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L.V. Fil’kov, V.L. Kashevarov
Lebedev Physical Institute
Dipole and quadrupole polarizabilities of the pion
NSTAR 2007
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1. Introduction
2.
3. p n
4.
5. A A
6. Discussion
7. Summary
NSTAR 2007
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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)
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→ 0 0
L. Fil’kov, V. Kashevarov, Eur. Phys. J. A5, 285 (1999); Phys. Rev. C72, 035211 (2005)
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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.
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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
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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.
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+ p → + + + n (MAMI)J. Ahrens et al., Eur. Phys. J. A 23, 113 (2005)
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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.
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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.
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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
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→+ -
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)±
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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).
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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
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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
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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
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Charged pion dipole polarizabilities
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Dispersion sum rules for the pion polarizabilities
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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
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Contribution of vector mesons
ChPT
DSR
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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.
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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.
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and contributions to 1–1
(11)±
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contribution to DSR