1. The correction of hadronic nucleus polarizability to hyperfine structure of light muonic atoms
A.V. Eskin (Samara University)
In collaboration with A.P. Martynenko
QFTHEP

2. PURPOSE AND TASKS
The purpose of this work is to investigate hadronic nucleus polarizability contribution to the hyperfine structure of the energy spectrum of light muonic atoms (muonic deuterium, muonic tritium, muonic helium-3).
Hadronic deuteron polarizability is determined by numerous nuclear reactions of π-, η-meson production on proton or neutron and the excitation of nucleon resonances.
Tasks:
The calculation of contribution due to hadronic polarizability of the nucleus to hfs of muonic hydrogen, deuterium, helium, and tritium contains several steps:
1. The calculation of virtual photoabsorption cross sections on nucleons using the MAID program in the resonance region;
2. The construction of the structure functions of polarized lepton-nucleon scattering ;
3. The calculation of the resonance and nonresonance contributions to the correction of the nucleus polarizability; 2

3. TWO PHOTON AMPLITUDES OF LEPTON-NUCLEON SCATTERING
Inelastic γ*+p scattering
Elastic γ*+p scattering b) N N N N
The amplitude of the elastic scattering is determined by the electromagnetic form factors of the proton (deuteron).
We take into account the following resonances N* in MAID:
P33(1232), P11(1440), S11(1650), F35(1905), P31(1910), D15(1675), D13(1520), F15(1680), F37(1950) , S11(1535), D33(1700), S31(1620), P13(1720).
D. Drechsel,  S.S. Kamalov, L. Tiator „Unitary isobar model - MAID2007”
/Eur. Phys. J., Vol. A34, 2007, Pp 3

4. AMPLITUDE OF COMPTON SCATTERING
The polarized structure functions g1(ν,Q2) and g2(ν,Q2) enter in the antisymmetric part of hadronic tensor Wμν , describing lepton–nucleon deep-inelastic scattering:
where εμναβ is the totally antisymmetric tensor in four dimensions, g1(ν,Q2) = m22νG1(ν,Q2), g2(ν,Q2) = m2ν2G2(ν,Q2), P is the four-momentum of the nucleon, x = Q2/2m2ν is the Bjorken variable, S is the nucleon spin four-vector, normalized to S2 = −1, q2 = −Q2 is the square of the four-momentum transfer. The invariant quantity P · q is related to the energy transfer ν in the nucleon rest frame: P · q = m2ν. The invariant mass of the electroproduced hadronic system W is then W2 = m22+ 2m2ν − Q2 = m22+Q2(1/x − 1). Here W1 and W2 are the structure functions for unpolarized scattering. In the DIS regime the invariant mass W must be greater than any resonance in the nucleon. The threshold between the resonance region and the deep-inelastic region is not well defined, but it is usually taken to be at about W2 = 4 GeV2.

5. NUCLEUS POLARIZABILITY CONTRIBUTION TO HFS
We deal only with the case where the nucleons are assumed to contribute independently to the scattering. This means we neglect bound state effects in nuclei.
The polarized structure functions g1(ν,Q2) and g2(ν,Q2) give the contribution to hfs which is determined by the following expressions:
EF is the Fermi energy;
F2(Q2) is the Pauli form factor;
m2 is the nucleon mass.
Where ν0 determines the pion-nucleon threshold:
G.M. Zinov’ev, B.V. Struminsky, R.N. Faustov, et al., Sov. J. Nucl. Phys., Vol. 11, 1970, P. 715.

6. POLARIZED STRUCTURE FUNCTIONS
The nucleon polarized structure functions can be measured in the inelastic scattering of polarized electrons on polarized protons. Recent improvements in polarized lepton beams and nucleon targets have made it possible to make accurate measurements of nucleon polarized structure functions g1,2 in experiments at SLAC, CERN and DESY. The polarized structure functions can be expressed in terms of virtual photon-absorption cross sections:
where K = ν − Q2/2m2 is the Hand kinematical flux factor for virtual photons, σ1/2, σ3/2 are the virtual photoabsorption transverse cross sections for the total photon–nucleon helicity of 1/2 and 3/2 respectively, σTL is the interference term between the transverse and longitudinal photon–nucleon amplitudes. In this work we calculate contribution on the basis of the latest experimental data on structure functions g1,2(ν,Q2) and theoretical predictions for the cross sections σ1/2,3/2,TL obtained in MAID.

