1. Tanja Horn Jefferson Lab
The Pion Form Factor
Tanja Horn
Jefferson Lab
BNL Nuclear Physics Seminar, 27 Feb, 2007

2. JLab Fπ-2 Collaboration
R. Ent, D. Gaskell, M.K. Jones, D. Mack, D. Meekins, J. Roche, G. Smith, W. Vulcan, G. Warren, S. Wood
Jefferson Lab, Newport News, VA , USA
E. Brash, G.M. Huber, V. Kovaltchouk, G.J. Lolos, C. Xu
University of Regina, Regina, SK, Canada
H. Blok, V. Tvaskis
Vrije Universiteit, Amsterdam, Netherlands
E. Beise, H. Breuer, C.C. Chang, T. Horn, P. King, J. Liu,
P.G. Roos
University of Maryland, College Park, MD, USA
W. Boeglin, P. Markowitz, J. Reinhold
Florida International University, FL, USA
J. Arrington, R. Holt, D. Potterveld, P. Reimer, X. Zheng
Argonne National Laboratory, Argonne, IL, USA
H. Mkrtchyan, V. Tadevosyan
Yerevan Physics Institute, Yerevan, Armenia
S. Jin, W. Kim
Kyungook National University, Taegu, Korea
M.E. Christy, L.G. Tang
Hampton University, Hampton, VA, USA
J. Volmer
DESY, Hamburg, Germany
T. Miyoshi, Y. Okayasu, A. Matsumura
Tohuku University, Sendai, Japan
B. Barrett, A. Sarty
Saint Mary’s University, Halifax, NS Canada
K. Aniol, D. Margaziotis
California State University, Los Angeles, CA, USA
L. Pentchev, C. Perdrisat
College of William and Mary, Williamsburg, VA, USA
I. Niculescu
James Madison University, Harrisonburg, VA, USA
V. Punjabi
Norfolk State University, Norfolk, VA, USA
E. Gibson
California State University, Sacramento, CA, USA

3. Hadronic Form Factors in QCD
Quantum Chromo-Dynamics (QCD) is very successful describing strong interactions
BUT, we are unable to construct a quantitative description of hadrons in terms of the underlying constituents, quarks and gluons.
We know that there is an asymptotic limit, but how do we get there and what governs the transition? - JLAB
Short Distance
Asymptotic Freedom
Perturbative QCD
Long Distance
Binding
Collective degrees of Freedom Fπ
Form factors provide important information about the transition between non-perturbative and perturbative regimes
Meaningful constraint on theoretical models requires measurements at high precision

4. Scientific Motivation – why pions?
The pion form factor is one of the simplest objects available for experimental studies
Simple qq valence structure
Expect pQCD description to be valid at lower values of Q2 relative to nucleon
Test case for all models of hadronic structure
At large Q2 one can calculate Fπ in pQCD – in the Q2→∞ this reduces to the well known form
One hard gluon exchange
where f2π=93 MeV is
the π+→μ+ν decay constant

5. Modeling QCD – the transition from the non-perturbative regime
Fundamental question: what is the structure of π+ at finite values of Q2? – i.e. at experimentally accessible energies
At what value of Q2 will pQCD dominate over soft contributions?
Theoretical challenge: at moderate values of
Q2 both hard and soft contributions must be taken
into account
Braun et al. calculate ~30% at Q2=1 GeV2
Feynman Mechanism
The limits on Fπ are well defined and many treatments for soft physics are available,
BUT experimental data are needed to de-convolute the role of soft and hard contributions at intermediate values of Q2 – study of space-like Fπ unique to JLab

6. More modeling or implications
Following slides are in progress … mostly placeholders at the moment

7. Fπ via Pion Electroproduction
Fπ can be measured directly from π+e scattering (S.R. Amendolia et al., NP B277 (1986)) up to Q2~0.3 GeV2
No “free pion” target – to extend measurement of Fπ to larger Q2 values use “virtual pion cloud” of the proton
The t-channel diagram dominates sigL at small -t
Method check - Extracted results are in good agreement with π+e data

