2. Hadronic Form Factors Or: We Thought the Matter was Closed. . . - PowerPoint PPT Presentation

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

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

(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 = − 2 k · k ′ = − 2 E 2 = B ( 1 − cos θ B ) � � t = − 2 k B · � q B = + 4 E 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

(c) Rosenbluth Separation � � � d σ ��� d σ � � G 2 E + τ G 2 tan 2 θ � M 2 τ G 2 = + � M 1 + τ d Ω d Ω � 2 Mott lab � �� � � �� � intercept A ( q 2 ) slope B ( q 2 ) � d σ ��� d σ � τ = − q 2 E ( q 2 ) → 1 − q 2 For q 2 → 0 : → G 2 3! � r 2 4 M 2 → 0 = ⇒ E � d Ω d Ω Mott � d σ ��� d σ � � � For q 2 → −∞ : τ = − q 2 1 + 2 τ tan 2 θ G 2 M ( q 2 ) 4 M 2 → + ∞ = ⇒ → d Ω d Ω 2 Mott = ⇒ Each limit has 1 FF which is difficult to measure, and 1 easy one. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.3

Example: ep → ep at E lab = 529 . 5 MeV [Thomson lecture]; exps: MAMI, JLab, SLAC,. . . q 2 = − 2 EE ′ ( 1 − cos θ lab ) ≤ 0 PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.4

Form Factors at “Any” Q 2 from Polarisation Transfer MAMI, JLab, SLAC,. . . � � � d σ ��� d σ � � G 2 E + τ G 2 M tan 2 θ unpolarised beam & target � M + 2 τ G 2 = � 1 + τ d Ω d Ω � 2 outgoing spins undetected Mott lab ⇒ 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. 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 � right � transverse (“real”) photon: J z = ± 1 = left γ -polarisations P ( γ ) long / trans by e -spin, kinematics. [cf. Tho 8.6] mostly ∝ P ( γ ) ∝ P ( γ ) trans G M long G E P ( γ ) trans tan θ ⇒ Spin-dep. measurement uses QM interference of amplitudes: G E ( Q 2 ) G M ( Q 2 ) = − E + E ′ 2 = P ( γ ) 2 M long No absolute cross section, no absolute beam & recoil polarimetry. = ⇒ Many systematics cancel. So accurate that discrepancies to Rosenbluth led to theory update ( 2 γ exchange) [Afanasev/. . . 2008] . PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.2.5

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

(e) Proton Form Factors: Why So Simple? [Tho, Fig 7.8; much more data available] � − 2 � 1 + Q 2 G M with a = 4 . 27 fm − 1 = 0 . 84 GeV Exp. at low Q 2 : dipole G E = µ p = 2 . 79 ... = a 2 � Ep � = − 3!d G E Q = 0 = 12 ⇒ ρ ( r ) = ρ 0 e − ra exponential � r 2 a 2 ≈ ( 0 . 82 fm ) 2 � = � d Q 2 high-accuracy data at Q 2 → 0 : � r 2 Ep � = ([ 0 . 8775 ± 0 . 0051 ] fm ) 2 high-accuracy data at Q 2 → 0 : � r 2 Mp � = ([ 0 . 777 ± 0 . 013 ± 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

There Is Some Deviation from Simple 1-Dipole Form at High Q 2 � � � � d σ � G 2 E + τ G 2 , τ = Q 2 d σ M tan 2 θ � M + 2 τ G 2 = × � 1 + τ 4 M 2 d Ω � 2 d Ω Mott lab Ratio electric-to-magnetic proton FF Magnetic proton FF: deviation from dipole � d σ � � d σ � ( Q 2 → ∞ , i.e. also τ → ∞ ) ∝ tan 2 θ 1 2 � 2 largely ok = ⇒ × Dipole � Q 6 d Ω d Ω 1 + Q 2 Mott elastic a 2 ⇒ 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

