andrei afanasev, lepton scattering… hep school, january 11-13, 2010, valparaiso, chile lepton...

Andrei Afanasev, Lepton Scattering…

HEP School, January 11-13, 2010, Valparaiso, Chile

Lepton Scattering as a Probe of Hadronic Structure

Andrei Afanasev

Jefferson Lab/Hampton U

USA

Part 2


Plan of Lectures

• Basic formulae

• Scattering cross section, spin asymmetries

• Experimental data on nucleon form factors

• Overview of lepton scattering processes, experimental results and future programs


Form Factors

Exercise: Calculate the charge distribution ρ(x) is the form factor is described by a dipole formula

F(q2)=1/(1+q2/μ2)2


Fourier Transform Excercise

• Magnetic form factor of the proton follows the dipole formula up to Q2~25GeV2:

• Exponential fall-off of magnetic moment distribution with radius

0

32

22

22

2

22

23

22222

22

22

22

)1(

|)||sin(|4(...)cos2

(...)

)1(

]exp[/)(

)frameBreitin0(

12/,71.0,)1(

1/)(

x

B

BBBBBB

BBB

B

BBpMp

B

pMp

eq

xqdqq

xddqq

ddqqq

xqiqdxG

qqqQ

chGeVQ

QG


Mean Square Radius

• Defines characteristic size of the charge and magnetic moment distribution

• Believed to be the same for charge and magnetic moment

• But is it really the same?

• JLAB experiments proved otherwise

23322 )81.0()(/)( fmxdxGxdxGxr MpMpMp


Elastic Nucleon Form Factors

pm

q

peefi

fifi

uqFqFuuuq

ieM

MM

N))()(( 2

222

12

21

1

• Based on one-photon exchange approximation

)0(

)2

tan21(,4

,

:1

)1(2

:)(

122

2

2

2

2121

2

220

y

elab

N

ME

M

E

z

x

z

x

EM

P

q

q

m

q

FFGFFG

techniqueonPolarizatiG

G

A

A

P

P

techniqueRosenbluthGG

• Two techniques to measure

Latter due to: Akhiezer, Rekalo; Arnold, Carlson, Gross


Deriving the Rosenbluth formula

• Matrix element of electron-proton scattering in the first Born approximation:

• For scattering ep->ep, the electromagnetic current is Hermitian and form factors F1,2(q2) are real functions

• Squaring the matrix element, we obtain for the elastic contribution to the hadronic tensor:

)())()()(()()( 12

222

12122

21

1

puqFqFpukukuq

ieM

MM

pm

q

peefi

fifi

N

pp

mpq

pm

FFFmp

qp

m

FFFtrJJW N

NN

Nspin

ˆ

),ˆ()2

()()ˆ()2

()(2

121

22111

221


Rosenbluth Formula (cont.)

• Computing the trace over spinor indices, we obtain structure functions of elastic scattering:

• Differential cross section in terms of structure functions (Rosenbluth formula:

)4

1/()4

(4)(

)()(

)()2

)(2

(1

)()(

2

22

2

2222

2

22221

221

22112

212

NMp

NEpN

el

Mpel

el

N

el

m

qG

m

qGmqW

GqFFqqW

qWq

pq

pm

qWq

qqgW

2

2

1

22

1

22

14

2

0

21

2

1

2

21

22

22

1

2022

2

2

22

22

21

0

22

122

1212

2

4,

21)

