Hadron Structure with DWF (II)

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1 Yale University 3rd International Lattice Field Theory Network Workshop Jefferson Lab, Newport News, VA

2 LHPC Hadron Structure project on USQCD resources Dru Renner University of Arizona Ronald Babich, Richard Brower, James Osborn Boston University Rebecca Irwin, Michael Ramsey-Musolf California Institute of Technology Robert Edwards, Kostas Orginos (W&M), David Richards Thomas Jefferson National Accelerator Facility Bojan Bistrović, Jonathan Bratt, Patrick Dreher, Oliver Jahn, John Negele, Andrew Pochinsky, Dmitry Sigaev Massachusetts Institute of Technology Matthias Burkardt, Michael Engelhardt New Mexico State University George Fleming Yale University

3 LHPC International Collaborators Constantia Alexandrou, Antonios Tsapalis University of Cyprus Wolfram Schroers NIC/DESY, Zeuthen Philippe de Forcrand ETH-Zürich and CERN Philipp Hägler Vrije Universiteit, Amsterdam

4 PDF s and Generalized PDF s (GPD s) e + k=(e e,p e ) x P P=(E p,p p ) p e + / ν e k q=-q X q X p Thanks to QCD factorization, very little knowledge of hadron structure needed for inclusive DIS reactions. A lot of information about hadron structure is lost in the inclusive sum over final states of remnant. Semi-inclusive and exclusive processes build two-dimensional pictures of hadrons ( DVCS, DVMP,... ) GPD s are PDF s with a transverse kick to struck parton before putting it back into the hadron. More kinematic variables: x= 2 (x f + x i ), ξ= 2 (x f x i ), t (or Q 2 ). GPD s are form factors of the collection of partons with same fixed x. As ξ and t, GPD s must reduce to ordinary PDF s. t dependence at fixed x should contain information about spatial distribution of all quarks with same x.

5 GPD s and Generalized Form Factors (GFF s) Experimentalists measure matrix elements of light cone operators D O P S q Γ PS E fi = P S q x Γ P exp 2 They can be written in terms of GPD s R dx 2π eix p + D O x P S q Γ PS E h = U(P, S ) ig R x /2 x A + (y)dy /2 H q (x, ξ, 2 )Γ + E q (x, ξ, 2 ) iσ 2M fl q x2 PS i U(P, S) Eight GPD s in all: H, E, H, Ẽ, HT, E T, HT, Ẽ T Using OPE, light cone operators replaced by tower of local twist two operators D P S O µ µn Γ PS E = DP q(x)id E S (µ µ id n Γ µn) q(x) PS They can be parameterized by generalized form factors (GFF s). D O P S µ µ 2 q " E = PS U(P, S ) A q 2 (Q2 )γ (µ µ 2 ) + B q 2 (Q2 ) iσ(µ α α µ 2 ) + C q 2 2M (Q2 ) (µ µ 2 ) # U(P, S) 2M Nine GFF s: A ni, B ni, C n, Ã ni, Bni, A Tni, B Tni, Ã Tni, BTni X. D. Ji, hep-ph/ Recent review: M. Diehl, Phys. Rept. 388, (23).

6 Equivalence of GPD s and GFF s GPD s and GFF s are formally equivalent by Mellin transformation R dx x n H q (x, ξ, Q 2 ) = R dx x n E q (x, ξ, Q 2 ) = P n i=, even Aq ni (Q2 )( 2ξ) i + δ n, evencn q (Q2 )( 2ξ) n P n i=, even Bq ni (Q2 )( 2ξ) i δ n, evencn q (Q2 )( 2ξ) n Choice of GPD s vs. GPD: PDF s and transverse PDF s GFF: elastic form factors and nucleon spin GFF s depends on physics. In Euclidean lattice QCD, only GFF s can be computed directly. Many GFF s are familiar experimental quantities: A q (Q2 ) = F q (Q2 ), B q (Q2 ) = F2 2 (Q2 ) A eq (Q2 ) = G q A (Q2 ), B eq (Q2 ) = G q P (Q2 ), J q = `Aq 2 2 () + Bq 2 (), 2 Σ q = A eq () L q = J q 2 Σq D x n E D = q Aq n (), x n E = A eq q n Dx (), n E = δq Aq Tn ()

7 Long term program to compute all n 4 GFF s in dynamical lattice QCD. Current pion masses m π 3 75 MeV and lattice spacing a 8 fm. Status of the calculation Matrix Operator GFF Operators elements renorm. extraction Analysis qγ µq Done! Done! Done! Preliminary qγ (µ D ν) q Done! Done! Done! Preliminary qγ (µ D ν D ρ) q Done! Done! Done! Very Preliminary qγ (µ D ν D ρd σ) q Not yet Done! Not yet Not yet Only isovector flavor combinations for GFF s in this round. Finite perturbative renormalization needed to quote results in MS scheme. D D E EMS = Z P S O µ µn Γ PS P S O µ µn Γ PS latt Lighter pion masses m π 25 MeV finished by next year.

