1. University of Maryland
The Pion Form Factor
Tanja Horn
University of Maryland
Calculation of the Pion Form Factor
-> The road to a hard place
Fπ measurement via pion electroproduction
-> Extracting Fπ from H(e,e’π+) data
-> Fπ-1/Fπ-2 in Hall C
Results and new directions
JLab Seminar, 19 April 2006

2. Hadronic Form Factors in QCD
Fundamental issue: quantitative description of hadrons in terms of underlying constituents.
Theory: Quantum Chromo-Dynamics (QCD) describes strong interactions.
Degrees of freedom: quarks and gluons.
Short Distance
Asymptotic Freedom
Long Distance
Binding, collective dof Fπ
Studies of short/long distance scales:
Theory – QCD framework, GPD’s, lattice, models.
Experiments – form factors, neutral weak nucleon structure.

3. Pion Form Factor Limits
Why pions? - Simplest QCD system (qq)
“Hydrogen Atom of QCD”
Large Q2 behavior calculated by Brodsky-Farrar (PRL 43 (1976) 246) in perturbative QCD
At small Q2 vector meson dominance gives accurate description with normalization Fπ(0)=1 by charge conservation

4. Nonperturbative Physics
Feynman Mechanism
At moderate Q2 nonperturbative corrections to pQCD can be large – Braun et al calculate ~30% at Q2=1 GeV2 to Fπ
Need a description for transition to asymptotic behavior
Variety of effective Fπ models on the market – how do we know which one is “right”?
V.A. Nesterenko and A.V. Radyushkin, Phys. Lett. B115 (1982) 410
P. Maris and P. Tandy Phys Rev C62 (2000)
F. Cardarelli et al. Phys. Lett. B357 (1995) 267.

5. Dynamical Approach: DSE+BSE
Maris and Tandy use Dyson-Schwinger equations (Phys.Rev.C62(2000)055204)
Dressed quark propagator input to BSE
Model parameters fixed by agreement of fπ, mπ with data
Predict Fπ and rπ with no further adjustment
Higher Q2: No solution yet, try to use lattice input?
P. Maris and P.C. Tandy, Phys. Rev. C62(2000)055204,.

6. Summary Fπ Calculations
Limits on Fπ well defined and many model calculations available for transition region
Key Point: Know there is asymptotic limit, but how to get there and what governs transition? ??
Long distance
Low Q2
Short distance
High Q2
Need experimental data to study behavior of QCD in transition from long distance to short distance scales

7. Pion Form Factor via Electroproduction
Charge radius well known from p+e scattering
No “free pion” target – to extend measurement of Fp to larger Q2 values use “virtual pion cloud” of the proton
Method check - Extracted results are in good agreement with p+e data
J. Volmer et al. Phys Rev. Lett. 86 (2001)

8. Pion Electroproduction Kinematics
Hadronic information determined by Lorentz invariants
Q2= |q|2 – ω2
W=√-Q2 + M2 + 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
Unpolarized pion electroproduction cross section
In general, LT and TT → 0 by integrating over φ acceptance

10. “Simple” Longitudinal-Transverse Separation
For uniform φ-acceptance, σTT, σLT –›0 when integrated over φ
Determine σT+ ε σL for high and low ε in each t-bin for each Q2
Isolate σL, by varying photon polarization, ε

11. “Real” Cross Section Separation
Cross Section Extraction
φ acceptance not uniform
In reality, must measure σLT and σTT
Extract σL by simultaneous fit using measured azimuthal angle (φπ) and knowledge of photon polarization (ε)

12. Extracting Fπ from σL data
In t-pole approximation:
Want smallest possible –t for proximity to π-pole
Need to know t- dependence
Extraction of Fπ requires a model incorporating pion electroproduction

13. Pion Electroproduction Models
MAID
Only useful for W < 2 GeV, Fπ-2 kinematics above this region, too many free parameters
Born term models
Do not describe t-dependence well away from pole
VGL/Regge (PRC 57 (1998) 1454)
Appropriate at W > 2 GeV
Fit to photo-production data, one free parameter
Constituent Quark Model (Obukhovsky et al. Phys.Lett.B634: )
Same kinematic range as VGL/Regge, two free parameters
Not previously used in data analysis

