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Chiral Extrapolations for NN, NDelta Form Factors --- Update

Chiral Extrapolations for NN, NDelta Form Factors --- Update. Thomas R. Hemmert Theoretische Physik T39 Physik Department, TU München. Workshop on Computational Hadron Physics University of Cyprus, Nicosia Sep 14 -17, 2005. Outline.

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Chiral Extrapolations for NN, NDelta Form Factors --- Update

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  1. Chiral Extrapolations for NN, NDelta Form Factors --- Update Thomas R. Hemmert Theoretische Physik T39 Physik Department, TU München Workshop on Computational Hadron Physics University of Cyprus, Nicosia Sep 14 -17, 2005

  2. Outline • Basics regarding covariant BChPT versus non-relativistic BChPT (HBChPT) • Covariant analysis of the isovector magnetic moment of the nucleon • Momentum-dependence of the NDelta-transition form factors revisited • On the road to a quantitative chiral extrapolation of NDelta-form factors • Outlook T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  3. Baryon ChPT • Low energy effective theory of QCD • Pions as (Quasi-) Goldstone Bosons of the spontaneously broken chiral symmetry of QCD • In addition explicit breaking of chiral symmetry due to finite quark masses • Baryons added as matter fields in a chirally invariant procedure (CCWZ) • Perturbation Theory organised in powers pn • Careful: Baryon ChPT has 2 scales: MN, Λχ ~ 1 GeV (at physical point) T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  4. NR-BChPT • Non-relativistic framework (HBChPT) (Jenkins, Manohar 1991) → only terms ~ 1/MN appear in calculation • Organise perturbative calculation as a simultaneous expansion in 1/MN and 1/ Λχ(Bernard et al. 1992) 1/Mn-1 T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  5. Baryon ChPT • NR-BChPT/HBChPT very successful for near-threshold scattering experiments (mπ=140 MeV) • Chiral extrapolation rather tedious in HBChPT: Low order polynomial in mπ compared to smoothly varying lattice results • Idea: Utilize covariant BChPT • At each order pn the result to that order is given in terms of smoothly-varying analytic functions f(μ) with μ=mπ /MN • Different organization of the perturbative expansion T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  6. Regularization in BChPT • Ultraviolet divergences can be absorbed via counterterms in the effective theory; UV is not the (main) issue • Troublesome are terms ~ MN/Λχ ~O(1), (which for example appear in MS-scheme) → uncontrolled finite renormalization of coupling constants 1/Mn-1 T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  7. Infrared Regularization • Use a regularization scheme that avoids terms ~ (mq)a (MN/Λχ)b • e.g. Infrared Regularization (IR) (Becher, Leutwyler 1999) • Idea: Add an extra integral in Feynman-parameter space that contains an infinite string of quark-mass insertions which cancel these terms Regulator Integral IR MS T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  8. Covariant BChPT with IR • Controlled coupling constant renormalization • Reorganization of perturbative expansion with exact HBChPT limit • Successful scheme at physical point (e.g. nucleon spin structure, neutron form factor, …) • Promising results for chiral extrapolation of nucleon mass in finite volume (QCDSF collaboration, Nucl. Phys. B689, 175 (2004)) • However: Can be problematic in large mπ , large Q2 behaviour 1/Mn-1 T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  9. Outline • Basics regarding covariant BChPT versus non-relativistic BChPT (HBChPT) • Covariant analysis of the isovector magnetic moment of the nucleon • Momentum-dependence of the NDelta-transition form factors revisited • On the road to a quantitative chiral extrapolation of NDelta-form factors • Outlook T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  10. Magnetic Moments on the Lattice • Isovector anomalous magnetic moment of the nucleon: κv • extrapolated to q2=0 via Dipole-fits • Slope ? Caldi-Pagels ↓ ! Turning-points in quark-mass dependence ? Breakdown of ChEFT ? Quenched (improved) Wilson Data: QCDSF collaboration; Phys. Rev. D71, 034508 (2005) κproton-κneutron = 3.71 n.m. T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  11. Magnetic Moments ? • Nucleon isovector electromagnetic current: • Dirac FF: F1v(0)=1; • Pauli FF: F2v(0)=κv=κp-κn=3.71 [n.m.] with [n.m.]