1. Tanja Horn Jefferson Lab 1 The Pion Form Factor Nuclear Physics Seminar, University of Virginia 25 September, 2007 Present status and future outlook

2. 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? 2 Form factors provide important information about the transition from collective degrees of freedom to quarks and gluons –i.e., from the non-perturbative to the perturbative regime Short Distance Asymptotic Freedom Perturbative QCD Long Distance Binding Collective DOF ? Hadronic Form Factors

3. The study of F π is one of the most fundamental experimental studies for understanding hadronic structure –Simple qq valence structure of π + In QFT, F π is the overlap integral, –The pion wave function can be separated into φ soft with low momentum contributions (k<k 0 ) and a hard tail, φ hard Hard scattering part can be calculated in pQCD The Pion Charge Form Factor

4. QCD Hard Scattering Picture At large Q 2, perturbative QCD (pQCD) can be used, at asymptotically high Q 2, only the hardest portion of the wave function remains and F π reduces to the factorized form G.P. Lepage, S.J. Brodsky, Phys.Lett. 87B(1979)359. where f 2  =93 MeV is the  + →  + decay constant.

5. At experimentally accessible Q 2 both hard and soft components contribute –Transverse momentum effects The interplay of hard and soft components is not well understood –Non-perturbative hard components of higher twist cancel soft components [V. Braun et al., PRD 61 (2000) 07300] 5 Intermediate Q 2 regime

6. Meson Form Factors and QCD The simple valence quark structure of mesons presents the ideal laboratory for testing our understanding of bound quark systems –All hadronic structure models use π + as a test case Excellent opportunity for studying the transition from effective degrees of freedom to quarks and gluons, i.e., from the soft to hard regime Situation for nucleon is even more complicated Many studies of F π, but the interplay of hard and soft contributions is not well understood –Constraints on theoretical models require high precision data –JLab is the only experimental facility capable of the necessary measurements

7. At low Q 2, F π can be measured directly from π +e scattering up to Q 2 ~0.3 GeV 2 [S.R. Amendolia et al., NP B277 (1986)] –Accurate measure of the π + charge radius, r π =0.657 ± 0.012 fm At larger Q 2 values, one must use the “virtual pion cloud” of the proton to extend the F π measurement –t-channel diagram dominates σ L at small –t In the Born term model: 7 Measuring F π Measuring F π Pion electroproduction

8. Extraction of F π relies on the pole dominance of σ L –For maximum contribution from the π + pole to σ L, need data at the smallest possible –t –At fixed Q 2, a higher value of W allows for smaller -t min 8 Extraction of F π requires knowledge of the -t dependence of σ L –Only three of Q 2, W, t, and θ π are independent –Must vary θ π to measure the -t dependence (off-parallel) –In off-parallel kinematics, LT and TT must also be determined Q 2 = |q| 2 – ω 2 t=(q - p π ) 2 W=√-Q 2 + M 2 + 2Mω scattering plane reaction plane Pion Electroproduction Kinematics

9. 9 Extraction of F π from p(e,e’ π + )n data Extraction of F π from p(e,e’ π + )n data  + production data are obtained at –t>0 (away from the -t=m π 2 pole) Early experiments used “Chew- Low” extrapolation technique –Need to know the –t dependence through the unphysical region –A reliable extrapolation is not possible More reliable technique is to use a model including the π + reaction mechanism and extract F π from σ L data –Fit data in the physical region

10. Electroproduction starts from a virtual pion –Can this method yield the physical form-factor? Test the method by comparing F π values extracted from p(e,e’π + )n data with those obtained from π+e elastic scattering at the same kinematics DESY electroproduction data at Q 2 = 0.35 GeV 2 consistent with extrapolation of elastic data [Ackerman et al., NP B277 (1986) 168] 10 An improved check will be performed after the JLab Upgrade –Lower Q 2 (Q 2 =0.30 GeV 2 ) –Lower –t (-t=0.005 GeV 2 ) Electroproduction Method Test Electroproduction Method Test

11. 11 ? F π before 1997 Large Q 2 data from Cornell –Use extrapolation of σ T fit at low Q 2 to isolate σ L –Extract F π from unseparated cross sections Largest Q 2 points also taken at large –t –Carlson&Milana predict M pQCD /M pole grows significantly for –t min >0.2 GeV 2 [PRL 65 (1990) 1717] –Pole term may not dominate