14. VGL Regge Model
Pion electroproduction in terms of exchange of π and ρ like particles (PRC 57 (1998) 1454)
Model parameters fixed from pion photoproduction
Free parameters: Fπ
ρ exchange does not significantly influence σL at small –t Q2
(GeV/c)2
|t|
(Gev/c)2
ρ exchange effect
1.60 %
2.45 %
Fit to σL to model gives Fπ at each Q2

15. Competing Reaction Channels
Extraction of Fπ relies on dominance of t-channel (pole dominance)
BUT there are other diagrams π γ* π* p n
Pole dominance tested using π-/π+ from D(e,e’p)
G-parity: If pure pole then necessary R=1
Isoscalar background
t-channel process
(purely isovector)

16. Preliminary π+/π- Ratios (Fπ-1)
Preliminary analysis of Fπ-1 data suggests pole process dominates
Gives confidence that can extract Fπ from σL data

17. Fπ-1 (E93-021)
Data taken in 1997, thesis students: J. Volmer and K. Vansyoc
Q2= GeV2 at W=1.95 GeV
Steeper t-dependence for σL at low Q2 than in Regge model
attributed to resonance background contribution
VGL under-predicts σT
J. Volmer et al PRL 86 (2001)

18. Fπ-2 (E01-004) Higher W Repeat Q2=1.60 GeV2 closer to t=mπ pole
Goal:
Separate σL/σT via Rosenbluth separation for extracting Fπ using pole dominance
Experiment
Successfully completed in Hall C in the summer of 2003
Measure pion electroproduction from H and D
Extension of Fπ-1
Higher W
Repeat Q2=1.60 GeV2 closer to t=mπ pole
Highest possible value of Q2 at JLab with 6 GeV beam
Exp Q2
(GeV/c)2
W (GeV)
|t|
(Gev/c)2 Ee
(GeV)
Fπ-1
1.95
Fπ-2
1.6,2.5
2.22
0.093,0.189

19. 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.J. Brash, G.M. Huber, V. Kovaltchouk, G.J. Lolos, S. Vidakovic, 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, C. Keppel, L.G. Tang
Hampton University, Hampton, VA, USA
J. Volmer
DESY, Hamburg, Germany
A. Matsumura , T. Miyoshi, Y. Okayasu
Tohuku University, Sendai, Japan
B. Barrett, A. Sarty, K. Aniol, D. Margaziotis, L. Pentchev, C. Perdrisat, I. Niculescu, V. Punjabi, E. Gibson

20. Fπ-2 Experimental Details
Hall C spectrometers
Coincidence measurement
SOS detects e-
HMS detects π+
HMS Aerogel
First use in 2003
Built by Yerevan group (Nucl. Instrum. Meth. A548(2005)364)
Targets
Liquid 4-cm H/D cells
Al (dummy) target for background measurement
12C solid targets for optics calibration

21. Fπ-2 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

22. Good Event Selection
Coincidence measurement between charged pions in HMS and electrons in SOS
Coincidence time resolution ~ ps.
π+ detected in HMS – Aerogel Cerenkov and Coincidence time for PID
Electrons in SOS – identified by Cerenkov /Calorimeter
Exclusive neutron final state selected with missing mass cut: 0.92 ‹ MM ‹ 0.98 GeV
2π threshold
After PID cuts almost no random coincidences.

23. Good Event Selection: Particle Identification (Pion)
Cannot distinguish p/π+ at momenta ~ 3 GeV by TOF or gas Cerenkov
Tune detection threshold for particle of interest
n=1.03 for proton rejection in Fπ-2 (n=1.015 available)
Aerogel Cut Efficiency (npe=3) 99.5%
Proton Rejection 1:300
Note: cannot distinguish K+/π+ here

24. 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

25. Data Analysis Issues
Kinematic Offsets – directly affect physics quantities
Using the H(e,e’)p and over-constrained H(e,e’p) reactions, one can fit angle and momentum offsets.
SOS saturation requires momentum dependent modification to reconstruction
~1.5% effect at pSOS=1.74 GeV
HMS optics effects: Correlations in reconstructed on focal plane variables
HMS Q3 current/field offset
~4 MeV across acceptance

