1. Nucleon Axial and Nucleon-to-Delta Axial Transition Form Factors from Lattice QCD A. Tsapalis Institute of Accelerating Systems and Applications University of Athens in collaboration with C. Alexandrou (Univ. of Cyprus) G. Koutsou (Univ. of Cyprus) Th. Leontiou (Univ. of Cyprus) J. W. Negele (MIT)

2. outline Nucleon Axial Form Factors G A and G P PCAC and pion pole dominance Nucleon-to-Delta Axial Transition FFs Lattice Evaluation of the FFs Results – Checking the Pion Pole dominance & Goldberger-Treiman (GT) relations in N-N & N-Δ Conclusions arXiv:0706.3011, to appear in PRD

3. Nucleon Axial Form Factors axial isovector current axial vector form factorinduced pseudoscalar G A (q 2 ) – from neutrino scattering & pion electroproduction G P (q 2 ) – from muon capture experiments theoretically studied in chiral effective theories axial charge G A (0) = 1.2695 (29) from nuclear β decay pioneering lattice study in PRL 74 2172 (1995) ( K.F. Liu, S.J. Dong, T. Draper, W. Wilcox) recent study by LHPC+MILC, arXiv:0705.4295

4. Pseudoscalar Form Factor & PCAC PCAC in hadron world Axial WT identity in QCD pseudoscalar current πΝΝ form factor defined via connected to πΝΝ strong coupling constant g πΝΝ = G πΝΝ (m π 2 ) PCAC

5. Pion Pole dominance & GT relations Pion pole on RHS constraints the induced pseudoscalar and leads to Goldberger-Treiman relation at q 2 = 0 satisfy to 5% accuracy from low energy πΝΝ dynamics also fixes the ratio

6. Nucleon to Δ(1232) Axial Transition Form Factors transverse part Adler parameterization small ≈ 0 dominant FFs C 5 A analogous to G A (q 2 )C 6 A analogous to G P (q 2 ) not much known experimentally electroproduction experiments at JLab will measure N to Δ parity violating asymmetry connected to C 5 A theoretical arguments indicate that C 3 A, C 4 A are small

7. Lattice study in PRL 98,052003 (2006) established smallness of C 3 A and C 4 A, predicted q 2 dependence of dominant form factors C 5 A and C 6 A

8. Pseudoscalar πΝΔ Form Factor & PCAC πΝΔ form factor defined via connected to πΝΔ strong coupling constant g πΝΔ = G πΝΔ (m π 2 ) PCAC Non-diagonal Goldberger-Treiman relation Pion pole dominance relates:..and fixes the ratio

9. Evaluating Form Factors from Lattice QCD measure 3-point-functions of axial & pseudoscalar currents form ratios where t- and Z- dependence cancels determine the optimal linear combination of 3pts kinematics: X maximal number of momentum vectors contribute in rotationally symmetric fashion

10. optimizing the measurement sequential inversions through the sink X only one sequential inversion for G A (Q 2 ), G P (Q 2 ), G πNN (Q 2 ) all operators and momentameasured at small cost look for plateau in t 1 / Smear source & sink quarks to damp fast the excited states simultaneous overconstrained analysis of all data maximal accuracy for the form factors – Q 2 dependence

11. Lattice parameters Wilson N F = 0 β=6.0 32 3 x64 a=0.09 fm m π = 0.56 GeV m π = 0.49 GeV m π = 0.41 GeV Wilson N F = 2 β=5.6 a=0.08 fm 24 3 x40 m π = 0.69 GeV (TXL) 24 3 x40 m π = 0.51 GeV (TXL) 24 3 x32 m π = 0.38 GeV (DESY) Nucleon Axial N-to-Δ Axial + Hybrid scheme MILC N F = 2 + 1 Domain Wall valence (L 5 =16) a=0.125 fm am s am u m π 0.05 0.03 0.59 GeV 20 3 x64 0.05 0.02 0.50 GeV 20 3 x64 0.05 0.01 0.36 GeV 28 3 x64

12. plateaus for G πΝΝ Wilson N F =0, 32 3 x64, m π =0.49 GeV N F =2, 24 3 x40, m π =0.69 GeV Wilson N F =0, 32 3 x64, m π =0.41 GeV G A (Q 2 ), G P (Q 2 ), G πNN (Q 2 ) C 5 A (Q 2 ), C 6 A (Q 2 ), G πNΔ (Q 2 ) MILC(DWF) 0.01/0.05, m π =0.36 GeV 20 3 x64 vs 28 3 x64, source-sink distance 11a vs 13a Volume (2.5fm) 3 vs (3.5fm) 3 Ground state dominance Checking the parameters

13. Results (I) – Nucleon Axial Form Factors Hybrid results from 0705.4295 LHPC & MILC (Hägler etal) dipole fit describes well G A m A >=1.5 GeV (solid / fit) m A =1.1 GeV (dotted / exp) pion pole dominates G p (dash) monopole fit (solid)

14. Results (II) – N to Δ Axial Transition FFs dipole fit describes well C A 5 m A >=1.5 GeV (solid / fit) m A =1.28 GeV (dotted / exp) pion pole dominates C A 6 (dash: wilson) (dot: MILC) monopole fit (solid)

15. Results (III) – Checking Ratios of GT relations pion pole dominance renormalization constants, f π, m q cancel 1.63(1) 1.60(2) 1.73(3) weak Q 2 and m q dependence

16. Conclusions momentum dependence of the NN & NΔ axial form factors is evaluated optimally in Lattice QCD dipole dependence of G A and C 5 A is verified – requires larger axial mass at the 410 MeV pion lattices monopole behavior of G p and C 6 A is verified unquenching effects are visible at low Q 2 and m π = 360 MeV in the Hybrid scheme (MILC+DWF) – G A approaches expected behavior ratios of GT relations in NN & NΔ systems are satisfied – show very weak quark mass and Q 2 dependence