Nucleon form factors and quark charge densities

Transcription

1 Nucleon form factors and quark charge densities Carl E. Carlson The College of William and Mary in Virginia MAMI and Beyond 30 March 03 April 2009

2 Nucleon form factors and quark charge densities Show some data, and comment Application to hadronic corrections to atomic processes: status and possibilities for improvement Quark charge densities from form factors and jour view of the nucleon End 2

3 Form factors Definition (for nucleon) : p 2 λ 2 J µ p 1 λ 1 = ū(p 2,λ 2 ) = ū(p 2,λ 2 ) q = p 2 p 1 G M = F 1 + F 2 { { G E = F 1 q2 4M 2 F 2 γ µ F 1 (q 2 ) + γ µ G M (q 2 ) + 1 2M i } 2M σµν q ν F 2 (q 2 ) u(p 1,λ 1 ) } ( ) µ p1 + p 2 F2 (q 2 ) u(p 1,λ 1 ) 3

4 Form factor data 1.2 p GM /µ G p D 1.0 p 1.5 p M 1.0 µ G /G E p Borkowski (Mainz) Walker (SLAC) Bosted (SLAC) Andivahis (SLAC) Sill (SLAC) Christy (Hall C) Qattan (Hall A) Dieterich (Hall A) Punjabi (Hall A) Gayou (Hall A) Perdrisat (Hall C) Milbrath (MIT-Bates) Arrington (reanalysis of world data from Bonn, CEA, DESY, SLAC) Q 30 2 [GeV ] Q [GeV 10 2 ] 1.2 n GM /µ G n D 1.0 G E n 0.05 Passchier (NIKHEF) Seimetz (Mainz) Herberg (Mainz) Warren (Hall C) Madey (Hall C) Wojtsekhowski (Hall A) Schiavilla Lung (SLAC) Rock (SLAC) Anklin (Mainz) Kubon (Mainz) Xu Lachniet Q 2 [GeV ] Q 2 [GeV 2 ]

5 Form factor data Higher Q 2 GEn data relatively new GEp/GMp plot shows famous discrepancy between measurements using Rosenbluth and polarization transfer methods. Attributed to significant 2-photon corrections needed but not included in the Rosenbluth method. Plot as shown has polarization transfer points at Q 2 = 5.2, 6.8 and 8.54 GeV 2, from GEP - III experiment at Hall C at JLab. Data taken between October 2007 and June Analysis not final. Points on plot valid for error bars only. 5

6 Form factor data As service, plot shown at PANIC 08 (November 2008), definitely and clearly labeled Preliminary. PRELIMINARY Final analysis expected soon Most recent analysis shown at a talk away from JLab has data for ratio still falling but not crossing zero. 6

7 Form factor data There is new low Q2 proton data from Mainz Very precise in this region. Important for obtaining charge radii and for applications. Thanks to Jan Bernauer 7

8 Form factor data Polynomial Dipole FW Arrington 07 Elab(e ) = 315 GeV σ exp /σ dipole PRELIMINARY Q 2 (GeV/c) 2 8

9 Form factor data σ exp /σ dipole PRELIMINARY Polynomial Dipole FW Arrington 07 Elab(e ) = 450 GeV Q 2 (GeV/c) 2 9

10 Form factor data Polynomial Dipole FW Arrington 07 Elab(e ) = 720 GeV PRELIMINARY σ exp /σ dipole Q 2 (GeV/c) 2 10

11 Form factor data Q [GeV/c] Q 2 [(GeV/c) 2 ] A limit B limit ε E<180 MeV E>1.53 GeV 11 E=855 MeV

12 Hydrogen hyperfine splitting and proton FF subject: atomic physics from viewpoint of this conference: applied physics connection: precision atomic calculations description of proton structure form factor and structure function data 12

13 Hhfs and proton FF Hyperfine splitting = energy splitting from proton magnetic moment and electron magnetic moment interaction (in given spatial state, such as ground state). μ p μ e μ p μ e electron slightly higher energy than electron proton (spin-1) proton (spin-0) Measured energy splitting (13 figures): Ehfs(e p) = (9) MHz LO calculation (course material) gives Fermi energy, E p F = 8α3 m 3 r 3π µ Bµ p convention: measured mag. mom. for p, Bohr magneton for e. 13

