Hadron Structure with DWF (II)Summary of LHPC hadron structure program Preliminary results Hadron...

GPD’s and generalized form factors (GFF’s)Summary of LHPC hadron structure program

Preliminary results

Hadron Structure with DWF (II)

George T. Fleming

Yale University

3rd International Lattice Field Theory Network WorkshopJefferson Lab, Newport News, VA

George T. Fleming Hadron Structure with DWF (II)


Preliminary results

LHPC Hadron Structure project on USQCD resources

Dru RennerUniversity of Arizona

Ronald Babich, Richard Brower, James OsbornBoston University

Rebecca Irwin, Michael Ramsey-MusolfCalifornia Institute of Technology

Robert Edwards, Kostas Orginos (W&M), David RichardsThomas Jefferson National Accelerator Facility

Bojan Bistrovic, Jonathan Bratt, Patrick Dreher, Oliver Jahn,John Negele, Andrew Pochinsky, Dmitry Sigaev

Massachusetts Institute of TechnologyMatthias Burkardt, Michael Engelhardt

New Mexico State UniversityGeorge FlemingYale University



Preliminary results

LHPC – International Collaborators

Constantia Alexandrou, Antonios TsapalisUniversity of Cyprus

Wolfram SchroersNIC/DESY, ZeuthenPhilippe de Forcrand

ETH-Zurich and CERNPhilipp Hagler

Vrije Universiteit, Amsterdam



Preliminary results

PDF’s and Generalized PDF’s (GPD’s)

p

P=(Ep,p→

p) Xp

x•PXq

q=-Q

e+

k=(Ee,p→

e)

e+ / νe

k’

I Thanks to QCD factorization, very little knowledge ofhadron structure needed for inclusive DIS reactions.

I A lot of information about hadron structure is lost inthe inclusive sum over final states of remnant.

I Semi-inclusive and exclusive processes buildtwo-dimensional pictures of hadrons ( DVCS, DVMP, . . . )

I GPD’s are PDF’s with a transverse kick to struckparton before putting it back into the hadron.

I More kinematic variables: x= 12(xf + xi ), ξ= 1

2(xf − xi ), t (or Q2).

I GPD’s are form factors of the collection of partons with same fixed x .I As ξ → 0 and t → 0, GPD’s must reduce to ordinary PDF’s.

I t dependence at fixed x should contain information about spatial

distribution of all quarks with same x .



Preliminary results

GPD’s and Generalized Form Factors (GFF’s)

I Experimentalists measure matrix elements of light cone operatorsD

P′S′˛Oq

Γ

˛PSE

=

fiP′S′

˛q

„− x−

2

«Γ P exp

»−ig

R−x−/2

x−/2A+(y)dy

–q“

x2

”˛PS

flI They can be written in terms of GPD’s 1

R dx−2π

e ix−p+xD

P′S′˛Oq

Γ

˛PSE

= U(P′, S′)hHq(x, ξ, ∆2)Γ + Eq(x, ξ, ∆2) iσ·∆

2M

iU(P, S)

I Eight GPD’s in all: H, E , H, E , HT , ET , HT , ETI Using OPE, light cone operators replaced by tower of local twist

two operatorsDP′S′

˛Oµ1···µn

Γ

˛PSE

=D

P′S′˛q(x)iD(µ1 · · · iDµn−1 Γµn)q(x)

˛PSE

I They can be parameterized by generalized form factors (GFF’s).DP′S′

˛Oµ1µ2

q

˛PSE

=

U(P′, S′)

"A

q20(Q2)γ(µ1 ∆µ2) + B

q20(Q2)

iσ(µ1α∆α

2M∆µ2) + C

q2 (Q2)

∆(µ1 ∆µ2)

2M

#U(P, S)