=e/2M=3.15 10-14 MeV T-1 at mπ=140 MeV • We are not interested in the quark-mass dependence of the nuclear magneton unit ! → express lattice results in units of the physical [n.m] Quark mass dependent magneton Quark mass dependent Pauli formfactor T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  12. Normalized Magnetic Moments • Measure Pauli form factor in physical [n.m.] • Not required, but elucidates quark mass dependence of the magnetic moment more clearly ! • Note: Similar complications occur in the NΔ-transition form factors! • Use lattice data for the normalization • Set lattice scale via MN • → normalization factor becomes 1 at physical point T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  13. Magnetic Moments II • Anomalous isovector magnetic moment of nucleon κv • Caldi-Pagels prediction ~ - mπ(= HBChPT O(p3) ) Quenched (improved) Wilson Data: QCDSF collaboration; Phys. Rev. D71, 034508 (2005) κv(mπ) measured in physical nuclear magnetons [n.m.]→ the remaining quark mass dependence is flat! (compare <x>, gA) T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  14. Magnetic Moments IIIa • Go beyond Caldi-Pagels ! • e.g. include explicit Delta(1232) degrees of freedom (NR SSE) (TRH, Weise, EPJA 15, 487 (2002)) • Physical point and lattice data in good agreement within assumptions for Delta parameters → Covariant BChPT ? Quenched (improved) Wilson Data: QCDSF collaboration; Phys. Rev. D71, 034508 (2005) T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  15. Magnetic Moments IIIb • O(p4) covariant BChPT with (modified) IR-regularization (T. Gail, TRH, forthcoming) • ci couplings fixed from πN and NN scattering • 2 unknown LECs κ0v, E1(λ) fit to QCDSF data → one free parameter less than in SSE calculation ci T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  16. Comparison Magnetic Moments Quenched (improved) Wilson Data: QCDSF collaboration; Phys. Rev. D71, 034508 (2005) • Comparable results BChPT - NR-SSE, need better data to study differences (T. Gail and TRH, forthcoming) • However: Results at finite Q2 in covariant BChPT still require work/thought (Q2 dependence in SSE very successful) NR-SSE Leading-one-loop O(p4) BChPT O(p3) BChPT T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  17. NN Form factors • Isovector NN form factors can be analyzed up to Q2 < 0.4 GeV2 and mπ< 600 MeV in SSE to O(ε3)(M. Göckeler et al., Phys. Rev. D71, 034508 (2005)) • Note: Direct comparison with simulation data at finite Q2 possible ! → We can even avoid extra uncertainties from dipole fits in this window Physics beyond the radii !! T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  18. Outline • Basics regarding covariant BChPT versus non-relativistic BChPT (HBChPT) • Covariant analysis of the isovector magnetic moment of the nucleon • Momentum-dependence of the NDelta-transition form factors revisited • On the road to a quantitative chiral extrapolation of NDelta-form factors • Outlook T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  19. NDelta Form Factors • 3 complex (isovector) transition form factors: G1(Q2), G2(Q2), G3(Q2) (real for mπ> MΔ-MN) • Known to O(ε3) in SSE (G.C. Gellas et al., Phys. Rev. D60, 054022 (1999)) mπ=140 MeV T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  20. Multipole Basis • 3 complex NDelta transition form factors: G1(Q2), G2(Q2), G3(Q2) → GM1*(Q2), GE2*(Q2), GC2*(Q2) • 2 free parameters at O(ε3) in SSE: G1(0), G2(0) → Fix at GM1*(Q2=0) and at EMR(Q2=0) = Re[GM1*(Q2)GE2(Q2)]/|GM1(Q2)|2 Abs! mπ=140 MeV G.C. Gellas et al., PRD 60, 054022 (1999), T. Gail and TRH, forthcoming T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  21. EMR(Q2) ? • O(ε3) SSE: EMR(Q2) in multipole basis ? (% effects !) • Problem results from G2(Q2): Rising with Q2 !??! • , no c.t. at this order! → check effect of extra c.t. in radius mπ=140 MeV T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors" Abs!