21(4

)421(

2/

12

12

tan2

41

4

)2

sin21()(

2cot

2

1

)2

sin21(

1

2sin8

NNN

N

MpMpEp

N

Mpe

N

MpN

Ep

e

N

e

eleel

Ne

N

ee

m

q

mE

qq

mE

qE

q

E

q

mE

qEq

GGG

mE

qG

m

q

Gm

qG

m

Eepep

d

d

WWmE

m

Ed

d


Polarization formula

)(2

1

4)(

protons outgoing of Pon vectorPolarizati

tensorleptonicLtarget,protontheofdirectiononpolarizatispinalongvectorunitaisn

2)(2),(2

1

4)(

sectioncrossdependentonPolarizati

2,where,

)(2)(,

)(2)(

:frameBreitinspinorscomponent)(4Dirac

)())()()((),()(

ionapproximatBorninmatrixscatteringofelementMatrix

2

2

1

24

2

212

12212

2

1

24

2)(

012

2

2

21

1

1

12

222

1212

2

2

fftrm

L

E

E

qepep

d

dP

kkiqgkkkkLwherefnftrm

L

E

E

qepep

d

d

GmfGqiffJ

mE

qmEpumE

qmEpu

puqFqFpuJkukuj

Jjq

ieM

Ne

Ne

n

EpNMpBp

NB

BBB

NB

BBB

pm

q

pp

eee

pefi

N


Polarization formula: result

)ionapproximatexchangephotononein0(

)2

tan21(,4

,

:1

)1(2

:)(

122

2

2

2

2121

2

22

y

elab

N

ME

M

E

z

x

z

x

EMR

P

q

q

m

q

FFGFFG

formulaonPolarizatiG

G

A

A

P

P

formulaRosenbluthGG


The Pre-JLab Era• Stern (1932) measured the proton magnetic moment µp ~ 2.5

µDirac

indicating that the proton was not a point-like particle• Hofstadter (1950’s) provided the first measurement of the

proton’s radius through elastic electron scattering• Subsequent data (≤ 1993) were based on:

Rosenbluth separation for proton, severely limiting the accuracy for GE

p at Q2 > 1 GeV2

• Early interpretation based on Vector-Meson Dominance

• Good description with phenomenological dipole form factor:

GD 2

2 Q2

2

with 0.84GeV

corresponding to r (770 MeV) and w (782 MeV) meson resonances in timelike region and to exponential distribution in coordinate space


Global Analysis

P. Bosted et al. PRC 51, 409 (1995)

Three form factors very similar

GEn zero within errors ->

accurate data on GEn early goal

of JLab

First JLab GEp proposal rated B+!


Magnetic Form Factor of the Proton at High Momentum Transfers

• In perturbative QCD description, GMp~1/Q4 at large

values of Q2. How large?

• Data from SLAC, (Andivahis et al ,1986) show scaling behavior


Modern Era

Akhiezer et al., Sov. Phys. JETP 6 (1958) 588 andArnold, Carlson and Gross, PR C 23 (1981) 363showed that:

accuracy of form-factor measurements can be significantly improved by measuring an interference term GEGM through the beam helicity asymmetry with a polarized target or with recoil polarimetry

Had to wait over 30 years for development of

• Polarized beam with high intensity (~100 µA) and high polarization (>70 %)

(strained GaAs, high-power diode/Ti-Sapphire lasers)• Beam polarimeters with 1-3 % absolute accuracy• Polarized targets with a high polarization or• Ejectile polarimeters with large analyzing powers


Pre-Jlab Measurements of GEn

Elastic e-d scattering (Platchkov, Saclay)

dd

{A Btan2(e / 2)} (GEp GE

n )2 [u2(r) w2 (r)] j0 (Qr2

)dr ....

No free neutron target available, early experiments used deuteronLarge systematic errors caused by subtraction of proton contribution

Yellow band represents range of GE

n-values resulting from the use of different NN-potentials


Double Polarization Experiments to Measure GnE

• Study the (e,e’n) reaction from a polarized ND3 targetlimitations: low current (~80 nA) on target

deuteron polarization (~25 %)• Study the (e,e’n) reaction from a LD2 target and

measure the neutron polarization with a polarimeter limitations: Figure of Merit of polarimeter• Study the (e,e’n) reaction from a polarized 3He targetlimitations: current on target (12 µA)

target polarization (40 %)nuclear medium corrections

)0(

,

:1

)1(2

2121

2

y

ME

M

E

z

x

z

x

P

FFGFFG


G

A

A

P

P


Neutron Electric Form Factor GEn

Galster: a parametrization fitted to old (<1971) data set of very limited quality

For Q2 > 1 GeV2

data hint that GEn has

similar Q2-behaviour as GE

p


Measuring GnM

• Measure (en)/(ep) ratioLuminosities cancelDetermine neutron detector efficiency

• On-line through e+p->e’+π+(+n) reaction (CLAS)• Off-line with neutron beam (Mainz)

• Measure inclusive quasi-elastic scattering off polarized 3He (Hall A)

RD

d3(eD e 'n(p))

dE ' de 'dn

d3(eD e ' p(n))

dE 'de'd p

Old method: quasi-elastic scattering from 2Hlarge systematic errors due to subtraction of proton contribution

