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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

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 OutlineOutline 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

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 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 p n Careful: Baryon ChPT has 2 scales: M N, Λχ ~ 1 GeV (at physical point)

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 NR-BChPTNR-BChPT Non-relativistic framework (HBChPT) (Jenkins, Manohar 1991) → only terms ~ 1/M N appear in calculation Organise perturbative calculation as a simultaneous expansion in 1/M N and 1/ Λχ (Bernard et al. 1992) 1/M n-1

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Baryon ChPT NR-BChPT/HBChPT very successful for near- threshold scattering experiments (m π =140 MeV) Chiral extrapolation rather tedious in HBChPT: Low order polynomial in m π compared to smoothly varying lattice results Idea: Utilize covariant BChPT –At each order p n the result to that order is given in terms of smoothly-varying analytic functions f(μ) with μ=m π /M N –Different organization of the perturbative expansion

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Regularization in BChPT Ultraviolet divergences can be absorbed via counterterms in the effective theory; UV is not the (main) issue Troublesome are terms ~ M N /Λχ ~O(1), (which for example appear in MS-scheme) → uncontrolled finite renormalization of coupling constants 1/M n-1

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Infrared Regularization Use a regularization scheme that avoids terms ~ (m q ) a (M N /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 MS IR Regulator Integral

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 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 m π, large Q 2 behaviour 1/M n-1

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 OutlineOutline 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

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Magnetic Moments on the Lattice Isovector anomalous magnetic moment of the nucleon: κ v –extrapolated to q 2 =0 via Dipole-fits Slope ? Caldi-Pagels ↓ ! Quenched (improved) Wilson Data: QCDSF collaboration; Phys. Rev. D71, 034508 (2005) Turning-points in quark-mass dependence ? Breakdown of ChEFT ? κ proton -κ neutron = 3.71 n.m.

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Magnetic Moments ? Magnetic Moments ? Nucleon isovector electromagnetic current: Dirac FF: F 1 v (0)=1; Pauli FF: F 2 v (0)=κ v =κ p -κ n =3.71 [n.m.] with [n.m.]=e/2M=3.15 10 -14 MeV T -1 at m π =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

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 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 M N → normalization factor becomes 1 at physical point

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Magnetic Moments II Anomalous isovector magnetic moment of nucleon κ v –Caldi-Pagels prediction ~ - m π (= HBChPT O(p 3 ) ) κ v (m π ) measured in physical nuclear magnetons [n.m.] → the remaining quark mass dependence is flat! (compare, g A ) Quenched (improved) Wilson Data: QCDSF collaboration; Phys. Rev. D71, 034508 (2005)

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 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)

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Magnetic Moments IIIb O(p 4 ) covariant BChPT with (modified) IR- regularization (T. Gail, TRH, forthcoming) c i couplings fixed from πN and NN scattering 2 unknown LECs κ 0 v, E 1 (λ) fit to QCDSF data → one free parameter less than in SSE calculation cici

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Comparison Magnetic Moments Comparable results BChPT - NR-SSE, need better data to study differences (T. Gail and TRH, forthcoming) However: Results at finite Q 2 in covariant BChPT still require work/thought (Q 2 dependence in SSE very successful) Quenched (improved) Wilson Data: QCDSF collaboration; Phys. Rev. D71, 034508 (2005) O(p 4 ) BChPT NR-SSE Leading-one-loop O(p 3 ) BChPT

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 NN Form factors Isovector NN form factors can be analyzed up to Q 2 < 0.4 GeV 2 and m π < 600 MeV in SSE to O(ε 3 ) (M. Göckeler et al., Phys. Rev. D71, 034508 (2005)) Note: Direct comparison with simulation data at finite Q 2 possible ! → We can even avoid extra uncertainties from dipole fits in this window Physics beyond the radii !!

