University of Regina, Regina, Canada.

Transcription

1 Nucleon Electromagnetic Form Factors Dr. E.J. Brash University of Regina, Regina, Canada. March 8, 22

2 Outline ffl Introduction: elastic nucleon form factors ffl Proton Form Factors - Experimental Techniques ffl Proton Form Factors - New Results/Reanalysis ffl Neutron Form Factors - New Results ffl Conclusion

3 Dirac Magnetic Moments It has long been known that the proton and neutron are not point-like particles. As fermions, their wavefunctions should be solutions to the Dirac equation, and if point-like, they should each posess a Dirac magnetic moment, given by μ Dirac = q mc j~ Sj μ proton = μ N μ neutron = ffl In 1932, Stern measured μ p =2.79 μ N. ffl Later, neutron magnetic moment measured! μ N.! Strong indication of substructure of these particles.

4 Electron scattering k p γ Q 2 F,F 1 2 k p 8 >< >: ~q = ~ k ~ k! = E e E e Q 2 =! 2 ~q 2 ffl In en scattering, lepton vertex known from QED. ffl However, important to include vaccuum polarization loop diagrams. ffl Information on the nucleon vertex from cross section and spin-dependent properties of the reaction.

5 Elastic form factors k p γ Q 2 F,F 1 2 k p J μ = μ U (p ) ρ ff fl μ F 1 (Q 2 ) + iffμν q ν F 2 (Q 2 ) U (p) 2M p F 1 (Q 2 ): non spin-flip Dirac form factor F 2 (Q 2 ): spin-flip Pauli form factor F p 1 () = 1; F n 1 () = F p 2 () =» p; F n 2 () =» n ffl Details of nucleon substructure are contained in the Q 2 evolution of F 1 (Q 2 ) and F 2 (Q 2 ). ffl Provides a convenient meeting place between theory and experiment.

6 Sachs form factors In the Breit frame (infinite momentum frame), we can relate F 1 and F 2 to the charge and spatial current densities as: ρ = J = 2eM [F 1 fif 2 ] J i = eμu(p)fl i u(p)[f 1 + F 2 ] (i=1;2;3) This leads us to define the following linear combinations: Electric form factor: G E (Q 2 ) = F 1 (Q 2 ) fif 2 (Q 2 ) Magnetic form factor: G M (Q 2 ) = F 1 (Q 2 ) + F 2 (Q 2 ) where fi = Q2 4M 2 p ffl G E and G M are images of charge and current distributions inside the nucleon. G Ep () = 1; G En () = G Mp () = μ p ; G Mn () = μ n μ p G Ep ()=G Mp () = 1

7 Proton Form Factor Data (pre-1998) ffl Found to follow the dipole approximation G Ep ο G M p μ p ο G d = Q2 :71 2 ffl G Mp well measured with Rosenbluth separation, but not G Ep dff h G fi i 2 1 E (Q2 ) + ffl G2 M (Q2 ) dff dω = dωmott 1 + fi with 1 ffl = 1 + 2(1 + fi )tan2 e G E /G D.5 1. G M /µg D Q 2 (GeV 2 ) Q 2 (GeV 2 )

8 Recoil Polarization Method ffl Elastic ~ep! e~p (A. I. Akhiezer and M. P. Rekalo, Sov. J. Part. Nuc. 3, (1974) 277; and Arnold, Carlson and Gross, Phys. Rev. C23 (1981) 363): e θ e T^ ^ L e h γ p N^ P n = I P t = 2p fi (1 + fi )GEp G Mp tan e 2 p I P` = M 1 p (E e + E e ) fi (1 + fi )G 2 Mp tan 2 e 2 where I = G 2 Ep + fig2 Mp 1 + 2(1 + fi )tan 2 Λ e 2

9 Recoil Polarization Method ffl Direct measurement of the form factor ratio by measuring the ratio of transferred polarization components at target, P t and P` G Ep G Mp = P t P` (E e + E e ) 2M p e tan 2 ffl G Ep G Mp is extracted in a single measurement! reduces systematic errors

