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Spin structure calculations from ChPT

Spin structure calculations from ChPT. Marc Vanderhaeghen Johannes Gutenberg Universität, Mainz. Workshop on “Spin Structure at Long Distance” JLab, March 12-13, 2009. Outline. Forward double virtual Compton scattering : brief review Inclusion of  in chiral EFT framework :

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Spin structure calculations from ChPT

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  1. Spin structure calculations from ChPT Marc Vanderhaeghen Johannes Gutenberg Universität, Mainz Workshop on “Spin Structure at Long Distance” JLab, March 12-13, 2009

  2. Outline Forward double virtual Compton scattering : brief review Inclusion of  in chiral EFT framework : masses, NΔ transition form factors chiral EFT used in dual role : as a framework to predict e p -> e p π observables (Q2 dependence of structure functions) and to perform chiral extrapolation of lattice data Sum rule / analyticity constraints on EFT calculations evaluation of sum rules in perturbation theory

  3. Forward double virtual Compton scattering (VVCS) optical theorem nucleon parton distribution

  4. Dispersion relation for fT subtracteddispersionrelation subtraction elastic contribution singularities at :  = §B with B = Q2/2MN ( $ xB=1) 0:  threshold low energy expansion for the non-pole part

  5. Generalized Baldin sum rule Q2 >> DIS (MRST 01) (F2)   +  +  +  +  “DIS” (W < 2 GeV) resonance estimate (W < 2 GeV, MAID  +  +  ) Drechsel, Pasquini, Vdh (2003) CLAS data: Osipenko et al. (2003)

  6. Spin dependent forward double VCS link between S1, S2 & gTT, gLT optical theorem

  7. Dispersion relations for gTT and S1 UNsubtracted dispersion relation low energy expansion for inelastic part

  8. Generalized GDH sum rule for proton HBChPt O(p4): Ji, Kao, Osborne (2000) rel. BChPt O(p4) : Bernard et al. (2002) DIS: Bluemlein, Boettcher (2003) resonance estimate + DIS (VMD) Anselmino, Ioffe, Leader / Burkert, Ioffe “DIS”: W < 2 GeV resonance estimate W < 2 GeV:  +  +  Drechsel, Pasquini, Vdh (2003)

  9. GDH sum rule for p - n DIS: Bluemlein, Boettcher (2003) “DIS”: W < 2 GeV HBChPT O(p4): Ji, Kao, Osborne (2000) rel. BChPT O(p4): Bernard, Meissner, Hemmert (2002) resonance estimate (W < 2 GeV):  N (MAID) Drechsel, Pasquini, Vdh (2003) resonance + DIS estimate (VMD)

  10. Forward spin polarizability g0 ofproton JLab/CLAS data : Prok et al. (2008) resonance estimate (W < 2 GeV) :  (MAID)

  11. Dispersion relation for gLT UNsubtracted dispersion relation low energy expansion for inelastic part first moment of g2 : Longitudinal – Transverse spin polarizability

  12. Forward spin polarizability of neutron JLab/Hall A data : E94010 (2004) E94010 MAID LT HBChPT: p3 HBChPT: p4 E94010 MAID 0 HBChPT: p3 HBChPT: p4 resonance estimate (MAID) : Drechsel, Pasquini, Vdh (2003) HBChPT: Kao, Spitzenberg, Vdh (2003)

  13. Burkhardt-Cottingham sum rule for proton E155 : 0.02 · x · 0.8 Burkhardt – Cottingham sum rule resonance estimate (W < 2 GeV):  (MAID) HBChPT O(p4): Kao, Spitzenberg, Vdh (2002)

  14. Burkhardt-Cottingham sum rule for neutron inelastic part Burhardt-Cottingham sum rule is satisfied for the neutron elastic part

  15. L T 0 ±1 Lz =1 Twist-2 : Wandzura-Wilczek Twist-3 :quark-gluon correlations in nucleon scale dependence scaling limit, Q2 -> ∞ : arbitrary Q2 : enter in low energy expansion of VVCS low Q2 : ChPT HBChPT : p3 p4

  16. Scale dependence of d2 ELASTIC HBChPT O(p4) HBChPT O(p3) incl. Δ HBChPT O(p3) E155 (2002) resonance estimate : π (MAID) Kao, Drechsel, Kamalov, Vdh (2003) JLab/Hall C RSS Coll. (2008) JLab/Hall A (2003)

  17. (1232)-resonance in chiral EFT N and Δ masses : chiral EFT calculation chiral extrapolation of lattice data NΔ transition form factors / e p -> e p π observables chiral EFT used in dual role : as a framework to predict e p -> e p π observables (extraction of NΔ form factors) and to perform chiral extrapolation of lattice data work done in coll. withV. Pascalutsa -> N and Δ masses : PLB 636 (2006) 31 -> NΔ transition : PRL 95 (2005) 232001 and PRD 73 (2006) 034003 -> Δ MDM : PRL 94 (2005) 102003 and PRD 77 (2008) 014027 ->Pascalutsa, Vdh, Yang: Phys. Rept. 437 (2007) 125

  18. N and Δchiral Lagrangians πNN πΔΔ πNΔ Δ part is such that # spin d.o.f. is constrained to physical number Couplings involving Δ are consistent : respect spin-3/2 gauge symmetry Pascalutsa (1998) Pascalutsa, Timmermans (1999) Chiral behavior of masses