8. Pion Electroproduction Kinematics
Q2= |q|2 – ω2
t=(q - pπ)2
W=√-Q2 + M2 + 2Mω
scattering plane
reaction plane
Hadronic information determined by Lorentz invariants
Q2= |q|2 – ω2
W=√-Q2 + M + 2Mω
t=(q - pπ)2
pπ = momentum of scattered pion
θπ = angle between scattered pion and virtual photon
φπ = angle between scattering and reaction plane

9. Pion Electroproduction Cross Section
The virtual photon cross section can be written in terms of contributions from transversely and longitudinally polarized photons.
In general, LT and TT → 0 by integrating over φ acceptance

10. “Simple” Longitudinal-Transverse Separation
From φ-dependence have σTT and σLT
Determine σT+ ε σL for high and low ε in each t-bin for each Q2
Isolate σL, by varying photon polarization, ε

11. Cross Section Separation
Cross Section Extraction
φ acceptance not uniform
Measure σLT and σTT
Extract σL by simultaneous fit using measured azimuthal angle (φπ) and knowledge of photon polarization (ε)

12. Extracting Fπ from π+ Electroproduction Data
Extraction of Fπ from σL relies on pion pole dominance
In this simple picture the t-channel process dominates σL
BUT no reliable phenomenological extrapolation to the t=mπ2 pole
More reliable technique: Extract Fπ from σL data using a model incorporating pion electroproduction
Test of this technique: 12 GeV

13. Kinematic Considerations
For maximal contribution from the π+ pole need data at smallest possible –t
For a given Q2, -tmin gets smaller as W increases
Smaller -tmin means Fπ measurements with reduced model uncertainty in the Fπ extraction
The t-dependence of σL must be known for the extraction of
Only three of W, Q2, -t and θ are independent
To measure the t-dependence we vary θ - for nonzero values of θ the interference terms contribute and must be measured explicitly

14. Pion Electroproduction Models
MAID
Only useful for W < 2 GeV, Fπ-2 kinematics above this region
Born term models
Do not describe t-dependence well away from pole
VGL/Regge
Appropriate at W > 2 GeV
Seems to under-predict σT consistently
Constituent Quark Model
Same kinematic range as VGL/Regge, two free parameters
Not previously used in data analysis

15. VGL Regge Model
Pion electroproduction in terms of exchange of π and ρ like particles
Model parameters fixed from pion photoproduction
Free parameters: Fπ and Fρ
ρ exchange does not influence σL at small -t
Fit to σL to model gives Fπ at each Q2

16. Precision Fπ data up to Q2=2.45 GeV2
Extension of 1997 run at highest possible value of Q2 with 6 GeV beam at JLab
New data at higher W
Repeat Q2=1.60 GeV2 closer to t=m2π to study model uncertainties
Successfully completed in Hall C at JLab 2003
Coincidence measurement between charged pions in HMS and electrons in SOS
Separate σL/σT via Rosenbluth separation
HMS: 6 GeV
SOS: 1.7 GeV
Exp Q2
(GeV2)
W (GeV)
|t|
(Gev)2 Ee
(GeV)
Fπ-1
1.95
Fπ-2
1.6,2.45
2.22
0.093,0.189

17. Experimental Details Hall C spectrometers Coincidence measurement
SOS detects e-
HMS detects π+
Targets
Liquid 4-cm H/D cells
Al (dummy) target for background measurement
12C solid targets for optics calibration
HMS Aerogel
Improvement of p/π+/K+ PID at large momenta, first use in 2003
Built by Yerevan group (Nucl. Instrum. Meth. A548(2005)364)

18. Good Event Selection
Coincidence measurement between charged pions in HMS and electrons in SOS
Coincidence time resolution ~ ps
Cut: ± 1ns
Protons in HMS rejected using coincidence time and Aerogel Cerenkov
Electrons in SOS identified by gas Cerenkov and Calorimeter
Exclusive neutron final state selected with missing mass cut: 0.92 ‹ MM ‹ 0.98 GeV
After PID cuts almost no random coincidences.

