2. Hadronic Form Factors

Transcription

1 PHYS 6610: Graduate Nuclear and Particle Physics I H. W. Grießhammer INS Institute for Nuclear Studies The George Washington University Institute for Nuclear Studies Spring 2018 II. Phenomena 2. Hadronic Form Factors Or: We Thought the Matter was Closed... References: [HM 8.2 (th); HG 6.5/6; Tho 7.5; Ann. Rev. Nucl. Part. Sci. 54 (2004) 217] and optional additional details in script. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.0

2 (a) Recap: Currents & Form Factors of Spin- 1 2 Target Most general current for spin- 1 2 target: J µ S,S = ie F 1 (q 2 ) ū S (p )γ µ u S (p) (I.7.5C) }{{} Dirac: modify point-form + e 2M F 2(q 2 ) q ν ū S (p )iσ µν u S (p) }{{} Pauli: anomalous mag. term F 1 (0) = Z charge; F 2 (0) = κ anom. mag. mom. Sachs FFs: G E = F 1 τf 2, G M = F 1 + F 2 ; τ = q2 4M 2 Rosenbluth formula/sachs cross section: ( )/( ) dσ dσ dω dω Mott= e on (I.7.5) point spin-0 lab = G2 E + τ G2 M 1 + τ spin-flip {}}{ + 2τ G 2 M tan 2 θ 2 [HG 6.11] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.1

3 (b) FF Interpretation in the Breit or Brick-Wall Frame Electric and magnetic are frame-dependent decompositions. = Careful! One Can Show: The Sachs Form Factors G E (q 2 ) and G M (q 2 ) are indeed the form factors of electric charge and magnetic current inside the target in one particular frame: Breit/Brick-Wall Frame E = E = q 0 := k 0 k 0 = 0 No energy transfer. p = p Nucleon recoils like from brick wall. = t = (k k) 2 = 2k k = 2E 2 B (1 cosθ B) t = 2 k B q B = +4E B q B cos ( k B, q B ) θ B small = q B small, grazing shot θ B large = q B large, head-on collision Optional additional details in script. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.2

4 (c) Rosenbluth Separation ( )/( ) [ dσ dσ dω dω = Mott lab G 2 E + τ G2 M } 1 {{ + τ } intercept A(q 2 ) + 2τ G 2 M tan 2 θ }{{} 2 slope B(q 2 ) ] For q 2 0: ( τ = q2 dσ 4M 2 0 = dω For q 2 : τ = q2 + = 4M2 ( dσ dω )/( ) dσ G 2 dω E(q 2 ) 1 q2 Mott 3! r2 E )/( ) ( dσ 1 + 2τ tan 2 θ dω Mott 2 = Each limit has 1 FF which is difficult to measure, and 1 easy one. ) G 2 M(q 2 ) PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.3

5 Example: ep ep at E lab = MeV [Thomson lecture]; exps: MAMI, JLab, SLAC,... q 2 = 2EE (1 cosθ lab ) 0 PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.4

6 Form Factors at Any Q 2 from Polarisation Transfer ( )/( ) unpolarised beam & target dσ dσ outgoing spins undetected dω dω = Mott lab [ G 2 E + τ G2 M 1 + τ MAMI, JLab, SLAC, τ G 2 M tan 2 θ 2 = Each limit Q 2 0, has 1 FF which is difficult to measure, and 1 easy one: How to do better? Polarisation-Transfer Method: Use helicity conservation to separate electric and magnetic. ] mostly P (γ) trans G M [cf. Tho 8.6] P (γ) long G E = Spin-dep. measurement uses QM interference of amplitudes: Amplitudes have different spin-transfer e p: = Scatter polarised e with definite helicity, measure recoil p s polarisation (not easy). longitudinal ( Coulomb ) photon: J z = 0 ( ) transverse ( real ) photon: J z = ±1 = right left γ-polarisations P (γ) long/trans by e-spin, kinematics. G E (Q 2 ) + E P (γ) trans tan θ 2 G M (Q 2 = E ) 2M P (γ) long No absolute cross section, no absolute beam & recoil polarimetry. = Many systematics cancel. So accurate that discrepancies to Rosenbluth led to theory update (2γ exchange) [Afanasev/ ]. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.5

7 (d) Experiments: Magnetic Spectrometers [PRSZR] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 SLAC, MAMI, Jlab,... MAMI-A1 (URL) Spectrometers H. W. Grießhammer, INS, George Washington University II.2.6

8 (e) Proton Form Factors: Why So Simple? Exp. at low Q 2 : dipole G E = [Tho, Fig 7.8; much more data available] G M µ p = = (1 + Q2 a 2 ) 2 with a = 4.27 fm 1 = 0.84 GeV = ρ(r) = ρ 0 e ra exponential rep 2 = 3! dg E Q=0 dq 2 = 12 (0.82 fm)2 a2 high-accuracy data at Q 2 0: r 2 Ep = ([ ± ] fm) 2 high-accuracy data at Q 2 0: r 2 Mp = ([0.777 ± ± 0.010] fm) 2 [PDG 2012] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.7