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 OutlineOutline 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

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 NDelta Form Factors 3 complex (isovector) transition form factors: G 1 (Q 2 ), G 2 (Q 2 ), G 3 (Q 2 ) (real for m π > M Δ -M N ) Known to O(ε 3 ) in SSE (G.C. Gellas et al., Phys. Rev. D60, 054022 (1999)) m π =140 MeV

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Multipole Basis 3 complex NDelta transition form factors: G 1 (Q 2 ), G 2 (Q 2 ), G 3 (Q 2 ) → G M1* (Q 2 ), G E2* (Q 2 ), G C2* (Q 2 ) 2 free parameters at O(ε 3 ) in SSE: G 1 (0), G 2 (0) → Fix at G M1* (Q 2 =0) and at EMR(Q 2 =0) = Re[G M1 * (Q 2 )G E2 (Q 2 )]/|G M1 (Q 2 )| 2 G.C. Gellas et al., PRD 60, 054022 (1999), T. Gail and TRH, forthcoming Abs! m π =140 MeV

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 EMR(Q 2 ) ? O(ε 3 ) SSE: EMR(Q 2 ) in multipole basis ? (% effects !) Problem results from G 2 (Q 2 ): Rising with Q 2 !??! –, no c.t. at this order! → check effect of extra c.t. in radius Abs! m π =140 MeV

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Prediction for CMR(Q 2 ) G M1* (Q 2 ) still okay, small radius correction in G 2 shows large effects CMR(Q 2 ) = Re[G M1 * (Q 2 )G C2 (Q 2 )]/|G M1 (Q 2 )| 2 is a prediction → Comparison to new Mainz data ? Abs! m π =140 MeV Abs!

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 OutlineOutline 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

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Chiral Extrapolation I (Quenched) Lattice Data for G M1* (Q 2 ) at low Q 2 (C. Alexandrou et al., [hep-lat/0307018]) Note: G M1* (Q 2 ) increases with increasing quark mass → similar to situation in G M (Q 2 ) !

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Chiral Extrapolation II First step: Extrapolate all dynamical factors in NDelta transition current to the physical point m π =140 MeV (not just the magneton!) –strong rise in m π is gone ! –lattice data are now lower than the m π =140 MeV curve Here: Data at Q 2 =0.135 GeV 2 from C. Alexandrou et al., [hep- lat/0307018] → Need M N (m π ) and M Δ (m π ) with correct extrapolation to the physical point to do this ! T. Gail and TRH, forthcoming

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Nucleon and Delta Mass We are utilizing the covariant O(ε 4 ) SSE result: V. Bernard, TRH, U.-G. Meißner, [hep-lat/0503022] 3-flavour data ! MILC [hep-lat/0104002] Rising NDelta mass-splitting near the chiral limit ?

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Chiral Extrapolation II Study quark mass dependence of the form factors with physical point kinematics –directly at low Q 2 and low m π region without dipole extrapolations Note: Before one can address the tiny EMR(m π ), CMR(m π ) ratios, one needs to get G M1* (m π,Q 2 ) right ! T. Gail and TRH, forthcoming

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 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 m π (no effect on Q 2 -dependence) –similar to κ v (m π ), but no steep slope due to Caldi Pagels ! T. Gail and TRH, in preparation !

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Outlook/SummaryOutlook/Summary κ v (m π ) to O(p 4 ) in covariant BChPT now under control –smooth m π dependence in modified IR-regularization –comparable to NR-SSE result –new data? (Cyprus-Athens?) Q 2 dependence of nucleon form factors in covariant BChPT still needs more work –Q 2 dependence of NR-SSE compares well with phenomenology –direct comparison with lattice data in low Q 2 window without dipole extrapolations Q 2 dependence of all 3 NDelta form factors now well under control –New compared to Gellas et al. calculation: Radius correction in G 2 (Q 2 ) –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 G M1* (m π,Q 2 ), direct comparison with small G E2* (Q 2 ), G C2* (Q 2 ) is step 2 –at the moment only qualitative results for EMR(m π ), CMR(m π ), 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(ε 3 ) SSE

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T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors"Sep 16, 2005 Magnetic Moment in naive IR O(p 3 ) BChPT in naive IR- regularization –Kubis, Meißner NPA 679, 698 (2001) –TRH, Weise EPJA 15, 487 (2002) Problems: –turning points in m π –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)

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