8 Perturbative renormalization of twist two matrix elements Tree level: Z =, One loop HYP corrections: < %. B. Bistrović, Ph. D. Thesis, MIT, 25

9 Proton Electromagnetic Form Factors C. Perdrisat (W&M), JLab Users Group Meeting, June, 25. Dramatic discrepancy between experimental techniques. Two photon exchange processes may be the cause. Data taking at Q 2 = 9 GeV 2 starting soon. Will Lattice QCD predict the vanishing of G p E (Q2 ) around Q 2 = 8 GeV 2?

10 omial in Q [ 3], but then the rms radius cannot be d because such a parametrization includes unphysiowers of Q. This problem Summary can beof circumvented LHPC hadron where structure both numerator program and denominator are polynomials in reciprocal of a polynomial in Q 2, but to obtain =QPreliminary 2 /4m 2 p and where results the degree of the denominator is larger for Q 2 in the several GeV/c 2 range one must use than that of the numerator to ensure that G Q!4 for large terms that the form factor falls too rapidly at large Q 2. For magnetic form factors we include a factor of on Yet another parametrization is based upon a the right-hand side, such that a Proton and Neutron Electromagnetic if the data for low QForm Factors 2 are -fraction expansion in Q 2 [5,6], but the limiting vior is usually not enforced because the required constraints become quite cumbersome. In Ref. [7] d a parametrization based upon charge and magneensities that was designed to enforce both condithe representations in terms of Fourier-Bessel or normalized accurately. With n= and a =, this parametrization provides excellent fits to G Ep, G Mp / p, and G Mn / n using only four parameters each. However, this approach is less successful for G En because the existing data are still too limited. Therefore, for G En I continue to use the Galster parametrization [8], Empirical fit of experimental data + b k k= G(Q 2 ) = τ = G n E (Q2 ) = G D (Q 2 ) = G() + b τ + b 2 τ 2 Q 2 4m 2 p Aτ + Bτ G D (Q 2 ) ( + Q 2 /Λ 2 ) 2 Λ 2 =.7 (GeV) 2 Fits to nucleon electromagnetic form factors. For G En, data using recoil or target polarization [6 22] are shown as filled circles obtained J. fromj. the Kelly deuteron (Maryland), quadrupole form Phys. factor Rev. [23] arec shown 7, 6822 as open circles. (24) Using covariance matrix 2 these fits can be tranformed into Dirac-Pauli form with isospin decomposition. /24/7(6)/6822(3)/$ The American Physical Society 2 J. J. Kelly, private communication

11 Isospin decomposition of nucleon form factors 3 (F p + F n ) / G D (F 2p + F 2n ) / G D (F 2p + F 2n ) / (F p + F n ) Q 2 (GeV 2 ) (F p - F n ) / G D (F 2p - F 2n ) / G D Q 2 (GeV 2 ) (F 2p - F 2n ) / (F p - F n ) (G Ep + G En ) / G D (G Mp + G Mn ) / G D (G Ep + G En ) / (G Mp + G Mn ) Q 2 (GeV 2 ) Q 2 (GeV 2 ) (G Mp - G Mn ) / G D (G Ep - G En ) / G D (G Ep - G En ) / (G Mp - G Mn )

12 p Nucleon Isovector F (or A ) Form Factor on the Lattice r 2 n ch ] (fm) sqrt[ r 2 ch expt m π = 775 MeV Dirac isovector charge radius m 2 π (GeV)2 Preliminary For Q 2 GeV 2, fitting lattice data to dipole ansatz gives the Dirac charge radius r 2 u d ch. Chiral extrapolation using leading analytic and non-analytic terms and finite range regulator (PRL 86 5). Dr 2E u d = a 2 ch Best fit: + 5gA 2 (4πf π) 2 Λ 74 MeV. 2 2 log mπ m 2 π + Λ2!

13 p Nucleon Isovector F (or A ) Form Factor on the Lattice r 2 n ch ] (fm) sqrt[ r 2 ch expt m π = 696 MeV m π = 775 MeV Dirac isovector charge radius m 2 π (GeV)2 Preliminary For Q 2 GeV 2, fitting lattice data to dipole ansatz gives the Dirac charge radius r 2 u d ch. Chiral extrapolation using leading analytic and non-analytic terms and finite range regulator (PRL 86 5). Dr 2E u d = a 2 ch Best fit: + 5gA 2 (4πf π) 2 Λ 74 MeV. 2 2 log mπ m 2 π + Λ2!

14 p Nucleon Isovector F (or A ) Form Factor on the Lattice r 2 n ch ] (fm) sqrt[ r 2 ch expt m π = 65 MeV m π = 696 MeV m π = 775 MeV Dirac isovector charge radius m 2 π (GeV)2 Preliminary For Q 2 GeV 2, fitting lattice data to dipole ansatz gives the Dirac charge radius r 2 u d ch. Chiral extrapolation using leading analytic and non-analytic terms and finite range regulator (PRL 86 5). Dr 2E u d = a 2 ch Best fit: + 5gA 2 (4πf π) 2 Λ 74 MeV. 2 2 log mπ m 2 π + Λ2!