I Nine GFF’s: Ani , Bni , Cn, Ani , Bni , ATni , BTni , ATni , BTni1

X. D. Ji, hep-ph/9807358. Recent review: M. Diehl, Phys. Rept. 388, 41-277 (2003).



Preliminary results

Equivalence of GPD’s and GFF’s

I GPD’s and GFF’s are formally equivalent by Mellin transformation

R 1−1 dx xn−1Hq(x, ξ, Q2) =

Pn−1i=0, even A

qni (Q

2)(−2ξ)i + δn, evenCqn (Q2)(−2ξ)nR 1

−1 dx xn−1Eq(x, ξ, Q2) =Pn−1

i=0, even Bqni (Q

2)(−2ξ)i − δn, evenCqn (Q2)(−2ξ)n

I Choice of GPD’s vs. GFF’s depends on physics.GPD: PDF’s and transverse PDF’sGFF: elastic form factors and nucleon spin

I In Euclidean lattice QCD, only GFF’s can be computed directly.

I Many GFF’s are familiar experimental quantities:I A

q10(Q2) = F

q1 (Q2), B

q10(Q2) = F2

2 (Q2)

I eAq10(Q2) = G

qA(Q2), eBq

10(Q2) = GqP

(Q2),

I Jq = 12

`A

q20(0) + B

q20(0)

´, 1

2∆Σq = eAq

10(0)

I Lq = Jq − 12∆Σq

ID

xn−1E

q= A

qn0(0),

Dxn−1

E∆q

= eAqn0(0),

Dxn−1

Eδq

= AqTn0

(0)



Preliminary results

Summary of LHPC hadron structure program

I Long term program to compute all n ≤ 4 GFF’s in dynamicallattice QCD.

I Current pion masses mπ ≈ 300 – 750 MeV and lattice spacinga ≈ 1

8 fm.

I Status of the calculationMatrix Operator GFF

Operators elements renorm. extraction AnalysisqΓµq Done! Done! Done! Preliminary

qΓ(µDν)q Done! Done! Done! Preliminary

qΓ(µDνDρ)q Done! Done! Done! Very Preliminary

qΓ(µDνDρDσ)q Not yet Done! Not yet Not yet

I Only isovector flavor combinations for GFF’s in this round.

I Finite perturbative renormalization needed to quote results in MSscheme. D

P′S′˛Oµ1···µn

Γ

˛PSEMS

= ZD

P′S′˛Oµ1···µn

Γ

˛PSElatt

I Lighter pion masses mπ ≈ 250 MeV finished by next year.



Preliminary results

Perturbative renormalization of twist two matrix elements

Tree level: Z = 1, One loop HYP corrections: < 10%.

B. Bistrovic, Ph. D. Thesis, MIT, 2005



Preliminary results

Proton Electromagnetic Form Factors

C. Perdrisat (W&M), JLab UsersGroup Meeting, June, 2005.

I Dramatic discrepancybetween experimentaltechniques. Two photonexchange processes may bethe cause.

I Data taking atQ2 = 9 GeV2 starting soon.

I Will Lattice QCD predictthe vanishing of Gp

E (Q2)around Q2 = 8 GeV2 ?



Preliminary results

Proton and Neutron Electromagnetic Form Factors

Empirical fit of experimental data

Simple parametrization of nucleon form factors

J. J. KellyDepartment of Physics, University of Maryland, College Park, Maryland 20742, USA

(Received 29 September 2004; published 8 December 2004)

This Brief Report provides simple parametrizations of the nucleon electromagnetic form factors using

functions of Q2 that are consistent with dimensional scaling at high Q2. Good fits require only four parameters

each for GEp, GMp, and GMn and only two for GEn.

