Hadrons in the Nuclear Medium

Steffen StrauchUniversity of South Carolina

26th Annual Hampton University Graduate Studies ProgramJefferson Lab, Newport News, Virginia

May 31 - June 17, 2011 51

Proton Form Factors

• Elastic cross section

• Form factor

52

Form Factors

Form factors characterize internal structure of particles

The form factor as a Fourier transformation of the charge distribution is a non-relativistic concept.

F (q2) =

eiq·x/ρ(x)d3x

53

Size of the Proton

• Exponential model, dipole fit:

• Effective charge rms radii:‣ rp = 0.879(8) fm from electron-scattering experiments.‣ rp = 0.84184(67) fm from recent muonic hydrogen data.

phen

omen

olog

ical F

F sq

uare

d

F(q2 ) = 1+ q2

λ2⎛⎝⎜

⎞⎠⎟

2

F(q2 ) = eiq ·x /ρ(x)d 3x∫

= F(0) 1− 16q2r2 +…⎛

⎝⎜⎞⎠⎟

r2 = −6

F(0)dFdq2 q2 =0

R. Hofstadter, Rev. Mod. Phys. 28, 214 (1956); Bernauer et al., Phys. Rev. Lett. 105, 242001 (2010); R. Pohl et al., Nature 466, 213 (2010).

1956

2010q2 (1026 cm-2)

54

Electric and Magnetic Proton Form Factors

• Dirac and Pauli form factors are matrix elements of the electromagnetic current operator

• ep cross section can be written without interference term with Sachs form factors Electric form factor, GE(Q2), with GE(0) = 1 Magnetic form factor, GM(Q2), with GM(0) = µp

55

Form Factors in the Breit Frame

• The physical meaning of GE and GM are best understood in the Breit frame: Electron transfers momentum but no energy

• Time and space components of the hadronic current in the Breit frame:

GE is related to electric charge distribution and GM is related to magnetic current density distributions

through a Fourier transformation.

Problem of Fourier-transformation interpretation: there is a Breit frame for every Q2 value; need to transform from the Breit to the Lab frame.

Brick-wallframe

Transverse Charge Densities of Partons

• In the infinite momentum frame the transverse charge density is the two-dimensional Fourier transform of the form factor F1(Q2)

• Proton: density is peaked at low values of b but has a long positive tail, suggesting a long-ranged, positively charged pion cloud

• Neutron: central charge density is negative

• Down quark density is larger than that for the up quark by about 30%

56G.A. Miller, Phys. Rev. Lett. 99, 112001 (2007); G.A. Miller, Ann. Rev. Nucl. Part. Sci. 60, 1 (2010).

ρ(b) =0

∞

∫dQQ2π

J0 (Qb)GE (Q

2 ) + τGM (Q2 )

1+ τ

+ --

+ Proton

Neutron

Neutron

np

π-

π-

pn

57

Form Factor Extraction – Rosenbluth Method

• Kinematic variables ε and τ

• GE is difficult to extract at large Q2 (τ = Q2/4M2).

• Multiple measurements needed per Q2 point

• Significantly affected by radiative corrections

dσ

dΩ∝ τ G2

M (Q2) + G2E(Q2)

Q2 = const

backwardangles

forwardangles

58

Proton Electromagnetic Form Factors

• Early data suggest

• Dipole fit

Rosenbluth data

Figures from: C.F. Perdrisat et al., Prog. in Part. and Nucl. Phys. 59, 694 (2007)

GEp ≈ GM

p / µp ≈ GD

GD = 1+ Q2

0.71 GeV2

⎛⎝⎜

⎞⎠⎟

2

59

Form Factor Extraction – Recoil-Polarization

• The ratio GpE / GpM is obtained from a single measurement

• Small systematic uncertainties (beam helicity, Ac, … cancel)

• Minimally affected by radiative correctionsA.I. Akhiezer and M.P. Rekalo, Sov. J. Part. Nucl. 3, 277 (1974)R. Arnold, C. Carlson, and F. Gross, Phys. Rev. C 23, 363 (1981)

P x = −2

τ(1 + τ)

GEp

GMp

( GEp

GMp)2 + τ

tanθe

2

P z =

1m

(Ei + Ef )

τ(1 + τ)1

( GEp

GMp)2 + τ

tan2 θe

2

GEp

GMp= −P

x

P z

(Ei + Ef )2m

tanθe

2

60

Proton Elastic Form-Factor Ratio

• Surprising Hall A discovery: Systematic decrease of GE/GM

• Difference in spatial distribution of charge and magnetization currents in the proton

• Inconsistency between cross-section and polarization measurements (two-photon exchange)

M.K. Jones et al., Phys.Rev.Lett., 84 (2000) 1398; O. Gayou et al., Phys.Rev.Lett., 88 (2002) 092301. Figure from: I.A. Qattan, Phys. Rev. Lett. 94, 142301 (2005) [Hall A E01-001]

Cross section dataPolarization data

61

Two-Photon Exchange

Real part of the two-photon amplitude Imaginary part of the two-photon amplitude

• Measurement of the e-p and e+p cross-section ratio

• Measurement of GEp/GM

p ratio at fixed Q2 = 2.5 GeV2 as a function of ε.

• High-statistics search of non-linearity in the Rosenbluth plot in ep scattering.

• Measurement of the induced polarization in ep → ep

• Measurement of the single-spin target asymmetry in quasi-elastic scattering on the neutron in 3He

Experiments planned at JLab to study possible two-photon effects:

P. G. Blunden, W. Melnitchouk, and J. A. Tjon, Phys. Rev. C 72, 034612 (2005)

Two-photon exchange box and crossed box diagrams for elastic electron-proton scattering

62

Recent High-Q2 Data

• Recoil Polarization Measurements of the Proton Electromagnetic Form Factor Ratio to Q2 = 8.5 GeV2

• An important consequence of pQCD is hadron helicity conservation; in terms of the non-spin flip (F1) and spin flip (F2) form factors (Dirac and Pauli).

• The data do not yet satisfy the leading-twist, leading order pQCD ‘‘dimensional scaling’’ relation

A.J.R. Puckett et al., Phys. Rev. Lett. 104, 242301 (2010).

F2p ∝ F1

p /Q2