7. UNITARY ISOBAR MODEL
Electromagnetic γNN and γππ interaction vertices are defined by following Lagrangians:
where electromagnetic vector potential, ψ and π-nucleon and pion field operators.
When describing the hadron system πNN, two options are possible for constructing the Lagrangian interaction:
Pseudoscalar
Pseudovector
Effective Lagrangians for ω and ρ exchange:
ω and ρ are the operators of meson fields, λV is a radiative coupling, determined from the decay V→πγ. 7

8. BREIT-WIGNER PARAMETERIZATION OF THE CROSS SECTIONS
GeV
GeV
For resonance P33(1232)
GeV
For resonances D13(1520),P11(1440), F15(1680).
δ-the angular momentum of the pion
- branching ratio for decay S11(1535) πN and ηN
For resonance S11(1535)
- Normalization factor
pf - three-momentum resonance r in the rest frame of resonance R. mr mass of meson resonance. q and qr - three-momentum pion in the rest system with masses μ and mr. Jr and Гr spin and resonance width with mass mr
GeV 8

9. THE PROGRAM OF CALCULATION OF CROSS SECTIONS IN MAID 9

10. THE PROGRAM OF CALCULATION OF CROSS SECTIONS IN MAID 10

11. NONRESONANCE CONTRIBUTION
The parametrization of the structure function g1 in the nonresonance region is written as follows (SLAC E80, SLAC E130, EMC, SMC):
pQCD behavior.
R.D. Erbacher “A Precision Measurement of the Spin Structure of the Proton at SLAC”/ SLAC-R-546 ,1999. 

12. NONRESONANCE CONTRIBUTION
In the nonresonance region, there are a number of parameterizations for the structure function F2 (W, Q2), which were obtained on the basis of experimental data on deeply inelastic lepton-nucleon scattering. F2(х,Q2) is represented as the sum of the Reggeon and Pomeron contributions:
A. Airapetian “Inclusive Measurements of Inelastic Electron and Positron Scattering from Unpolarized Hydrogen and Deuterium Targets”/JHEP, Vol. 1105, 2011, P   12

13. DEUTERON STRUCTURE FUNCTION g1(ν,Q2) IN RESONANCE REGION

14. DEUTERON STRUCTURE FUNCTION g2(ν,Q2) IN RESONANCE REGION

15. GDH SUM RULE proton,mcb neutron,mcb 15
Contribution to GDH integral
Contribution from Nπ states
165,885
133,227
Contribution from Nη states
-8,854
-5,692
Contribution from K meson
-1,801
-3,003
Contribution from Nππ state
47,843
50,503
The total contribution
203,072
175,034
The GDH value
204,784
231,783
proton,mcb
neutron,mcb
D. Drechsel, T. Walcher “Hadron structure at low Q2”/Rev.Mod.Phys., Vol. 80, 2008, Pp  
D. Drechsel, B. Pasquini, M. Vanderhaeghen “Dispersion relations in real and virtual Compton scattering”/Phys. Rev., Vol. 378, 2003, Pp 15

16. The contribution of the polarizability of the nucleus to HFS
RESULTS
For He3 the wave-function is almost entirely an S-state with the two protons having opposite spins. Thus all the spin is carried by the neutron neglecting the motion of the nucleons inside the nucleus and their binding:
The contribution of the polarizability of the nucleus to HFS
Proton:
Triton
Deuteron:
Helion

17. GENERAL RESULTS
In the course of the work, complete numerical values for the contributions of the nucleus polarizability to the hyperfine structure of the light muonic atoms were obtained: 17

18. THANK YOU FOR YOUR ATTENTION10