DOI: 10.1103/PhysRevC.70.068202 PACS number(s): 14.20.Dh, 13.40.Gp

Nucleon electromagnetic form factors are needed formany calculations in nuclear physics. Hence, it would beuseful to have a simple parametrization that accurately rep-resents the data over a wide range of Q2 with reasonablebehavior for both Q2→0 and Q2→!. To obtain reasonablebehavior at low Q2 the power-series representation shouldinvolve only even powers of Q. At high Q2 dimensional scal-ing rules require G"Q!4 apart from slowly varying logarith-mic corrections that can be ignored safely for most applica-tions. However, the most common parametrizations violateone or both of these conditions. Often one uses the reciprocalof a polynomial in Q [1–3], but then the rms radius cannot bedetermined because such a parametrization includes unphysi-cal odd powers of Q. This problem can be circumventedusing the reciprocal of a polynomial in Q2, but to obtaingood fits for Q2 in the several !GeV/c"2 range one must useso many terms that the form factor falls too rapidly at largeQ2 [4]. Yet another parametrization is based upon acontinued-fraction expansion in Q2 [5,6], but the limitingQ!4 behavior is usually not enforced because the requiredparameter constraints become quite cumbersome. In Ref. [7]I proposed a parametrization based upon charge and magne-tization densities that was designed to enforce both condi-tions, but the representations in terms of Fourier-Bessel or

Laguerre-Gaussian expansions require a fairly large numberof parameters and are somewhat difficult to implement incalculations that are not based upon densities. In this BriefReport, I propose a much simpler parametrization that issuitable for a wide variety of calculations.Perhaps the simplest parametrization takes the form

G!Q2" "

#k=0

n

ak#k

1 +#k=1

n+2

bk#k

, !1"

where both numerator and denominator are polynomials in#=Q2 /4mp

2 and where the degree of the denominator is largerthan that of the numerator to ensure that G"Q!4 for largeQ2. For magnetic form factors we include a factor of $ onthe right-hand side, such that a0$1 if the data for low Q2 arenormalized accurately. With n=1 and a0=1, this parametri-zation provides excellent fits to GEp, GMp /$p, and GMn /$n

using only four parameters each. However, this approach isless successful for GEn because the existing data are still toolimited. Therefore, for GEn I continue to use the Galster pa-rametrization [8],

FIG. 1. Fits to nucleon electromagnetic form factors. For GEn, data using recoil or target polarization [16–22] are shown as filled circleswhile data obtained from the deuteron quadrupole form factor [23] are shown as open circles.

PHYSICAL REVIEW C 70, 068202 (2004)

0556-2813/2004/70(6)/068202(3)/$22.50 ©2004 The American Physical Society068202-1

J. J. Kelly (Maryland), Phys. Rev. C 70, 068202 (2004)

G(Q2) =G(0)

1 + b1τ + b2τ2

τ =Q2

4m2p

GnE (Q2) =

Aτ

1 + BτGD (Q2)

GD (Q2) =1

(1 + Q2/Λ2)2

Λ2 = 0.71 (GeV)2

I Using covariance matrix2 these fits can be tranformed intoDirac-Pauli form with isospin decomposition.

2J. J. Kelly, private communicationGeorge T. Fleming Hadron Structure with DWF (II)


Preliminary results

Isospin decomposition of nucleon form factors

0 2 4 6 8 100

0.5

1

(F1p

+ F

1n)

/ GD

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

(F1p

- F

1n)

/ GD

0 2 4 6 8 10

-0.2

-0.1

0

0.1

0.2

0.3

(F2p

+ F

2n)

/ GD

0 2 4 6 8 1000.511.522.533.5

(F2p

- F

2n)

/ GD

0 2 4 6 8 10

Q2 (GeV

2)

-0.2

-0.1

0

0.1

0.2

0.3

(F2p

+ F

2n)

/ (F

1p +

F1n

)

0 2 4 6 8 10

Q2 (GeV

2)

00.511.522.533.5

(F2p

- F

2n)

/ (F

1p -

F1n

)

0 2 4 6 8 100

0.5

1

(GE

p + G

En)

/ GD

0 2 4 6 8 10-0.5

0

0.5

1

(GE

p - G

En)

/ GD

0 2 4 6 8 100

0.5

1

(GM

p + G

Mn)