  22. Prediction for CMR(Q2) • GM1*(Q2) still okay, small radius correction in G2 shows large effects • CMR(Q2) = Re[GM1*(Q2)GC2(Q2)]/|GM1(Q2)|2is a prediction → Comparison to new Mainz data ? mπ=140 MeV T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors" Abs! Abs!

  23. Outline • Basics regarding covariant BChPT versus non-relativistic BChPT (HBChPT) • Covariant analysis of the isovector magnetic moment of the nucleon • Momentum-dependence of the NDelta-transition form factors revisited • On the road to a quantitative chiral extrapolation of NDelta-form factors • Outlook T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  24. Chiral Extrapolation I • (Quenched) Lattice Data for GM1*(Q2) at low Q2 (C. Alexandrou et al., [hep-lat/0307018]) Note: GM1*(Q2) increases with increasing quark mass → similar to situation in GM(Q2) ! T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  25. Chiral Extrapolation II • First step: Extrapolate all dynamical factors in NDelta transition current to the physical point mπ=140 MeV (not just the magneton!) • strong rise in mπ is gone ! • lattice data are now lower than the mπ=140 MeV curve Here: Data at Q2=0.135 GeV2 from C. Alexandrou et al., [hep-lat/0307018] → Need MN(mπ) and MΔ(mπ) with correct extrapolation to the physical point to do this ! T. Gail and TRH, forthcoming T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  26. Nucleon and Delta Mass • We are utilizing the covariant O(ε4) SSE result: 3-flavour data ! MILC [hep-lat/0104002] Rising NDelta mass-splitting near the chiral limit ? V. Bernard, TRH, U.-G. Meißner, [hep-lat/0503022] T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  27. Chiral Extrapolation II • Study quark mass dependence of the form factors with physical point kinematics • directly at low Q2 and low mπ region without dipole extrapolations Note: Before one can address the tiny EMR(mπ), CMR(mπ) ratios, one needs to get GM1*(mπ,Q2) right ! T. Gail and TRH, forthcoming T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  28. Chiral Extrapolation III • Essential NDelta intermediate state needs to be added for chiral extrapolation (see TRH, Weise, EPJA 15, 487 (2002)) • formally NLO, but essential at intermediate mπ (no effect on Q2-dependence) • similar to κv(mπ), but no steep slope due to Caldi Pagels ! ! T. Gail and TRH, in preparation T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  29. Outlook/Summary • κv(mπ) to O(p4) in covariant BChPT now under control • smooth mπ dependence in modified IR-regularization • comparable to NR-SSE result • new data? (Cyprus-Athens?) • Q2 dependence of nucleon form factors in covariant BChPT still needs more work • Q2 dependence of NR-SSE compares well with phenomenology • direct comparison with lattice data in low Q2 window without dipole extrapolations • Q2 dependence of all 3 NDelta form factors now well under control • New compared to Gellas et al. calculation: Radius correction in G2(Q2) • comparison to new Mainz data ? • Chiral extrapolation of NDelta form factors • evaluate kinematical factors at physical point → most headaches seem to be gone • focus first on GM1*(mπ,Q2), direct comparison with small GE2*(Q2), GC2*(Q2) is step 2 • at the moment only qualitative results for EMR(mπ), CMR(mπ), comparable to Pascalutsa, Vanderhaghen (same diagrams as in Gellas et al.) • Guess: Quantitative chiral extrapolation of quadrupole form factors will require a lot more effort than O(ε3) SSE T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

  30. Magnetic Moment in naive IR • O(p3) BChPT in naive IR-regularization • Kubis, Meißner NPA 679, 698 (2001) • TRH, Weise EPJA 15, 487 (2002) • Problems: • turning points in mπ • artefacts of naive IR • Alternative: Correct for the quark-mass dependence of the analytic structures in f(μ) „by hand“ to soften the curve (see e.g. MIT-meeting, Hemmert 2004) Quenched (improved) Wilson Data: QCDSF collaboration; Phys. Rev. D71, 034508 (2005) T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"

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