9 There Is Some Deviation from Simple 1-Dipole Form at High Q 2 dσ dω = lab [ G 2 E + τ G2 M 1 + τ + 2τ G 2 M tan 2 θ 2 ] ( ) dσ, τ = Q2 dω Mott 4M 2 Ratio electric-to-magnetic proton FF Magnetic proton FF: deviation from dipole 1 Dipole ( ) 2 largely ok = 1 + Q2 a 2 ( ) dσ (Q 2, i.e. also τ ) tan2 θ ( ) 2 dσ dω elastic Q 6 dω Mott = Q 2 dominated by G M. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.8

10 (f) Neutron Form Factors: Why So Similar to Proton? é No neutron targets. = d(e,e ) & subtract binding effects; or at Q 2 0: scatter n off atomic e cloud. Low Q 2 : nearly same dipole as proton for G n ( M µ n = Q 2 ) (0.84 GeV) 2 high-accuracy data: r 2 Mn = ([0.862 ± 0.009] fm) 2 r 2 Ep r 2 Mp [PDG 2012] high-accuracy data: ren 2 = [ ± ] fm 2 < 0!! d 3 r [ high-accuracy data: This is allowed: (2π) 3 r2 ρ + (r) ρ (r) ] high-accuracy data: This is allowed: = r 2 + r 2 > < 0! ➋ ➋➌ = On average, negative-charged neutron constituents farther from centre than positive-charged ones. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.9

11 (g) Not Even A Model: Meson-Cloud Argument [PRSZ 6.3] [HG 6.6] QFT: Every particle has a virtual cloud. = Even point-particle has F(Q 2 ) 1. 1 RMS of hadron FFs set by 2 mass of lightest constituent of cloud typically m π = r 2 hadron (0.7fm) 2 PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.10

12 Application: Meson Form Factors Still Quite Simple [HG 6.7] Expect r 2 (0.7 fm) 2 of all hadrons still set by pion cloud. Pion, Kaon: spin 0 = only electric F(q 2 ), no magnetic FF. Unstable Particle = Experiment in inverse kinematics : (cf. neutron) scatter secondary beam on electron cloud of atoms, detect recoil electron (not meson) monopole: F(Q 2 ) = ) 1 (1 + Q2 a 2 [PRSZR] r 2 π = 6 a 2 = ([0.67 ± 0.02] fm)2 r 2 K = ([0.58 ± 0.04] fm) 2 (s-quark!) PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.11

13 (h) List of Accomplishments [PRSZR 6.3, HG 6.6] Measurements so accurate that one has to go beyond One-Photon Approximation: Example contributions at O(α 3 ): [Afanasev, Koshchii, Solyanik] Nucleons have common dipole form: G p E Gp M /µp G n M/µ n = ρ(r) = ρ 0 e ra, a = 4.27 fm 1. r 2 Ep r 2 Mp r 2 Mn (0.8 fm) 2 1 (2m π ) 2. Distribution of charges similar, but different to that of currents. Proton: positive charges more on surface; mag. currents less spread. ( Q 2 ) (0.84 GeV) 2 Neutron: r 2 En 0: charges about equally distributed, but negative charges more on surface. r 2 E r2 M proton ([ ± ] fm) 2 ([0.777 ± ± 0.010] fm) 2 neutron ([ ± ] fm 2 ([0.862 ± 0.009] fm) 2 Mesons: Monopole FFs with small dependence on constituent quark content. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.12

14 (i)... Then Someone Had To Do An Experiment Downie Briscoe ] New Method in 2000: Hyperfine Splitting σ e σ p [δ (3) ( r) + rp 2 2 δ (3) ( r) +... [Hänsch et al.] = Atomic Precision Spectroscopy: r 2 p from Hydrogen-atom near-identical, compatible error bars. = Until 2010: static properties of proton very well known. Idea: Muonic Hydrogen µh: m µ 200m e = Bohr-radius of µh is a B (µ) = µ closer to proton = Better signal. Indeed, much smaller error bars. a B m µ /m e 200 : µh result is 7 standard-deviations off accepted value!! [PDG since 2014] The µp and ep results for the charge radius are much too different to average them. The disagreement is not yet understood. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.13

15 Theorists Speculate... But Be Careful! slide: Downie Beyond-The-Standard Model: Break Lepton Universality: An interaction which is seen by µ but not by e?? [Afanasev, Koshchii, Solyanik] And, of course: check & recheck Theory of previous analyses!! PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.14

16 The MUon Proton Scattering Experiment MUSE at PSI Link to proposal here. GW: Downie (spokesperson), Briscoe, Afanasev, Lavrukhin,... Idea: Use PSI mixed meson/muon/electron beam at E beam = 115,153,210 MeV Idea: to simultaneously measure ep and µp and πp scattering. = Simultaneous determination of proton radius from e p and e + p and µ p and µ + p. present & projected total uncertainties Goals: Test theory understanding of two-photon effects. Test Lepton Universality. Funding by NSF: US$2.5M PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.15

17 Next: 3. Resonance Region, Isospin Familiarise yourself with: [PRSZR 2.4, 6.2, 7.1/4; HG 6.8, 14.2, 8.4-7; Per 3.12; HM 2.6/7; PDG 47] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.16