Hadronic Form-Factors Robert Edwards Jefferson Lab Abstract: A TECHNOLOGY TALK!! Outline a known but uncommon method in 3-pt function calculations that avoids sequential sources Demonstrate efficacy on some hadronic form-factors Particularly suitable for overlap quarks

Motivation Motivation for various electromagnetic form-factors – why do all of them together?? Pion -> Pion : transition from perturbative to non-perturbative regimes Rho -> Pion : isolate isovector meson exchange currents within deuterons, etc. Rho -> Rho : elucidate dominant exchange mechanisms in nuclei Nucleon -> Nucleon : fundamental, intensive experimental studies Delta -> Nucleon : info on shape/deformation of nucleon Delta -> Delta : allows access to Q 2 =0 and determine magnetic moments Similarly considerations apply to mixed valence form-factors and structure functions

Anatomy of a Matrix Element Calculation J f,i y : Current with desired quantum numbers of state A,B Normalize: Compute ratio: Problem: need propagator from t ! t 2 Want | h 0|J y |n i | 2 »  n,0 for best plateau

Method How to get the backward propagator in 3-pt: –Sequential inversion through insertion: Pros: can vary source and sink fields Cons: insertion momenta and operator fixed –Sequential inversion through sink: Pros: can vary insertion operator & momenta Cons: sink operator& momenta fixed. Baryon spin projection fixed Common problem is one vertex have a definite momentum Instead, make a sink (or source) propagator at definite momentum, but not sequential

Wall-sink(source) Method Put sink(source) quark at definite momentum (e.g., 0): Build any (accessible) hadron state at source/sink Avoid sequential inversions computing h B(t 2 ) O (t) A(t 1 ) i Need to gauge fix Known tricks: – Improve statistics with time-reversal in anti-periodic BC Method does work for Dirichlet boundary conditions: –Maintain equal source & sink separation from Dirichlet wall –Use time-reversal – then do wall source Overlap: can use multimass inversion both source/sink

Comparisons How does a wall sink (or source) method compare to say a sequential-through-sink method? Examples: Electromagnetic form-factors of –Pion -> Pion –Rho -> Pion –Nucleon -> Nucleon –Rho -> Rho (not presented) –Delta -> Nucleon (not presented) –Delta -> Delta (not presented)

Ratios Need new ratio method of correlation functions (e.g. for   ! N): where A, B, C are generic smearing labels, L is local, J  =    Similarly, R  N = R N  where  $ N. Note, momenta and smearing labels not interchanged The combination (R  N R N  ) 1/2 cancels all wave-function factors and exponentials

Computational Strategies Dynamical (full QCD) –N f = –Asqtad staggered sea quarks –Domain Wall valence quarks, 616MeV – 320MeV –Use partially quenched chiral perturbation theory –Low energy Gasser-Leutwyler constants are those of QCD! Other calculations presented by G. Fleming, D. Renner, W. Schroers

Partially Quenched Chiral Perturbation Theory Full QCD expensive! –Leverage off cheap(er) valence calcs Correct low-energy constants, in principle Must be in domain of validity Extend partially quenched  PT to include O(a) terms –Mixed actions Bär, Rupak, Shoresh, 2002, 2003

Asqtad Action: O(a 2 ) Perturbatively Improved MILC collab: computationally tractable full QCD Symanzik improved glue Smeared staggered fermions: S f (V,U) –Fat links remove taste changing gluons –Lepage term: 5-link O(a 2 ) correction of flavor conserving gluons –Third-nearest neighbor Naik term (thin links) –All terms tadpole improved

“Decay” in Quenched Approximation Dramatic behavior in Isotriplet scalar particle a 0 !  intermediate state Loss of positivity of a 0 propagator from missing bubble insertions Quenched a 0 has double pole in  PT Also appears in m  0 Bardeen, Duncan, Eichten, Thacker, 2000

Partially Quenched Singularity Non-positivity of a 0 correlator (Partially) Quenched singularity (still) present at m , valence a = m , sea a. Suggests not single staggered pion in chiral loops – taste breaking not neglible Need complete partial  PT –Vary valence and sea masses –Theory under development…

Pion Electromagnetic Form Factor F  (Q 2 ) Considered a good observable for studying the interplay between perturbative and non-perturbative descriptions of QCD –Large Q 2 scaling as predicted by Brodsky-Farrar –For small Q 2, vector meson dominance gives an accurate description – F  (0) = 1 by charge conservation –No disconnected diagrams –Experimental results are coming for Q 2 ¸ 1 GeV 2

Experimental Results Existing data fit VMD monopole formulae too well. Where’s perturbative QCD? Dispersion relation estimates – correct asymptotics but suggest a slow approach to perturbative behavior The introduction in many experimental papers read: –The valence structore of the pion is relatively simple. Hence, it is expected that the value of Q 2 down to which pQCD can be applied is lower than e.g. for the nucleon Results from Lattice QCD simulations can shed light on the debate

Comparing techniques for extracting F  (Q 2 ) Form factor definition Compare sequential-sink and wall-sink methods: Forward: APE smeared Sequential-sink: APE smeared Wall-sink: gauge-fixed wall smeared (zero sink momentum) Conclusion: wall-sink compares favorably

Partially Quenched DWF Form Factor DWF F  (Q 2,t) –Smaller mass close to experimental VMD. Charge radius (crude analysis): –Exp. h r 2 i = 0.439(8)fm 2, VMD ! 0.405fm 2 –Statistical: 0.156(5)fm 2 [m  =730MeV], 0.310( 6)fm 2 [m  =300MeV] strong mass dependence

Proton Electric Form-Factor Plateaus and Q 2 dependence reasonable: limited statistics All proton spin polarizations computed – can average

Proton E&M Form-Factors Comparison at fixed mass with experiment: reasonable agreement GMpGMp GEpGEp

Neutron Magnetic Form-Factor Comparison at fixed mass with experiment: reasonable agreement GMpGMp

Rho ! Pion Transition Form-Factor Electro-disintegration of deuteron intensively studied –Isovector exchange currents identified –Isoscalar exchange currents not clear h  (p f )|J  |  k (p i ) i » V(Q 2 ) First lattice measurement Ito-Gross 93 8 GeV 2

Conclusions Work in progress! –  !  –  ! N –  !  Wall-sink method (at least so far) appears competitive with sequential-sink method. Need tests at non-zero sink momenta Should probably use the wall-source method Cheaper! [greater reuse of propagators] Well-suited to multi-mass systems (e.g., overlap)