Hermann Krebs Ruhr-Universität-Bochum September 27, 2010, Spin 2010, Jülich

1. Spin Structure of the Nucleon at Low Energies Hermann Krebs Ruhr-Universität-Bochum September 27, 2010, Spin 2010, Jülich With V. Bernard, E. Epelbaum, U.-G. Meißner TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAA

2. Outline ChPTandlowenergy QCD Nucleon spin structure and sum rules V2CS within ChPT and the role of Δ-isobar Δ-isobar in ChPT Preliminarynumericalresults Summary & Outlook

3. ChPT and low energy QCD Spontaneous + explicit (by small quark masses) breaking of chiral symmetry in QCD Existence of light weakly interacting Goldstone bosons Chiral Perturbation theory (ChPT) Expansion in small momenta and masses of Goldstone bosons Systematic description of QCD by ChPT in low energy sector ( low momenta )

4. Deep Inelastic Scattering related to unpolarized DIS related to polarized DIS Structure functions and describe internal spin structure of the nucleon Recently available polarized beams and targets Study of nucleon spin structure and are related by various sum rules to double virtual Compton scattering connecting information at all energy scales One of the main goals of JLab activities to provide experimental mapping of Spin-dependent observables from low-momentum transfer to the multi-Gev region

5. Nucleon spin structure and sum rules Spin-dependent forward Compton-Amplitude (V2CS) and related to spin-structure functions and - Expansion at low energies of reduced amplitudes and Generalized Gerasimov-Drell-Hearn (GDH) sum rule Ji, Osborne ‘00 Two different limits Ellis-Jaffe S.R. GDH S.R.

6. ChPT calculations of V2CS up to q4 for low virtualities calculated within ChPT HBChPT: Ji, Kao, Osborne, Spitzenberg, Vanderhaeghen, Birse, McGovern, Kumar Expansion of amplitude in one over nucleon mass IRChPT: Bernard, Hemmert, Meißner Systematic resumation of one over nucleon mass contributions Contributions up to All tree diagrams with insertion from All loop diagrams with insertion from at most Two well known low energy constants: Elastic contributions to subtracted

7. Different regularization schemes Naive dimensional regularization in nucleon sector breaks power counting Gasser et al. NPB307 (1988) 779 Infrared regularization Heavy Baryon expansion Jenkins & Manohar ‘91 Becher & Leutwyler ‘99 Extended on-mass-shell renormalization and related schemes Fuchs et al. ‘03, Pascalutsa et al. ‘04, … Power counting violating contributions can be absorbed by counter terms First possible spin-dependent counter term is of order Ways out Spin independent terms such as Naive dimensional regularization does not violate power counting for spin-dependent observables up to the order

8. Validity range of IRChPT Change in integral boundaries of Feynman-parameter integration Two-point function examplified Becher, Leutwyler ‘99 Change in integral boundaries Unphysical cuts at finite Restriction in range of In V2CS unphysical cut at far beyond the range of validity: However Calculations of derivatives of loop functions Effects at

9. Numerical Results First moment for proton and neutron Kuhn, Chen, Leader ’08, Chen ‘10 Reasonable agreement of IRChPT results (with large uncertainty) with data at

10. Generalized spin-polarizabilities Bernard, Hemmert, Meißner ‘03 Forward spin polarizabilities at : Chiral expansion Poor convergence of due to large prefactors Much better convergence of Longit.-transv. spin-polarizabilities at : Well behaved chiral expansion of No large prefactors in All numbers given in

11. Generalized spin-polarizabilities Forward spin-polarizability of the proton CLAS Colaboration ‘09 IRChPT contributions Positive chiral loop contr. increasing with Born-graph contr. Is Negative and flat Serious disagreement with the data for Hope for improvement From loop-contr.

