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' Domain wall valence on staggered sea: Hadron Structure $ George T. Fleming c & Yale University [Lattice Hadron Physics Collaboration] • Summary of LHPC hadron structure proposal. • Pion and nucleon electromagnetic form factors. • Nucleon magnetic moments, gA, . . . % LHPC Hadron Structure project on USQCD resources Dru Renner University of Arizona Richard Brower, James Osborn Boston University Rebecca Irwin, Michael Ramsey-Musolf California Institute of Technology Robert Edwards, David Richards Thomas Jefferson National Accelerator Facility Bojan Bistrovic, Jonathan Bratt, Patrick Dreher, Oliver Jahn, John Negele, Kostas Orginos, Andrew Pochinsky, Dmitry Sigaev Massachusetts Institute of Technology Matthias Burkardt, Michael Engelhardt New Mexico State University George Fleming Yale University Constantia Alexandrou, Antonios Tsapalis Philippe de Forcrand University of Cyprus ETH-Zürich and CERN Wolfram Schroers Philipp Hägler DESY, Zeuthen Vrije Universiteit, Amsterdam LHPC Hadron Structure project on USQCD resources • This years plan for valence DWF propagator computation: – a = 0.13 fm, mπ = 250 – 350 MeV – a = 0.09 fm, mπ = 250 – 775 MeV – a = 0.06 fm, mπ = 350 – 500 MeV • Available USQCD Resources: – 65% of time on gigE mesh clusters at JLab – 9% of time on USQCD’s QCDOC at BNL • Planned observable calculations include: Form factors at high Q2 : F1 , F2 , GA , GP , Fπ , . . . Transition form factors: ∆ → N , ρ → π, π ∗ → π, π ∗ → ρ Moments of structure functions for nucleon and pion: hxn iq , hxn i∆q , hxn iδq Generalized form factors which give access to transverse structure, quark spin ∆Σ and orbital angular momentum Lq of nucleon. – Non-perturbative renormalization of hxi moments in RI/MOM scheme. – Starting to incorporate disconnected contributions in all of the above . . . – – – – 2) The Pion Electromagnetic Form Factor F (Q π • Widely considered a good observable for studying the interplay between perturbative and non-perturbative descriptions of QCD – Asymptotic normalization fixed by precise measurement of pion decay 8παs(Q2)fπ2 Fπ (Q ) = Q2 2 as Q2 → ∞ – For small Q2, vector meson dominance (VMD) gives an accurate description, with normalization Fπ (0) = 1 given by charge conservation Fπ (Q2) ≈ 1 2 1 + Q m2ρ for Q2 m2ρ • Not too hard to calculate on the lattice since there are no disconnected diagrams. • Experimental results are coming for Q2 ≥ 1 GeV2 Pion Form Factor: A QCD Laboratory? Presented by G. M. Huber at 2004 JLab User’s Meeting. The Experimental Situation • Existing data fit VMD monopole formulae too well. Where’s perturbative QCD? • Results from Lattice QCD simulations will have a huge impact on the debate. Experimental Methods (I) ± Elastic Scattering of 300 GeV pions on atomic electrons − • Elastic π +e scattering provides direct measurement of Fπ , (One Photon Exchange). 2 0.2 2 Q Fπ (GeV) • Kinematics of pion backscatter in CM frame limits maximum Q2. CERN NA7: Amendolia et al., Nucl. Phys. B 277, 168 (1986) • CERN NA7: Even with 300 GeV pions from SPS incident on liquid hydrogen target, Q2max . 0.288 GeV2. 0.1 0 0 0.1 0.2 2 2 Q (GeV) • Hypothetically, for 1 TeV pions from Tevatron, Q2max . 1 GeV2. • For 7 TeV pions from LHC, Q2max . 7.1 GeV2. − Experimental Methods (II) e + p → e− + π + + n (One Photon Exchange) • Differential cross section depends on Q2, s and t p d2 σ = σL + σT + 2( + 1)σLT cos φπ dΩ +σTT cos 2φπ !−1 • Interference terms σLT and σTT extracted from fit and subtracted from dσ/dΩ. 2 15 10 5 2 0 2 Q = 1.0 (GeV/c) 2 -t = 0.09 - 0.13 (GeV/c) o o θπq = 1.1 - 6.8 2 = 2 |q|2 θe 1+ tan2 2 Q 2 Phys. Rev. Lett. 86, 1713 (2001) 2π d σ/(dt dφπ) (µb/(GeV/c) ) • Experiments hold Q2, s and t fixed and measure dσ/dΩ vs. φπ and virtual photon polarization . 