10 Spin-Dependent Measurements (JLab, Bates, Mainz) Lower Q 2 Measurements ffl Milbrath et al: - First measurement using FPP at Bates. Measured μg Ep =G Mp at Q 2 = :38; :5 GeV 2. ffl Dieterich et al: - Measurements using new FPP at Mainz. Measured μg Ep =G Mp at Q 2 = :4 GeV 2. JLab Measurements ffl Jones et al: (E9327)- First measurement using FPP at JLab. Measured μg Ep =G Mp at Q 2 = :5 3:5 GeV 2. ffl Gayou,Wijesooriya et al: (d,p(fl,p))- Series of Measurements. ffl Gayou et al: (E997)- Extended to Q 2 = 5:6 GeV 2.

11 Current World Data

12 Comparison With Theory

13 Comparison With Theory

14 Proton Magnetic Form Factor ffl Reanalysis of Rosenbluth cross section data using new ratio data from JLab as a constraint: 8 >< >: r = 1; < Q 2 < :4 GeV 2 r = 1: :13 Q2 :4Λ ; :4 < Q2 < 7:7 GeV 2 r = ;Q 2 > 7:7 GeV Q 2 =1.75 GeV 2.22 Q 2 =2.5 GeV σ R Q 2 =3.25 GeV 2.46 Q 2 =4. GeV ε

15 Proton Magnetic Form Factor a) 1.9 G Mp /µg D Andivahis [4] Bartel [5] Berger [6] Litt [7] Janssens [1] Walker [9] Sill [18] Bosted [16] Q 2 [ (GeV/c) 2 ] 1.2 b) G Mp /µg D Andivahis [4] Bartel [5] Berger [6] Litt [7] Janssens [1] Walker [9] Sill [18] Bosted [16] This work Q 2 [ (GeV/c) 2 ]

16 Rosenbluth Difficulties ffl The electric part of the ep cross section becomes exceedingly small with increasing Q 2! much smaller than previously thought 1 Electric Cross Section [%] % 1.1% Q 2 [GeV 2 ]

17 Theoretical models Vector Meson Dominance (VMD) ffl Virtual photon couples to nucleon via vector mesons. e p ρ,ω γ e p ffl Valid at small momentum transfers.

18 Perturbative QCD ffl Strong coupling constant ff is a decreasing function of Q 2! at high Q 2, one can solve equations perturbatively when ff is small enough.! valence quarks and sea quark-antiquark pairs interacting via gluon exchange, treated perturbatively the same way as charges and photons in QED ffl p-qcd prediction: F 1! 1 Q 4 F 2! 1 Q 2 F 1 9 = ; ) Q2 F 2 F 1 ο const Constituent quark model ffl Nucleon = bound state of 3 quarks. ffl All other degrees of freedom absorbed in these 3 quarks. ffl Successfull at describing baryon mass spectrum

19 Q F 2 /F 1 Scaling ffl Miller and Frank have considered the notion of Poincaré Invariance and Non-conservation of helicity in a simple Relativisitic Constituent Quark Model. ffl Replace the normal Pauli spinors with spinors described by Melosh rotations of the Pauli spinors! important relativistic spin effect. ffl J.Ralston, P. Jain and V. Buniy,! on the basis of skewed parton distribution, take into account quark orbital angular momentum and predict Q F 2 F 1 to be a constant.