15 p Nucleon Isovector F (or A ) Form Factor on the Lattice r 2 n ch ] (fm) sqrt[ r 2 ch expt m π = 498 MeV m π = 65 MeV m π = 696 MeV m π = 775 MeV Dirac isovector charge radius m 2 π (GeV)2 Preliminary For Q 2 GeV 2, fitting lattice data to dipole ansatz gives the Dirac charge radius r 2 u d ch. Chiral extrapolation using leading analytic and non-analytic terms and finite range regulator (PRL 86 5). Dr 2E u d = a 2 ch Best fit: + 5gA 2 (4πf π) 2 Λ 74 MeV. 2 2 log mπ m 2 π + Λ2!

16 p Nucleon Isovector F (or A ) Form Factor on the Lattice r 2 n ch ] (fm) sqrt[ r 2 ch expt m π = 359 MeV m π = 498 MeV m π = 65 MeV m π = 696 MeV m π = 775 MeV Dirac isovector charge radius m 2 π (GeV)2 Preliminary For Q 2 GeV 2, fitting lattice data to dipole ansatz gives the Dirac charge radius r 2 u d ch. Chiral extrapolation using leading analytic and non-analytic terms and finite range regulator (PRL 86 5). Dr 2E u d = a 2 ch Best fit: + 5gA 2 (4πf π) 2 Λ 74 MeV. 2 2 log mπ m 2 π + Λ2!

17 p Nucleon Isovector F (or A ) Form Factor on the Lattice r 2 n ch ] (fm) sqrt[ r 2 ch Dirac isovector charge radius expt m π = 3 MeV m π = 359 MeV m π = 498 MeV m π = 65 MeV.2 m π = 696 MeV m π = 775 MeV m 2 π (GeV)2 Preliminary For Q 2 GeV 2, fitting lattice data to dipole ansatz gives the Dirac charge radius r 2 u d ch. Chiral extrapolation using leading analytic and non-analytic terms and finite range regulator (PRL 86 5). Dr 2E u d = a 2 ch Best fit: + 5gA 2 (4πf π) 2 Λ 74 MeV. 2 2 log mπ m 2 π + Λ2!

18 Nucleon Isovector F at Higher Q 2 Preliminary Even at m π 6 MeV, extraction at higher Q 2 is noisy. Both polarization transfer experiments and lattice results still consistent with dipole ansatz up to 4 GeV 2. Lattice QCD simulations will need much more statistics to reach 8 GeV 2 before experiments.

19 Nucleon Isovector G E at Higher Q 2 Preliminary Fits of experimental data suggest G u d E vanishes around Q 2 4 GeV 2. With m π 6 MeV on 282 configs, the lattice points are consistent with something interesting happening at 4 GeV 2. Lattice QCD simulations will need more statistics to say anything more conclusive.

20 Nucleon Isovector Ratio F 2 /F Preliminary

21 Axial Charge: g A = u d = Ãu d ().2 g A DWF / MILC L = 3.5 fm DWF / MILC L = 2.5 fm WILSON / SESAM L =.4 fm Experiment m π (GeV ) Dru B. Renner, Lattice 25

22 Tensor Charge: g T = δu δd = A u d T ().2 g T DWF / MILC L = 3.5 fm DWF / MILC L = 2.5 fm WILSON / SESAM L =.4 fm m π (GeV ) Dru B. Renner, Lattice 25

23 Momentum Fraction: x u d = A u d 2 ().3.25 <x> u-d.2.5 DWF / MILC L = 3.5 fm DWF / MILC L = 2.5 fm WILSON / SESAM L =.4 fm Experiment m π (GeV ) Dru B. Renner, Lattice 25

24 Polarized Momentum Fraction: x u d = Ãu d 2 () <x> u-d / <x> u- d DWF / MILC L = 3.5 fm DWF / MILC L = 2.5 fm WILSON / SESAM L =.4 fm Experiment m π (GeV ) Dru B. Renner, Lattice 25

25 Transverse quark distributions x.5 b? -.5 (fm) - M. Burkardt hep-ph/ A q n ( 2 ) = Z d 2 b e i b Z x n q(x, b ) D b 2 E q = 4 A q n () (n) A q n ().25! = 28 3 x32, m " =352 MeV lim q(x, b ) δ(b 2 x ) A u-d (-Q 2 ) u-d 2 A 2 (-Q ) Scaled GFFs Higher moments A n weight x. Slope of A q n decreases as n increases Q 2 / GeV 2 W. Schroers, Lattice 25

26 Summary and outlook Large scale computation of isovector matrix elements (n 3) is done. Data analysis is proceeding rapidly. Expect published results soon. Perturbative renormalization complete. B. Bistrović (MIT) Reaching higher Q 2 is high priority for nucleon form factors. Isoscalar and strange matrix elements are O() O() times harder to compute due to statistical noise. We re making our first serious attempt this year.