L.V. Fil’kov, V.L. Kashevarov

Lebedev Physical Institute

Dipole and quadrupole polarizabilities of the pion

NSTAR 2007

1. Introduction

2.

3. p n

4.

5. A A

6. Discussion

7. Summary

NSTAR 2007

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

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)J. Ahrens et al., Eur. Phys. J. A 23, 113 (2005)

where t = (pp –pn )2 = -2mp Tn

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), P11(1440), D13(1520), S11(1535) resonances,

and σ-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

(α1 – β1)+= (11.6 ± 1.5st ± 3.0sys ± 0.5mod) 10-4 fm3

ChPT (Gasser et al. (2006)): (α1 –β1 (5.7±1.0) 10-4 fm3

→+ -

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

A→ A

t 10(GeV/c)2dominance of Coulomb bremsstrahlung t 10 Coulomb and nuclear contributions are of similar

size t 102dominance of nuclear bremsstrahlungSerpukhov (1983): Yu.M. Antipov et al., Phys.Lett. B121, 445(1983)

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

|t| < 6x104 (GeV/c)2

:

13.6 2.82.4

2 /E1

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)±

contribution to DSR