Hadronic Form-FactorsRobert 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 Q2=0 and determine magnetic

moments

• Similarly considerations apply to mixed valence form-factors and structure functions

Anatomy of a Matrix Element Calculation

Jf,iy : Current with desired quantum numbers of state A,B

Normalize:

Compute ratio:

Problem: need propagator from t ! t2

Want |h 0|Jy|ni|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 hB(t2) O(t) A(t1)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 = RN where $ N . Note, momenta and smearing labels not interchanged

• The combination (R N RN)1/2 cancels all wave-function factors and exponentials

Computational Strategies

• Dynamical (full QCD)– Nf = 2 + 1– 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(a2) Perturbatively Improved• MILC collab: computationally tractable full QCD• Symanzik improved glue

• Smeared staggered fermions: Sf(V,U)– Fat links remove taste changing gluons– Lepage term: 5-link O(a2) 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 a0!intermediate state

• Loss of positivity of a0 propagator from missing bubble insertions

• Quenched a0 has double pole in PT• Also appears in m0

0a 0a

0a0a

0a0a

0 Correlation Fna

Bardeen, Duncan, Eichten, Thacker, 2000

Quark lines

Partially Quenched Singularity

• Non-positivity of a0 correlator • (Partially) Quenched

singularity (still) present at m, valencea = m, seaa .

• 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(Q2)

• Considered a good observable for studying the interplay between perturbative and non-perturbative descriptions of QCD– Large Q2 scaling as predicted by Brodsky-Farrar

– For small Q2 , vector meson dominance gives an accurate description – F(0) = 1 by charge conservation

– No disconnected diagrams– Experimental results are coming for Q2 ¸ 1 GeV2

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 Q2 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(Q2)• 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(Q2,t)

– Smaller mass close to experimental VMD.

• Charge radius (crude analysis):– Exp. h r2i = 0.439(8)fm2 , VMD ! 0.405fm2

– Statistical: 0.156(5)fm2 [m=730MeV], 0.310( 6)fm2 [m=300MeV] strong mass dependence

Proton Electric Form-Factor

• Plateaus and Q2 dependence reasonable: limited statistics

• All proton spin polarizations computed – can average

Proton E&M Form-FactorsComparison at fixed mass with experiment: reasonable agreement

GMp

GEp

Neutron Magnetic Form-FactorComparison at fixed mass with experiment: reasonable agreement

GMp

Rho!Pion Transition Form-Factor• Electro-disintegration of deuteron intensively studied

– Isovector exchange currents identified– Isoscalar exchange currents not clear

• h (pf)|J|k(pi)i » V(Q2)• First lattice measurement

Ito-Gross 93

8 GeV2

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)