0 π/2 π 3π/2 2π φπ Experimental Methods (III) L/T (or Rosenbluth) separation • Subtracted dσ/dΩ is σL + σT. Separation of σL and σT by dependence. • Near the unphysical pion pole : (t → Mπ2 ) −tQ2 2 2 2 gπN σL ∼ N (t)Fπ (Q , t) 2 2 (t − Mπ ) • Experiments access t < 0. Extrapolation introduces model dependence. dσ/dt (µb/(GeV/c)2) Phys. Rev. Lett. 86, 1713 (2001) 40 2 Q =0.60 25 2 Q =0.75 20 30 15 20 10 10 0 5 0 0.1 0.2 0 0 0.1 0.2 10 20 2 Q2=1.60 Q =1.00 10 0 0 0.1 0.2 0 0 0.1 0.2 -t (GeV/c)2 • Figure shows fits of VGL Regge model to JLab data where Fπ (Q2) is a free parameter. Model fits σL reasonably well. 2) Lattice techniques for extracting F (Q π • Form factor definition. π(pf ) Vµ(pf − pi, t) π(pi) t,q cont = ZV π(pf ) Vµ(pf − pi, t) π(pi) lat t i ,pi t ,p f f 2 = Fπ (Q )(pf + pi)µ • We use the sequential source method. • Main disadvantage: Momentum pf and pion quantum numbers are fixed at the sink. • Main advantages: Variety of meson form factors, e.g. ρ → γπ computable from same set of propagators. • Largest Q2 available in Breit frame (pf = −pi). Sequential propagators for several pf . • Fπ (Q2) extracted either from simultaneous fit to several two and three-point functions or by constructing ratios. Three point functions, smearings, and ratios • The three point correlator with local (L) or smeared (S) source A and sink B: ΓAB (t , t, t , p , p ) = e−i(xf −x)·pf π B (x , t ) |V (x, t)| π A(x , t ) e−i(x−xi)·pi π4π i f i f f f 4 i i • After inserting complete set of states and taking large Euclidean times ti t tf : ΓAB π4π ZB (pf )ZA∗ (pi) −(tf −t)E(p ) −(t−ti)E(p ) f e i → e π(pf ) V4(pf − pi, t) π(pi) 4E(pf )E(pi) • For the two point correlator: ΓAB ππ (ti , tf , p) ZB (p)ZA∗ (p) −(tf −ti)E(p) → e 2E(p) ti tf • Crucial point: ZL(p) = ZL. Also ZS (p) = ZS (“|p|”) and E(p) = E(“|p|”). • Ratio gives Fπ (Q2, t) independent of Z’s and exponentials: AB CL Γ (t , t, t , p , p ) Γ (t , t, p ) 2 Z E(|p |) f V i f f f π4π i ππ i Fπ (Q2, t) = CB ΓAL E(|pi|) + E(|pf )| ππ (ti , t, pi ) Γππ (ti , tf , pf ) Strategy for data analysis • When operators are smeared, a gauge-invariant Gaussian smearing function is used. • Smearing optimal for ground state overlap of nucleon operators at rest. • Simultaneous fit of smeared-smeared (SS) and smeared-local (SL) hP P i and hA4A4i correlators with constraints ZL(p) = ZL, ZS (p) = ZS (|p|) and E(p) = E(|p|). • For each Q2, compute Fπ (Q2, t) ratios using extracted energies E(|p|) and choose plateau ranges [tmin, tmax]. • For each Q2, simultaneously fit two and three point functions to extract constant Fπ (Q2) over range [tmin, tmax]. • Need small pi, pf to minimize statistical errors but need large Q2 to address physics. – Signal to noise ∝ exp {− [Eπ (p) − Eπ (0)] t}. • Staying in Breit frame is an important part of overall program. DWF Results: Ratio Method The Nucleon Electromagnetic Form Factors • Brodsky-Lepage predicted F1 ∼ Q−4 and F2 ∼ Q−6 as Q2 → ∞. • One Photon Exchange approximation may not be justified. • Softer scaling also possible (hep-ph/0212351) Q2 F2(Q2) ∼ const log2(Q2/Λ2) F1(Q2) 2 1.5 (a) CQM SU(6) Soliton SU(6) + CQ ff PFSA VMD 1 2 Q F2p/F1p • New polarization transfer experiments do not see expected scaling. Phys. Rev. Lett. 88, 092301 (2002) 0.5 0 (b) 0.75 QF2p/F1p • Older L/T separation experiments observed predicted behaviour. 0.5 Jones [11] This work 0.25 0 0 1 2 3 4 2 2 Q (GeV ) 5 6 Nucleon F2/F1 on the Lattice (I) • But, . . . We don’t control asqtad lattice generation. 2 (GeV ) (I=1) 0.5 Q F2 (I=1) 1.5 2 / κF1 2 (GeV) 0 mπ = 775 MeV mπ = 498 MeV mπ = 359 MeV 1.5 n expt: Q F2p / F1p (I=1) • Exploring sink momenta up to pf =(2,0,0). May reach Q2 ∼ 4 GeV2 with sufficient statistics. 2.5 1 / κF1 • Errors shown do not account for data covariance, are not jackknifed and are likely overestimated. 3 (I=1) • Our normalization is F2(Q2) → κ as Q2 → 0. VERY PRELIMINARY!!! Q F2 • Only I = 1 form factors computed so far to avoid disconnected diagrams. F1I=1 = F1p − F1n but F1n, F2n not known accurately for Q2 & 1 GeV2. 1 0.5 0 0 1 2 3 4 2 2 Q (GeV ) 5 6 Nucleon F2/F1 on the Lattice (II) VERY PRELIMINARY!!! 4 • F2I=1/F1I=1 → κp −κn as Q2 → 0. • PDG: κp = 1.792847351(28) (I=1) / F1 (I=1) • So, comparison of I = 1 with p − n could be OK with proper chiral extrapolation. 2 F2 • PDG: κn = −0.91304273(45) 3 1 0 0 mπ = 775 MeV mπ = 498 MeV mπ = 359 MeV Expt: κp - κn 0.5 1 2 1.5 2 2 Q (GeV ) 2.5 3 Nucleon axial charge PRELIMINARY Philipp Hägler (Vrije Universiteit) Summary and outlook • Lots of propagator and observable calculations already completed. Data analysis is proceeding rapidly. Expect published results soon. • Hashimoto’s talk has inspired me to look at pion scalar form factor, K`3, etc. Data is on disk . . . • Plan to measure mixed asqtad-DWF pion mass discussed in Baer’s talk. • Reaching higher Q2 is high priority. • Looking forward to dynamical chiral fermion lattices. . .