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’kovLebedev 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=(q1+k1)2, u=(q1–k2)2, t=(k2–k1)2

M++(s=μ2,t 2(α1 - β1) + 1/6(α2 - β2)t ] + O(t2)

M+-(s=μ2,t 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)

→ 0 0

L. Fil’kov, V. Kashevarov, Eur. Phys. J. A5, 285 (1999); Phys. Rev. C72, 035211 (2005)

s-channel: ρ(770), ω(782), φ(1020);

t-channel: σ, f0(980), f0(1370), f2(1270), f2(1525)

Free parameters: mσ, Γσ, Γσ→,

(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→=(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.

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

(α2 –β2)0 less by 5%

(α2 –β2)0 bigger by 5%

(α2 +β2)0 bigger by 5%

(α2 +β2)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, 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+β,

in contrast to experimental result.One expects substantial correction to it from three-loop

calculations.

+ p → + + + n (MAMI)

where t = (pp –pn )2 = -2mp Tn

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), P11(1440), D13(1520), S11(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 (α1 – β1 =0.

I. The kinematical region where the contribution of (α1 – β1)+ is

small: 1.5 2 < s1 < 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 (α1 – β1)+ contribution

is substantial:

< s1 < 152, -122 < t < -22

(1- 1)=0

model-11

model-2

()= 11.6 1.5st3.0syst0.5mod

ChPT (Gasser et al. (2006)): ()= 5.7 1.0

→+ -

L.V. Fil’kov, V.L. Kashevarov, Phys. Rev. C 73, 035210 (2006)

Old analyses: energy region 280 - 700 MeV (α1-β1)± = 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 pionpolarizabilities.

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)±, (α1+β1)±, (α2-β2)±, (α2+β2)±

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).

Total cross section of the process →

our best fit

Born contribution

calculations with α1 and β1 from ChPT fit with α1 and β1 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)

(n

b)

: 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)2dominance of Coulomb bremsstrahlung t 10 Coulomb and nuclear contributions are of similar

size t 102dominance of nuclear bremsstrahlung

0:

5.60 Born + 5.6

Serpukhov (1983): Yu.M. Antipov et al., Phys. Lett. B121, 445 (1983)

E1=40 GeV, Be, C, Al, Fe, Cu, Pb

tmin=6x108

t < 6x104(GeV/c)2

t x 103

Maximum at t=2 tmin

t=(2 – 4)x103 estimation of the strong

interactions

1. (11)=0

2/E1 (lab. syst.)

1.2

2. (11)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 10-4 fm3 for dipole and 10-4 fm5 quadrupole polarizabilities.

The DSR predictions for the meson polarizabilities

Contribution of vector mesons

ChPT

DSR

Discussion

1. (α1 - β1)±

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-loop two-loops experiment

(α2-β2)± = 11.9 16.2 [21.6] 25 +0.8-0.3 The LECs at order p6 are not well known. The two-loop contribution is very big (~100%).

3. (α1,2+β1,2)±

Calculations at order p6 determine only the leading order term in the chiral expansion.

Contributions at order p8 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 (α1± β1)0 and (α2 –β2)0 do not conflict within the errors

with the ChPT prediction.

3. Two-loop ChPT calculations have given opposite sign for (α2+β2)0.

4. The value of (α1 –β1)± =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 (α2 ±β2 )± 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

(11)±