12. Generalized spin-polarizabilities Spin-polarizabilities for the neutron Bernard’s review PPNP 60 (2008) 82 is sensitive to dof Nice agreement with exp. point at is insensitive to dof Very well behaved chiral Expansion of at Strong curvature due to unphysical cut in IR results Comparisons with exp. data at lower are desirable

13. SSE calculation of V2CS 14 diagrams contribute to the order -vertex starts to contribute to the order Diagrams like suppressed Like in the nucleon case all divergencies from loop-diagrams cancel No counter-term contributions to the spin-dependent part of V2CS For additional - and - coupling we use fit to the width We explicitly checked the amplitudes for gauge-invariance

14. Considerations at photon point Dimensional vs infrared regularization for nucleon contributions Dim. reg. IR reg. Both regularization schemes coincide close to chiral limit Disagreement betw. two schemes increases with increasing pion mass Dim. reg. At physical point reasonable systematic uncertainty Much better convergence of chiral expansion for IR reg.

15. Considerations at photon point Dimensional vs infrared regularization for Δ contributions: Dim. reg. IR reg. Disagreement betw. two schemes at much smaller ! Dim. reg. Asymptotic behaviour at large IR reg. Additional Δ – scaleintroduces unnaturally large coefficient in large expansion

16. Longitudinal – transverse polarizability Preliminary Small sensitivity to variation of and Dominant contribution from pion – nucleon loops at order Weak dependence of on Δ – contributions is confirmed Consider the difference betw. and partial results as systematic error

17. Forward spin - polarizability Preliminary Larger sensitivity to variation of and Large pion-nucleon and pion - Δ contributions have opposite sign results closed to data point. Systematic errors are expected to be large due to smaller but still sizeable pion-nucleon loops corrections

18. Forward spin - polarizability Preliminary Δ-contributions are dominant: factor two stronger than nucleon-contr. Improvement at photon point Systematic errors are expected to be large due to smaller but still sizeable pion-nucleon loops corrections JLab data within the large systematic error estimate

19. First moment Preliminary Δ-contributionsslightlyimprovechiralprediction Curvature comes from pion-nucleon loops and Δ-contributions Full calculation needed to decrease systematic error estimate

20. Summary Spin-structure functions analyzed within ChPTupto order q4 No counter terms appear at this order Parameter free predictions Qualitative agreement between Theory & Experiment Systematic inclusion of - isobar for V2CS upto order Improvement of chiral predictions with large systematic error estimate Outlook Full calculation needed to decrease systematic error estimate Looking forward to more data at low

21. Generalized GDH – sum rule Preliminary JLab data error bars indicate systematical errors: statistical errors are not shown Δ-contributions are dominant at higher virtuality Δ-less theory describes the JLab data Sizeable pion-nucleon loops give rise to larger systematical error estimate

22. Theory with explicit Δdof Small scale expansion (SSE) is a systematic inclusion of dof in ChPT Hemmert, Holstein, Kambor ‘98 Expansion in low momenta, pion-masses and nucleon-delta mass splitting Two on mass-shell equivalent representations for covariant - fields Pascalutsa ’01, H.K., Epelbaum, Meißner ‘08 No unphysical spin-1/2 contr. But Gauge-invariance only modulo higher order All interactions are invariant under Gauge-invariance exact But Unphysical spin-1/2 contr. to be absorbed by LECs of higher order All interactions are Gauge-invariant Hamiltonian quantization of - fields in ChPT Wies, Gegelia, Scherer ‘06

23. Calculationof SSE loops Computational Effort Estimate propagator includes 5 times more terms than nucleon propagator Box-diagram has times more terms More complicated Spin-structures Automatization of calculation is necessary Mathematica & FORM used for tensor reduction in current calc. We developed our own code which is able to reduce tensor integrals of any rank in relativistic and Heavy Baryon formalism FORM used to reduce tensor integrals to series of scalar integrals with higher power of propagators and shifted dimensions Davydychev ‘91 Mathematica used for further Passarino-Veltman reduction (if needed)

24. LECs from Δ - width fitted to the Δ - width fitted to the electromagnetic Δ - width