/ GD

0 2 4 6 8 100

1

2

3

4

5

(GM

p - G

Mn)

/ GD

0 2 4 6 8 10

Q2 (GeV

2)

0

0.5

1

1.5

(GE

p + G

En)

/ (G

Mp +

GM

n)

0 2 4 6 8 10

Q2 (GeV

2)

-0.1

0

0.1

0.2

(GE

p - G

En)

/ (G

Mp -

GM

n)



Preliminary results

Nucleon Isovector F1 (or A10) Form Factor on the Lattice

0 0.2 0.4 0.6m

2

π (GeV)2

0

0.2

0.4

0.6

0.8

sqrt

[⟨r2 ⟩ chp

− ⟨r

2 ⟩ chn]

(fm

)

exptmπ = 775 MeV

Dirac isovector charge radius

Preliminary

I For Q2 . 1 GeV2, fitting lattice datato dipole ansatz gives the Dirac

charge radius⟨r2

⟩u−d

ch.

I Chiral extrapolation using leadinganalytic and non-analytic terms andfinite range regulator(PRL 86 5011).

Dr2Eu−d

ch= a0 − 2

“1 + 5g2

A

”(4πfπ)2

1

2log

m2

π

m2π + Λ2

!

I Best fit: Λ ≈ 740 MeV.



Preliminary results


0 0.2 0.4 0.6m

2

π (GeV)2

0

0.2

0.4

0.6

0.8

sqrt

[⟨r2 ⟩ chp

− ⟨r

2 ⟩ chn]

(fm

)

exptmπ = 696 MeV

mπ = 775 MeV


Preliminary


charge radius⟨r2

⟩u−d

ch.


Dr2Eu−d

ch= a0 − 2

“1 + 5g2

A

”(4πfπ)2

1

2log

m2

π

m2π + Λ2

!




Preliminary results


0 0.2 0.4 0.6m

2

π (GeV)2

0

0.2

0.4

0.6

0.8

sqrt

[⟨r2 ⟩ chp

− ⟨r

2 ⟩ chn]

(fm

)

exptmπ = 605 MeV

mπ = 696 MeV

mπ = 775 MeV


Preliminary


charge radius⟨r2

⟩u−d

ch.


Dr2Eu−d

ch= a0 − 2

“1 + 5g2

A

”(4πfπ)2

1

2log

m2

π

m2π + Λ2

!




Preliminary results


0 0.2 0.4 0.6m

2

π (GeV)2

0

0.2

0.4

0.6

0.8

sqrt

[⟨r2 ⟩ chp

− ⟨r

2 ⟩ chn]

(fm

)

exptmπ = 498 MeV

mπ = 605 MeV

mπ = 696 MeV

mπ = 775 MeV


Preliminary


charge radius⟨r2

⟩u−d

ch.


Dr2Eu−d

ch= a0 − 2

“1 + 5g2

A

”(4πfπ)2

1

2log

m2

π

m2π + Λ2

!




Preliminary results


0 0.2 0.4 0.6m

2

π (GeV)2

0

0.2

0.4

0.6

0.8

sqrt

[⟨r2 ⟩ chp

− ⟨r

2 ⟩ chn]

(fm

)

exptmπ = 359 MeV

mπ = 498 MeV

mπ = 605 MeV

mπ = 696 MeV

mπ = 775 MeV


Preliminary


charge radius⟨r2

⟩u−d

ch.


Dr2Eu−d

ch= a0 − 2

“1 + 5g2

A

”(4πfπ)2

1

2log

m2

π

m2π + Λ2

!




Preliminary results


0 0.2 0.4 0.6m

2

π (GeV)2

0

0.2

0.4

0.6

0.8

sqrt

[⟨r2 ⟩ chp

− ⟨r

2 ⟩ chn]

(fm

)

exptmπ = 301 MeV

mπ = 359 MeV

mπ = 498 MeV

mπ = 605 MeV

mπ = 696 MeV

mπ = 775 MeV


Preliminary


charge radius⟨r2

⟩u−d

ch.