26. Data Analysis
Charge normalized experimental yields, background subtracted, all corrections applied
Efficiencies:
Computer dead time: 1-11%
Tracking Efficiency: ~97% (HMS), ~99% (SOS)
Electronic dead time: 0.5-1% (HMS), ~ % (SOS)
Other (Coincidence Blocking, cuts etc.)
Data binned in t and φπ at fixed common values of W/Q2 for high and low epsilon
Compare Data to Hall C Monte Carlo – SIMC
Model for H(e,e’π+)
Radiative effects, pion decay, energy loss, multiple scattering
COSY model for spectrometer optics
Overall normalization confirmed to 2% based on e-p elastics using parameterization of world data (J.Arrington, Phys.Rev.C69:022201,2004)

27. SIMC – Radiative Corrections
H(e,e’p)
Charged particles radiate in presence of electric field
External: radiate independently – no interference terms
Internal: Coherent radiation in field of primary target nucleon
Description of elastic data looks good
Pion radiation important, changes SIMC yield ratio ~3.5%
Point-to-point uncertainty 0.5% based on radiative tail studies between ε points.

28. Cross Section Extraction
Cross Section Extraction requires Monte Carlo Model of spectrometer optics and acceptance
W,Q2,t,φ – kinematic dependence of cross section model
Experimental cross section extracted by iterating model until achieve agreement with data

29. Spectrometer Acceptance
Generally looks good, BUT small regions in Y’tar acceptance not well described
Likely no problem if integrating over Y’tar, but relevant for binning in –t

30. Y’tar Acceptance Correction
Using cut dependence as systematic estimate  error in one t-bin dominates. Partially correlated, so overestimates.
Assume part of discrepancy due to simulation of Y’tar acceptance - can correct for this.
Use elastic data and fit correction function
Fit correction

31. Y’tar Acceptance Correction
σL vs -t
Compare DATA/SIMC yields with Y’tar acceptance correction. Effect on other kinematical variables small.
Best estimate of uncertainty in acceptance using acceptance weight.
Total uncorrelated acceptance uncertainty: 1.3%

32. Separated Cross Sections and VGL/Regge Model
Fπ-2 Separated cross sections
Y’tar acceptance weight applied
Source
Pt-Pt
Scale
Acceptance
1.3%
0.7%
Radiative Corrections
0.5%
2.0%
Pion Absorption
0.1%
Pion Decay
0.03%
1.0%
Model Dependence -
Kinematics
HMS Tracking
Charge
0.3%
Target Thickness
0.2%
0.8%
Detection Efficiency
Λπ=0.516, (GeV/c)2

33. Fπ-2 Results – Monopole
Data point at Q2=1.60 GeV2 to check model dependence of mass pole extrapolation
Agreement between Fπ-1/Fπ-2 to ~3%
Confidence in reliability of method
Fπ precision data follow monopole form, Λ2π=0.54 GeV2 up to Q2=1.60 GeV2
At Q2=2.45 GeV2 deviates by ~1σ.
S.R. Amendolia et al., Nucl. Phys. B277 (1986)

34. Fπ-2 Results - Models
Several calculations describe data up to Q2=1.60 GeV2
Agreement at Q2=2.45 GeV2 with QCD sum rule and CQM
Small deviation from DSE/BSE ~1σ
Still far from pQCD prediction
V.A. Nesterenko and A.V. Radyushkin, Phys. Lett. B115 (1982) 410
P. Maris and P. Tandy Phys Rev C62 (2000)
C.-W. Hwang, Phys Rev D64 (2001)

35. Fπ at 12 GeV Summary
Significant progress on theoretical front expected in next 5 years – Lattice, GPD etc
Experiments need higher energy electron beam to reach the kinematic region where pQCD expectation may be approached
SHMS+HMS in Hall C will allow Fπ to be measured up to Q2=6 GeV2 – 12 GeV proposal
Small forward angle crucial for Fπ experiment since need to reach low –t values.

36. Summary Fπ good observable to study transition region to pQCD
Results from Fπ-1 shows Q2Fπ clearly not asymptotic yet
π-/π+ suggests dominance of pion exchange
Fπ-2 results in a region of Q2 where Fπ calculations begin to diverge
Data will constrain models describing the treatment of soft physics at higher Q2
Studies of Fπ at 12 GeV will allow us to reach the kinematic range where pQCD expectation may be approached

37. Effective Models QCD Sum Rules (Phys.Lett. B115(1982)410)
Use properties of Green functions – spectral function contains pion pole
Quark-Hadron Duality (Eur.Phys.J. A17(2003) )
Relate hadronic content of exclusive, elastic pion form factor and inclusive pion structure function, assumes duality holds
Bethe-Salpeter/Dyson-Schwinger (Phys.Rev. C62(2000)055204)
Systematic expansion in terms of dressed particle Schwinger equations
Constituent Quark Model (Phys. Rev. D64(2001)034001)
Constituent quarks and effective interaction potential (e.g. fit from experimental data)