19. Random Background Subtraction
Random e-π background subtracted using coincidence time
Contribution < 1% for H including all cuts
Separate real π and p events using coincidence time
Issue in random subtraction: π’s can interact in scintillators – produce slow protons
Eliminate with time of flight cut
Cut Efficiency: ~96%
Real protons

20. Kinematic Coverage Have full coverage in φ BUT acceptance not uniform
Measure σTT and σLT by taking data at three angles: θπ=0, +4, -3 degrees
W/Q2 phase space covered at low and high ε is different
For L/T separation use cuts to define common W/Q2 phase space
Θπ=0
Θπ=+4
Θπ=-3
-t=0.1
-t=0.3
Q2=1.60, High ε
Radial coordinate: -t, azimuthal coordinate: φ

21. Cross Section Extraction
Compare experimental yields to Monte Carlo of the experiment
Model for H(e,e’π+) based on pion electroproduction data
Radiative effects, pion decay, energy loss, multiple scattering
COSY model for spectrometer optics
Extract σL by simultaneous fit using measured azimuthal angle (φπ) and knowledge of photon polarization (ε).

22. Spectrometer Uncertainties
Uncertainty in separated cross sections has both statistical and systematic sources
Statistical uncertainty in ranges between 1 and 2%
Point-to-point errors amplified by 1/Δε in L-T separation
Scale errors propagate directly into separated cross section
Uncertainties in spectrometer quantities parameterized using over-constrained 1H(e,e’p) reaction
Beam energy and momenta to <0.1%
Spectrometer angles to ~0.5mrad
Spectrometer acceptance verified by comparing e-p elastic scattering data to global parameterization
Agreement better than 2%
Source
Pt-Pt
Scale
t-correlated
Acceptance
1.0(0.6)%
1.0%
0.6%
Radiative Corrections
0.1%
2.0%
0.4%
Pion Absorption -
Pion Decay
0.03%
Model Dependence
0.2%
1.1(1.3)%
Kinematics
HMS Tracking
Charge
0.5%
0.3%
Target Thickness
0.8%
Detection Efficiency

23. Models for Fpi Extraction

24. Regge Model

25. Comparison to VGL Model
Points include 1.0(0.6)% point-to-point systematic uncertainty (dominated by acceptance)
3.5 % normalization (correlated) uncertainty (radiative corrections, pion absorption, pion decay, kinematic uncertainties)
1.8(1.9)% “t-correlated” uncertainties, influence t-dependence at fixed ε
Compare to VGL/Regge model with Λπ2=0.513 (0.491)
σL well described, but σT under-predicted systematically

26. Fπ 1 results
First phase of the measurement ran in Hall C at JLab in 1997, thesis students: J. Volmer and K. Vansyoc
Measured pion electroproduction from H and 2H
Extracted σL, σT, σTT, and σLT at W=1.95 GeV, Q2 = 0.6, 0.75, 1.0, 1.6 GeV2
Fπ was determined by comparing σL to a Regge calculation by Vanderhaeghen, Guidal, Laget (VGL, PRC 57(1998)1454)
Model parameters fixed from pion photo-production, free parameters: Fπ and Fρ.
Fit to σL to model gives Fπ at each Q2

27. Outline Introduction Fπ from pion electroproduction data :
Fπ1 and Fπ2 in Hall C/Jefferson Lab
Some other comment:
Something else
A long explanation of something relevant:
A third point
Summary

28. Pion Electroproduction Unpolarized Cross Section
Q2= |q|2 – ω2
t=(q - pπ)2
W=√-Q2 + M2 + 2Mω
scattering plane
reaction plane
The lab cross section can be expressed in terms of virtual photon flux, Jacobian and virtual photon cross section.
The virtual photon cross section can be written in terms of contributions from transversely and longitudinally polarized photons.