14 Hhfs and proton FF with corrections, Ehfs(e p) = E F ( 1 + QED structure ) QED and... accurately known Want structure accurately enough for part-per-million (ppm) calculation Why? Challenge... New physics? Note: Hints of new physics in B-meson physics (BEACH 2008: Conference on Hyperons, Charm, and Beauty Hadrons) Appeared to be couple ppm discrepancy circa

15 Hhfs and proton FF Structure corrections come from 2-photon exchange e momenta of in and out e are essentially zero lots of energy short wavelength return energy here p Corrections involve two elastic form factors, or (via optical theorem) structure functions g 1 and g2 for inelastic case. structure = Zemach + Recoil + Polarizability = Z + R + pol structure (total) about 40 ppm, so need about 2% accuracy 15

16 Hhfs and proton FF Zemach corrections are NR part of elastic contributions Z = 8αm [ r dq GE (Q 2 )G M (Q 2 ] ) π Q 2 1 2αm r r 1 + Z κ p 0 Main contributions are from low Q 2 Using three modern FF fits, FF r Z (fm) Z (ppm) with small extra radiative correction AMT Arrington, Melnitchouk, Tjon AS Arrington, Sick Kelly Used middle value, and took uncertainty limit from spread 16

17 Hhfs and proton FF Results for other terms: R = fcn[ge,gm] = 5.85 ± 0.07 ppm pol= fcn[f2,g1,g2] Formulas known since 1966 (Drell and Sullivan) Need good g1, g2 data First non-zero results 2002 (Faustov and Martynenko) 2008 evaluation (Griffioen, Nazaryan, and me) pol = 1.88 ± (0.07)stat ± (0.60)syst ± (0.20)modeling ppm 17

18 Hhfs and proton FF For the record, pol = αm e 2(1 + κ p )πm p ( ) 1 = dq 2 Q 2 { F 2 2 (Q2 )+4m p ν th } dν ν2 β( ν2 Q 2 ) g 1(ν, Q 2 ), dq 2 = 2 dν 12m p 0 Q 2 ν th ν 2 β 2( ν2 Q 2 ) g 2(ν, Q 2 ) β(τ) = 4 [ ] 3τ + 2τ 2 + 2(2 τ) τ(τ + 1) 9 β 2 (τ) = 1 + 2τ 2 τ(τ + 1) Note: F2 2 term moved here to allow me 0 limit. 18

19 Hhfs and proton FF results Quantity value (ppm) uncertainty (ppm) (E hfs (e p)/e p F ) QED [aka, p µvp + p hvp + p weak ] 0.14 Z (using AMT) R (using AMT) pol (2008 evaluation) Total Deficit Agreement to ppm level no evidence of new physics To reduce error limits, improve low Q 2 elastic form factors systematic errors on g1 data for g2 on proton modeling where data lacking 19

20 Precision measurement of simple objects Currently, (g 2) µ has 3.6σ discrepancy between theory and experiment a µ = ( ) g 2 2 Measurement accurate to 0.54 ppm Theory (current) accurate to 0.44 ppm Within theory, hadronic corrections are 60 ppm ± 0.37 ppm I.e., bulk of theory error comes from hadronic corrections Fraction of ppm good goal for other precision quantities µ = α 2π

21 Matter distribution in coordinate space Commonplace: get charge and magnetic moment distributions for Fourier transform of corresponding FF For p or n, ρ C ( x) = (d 3 q)e i q x G E (q 2 ) r 2 ρ C (r ) r 2 ρ C (r ) r (fm) proton planar projection for neutron ρ C (b) = dz ρ C (r ) (b = x 2 + y 2 ) πb ρ C (b) (fm 1 ) neutron r (fm)

22 Matter distribution in coordinate space Derivation non-relativistic But usual wave function is frame dependent: usually equal time, and different points at equal time in one frame not a equal time in another Alternative: light front w. f. and light front coordinates, x + = t + z x = t z x = (x, y) Choice x + = 0, like observer moving at light speed in the z direction, is (longitudinal boost) Lorentz invariant. 22