A cos*vT 'RT ' 2sin*cos*vTL'RTL'

vLRL vTRT

RT’ directly sensitive to (GMn)2


Measurement of GnM at low Q2


New GnM Results from CLAS

J. Lachinet et al, [CLAS Collab.] Phys. Rev. Lett. 102: 192001, 2009


Early Measurements of GEp

• relied on Rosenbluth separation• measure ds/dW at constant Q2

• GEp inversely weighted with Q2, increasing the systematic error

above Q2 ~ 1 GeV2

At 6 GeV2 sR changes by only 8% from e=0 to e=1 if GE

p=GMp/µp

Hence, measurement of Gep

with 10% accuracy requires 1.6% cross-section measurements over a large range of electron energies

1

1 2(1 ) tan2 ( / 2)

R Q2 , 11

E

E '

E, Mott

{GMp Q2 }2

{GEp Q2 }2


Spin Transfer Reaction 1H(e,e’p)

No error contributions from• analyzing power• beam polarimetry

These errors cancel in the ratio of polarizations Px/Pz

)0(

,

:1

)1(2

2121

2

y

ME

M

E

z

x

z

x

P

FFGFFG


G

A

A

P

P


JLab Polarization-Transfer Data

• E93-027 PRL 84, 1398 (2000)Used both HRS in Hall A with FPP

• E99-007 PRL 88, 092301 (2002)used Pb-glass calorimeter for electron detection to match proton HRS acceptance

• Reanalysis of E93-027 (Pentchev)Using corrected HRS properties

→Clear discrepancy between polarization transfer and Rosenbluth data

→Investigate possible source, first by doing optimized Rosenbluth experiment


Charge and Magnetization Densities

• JLAB experiments demonstrated large differences in charge and magnetization densities of a nucleon

Kelly (2002)


Super-Rosenbluth (E01-001) 1H(e,p)J. Arrington and R. Segel• Detect recoil protons in HRS-L to

diminish sensitivity to:• Particle momentum and

angle• Data rate

• Use HRS-R as luminosity monitor• Very careful survey• Careful analysis of background

RosenbluthPol TransMC simulations


Rosenbluth Compared to Polarization Transfer

• John Arrington performed detailed reanalysis of SLAC data• Hall C Rosenbluth data (E94-110, Christy) in agreement with SLAC data• No reason to doubt quality of either Rosenbluth or polarization transfer

data→Investigate possible theoretical sources for discrepancy


Two-photon Contributions

Guichon and Vanderhaeghen (PRL 91 (2003) 142303) estimated the size of two-photon effects (TPE) necessary to reconcile the Rosenbluth and polarization transfer data

dd

˜ G M

2

˜ G E2

˜ G M2 2

˜ G E˜ G M

Y2 (,Q2)

Pt

Pl

2

(1 )

˜ G E˜ G M

12

1

˜ G E˜ G M

Y2 (,Q2 )

Need ~3% value for Y2g (6% correction to e-slope), independent of Q2, which yields minor correction to polarization transfer


Two-Photon Contributions (cont.)

Afanasev, Brodsky, Carlson et al,

PRD 72:013008 (2005); PRL 93:122301 (2004)

Model schematics: • Hard eq-interaction• GPDs describe quark

emission/absorption • Soft/hard separation• Assume factorization

Polarization transfer1g+2g(hard)1g+2g(hard+soft)

Blunden et al. have calculated elastic contribution of TPE

Resolves ~50% of discrepancy


Experimental Verification of TPE contributions

Experimental verification• non-linearity in e-dependence

(test of model calculations)• transverse single-spin asymmetry

(imaginary part of two-photon amplitude)

• ratio of e+p and e-p cross section (direct measurement of two-photon contributions)

CLAS experiment PR04-116 aims at a measurement of the e-dependence for Q2-values up to 2.0 GeV2


Reanalysis of SLAC data on GM

p

E. Brash et al., PRC submitted,have reanalyzed SLAC datawith JLab GE

p/GMp results

as constraint, using a similarfit function as BostedReanalysis results in 1.5-3% increase of GM

p data


Theory

→Vector Meson DominancePhoton couples to nucleon exchanging vector meson ( , , )r w fAdjust high-Q2 behaviour to pQCD scalingInclude 2π-continuum in finite width of r

• Lomon 3 isoscalar, isovector poles, intrinsic core FF• Iachello2 isoscalar, 1 isovector pole, intrinsic core FF• Hammer 4 isoscalar, 3 isovector poles, no additional FF