  19. Jenkins & Manohar (1991), Hemmert et al. (1998) … (2006) N and  propagators: Pascalutsa & Phillips (2003) Chiral Lagrangians withand power counting Include the  as an explicit d.o.f. , described by a spin-3/2 (Rarita-Schwinger) isospin-3/2 (isoquartet) field Power counting:  = + … = O(p3) = O(3 )

  20. N andΔ masses Chiral loops : depend on 2 light scales LEC (fit parameters)

  21. N andΔ masses: covariant chiral loops renormalizes MB(0) c1B 2 light scales :μandδ

  22. N andΔ masses: covariant chiral loops (contd.) in an expansion to third power in (mπ,Δ = MΔ – MN ) mπ< Δ mπ> Δ

  23. πN and πΔ chiral loops :leadingnon-analytic behavior Banerjee, Milana (1995) Young, Leinweber, Thomas, Wright (2002) Bernard, Hemmert, Meissner (2003) LNA terms agree with :

  24. N mass : mπ dependence full QCD lattice calculations : ETMC Alexandrou et al. (2008) MN(0) = 0.883 (29) GeV c1N = - 1.26 (9) GeV-1

  25. Δ mass : mπ dependence lattice : ETMC (2008) lattice : MILC (2001) MΔ(0) = 1.26 (54) GeV πN + πΔ : covariant, c2Δ= 0 c1Δ = - 1.16 (17) GeV-1 πΔ : mπ3 term (HBChPT)

  26. Pion-nucleon scattering in theΔregion Renormalized NLO propagator

  27. chiralEffectiveFieldTheory calculation ofe p -> e p π0inΔ(1232)region in threshold region : momentum p ~ mπ in Δ region : p ~ MΔ - MN Power counting scheme : LO calculation to NLO in δ expansion for e p -> e p π0 vertex corrections : unitarity & gauge invariance exactly preserved to NLO Pascalutsa, Vdh (2005)

  28. magnetic (M1) & electric (E2) N -> Δtransition 2 free parameters ! Δ pole + Born Δ pole + Born + vertex corr. Δ pole MAID (2003) SAID (2003) G*M = 2.97 G*E = 0.07 (E2/M1 = -2.3 %)

  29. DMT01 Sato-Lee DUO (Utrecht-Ohio) NLO χEFT  N -> π NinΔ(1232)region : observables

  30. Q2dependence ofE2/M1 and C2/M1 ratios data points : MIT-Bates (Sparveris et al., 2005) MAMI : Q2 = 0 (Beck et al., 2000) Q2 = 0.06 (Stave et al., 2006) Q2 = 0.2 (Elsner et al., 2005, Sparveris et al., 2006) EFT calculation predicts the Q2 dependence no pion loops pion loops included

  31. GDH sum rule in QED Compute both sides of the sum rule in perturbation theory. Is the GDH sum rule verified? for Dirac particle : electron anomalous magnetic moment (loop effect) to 1-loop accuracy : Schwinger experiment electron : g = 2.002319304374 ± 8 . 10-12 !

  32. GDH sum rule in QED: O(e4) 2 + at O(e4) : Altarelli, Cabibbo, Maiani (1972)

  33. GDH sum rule in QED : O(e6) + ... + tree level 1-loop diagrams 2 + + ... Dicus, Vega (2001) GDH sum rule is satisfied in QED to order O(e6)

  34. verification of GDH sum rule in Chiral Effective Field Theory (ChEFT) to lowest order in g NN κ =κ0+δκ O(g2): trial value loop contribution 2 0 = =

  35. Yes! However, only in the fully covariant calculation. Any “heavy-baryon” type of expansion does not do it. Reason: violation of analyticity… Verification in ChEFT (cont’d)

  36. Nucleon anomalous magnetic moments equivalent to a sideways dispersion relation agrees with direct calculation!

  37. Chiral behavior of nucleon anomalous magnetic moments HB LO Rel. corr. goes as 1/mq Exactly as expected from a naïve quark model. Coincidence?

  38. analyticity constraints on magnetic moments covariant chiral loops (SR) compared with heavy-baryon expansion (HB) or Infrared-Regularized ChPT(IR) lattice : Zanotti et al. (2004) Red curve is the single-parameter fit to lattice data The parametrization is based on SR result Pascalutsa, Holstein, Vdh (2004)

  39. The covariant ChPT calculation (usingInfrared Regularization) [Bernard, Hemmert, Meissner, PRD (2003)] agrees with sum rule calculation up to terms analytic in quark mass : BHM SR IR destroys analyticity (IR-regulated integrals do not satisfy DRs) Chiral behavior of spin polarizabilities forward spin polarizability Our sum rule (SR) evaluation, using the lowest order (Born) total cross-section, upon expansion in pion mass agrees with NLO heavy-baryon result of [Ji, Kao, Osborne (2000); Kumar, McGovern, Birse (2000)], not [Gellas, Hemmert, Meissner (2000)]

  40. Covariant chiral loops (RLO) should describe chiral behavior better than the heavy-baryon expansion (HBLO). Green dashed : HBLO + 1st rel. corr. “Exp.” is a fit to lattice QCD calculations : [Leinweber et al. (1999)]

  41. Conclusions & Outlook • (1232)-resonance in chiral EFT quantitative framework for pion electroproduction observables at low Q2 (moments of) structure functions • Sum rules of forward double VCS improve/extend chiral expansion : calculate/resum higher order terms (guidance by analyticity)

  42. The End…

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