12. –t min <0.2 GeV 2 constraint limits Q 2 reach of F  measurements Measurement of σ L for  0 could help constrain pQCD backgrounds –JLab PAC31 proposal In a GPD framework,  + and  0 cross sections involve different combinations of same GPDs – but  0 has no pole contribution 12 00 ++ VGG GPD-based calculation pole non-pole F π Backgrounds

13. 13 Two superconducting Linacs –Three experimental Halls operating concurrently E<~ 5.7 GeV –Hadron-parton transition region C.W. beam with currents of up to 100 uA –Luminosity ~10 38 F π measurements Jefferson Lab

14. ExpQ 2 (GeV 2 ) W (GeV) |t| (Gev) 2 E e (GeV) Fpi1 1.6 0.6-1.61.95 0.150 0.03-0.1502.445-4.045 Fpi2 1.6 1.6,2.452.22 0.093 0.093,0.1893.779-5.246 14 Fpi2 extends the earlier Fpi1 data to the highest possible value of Q 2 with 6 GeV beam at JLab –Fpi2 data at higher W, smaller -t –Repeat Q 2 =1.60 GeV 2 closer to t=m 2 π to study model uncertainties Full L/T/TT/LT separation in π + production Measurement of separated π + / π - ratio to test the reaction mechanism HMS: 6 GeVSOS: 1.7 GeV The F π Program at JLab

15. 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, 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 15 Jefferson Lab Fpi2 Collaboration

16. 16 Hall C spectrometers –Coincidence measurement –SOS detects e - –HMS detects π + Targets –Liquid 4-cm H/D cells –Al (dummy) target for background measurement – 12 C 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] Experimental Details

17. Coincidence measurement between charged pions in HMS and electrons in SOS –Coincidence time resolution ~200-230 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 17 After PID cuts almost no random coincidences p(e,e’ π +)n Event Selection

18. 18 W/Q 2 phase space covered at low and high ε is different For L/T separation use cuts to define common W/Q 2 phase space Have full coverage in φ BUT acceptance not uniform Measure σ TT and σ LT by taking data at three angles: θ π =0, +4, -3 degrees Θ π =0 Θ π =+4 Θ π =-3 -t=0.1 -t=0.3 Q 2 =1.60, High ε Radial coordinate: -t, azimuthal coordinate: φ Q 2 =1.60 GeV 2 Q 2 =2.45 GeV 2 Fpi2 Kinematic Coverage

19. SourcePt-PtScalet-correlated Acceptance1.0(0.6)%1.0%0.6% Radiative Corrections 0.1%2.0%0.4% Pion Absorption-2.0%0.1% Pion Decay0.03%1.0%- Model Dependence0.2%-1.1(1.3)% Kinematics0.2%-1.0% HMS Tracking0.1%1.0%0.4% Charge-0.5%0.3% Target Thickness-0.8%0.2% Detection Efficiency-0.5%0.3% 19 Uncertainties in spectrometer quantities parameterized using over- constrained 1 H(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% Statistical uncertainty in ranges between 1 and 2% Uncorrelated systematic uncertainty: 1.1(0.9)% Total correlated uncertainty: 3.5% Magnetic Spectrometer Calibration

20. 20 σ L is isolated using the Rosenbluth separation technique –Measure the cross section at two beam energies and fixed W, Q 2, -t –Simultaneous fit using the measured azimuthal angle (φ π ) allows for extracting L, T, LT, and TT Careful evaluation of the systematic uncertainties is important due to the 1/ ε amplification in the σ L extraction –Spectrometer acceptance, kinematics, and efficiencies Determination of σ L

21. MAID – unitary isobar model for pion photo- and electroproduction –Only useful for W < 2 GeV, F π -2 kinematics above this region –Too many free parameters Born term models –Do not describe t-dependence well away from pole VGL/Regge [ Vanderhaeghen, Guidal, Laget, PRC 57 (1998) 1454] –Appropriate at W > 2 GeV –Model parameters fixed from pion photoproduction data –F π is the only free parameter in the calculation of σ L Constituent Quark Model (Obukhovsky et al., Phys. Lett. B634 (2005) –Same kinematic range as VGL/Regge, two free parameters –Model still in development, not yet used in data analysis Pion Electroproduction Models