❃ ❬ ❞ ❝ ❜ ❛ ❵ ❴ ❫ ❪ ❭ ❃ → ❡ ❩ ❨ ✖ ❃ ✭ ❳ ❲ ➌ ➋ ❱ ❃ ❢ ❚ ❧ ❃ → ❃ → ❈ → ➚ ♠ ❃ ❃ ❦ ❣ ➋ ➥ ❥ ✐ ➋ ➋ ❃ ✭ ➘ ❤ ❯ ✭ → é ❇ ❃ ➧ ❆ ❅ õ ❄ ➋ → ❂ → ➧ ❁ ❀ ✿ ✾ ✽ ✼ ✻ ✺ ✹ ✸ ➐ ✖ ❚ ➹ ✖ ➋ ❙ ❘ → ➐ ◗ P ❖ ◆ ▼ ❈ ▲ ❑ ➋ ❏ ■ ❍ ● ❋ ❊ ❉ ➚ ❢ ➚ ✶ ê ❺ ❃ ❃ ➚ ❹ ➚ ➧ ❸ ⑩ ❷ P ❻ ➋ ➋ ❶ ⑩ → ⑨ ❃ → ➥ ⑧ → ❼ ➋ ➁ ⑩ → ❘ ❃ ➄ → ➚ ➃ ➂ ➁ ➀ ❽ ❥ ❿ ❾ → → → → ❼ ❃ ✇ ❃ ⑦ ➋ ♥ → ✉ ➋ ❛ ✖ ➘ ✉ t s ✖ r ñ ➋ q ♣ ❃ ❃ ♦ ❃ ➐ ❃ ❢ ❃ ✈ ✇ ❚ → ⑥ ➘ ⑤ ➴ ✭ ✖ ④ ➋ ❚ ③ → ① ❂ ➐ ➐ ➧ ➚ ❈ ➋ ✭ ✖ ② ➴ ✷ ✵ ➆ ➦ ➵ ➳ ➲ ➯ ➭ ➫ ➩ ➨ ➥ ➧ ➥ ➺ ➤ ➢ ➡ ➠ ➟ → ➞ ➝ ➜ ➛ ➸ ➻ ↕ ➷ Ð Ï ❰ ❮ ❒ ❐ ✃ ➱ ➮ ➬ ➴ ➼ → ➘ ➹ ➶ ➚ ➪ ➣ ➚ ➾ ➽ ➙ ↔ Ò ➋ ➉ ➊ ➋ ➧ ➚ ➋ → ❺ ➚ ❿ → ➇ ➌ ➍ ➅ ➎ ➅ → ➚ → → → ➈ ë ➣ ➌ → → ➔ ➓ ➒ ➑ ➐ ➏ ➎ ➍ ➋ ➋ ➊ ➉ ➈ ➇ ➆ ➅ ➄ ➃ ➂ ➁ ➀ Ñ Ó ✴ ☞ ➥ ✕ ✔ ✓ ✒ ✑ ✏ ✎ ✍ ✌ ➧ ✖ ù ☛ ✡ ✠ ✟ ✞ ✝ ✆ ☎ ✄ ✓ ✗ ✁ ✩ ✳ ✲ ✱ ✰ ✯ ✮ ✭ ✬ ✫ ✪ ★ ✘ ✖ ✧ ✦ ✥ ✤ ✣ ✢ ✜ ✛ ✚ ✙ ✂ Ô Û ä ➘ ã â á à ß Þ Ý Ü Ú æ ➚ ➣ Ù Ø ➧ × ➘ ➣ Ö Õ å ç ÿ ➥ þ ý ü û ú ù ø ÷ ö õ ô è ó ò ñ ð ï î í ì ë ê é ➅ (f) Neutron Form Factors: Why So Similar to Proton? ⇒ d ( e , e ′ ) & subtract binding effects; or at Q 2 → 0 : scatter n off atomic e − cloud. No neutron targets. = � − 2 � G n Q 2 Low Q 2 : nearly same dipole as proton for M µ n = − 1 . 91 ... ≈ 1 + ( 0 . 84 GeV ) 2 Mn � = ([ 0 . 862 ± 0 . 009 ] fm ) 2 ≈ � r 2 high-accuracy data: � r 2 Ep � ≈ � r 2 Mp � [PDG 2012] high-accuracy data: � r 2 En � = − [ 0 . 1161 ± 0 . 0022 ] fm 2 < 0 !! � d 3 r ( 2 π ) 3 r 2 � � | ρ + ( r ) |−| ρ − ( r ) | high-accuracy data: This is allowed: − � > high-accuracy data: This is allowed: = � r 2 + �−� r 2 < 0 ! = ⇒ 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

(g) Not Even A Model: Meson-Cloud Argument [PRSZ 6.3] [HG 6.6] ⇒ Even point-particle has F ( Q 2 ) � = 1 . QFT: Every particle has a virtual cloud. = 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