1. Introduction 2.     3.  p    n 4.     5.     A   A 6. Discussion 7. Summary Bosen Workshop 2007 Review on Experimental and Theoretical Results on the Pion Polarizabilities L.V. Fil’kov Lebedev Physical Institute

Pion polarizabilities characterize the behavior of the pion in an external electromagnetic field.

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 fm 3 and fm 5, respectively)

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

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 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 MeV. A repeated iteration procedure was used to obtain stable results.

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

The total cross section of the reaction  →     at E  <850 MeV our fit

The sensitivity of the cross section calculations to different values of the quadrupole polarizabilities (α 2 –β 2 )  0 less by 5% (α 2 –β 2 )  0 bigger by 5% (α 2 +β 2 )  0 bigger by 5% (α 2 +β 2 )  0 less by 5%

 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, (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.

 + p →  +  + + n (MAMI)

where t = (p p –p n ) 2 = -2m p T n

The pion polarizabilities can extracted from the experimental data on radiative pion photoproduction either by extrapolating these data to the pion pole or by comparing the experimental cross section with prediction of different theoretical models. Extrapolation: 1. Data with small errors in a wide region of t, in particular, very close to t=0. 2. The pion pole amplitude alone is not gauge invariant. The sum of the pion and nucleon pole amplitudes does not vanish at t=0. The cross section of  p→   + n has been calculated in the framework of two different models:

Model-1: Contribution of all the pion and nucleon diagrams

Model-2: Contribution of the pion and nucleon pole diagrams and  (1232), P 11 (1440), D 13 (1520), S 11 (1535) resonances, and  meson

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.

II. The kinematical region where the (α 1 – β 1 )  + contribution is substantial:    < s 1 < 15  2, -12  2 < t < -2  2 (  1 -  1 )=0      model-1    1       model-2

(     )  = 11.6  1.5 st  3.0 syst  0.5 mod ChPT (Gasser et al. (2006)): (     )  = 5.7  1.0

 →+ - →+ - L.V. Fil’kov, V.L. Kashevarov, Phys. Rev. C 73, (2006) Old analyses: energy region MeV (α 1 -β 1 )  ± = Our analysis: energy region 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 )  ±

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

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

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

Fits of the experimental data on total cross section of the different collaborations separately

   A  →    A        cm    (alab.s.)

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      0:      5.6      0 Born +      5.6

Serpukhov (1983): Yu.M. Antipov et al., Phys. Lett. B121, 445 (1983) E 1 =40 GeV, Be, C, Al, Fe, Cu, Pb t min =6 x 10  8 t < 6 x 10  4 ( GeV/c ) 2 t x 10 3 Maximum at t=2 t min t=(2 – 4) x 10  3  estimation of the strong interactions

1. (  1  1 )=0  2 /E 1 (lab. syst.)    (  1  1 )  0         (Yu.M. Antipov et al., Z. Phys. C 26, 495 (1985) )

Charged pion dipole polarizabilities

Dispersion sum rules for the pion polarizabilities

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

Contribution of vector mesons ChPT DSR

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 )  ± = [21.6] 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.

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 )  The value of (α 1 –β 1 )  ± = 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.

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