20 Gari-Krumpelmann Models ffl Vector Meson Dominance model at low Q 2. ffl PQCD constraints enforced at high Q 2. F p i = F i iv +F is i F n 2 i = F i iv F is i 2 F iv 1 = g ρ f ρ F iv 2 =» ρ g ρ f ρ m 2 ρ m F ρ 2 ρ +Q2 1 (Q2 ) + (1 g ρ f ρ )F D 1 (Q2 ) m 2 ρ m F ρ g 2 ρ +Q2 2 (Q2 ) + (» ν» ρ ρ f ρ )F D 2 (Q2 ) F is 1 = g! f! m 2! m 2! +Q2 F! 1 (Q2 ) + [ffi] + (1 g! f! )F D 1 (Q2 ) F is 2 =»! g! f! m 2! m 2! +Q2 F! 2 (Q2 ) + [ffi] + (» s»! g! f! )F D 2 (Q2 ) ffl Isoscalar and Isovector Meson Pole Terms with meson-nucleon FF s. ffl Asymptotic properties of QCD! F D i quark-nucleon FF s. are effectively F is=iv 1 ο F is=iv 2 ο F is=iv 1 Q 2 hq 2 ln( Q2 Λ 2 QCD ) i 2

21 Neutron Form Factors ffl Knowledge of neutron form factors is much poorer than for proton form factors since a) q= and b) there are no free neutron targets. ffl G En ο, and hence deviations from zero are a sensitive measure of complicated dynamical effects. ffl Slope of G En at Q 2 = is known from thermal neutron scattering. ffl Higher Q 2 measurements proceeded in the past from inclusive unpolarized e d scattering. ffl Rosenbluth separation is difficult due to the familiar magnetic/electric imbalance. ffl Proton subtraction which is necessary adds great difficulty, due to both experimental systematics and required detailed knowledge of target wavefunction.

22 Generalized Parton Distributions ffl Coherent pictures of structure functions from DIS and elastic form factors. ffl Nucleon elastic form factors are first moments of the GPD s. F 1 (t) = P q F 2 (t) = P q R 1 F q (x; t)dx R 1 Kq (x; t)dx ffl F 1p and F 1n together give information on (u-d) distribtion! Goeke, Polyakov, Vanderhaeghen ffl Second moment of F q Momentum Sum Rule! Ji (x; t) connected to Angular

23 Nuclear Form Factors 1 Electric Form Factor G Ep G En Q 2 [GeV 2 ] ffl At large Q 2, G Ep ο G En,soboth are required for an accurate description of deuteron form factors.

24 Spin-Dependent Measurements ffl Double polarization measurements, (beam+target) or (beam+recoil), have been performed at Bates, NIKHEF, MAINZ, and JLab. ffl Galster Parametrization! G En = μ nfi (1+5:6fi ) G D

25 High Resolution Spectrometer ffl 2 quadrupoles, 1 dipole (45 ffi upward), 1 quadrupole ffl Tracking in 2 Vertical Drift Chambers ffl Single proton trigger in 2 planes of scintillators ffl SigniÞcant background: p( ) is proton momentum calculated from proton scattering angle and elastic kinematics, p is momentum measured by HRS 4 Number of counts elastic pion electroproduction + target walls p(θ) p (MeV/c) ffl Necessity to detect the electron

26 Electron detected in other HRS Coincidence trigger between proton and electron missing energy (MeV) missing momentum (MeV/c) Conventional way to select elastic events in Hall A.

27 Other kinematics: the solid angle problem ffl Beam energy 4.6 GeV; HRS solid angle 7.2 msr Q 2 (GeV 2 ) p p (GeV/c) p e E. sol. ang. 4: 2:923 28:6 o 34:5 o 11:9 4:8 3:359 23:8 o 42:1 o 22: 5:6 3:88 19:4 o 51:4 o 42:5 ffl Assembled a 1:35 2:55 m 2 calorimeter, with 17 rows and 9 columns of cm 2 lead-glass blocks Hall A Beam line Target e HRS p Calorimeter

28 Software cuts ffl Timing cut: A TDC was connected to every block, triggered by a 1 mv signal. Number of counts 2 1 randoms coincidence peak TDC channel (1 channel =.5 ns) ffl Position cut: Difference between expected and measured position dycalo (m) dxcalo (m)

29 Result of selection 4 Number of counts Elastic events kept 5% missing Background <1% kept p(θ) p (MeV/c) ffl Very good selection ffl About 5% of elastic events are rejected, due to a mismatch between the geometrical shape of the calorimeter and the actual acceptance ffl Less than 1% of the background remains. Its contribution is taken into account in the polarization analysis.

