2-β2)p±, (α2+β2)p±

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 gg→p+p- our best fit calculations with α1 and β1 from ChPT Born contribution fit with α1 and β1 from ChPT

15. Angular distributions of the differential cross sections Mark II – 90 CELLO - 92 VENUS - 95 ╬ ds/d(|cosQ*|<0.6) (nb) Calculations using our fit a1, b1: Bürgi-97, a2, b2 : our fit a1, b1, a2, b2: Gasser-06 |cosQ*|

16. p- A→ p- g A t  10-4(GeV/c)2 dominance of Coulomb bremsstrahlung t  10-3 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) E1=40 GeV Be, C, Al, Fe, Cu, Pb w = w2/E1 |t| < 6x10-4 (GeV/c)2 (a1 + b1)=0: (a1 - b1)= 13.6  2.8  2.4

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 fm3 for dipole and 10-4 fm5 quadrupole polarizabilities. The DSR predictions for the p0 meson polarizabilities

20. Contribution of vector mesons DSR ChPT

21. Discussion • (α1 - β1)p± The σ meson gives a big contribution to DSR for (α1 –β1). However, it was not taken into account in the ChPT calculations. Different contributions of vector mesons to DSR and ChPT. 2. one-looptwo-loopsexperiment (α2-β2)p± = 11.9 16.2 [21.6] 25 +0.8-0.3 The LECs at order p6are not well known. The two-loop contribution is very big (~100%). • (α1,2+β1,2)p± Calculations at order p6 determine only the leading order term in the chiral expansion. Contributions at order p8could be essential.

22. Summary • The values of the dipole and quadrupole polarizabilities of p0 have been found from the analysis of the data on the process gg→p0p0. • The values of (α1± β1)p0 and (α2 –β2)p0 do not conflict within the errors with the ChPT prediction. 3.Two-loop ChPT calculations have given opposite sign for (α2+β2)p0. 4. The value of (α1 –β1)p± =13.0+2.6-1.9 found from the analysis of the data on the process gg→ p+p - is consisted with results obtained at MAMI (2005) (g p→ g p+ n), Serpukhov (1983) (p-Z → g p-Z), and Lebedev Phys. Inst. (1984) (g p→ g 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 (α2 ±β2 )p± disagree with the present two-loop ChPT calculations. 7. All values of the polarizabilities found agree with the DSR predictions.

23. pp and rr contributions to (a1– b1) D(a1b1)p± - 1.88

24. rr contribution to DSR