Dr2Eu−d

ch= a0 − 2

“1 + 5g2

A

”(4πfπ)2

1

2log

m2

π

m2π + Λ2

!




Preliminary results

Nucleon Isovector F1 at Higher Q2

PreliminaryI Even at mπ≈ 600

MeV, extraction athigher Q2 is noisy.

I Both polarizationtransfer experimentsand lattice results stillconsistent with dipoleansatz up to 4 GeV2.

I Lattice QCDsimulations will needmuch more statisticsto reach 8 GeV2

before experiments.



Preliminary results

Nucleon Isovector GE at Higher Q2

Preliminary

I Fits of experimental data suggestGu−d

E vanishes aroundQ2 ∼ 4 GeV2.

I With mπ ≈ 600 MeV on 282configs, the lattice points areconsistent with somethinginteresting happening at 4 GeV2.

I Lattice QCD simulations willneed more statistics to sayanything more conclusive.



Preliminary results

Nucleon Isovector Ratio F2/F1

Preliminary



Preliminary results

Axial Charge: gA = 〈1〉∆u−∆d = Au−d10 (0)

0.2 0.4 0.6 0.8

mπ2 (GeV

2)

0

0.2

0.4

0.6

0.8

1

1.2

g A

DWF / MILC L = 3.5 fmDWF / MILC L = 2.5 fmWILSON / SESAM L = 1.4 fmExperiment

Dru B. Renner, Lattice 2005



Preliminary results

Tensor Charge: gT = 〈1〉δu−δd = Au−dT10 (0)

0.2 0.4 0.6 0.8

mπ2 (GeV

2)

0

0.2

0.4

0.6

0.8

1

1.2

g T

DWF / MILC L = 3.5 fmDWF / MILC L = 2.5 fmWILSON / SESAM L = 1.4 fm




Preliminary results

Momentum Fraction: 〈x〉u−d = Au−d20 (0)

0 0.2 0.4 0.6 0.8

mπ2 (GeV

2)

0.1

0.15

0.2

0.25

0.3

<x>

u-d





Preliminary results

Polarized Momentum Fraction: 〈x〉∆u−∆d = Au−d20 (0)

0.2 0.4 0.6 0.8

mπ2 (GeV

2)

0

0.2

0.4

0.6

0.8

1

<x>

u-d /

<x>

∆u-∆

d





Preliminary results

Transverse quark distributions

0.20.4

0.60.8

1

-1-0.5

00.5

10

2

4

6

8

0

2

4

b?(fm) xM. Burkardt hep-ph/0207047

0 0.2 0.4 0.6 0.8

-Q2 / GeV

2

0.25

0.5

0.75

1

1.25

Scale

d G

FF

s

A1

u-d(-Q

2)

A20

u-d(-Q

2)

! = 283x32, m

"=352 MeV

W. Schroers, Lattice 2005

Aqn0(−∆2

⊥) =

Zd2b⊥e i∆⊥·b⊥

Z 1

−1xn−1q(x, b⊥)

Db2⊥

Eq

(n)= −4

Aq′n0(0)

Aqn0(0)

limx→1

q(x, b⊥) ∝ δ(b2⊥)

I Higher moments An0 weight x ∼ 1.

I Slope of Aqn0 decreases as n increases.



Preliminary results

Summary and outlook

I Large scale computation of isovector matrix elements (n ≤ 3)is done. Data analysis is proceeding rapidly. Expect publishedresults soon.

I Perturbative renormalization complete. B. Bistrovic (MIT)

I Reaching higher Q2 is high priority for nucleon form factors.

I Isoscalar and strange matrix elements are O(10)−O(100)times harder to compute due to statistical noise. We’remaking our first serious attempt this year.


Hadron Structure with DWF (II)Summary of LHPC hadron structure program Preliminary results Hadron...

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