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