30 Polarization observables extraction Principle ffl Interested in transfered polarization at target P t and P` ffl Focal Plane Polarimeter measures perpendicular components at the focal plane Pn fpp and Pt fpp ffl Precession through the magnetic elements: rotation matrix relates the two polarizations P = S:P fpp ffl Dipole approximation! Average spin transport approximation! Maximum likelihood approach

31 Focal Plane Polarimeter ffl How do we measure polarization at the focal plane? Front straw chambers Rear straw chambers θ φ CH2 analyzer ffl Selection of good FPP events: good tracking, good vertex reconstruction, cone test ffl Asymmetries in azimuthal distributions correspond to polarization components

32 Asymmetries N (#; ') = N (#)f1 + [A y (#)Pt fpp [A y (#)Pn fpp ffl Difference between two beam helicity states ) instrumental assymetries cancel to Þrst order ) access to polarization transfer components.2 Q 2 = 5.6 GeV 2 + a inst ]sin' + b inst ]cos'g.1 N + N ϕ (degrees) ffl Fit C cos(' + ffi)! tan ffi = Pt fpp =Pn fpp

33 Dipole approximation ffl Consider the case of a single perfect dipole χ l P fpp n B P P fpp n = sin χ:hp` P fpp t = hp t

34 ! Takes into account precession in non dispersive plane Snl Snt γ(g/2-1)(θfp-θtg-45 ), rad γ(g/2-1)(θfp-θtg-45 ), rad Stl Stt γ(g/2-1)(φfp-φtg), rad γ(g/2-1)(φfp-φtg), rad

35 Average spin transport ffl Ratio μg Ep =G Mp can not depend on p, ffi p, y p and ffi p Q 2 = 5.6 GeV µg Ep /G Mp δ θ tgt (rad) Full STM Dipole 2 φ tgt (rad).5 µg Ep /G Mp y tgt (mm) ffl To be more accurate, need yet another method to take into account dependance on these spin coefþcients distribution, analyzing power dependance on # fpp ) Need to relate polarization at the target directly to scattering in the analyzer, without calculating polarization at the focal plane

36 Maximum-likelihood procedure ffl Probability that a proton i scatters in analyzer with angle ' i : P i (' i ) = 1 + A y (# i )P fpp t sin ' i A y (# i )P fpp ` cos ' i = 1 + A y (# i )(S tt;i hp t + S t`;i hp`)sin' i A y (# i )(S nt;i hp t + S n`;i hp`)cos' i g ffl Likelihood function L(P t ;P`) = N Y p i=1 f1 ± A y (# i )(S tt;i hp t + S t`;i hp`) sin ' i A y (# i )(S nt;i hp t + S n`;i hp`)cos' i g ffl Angular dependence of analyzing power taken into account; magnitude cancels in the ratio ffl Parameters hp t and hp` G Ep G Mp = hp t hp` (E e + E e ) 2M p tan e 2

37 Current Status and Future Experiments ffl Based on reanalysis of Rosenbluth data for GMp,new empirical parametrization is 2-3% higher than Bosted parametrization. ffl New data for μgep=gmp, together with new GEn data, constrain phenomenological G-K type models. Future Measurements ffl Hall C Form Factor Ratio - continue to 9 GeV 2 with 6 GeV beam. Possible to continue to 12.4 GeV 2 with 12 GeV beam and current detectors. ffl Hall A Super Rosenbluth - measurements at ο 1,3,5 GeV 2. ffl Hall B measurement of GMn -uptoο 4.8 GeV 2. ffl Hall A Polarized 3 He GEn - up to 3.4 GeV 2.

38 G Mp /µg D Q 2 [ GeV 2 ] µg Ep /G Mp Q 2 [ GeV 2 ] G Mn /µg D Q 2 [ GeV 2 ] G En Q 2 [ GeV 2 ]