22. 22 Λ π 2 =0.513 (0.491) GeV 2, Λ π 2 =1.7 GeV 2 uses the VGL Regge model, which describes pion electroproduction in terms of the exchange of π and ρ like particles [Vanderhaeghen, Guidal, Laget, PRC 57 (1998), 1454] –Model parameters fixed from pion photoproduction –Free parameters: F π and F ρ - The error bars denote statistical and systematic uncertainties in quadrature (1.0 (0.6)%) - Yellow band denotes the normalization and –t correlated systematic uncertainty (3.5%, 1.8(9)%) Fit to σ L to model gives F π at each Q 2 T. Horn et al., Phys. Rev. Lett. 97 (2006) 192001. Extraction of F π from Fpi2 data

23. 23 Extract F π for each t-bin separately –F π values are insensitive (<2%) to the t-bin used This result gives confidence in the applicability of the VGL Regge model in the kinematic regime of Fpi2 data Fpi2 model test F π t-dependence

24. F π precision data deviate from 0.657 fm charge radius at Q 2 =2.45 GeV 2 by ~1σ –The monopole reflects the soft (VMD) physics at low Q 2 –The deviation suggests that the π+ “harder” at this Q 2 24 T. Horn et al., Phys. Rev. Lett. 97 (2006)192001. V. Tadevosyan et al., nucl-ex/0607007. JLab Experimental Results P. Brauel et al., Z. Phys. C3 (1979) 101 H. Ackermann et al., Nucl. Phys. B137 (1978) 294 S. R. Amendolia et al., Nucl. Phys. B277 (1986) 168 F π is still far from the pQCD prediction –Including transverse momentum effects has no significant impact New point from π CT (2004) in good agreement with Fpi experiments T. Horn et al., arXiv:0707.1794 (2007).

25. Fpi Interpretation Issues 25 t dependence of Fpi1 σ L is significantly steeper than VGL/Regge at the lowest Q 2 –May be due to resonance contributions not included in Regge –Linear fit to Λ π 2 to t min gives the best estimate of F π at each Q 2 Check the model dependence of the mass pole extrapolation –Good agreement between Fpi1 (t min =0.093 GeV2) and Fpi2 (t min =0.150 GeV2) gives confidence in the reliability of the method

26. 26 A.P. Bakulev et al. Phys. Rev. D70 (2004). Chernyak & Zhitnitsky (CZ) DA Asymptotic DA Bakulev et al. use analytic perturbation theory at the parton amplitude level –π DA is consistent to 1σ level with CLEO πγ transition data –Analytic perturbation theory at the parton amplitude level F π ˚ results taken as evidence that asymptotic π DA appropriate as low as Q 2 =1 GeV 2, BUT … pQCD LO+NLO Calculations (I)

27. For F π+ soft contributions from quark-hadron duality model need to be included to describe the data 27 A.P. Bakulev et al. Phys. Rev. D70 (2004). Hard component is only slightly larger than the one calculated with asymptotic DA in all considered schemes –To describe the data must include soft contribution – here, via local duality pQCD LO+NLO Calculations (II)

28. 28 T. Horn et al., Phys. Rev. Lett. 97 (2006) 192001. F π in 2007 QCD Sum Rules [V.A. Nesterenko and A.V. Radyushkin, Phys. Lett.B115 (1982)410] –Use properties of Green functions – spectral function contains pion pole Bethe-Salpeter/Dyson-Schwinger [P.Maris and P. Tandy, Phys.Rev.C62 (2000)055204] –Systematic expansion in terms of dressed particle Schwinger equations Anti de Sitter/Conformal Field Thry [S.J. Brosdky and G.F. de Teramond, arXiv:0707.3859] T. Horn et al., arXiv:0707.1794 (2007). [ [A.P. Bakulev et al, Phys. Rev. D70 (2004)]  Hard contribution to NLO with improved π DA  Soft contribution from local duality

29. F π time-like vs. space-like (2007) Expect same asymptotic prediction for both space-like and time-like data –The way one gets there may be different –pQCD under-predicts both cases Calculations in time-like region complicated by explicit resonances Timelike data from P.K. Zweber Ph.D. thesis (2006)