38. Electronic Dead Time
Probability for event to arrive at time t> τ after first event
“Measure” electronic dead time by varying gate width τ
Assume that PRETRIG gate is limiting (60 ns)

39. CPU Dead Time
Source: trigger supervisor “busy” – cannot accept pretrigger from 8LM, e.g. reading out event
Measure CPU DT: ratio of events coming from/going into 8LM (single event type)
Test measurement: normalized yield independent of CPU dead time

40. Y’tar Acceptance Correction – other variables
Compare DATA/SIMC yields with Y’tar acceptance correction. Total effect on other kinematical variables small.
No Y’tar acceptance weight
Y’tar acceptance weight – elastic fit

41. Fπ at 12 GeV Kinematics
Maximize t-channel contribution to σL, increase in both Q2 and W
Full φ coverage at given ε over extended –t range for separating L,T,LT and TT
σL/σT~1 near –t~0.6 GeV2 at Q2=6 GeV2
To reach tmin, need θπ~5º for all kinematic settings
Require good understanding of systematics – kinematic offsets
Sensitive to scattering angles, beam energy and momenta

42. Fπ at 12 GeV – Hardware Requirements
Hall C Spectrometers
Optics in HMS accumulated over 10 years, many successful L/T
SHMS design similar to HMS, QQ(QD), point-to-point focusing, minimum angle 5º in SSA tune
rigid optics -> good reproducibility of pointing and angular offsets
flat acceptance reduces systematic error
angle and momentum resolution
Kinematics: Full φ coverage at high ε using three angles

43. Constituent Quark Model
Obukhovsky et al. use microscopic description on basis of CQM
Calculate σL using both t-pole and s-, u-pole contributions
Free parameters: Fπ and strong FπNN
Calculation at Λπ2=0.54
Shape similar for σL
Significant difference in σT
I. Obukhovsky et al hep-ph/
M. Vanderhaeghen, M Guidal, J-M Laget Phys Rev C57 (1998)

44. Constituent Quark Model
Compare to pole exchange model (Obukhovsky et al. ) with Λπ2=0.54, Λρ2=0.54
Description of σT better – model difference: ρNN interaction
Sensitivity of σT from Fπ2 may be direct measurement of Fρπγ and ρNN vertex
Analogous to relation between σL and Fπ/πNN vertex
Λπ=0.540 (GeV/c)2

45. Good Event Selection: Cuts
Spectrometer Quantities
Restrict momentum to nominal acceptance region
HMS dp/p: ± 8%
SOS dp/p: -15% to +14%
Loose cuts on spectrometer angles, allow collimator to define acceptance
Missing Mass Cuts
Restrict missing mass to range 0.92 to 0.96 GeV (4σ)
PID
Coincidence Time, HMS Aerogel, SOS gas Cerenkov, shower counter
Acceptance Cuts
W/Q2 cut to define common phase space at high and low ε

46. HMS Kinematic Calibrations
Using H(e,e’)p one can fit HMS angle and momentum offsets
Assumes that beam energy known (3x10-4)
Compare reconstructed W to proton mass
Take into account:
Correlations in reconstructed on focal plane variables
Radiative effects and energy loss
Vertical beam, can look like momentum offset
Consistent with elastic singles data from Spring 2003 and 1999 (E. Christy) and 2004 (D. Gaskell)
Quantity
Value
In-plane angle
0.0 mrad
Momentum
-0.13 %

47. SOS Kinematic Offsets
Using the over-constrained H(e,e’p) reaction, one can fit SOS spectrometer and momentum offsets
Compare W to proton mass, missing energy and components of missing momentum
Input: best HMS kinematic offsets
From fitted momentum offsets, one can parameterize saturation effects
Effect at pSOS=1.74 GeV: ~1.5%
Result consistent with Fπ-1 (scaled) and data from 2004 (B. Clasie)
Quantity
Value
In-plane angle
0.0 mrad
Momentum
f(pSOS)

48. “Strange” Features of Data
Observe peak at ~1.16 GeV in missing mass spectrum
Only in high epsilon acceptance
Not part of regular analysis – eliminated by missing mass cut