→Relativistic chiral soliton model • Holzwarth one VM in Lagrangian, boost to Breit frame• Goeke NJL Lagrangian, few parameters

→Lattice QCD (Schierholz, QCDSF)quenched approximation, box size of 1.6 fm, mπ = 650 MeVchiral “unquenching” and extrapolation to mπ = 140 MeV (Adelaide)


Vector-Meson Dominance Model

charge magnetization

proton

neutron


Constituent quark model predictions

• Relativistic Constituent Quark Models: |p>=|uud>, |n>=|udd>Variety of q-q potentials (harmonic oscillator, hypercentral, linear)Non-relativistic treatment of quark dynamics, relativistic EM currents

• Miller: extension of cloudy bag model, light-front kinematicswave function and pion cloud adjusted to static parameters

• Cardarelli & SimulaIsgur-Capstick oge potential, light-front kinematicsconstituent quark FF in agreement with DIS data

• Wagenbrunn & Plessaspoint-form spectator approximationlinear confinement potential, Goldstone-boson exchange

• Giannini et al.gluon-gluon interaction in hypercentral modelboost to Breit frame

• Metsch et al.solve Bethe-Salpeter equation, linear confinement potential


Relativistic Constituent Quark Model

charge magnetization

proton

neutron


Form Factors and Generalized Parton Distributions

• Radyushkin, PRD58:114008 (1998); AA, hep- ph/9910565; Guidal et al, PRD 72: 054013 (2005); Diehl et al, EPJ 39, 1 (2005)

Fits to the data; additional constraintsfrom parton distributions measured in deep-inelastic scattering


High-Q2 behaviour

Basic pQCD scaling (Bjørken) predicts F1 µ1/Q4 ; F2 µ1/Q6 Þ F2/F1 µ1/Q2

Data clearly do not follow this trend

Schlumpf (1994), Miller (1996) andRalston (2002) agree that by• freeing the pT=0 pQCD condition• applying a (Melosh) transformation to

a relativistic (light-front) system• an orbital angular momentum

component is introduced in the proton wf (giving up helicity conservation) and one obtainsÞ F2/F1 µ1/Q

• or equivalently a linear drop off of GE/GM with Q2

Brodsky argues that in pQCD limit non-zero OAM contributes to F1 and F2


High-Q2 Behaviour (cont)

Belitsky et al. have included logarithmic corrections in pQCD limit

Solid: protonopen: neutron

They warn that the observed scaling could very well be precocious


Low-Q2 Behaviour

All EMFF allow shallow minimum (max for GEn) at Q ~ 0.5 GeV


Pion Cloud of a Nucleon

• Kelly has performed simultaneous fit to all four EMFF in coordinate space using Laguerre-Gaussian expansion and first-order approximation for Lorentz contraction of local Breit frame

˜ G E,M (k) GE ,M (Q2) 1 2 with k 2

Q2

1 and

Q

2M

2

• Friedrich and Walcher have performed a similar analysis using a sum of dipole FF for valence quarks but neglecting the Lorentz contraction

• Both observe a structure in the proton and neutron densities at ~0.9 fm which they assign to a pion cloud

• Hammer et al. have extracted the pion cloud assigned to the NN2π component which they find to peak at ~ 0.4 fm

_


Summary

• Very successful experimental program at JLab on nucleon form factors thanks to development of polarized beam (> 100 µA, > 75 %), polarized targets and polarimeters with large analyzing powers

• GEn 3 successful experiments, precise data up to Q2 = 1.5

GeV2

• GMn Q2 < 1 GeV2 data from 3He(e,e’) in Hall A

Q2 < 5 GeV2 data from 2H(e,e’n)/2H(e,e’p) in CLAS• GE

p Precise polarization-transfer data set up to Q2 =5.6 GeV2

New Rosenbluth data from Halls A and C confirm SLAC data

• The experimental discrepancy between polarization transfer and Rosenbluth data may be resolved with two-photon exchange contributions; further experimental tests coming up

• JLab at 12 GeV will make further extensions to even higher Q2 possible


Coming up next

• Electroweak form factors and parity-violating electron scattering

• Form factors of mesons

• Excitation of baryon resonances

• Virtual Compton scattering on a nucleon

• Primakoff production of mesons

• Search for mesons with exotic quantum numbers

andrei afanasev, lepton scattering… hep school, january 11-13, 2010, valparaiso, chile lepton...

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