23 Matter distribution in coordinate space Charge current component J+ : Each piece is positive operator : Quark transverse charge density in a nucleon : or : J + (x) =+2/3ū(x)γ + u(x) 1/3 d(x)γ + d(x) +... ρ N 0 ( b) = d 2 q (2π) 2 ei q b 1 2P + P +, q 2,λ J + (0) P +, q 2,λ from form factor definition : ρ N 0 (b) = qγ + q γ + q 2 ρ N 0 ( b) 1 2P + P +,R = 0,λ J + (x) P +,R = 0,λ x + =0,x =0,b 0 dq 2π QJ 0(bQ)F 1 (Q 2 ) 23

24 Unpolarized nucleon densities 2πb ρ C (b) (fm 1 ) proton b (fm) 2πb ρ C (b) (fm 1 ) neutron ρ C (b x,b y ) (units for b and ρn are fm. and fm 2, resp.) 24 Gerry Miller, summer 2007

25 Unpolarized nucleon densities Interpretation for neutron Known picture: higher x quarks have more compact distribution in transverse directions. Hence high x quarks contribute disproportionately to inner charge density Perturbative QCD idea: higher x quarks more likely to have same helicity of parent hadron than opposite. For neutron, this means d quarks. Hence d quarks give central charge density of neutron (in this light front view). 25 Arrington-Miller

26 Transversely polarized nucleon Polarize along x : ρ N T ( b) = 1 2P + P +,R = 0, s = ˆx J + (x) P +,R = 0, s = ˆx x+=0,x =0,x =b or, ρ N T ( b) = ρ N 0 (b) sinφ b 0 dq 2π Q 2 2M N J 1 (bq)f 2 (Q 2 ) ϕb is the azimuth angle of the impact parameter, 26

27 Transversely polarized nucleon b y fm b y fm pol b x 1.5 fm proton neutron Known: MDM in moving frame shows EDM (Einstein and Laub, 1908) d = v m b x fm 27

28 Transversely polarized nucleon Interpret: picture with spin pointing out of screen, observer on light front entering along -z direction z y For orbiting matter, current component J + = J 0 + J z is zero below and maximal above. I.e., we measure orbiting material above: deficit of + charge in proton means d- quarks are more likely to have non-zero angular momentum than u-quarks. (Cf., Guidal et al., or Brodsky and Gardiner) Reverse u and d for neutron. 28 M. Burkardt, G. Miller, M. Vanderhaeghen, B. Pasquini, S. Boffi...

29 Application to deuterons S and D spatial states, nucleon spins add to S = 1, total J = 1. Illustrations below are for m J = 1 and mj = 0 states. Plots are surfaces of fixed magnitude w. f., in coordinate space. Low fixed magnitude reached at high distances, where S state dominates and give spherical symmetry, or at hole in middle. Illustration is for a high fixed magnitude, showing toroidal shape for mj = 0 state. mj=1 mj=0 29 picture from Argonne theory group home page

30 check from deuteron FF data Same procedure ρ d m J =0 (b) = 0 ρ d m J =1 (b) = 0 dq 2π QJ 0(bQ) dq 2π QJ 0(bQ) 1 { G C + ηg M + η } 1 + η 3 G [ Q η = Q 2 /(4M 2 d )] { 1 (1 η)g C + 2ηG M 2η } 1 + η 3 (1 + 2η)G Q Standard GC, GM, GQ, analytic fit to data by Abbott et al. (2000) b y fm b y fm Ρ 0 d, Ρ 1 d fm b y fm b x fm dashed blue - mj = 1 solid red - mj = 0 mj = 1 mj = 0 b x fm

31 Summary new FF data, including data under analysis. Important for applications and for understanding of proton structure One application is to HHFS, where ppm calculation is possible with present proton structure data (form factors and structure functions). The ppm level significant in related contexts. Anticipate more accurate hadronic corrections with newer FF and g1, g2 data. Analytic fits to data useful. Many thanks to the people who do these fits. 2D density projections using light front ideas are relativistically exact and show a nucleon with compact high momentum fraction matter and quarks with non-zero angular momentum. The End 31

32 For FF talk at MAMI* Accuracy requirements for FF in HHFS calculation Lamb shift and Gep X Low Q^2 Mainz data FF and 2D charge distribution (w/matthias picture) Plans for Gen at Mainz X Charles s latest Gen precision and Qweak or parity violating expt. Highest Q^2 for each FF Radius with error for each FF X 32