30. 30 Hall C High Momentum Spectrometer and Short Orbit Spectrometer at present –Form Factors and simple quark systems –Color Transparency –Nuclei with strange quarks Add a Super-High Momentum Spectrometer for studies of –Form Factors and simple quark systems –Color Transparency –Semi-inclusive DIS SHMS HMS (QQQD) SOS (QQD) Hall C at 12 GeV

31. Significant progress on theoretical front expected in next 5 years –Lattice, GPDs etc. 31 The 11 GeV electron beam and the SHMS in Hall C with θ=5.5º allows for –Precision data up tp Q 2 =6 GeV 2 to study the transition to hard QCD –Test of the electroproduction method at Q 2 =0.3 GeV 2 with the upper limit of elastic scattering data –Most stringent test of the model dependence in the F π extraction by comparing data at several values of W F π after the JLab Upgrade

32. F π is a good observable to study the transition from collective degrees of freedom to quarks and gluons F π measurements from JLab yield high quality data – in part due to –Continuous electron beam provided by JLab accelerator –Magnetic spectrometers and detectors with well-understood properties The highest Q 2 JLab results indicate that Q 2 F π is still increasing, but ~1σ below the monopole parameterization of the charge radius –Still far from the QCD prediction Studies of F π at higher electron beam energies will allow to reach the kinematic range where hard contributions are expected to dominate –Planned measurement of F π at JLab after the upgrade to Q 2 =6 GeV 2 Further development of QCD techniques for the non-perturbative physics are anticipated 32 Summary

33. Measurement of K + Form Factor Similar to  + form factor, elastic K + scattering from electrons used to measure charged kaon for factor at low Q 2 [Amendolia et al, PLB 178, 435 (1986)] Can “kaon cloud” of the proton be used in the same way as the pion to extract kaon form factor via p(e,e’K + )L ? Kaon pole further from kinematically allowed region Can we demonstrate that the “pole” term dominates the reaction mechanism?

34. Kaon Form Factor at Large Q 2 JLAB experiment E93-018 extracted –t dependence of K + longitudinal cross section near Q 2 =1 GeV 2 A trial Kaon FF extraction was attempted using a simple Chew-Low extrapolation technique g KLN poorly known –Assume form factor follows monopole form –Used measurements at Q 2 =0.75 and 1 GeV 2 to constrain g KLN and F K simultaneously Improved extraction possible using VGL model?

35. Test Extraction of K + Form Factor -t dependence shows some “pole-like” behavior “Chew-Low” type extraction G. Niculescu, PhD. Thesis, Hampton U.

36. Lattice QCD allows for calculations from first principles –This is different from QCD-inspired models where confinement must be put in by hand 36 BUT LQCD requires a number of approximations –Lattice discretization errors – improved LQCD action helps –Chiral extrapolation of LQCD is used to obtain the pion mass –Quenching errors – need to include disconnected quark loops Advances in computational techniques have improved over the last two decades –Potential for precision predictions of hadronic properties F π on the Lattice F π on the Lattice

37. Unquenched (dynamical) domain-wall action calculation –Lattice Hadron Physics Collaboration (Jefferson Lab, Regina,Yale) –F. Bonnet et al., hep- lat/0411028 37 Lattice calculations are consistent with experimental data within large statistical and systematic errors, dominated by chiral extrapolation –Primary goal is to test proof-of-principle of different techniques For future calculations expect m π sufficiently small to yield small chiral extrapolation errors –Require higher Q 2 data to validate new LQCD methods pQCD → F π from a recent unquenched Lattice QCD calculation F π from a recent unquenched Lattice QCD calculation Fpi1 Fpi2

38. Extraction of F π relies on dominance of t-channel (pole dominance) –t-channel diagram is purely isovector 38 R consistent with the model predictions indicates t-channel dominance of the data Competing Reaction Channels Pole dominance tested using π - / π + from D(e,e’p) –G-parity: If pure pole then necessary R=1

39. Constituent Quark Model Obukhovsky et al. use microscopic description on basis of CQM –Calculate σ L using both t- pole and s-, u-pole contributions –Free parameters: F π and strong F πNN Calculation at Λ π 2 =0.54 –Shape similar for σ L –Significant difference in σ T I. Obukhovsky et al hep-ph/0506319 M. Vanderhaeghen, M Guidal, J-M Laget Phys Rev C57 (1998)