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### Dipole and quadrupole polarizabilities of the pion

L.V. Fil’kov, V.L. Kashevarov

Lebedev Physical Institute

NSTAR 2007

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

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)

g g→p0p0

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→ 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.

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

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.

g + p →g + 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 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.

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.

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

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±

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 gg→p+p-

our best fit

calculations with α1 and β1 from ChPT

Born contribution

fit with α1 and β1 from ChPT

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*|

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 102 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

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

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.

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.

pp and rr contributions to (a1– b1)

D(a1b1)p± - 1.88

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