7. Theory of Electron Scattering Or: Our Analysis Tool References: - 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 I. Tools 7. Theory of Electron Scattering Or: Our Analysis Tool References: [HM 4, 6.1/3-6/9/11/13, 8] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University I.7.0

Helicity Conservation: “Massless” Spin- 1 2 on Spin- 0 Target Coulomb interaction of spin- 1 2 projectile ( m e ≪ E ) on infinitely-heavy, extended spin- 0 target: � � 2 � � � d σ Z α 2 cos 2 θ � � q 2 ) � = × � F ( � × � � 2 E sin 2 θ d Ω � lab 2 2 � �� � � �� � � �� � charge helicity Rutherford form factor conservation [PRSZR] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University I.7.1

Helicity Conservation: Massless Spin- 1 2 on Massive Spin- 1 2 Target HW: Electromagnetic interaction of spin- 1 2 projectile ( m e ≪ E ) on massive, point-like spin- 1 2 target: W µν = 1 p ′ + M )] = 2 [ p µ p ′ ν + p ′ µ p ν − g µν ( p · p ′ − M 2 )] : L µν with M � = 0 2tr [ γ µ ( / p + M ) γ ν ( / Mott cross section � �� � � � 2 � E ′ q 2 � � d σ Z α cos 2 θ 2 M 2 tan 2 θ � = × × 1 − � 2 E sin 2 θ d Ω � lab E 2 2 2 ���� � �� � � �� � � �� � recoil helicity backscattering Rutherford conservation on target spin [cf. Tho 8.6] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University I.7.2

Inelastic Scattering [HG 6.17] [PRSZR 7.1] scattered electron looses energy to target excitation/breakup ⇒ In lab frame: E ′ 1 = E < 1 + E M ( 1 − cos θ lab ) ⇒ New kinematic variable E ′ = lab or Invariant mass-squared of all fragments: W 2 : = p ′ 2 = ( p + q ) 2 = M 2 + 2 p · q + q 2 = M 2 + 2 p · q ( 1 − − q 2 ≥ M 2 2 p · q ) � �� � Bjorken − x PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University I.7.3

� Summary: Electron Scattering Cross Section d σ � � [HM 8.4] d Ω � lab � � 2 Z α rel. Rutherford: Coulomb (I.7.1) ( s = 0; M = ∞ , S = 0 ) 2 E sin 2 θ 2 � � 2 e ± Coulomb on composite E ′ Z α cos 2 θ � � 2 q 2 ) � � � F ( � (I.7.2) ( s = 1 2 ; M = ∞ , S = 0 ) � 2 E sin 2 θ 2 E 2 e ± full elmag on composite � � 2 E ′ Z α cos 2 θ � � 2 ( s = 1 q 2 ) � � 2 ; M finite , S = 0 ) � F ( � = ⇒ recoil (I.7.3) � 2 E sin 2 θ 2 E called Mott for F ( q 2 ) = 1 2 � �� � � � 1 − q 2 e µ → e µ : no structure 2 M 2 tan 2 θ [. . . ] × (I.7.4) ( s = 1 2 ; M , S = 1 2 ) 2 e ± on composite spin- 1 �� � � + 2 τ ( F 1 + F 2 ) 2 tan 2 θ 2 F 2 1 + τ F 2 ( s = 1 2 ; M , S = 1 2 ) [. . . ] × (I.7.5) 2 2 F 1 : Dirac FF; F 2 : Pauli FF (moved Z from prefactor to F 1 ( 0 ) = Z = G E ( 0 ) , F 2 ( 0 ) = κ = G M ( 0 ) − Z anom. mag. mom.) � G 2 � E + τ G 2 τ = − q 2 M tan 2 θ M + 2 τ G 2 [. . . ] × 4 M 2 ; G E , M : Sachs FFs 1 + τ 2 e ± inelastic � � 2 α � 2 � F 2 ( Q 2 , x ) � + 2 F 1 ( Q 2 , x ) d σ cos 2 θ tan 2 θ � 2 E ′ 2 = inclusive � (I.7.6) d Ω d E ′ q 2 � lab ν M 2 x ∈ [ 0;1 [ Structure functions F 1 , 2 ( Q 2 , x ) have nothing to do with Pauli/Dirac FFs PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University I.7.4 (I.7.7)

Summary: Electron Scattering, Hadron Currents & Tensors [HM 8.4] hadr. tensor, current 1 e 2 W µν = J µ S , S ′ ( p , p ′ ) J ν † S , S ′ ( p , p ′ ) with J µ S , S ′ ( p , p ′ ) = � p ′ , S ′ | J µ | p , S � 2 S + 1 ∑ scatter off virtual γ S , S ′ q µ J µ = 0 = ⇒ q µ W µν = 0 = q ν W µν electromagnetic current conservation rel. Rutherford: Coulomb J µ = − i Ze δ µ 0 i.e. point charge at rest (I.7.1C) ( s = 0; M = ∞ , S = 0 ) J µ = − i Ze δ µ 0 F ( e ± Coulomb on composite q 2 ) with charge form factor: Fourier of ρ ( � � r ) ∞ ( s = 1 2 ; M = ∞ , S = 0 ) � q 2 ) : = 4 π d r r F ( � q sin ( qr ) ρ ( r ) Ze J µ = − i Ze F ( q 2 ) ( p µ + p ′ µ ) 0 e ± full elmag on composite (I.7.3C) ⇒ W µν = Z 2 ( p + p ′ ) µ ( p + p ′ ) ν � � 2 ( s = 1 2 ; M finite , S = 0 ) � F ( q 2 ) � � = (I.7.3W) � (most general for S = 0 ) J µ u S ′ ( p ′ ) γ µ u S ( p ) e µ → e µ : no structure S , S ′ = − i Ze ¯ (I.7.4C) ⇒ W µν = 2 Z 2 � � ( s = 1 2 ; M , S = 1 p µ p ′ ν + p ′ µ p ν − g µν ( p · p ′ − M 2 ) 2 ) = (I.7.4W) + eF 2 ( q 2 ) J µ e ± on composite spin- 1 S , S ′ = − i e F 1 ( q 2 ) ¯ u S ′ ( p ′ ) γ µ u S ( p ) u S ′ ( p ′ ) i σ µν u S ( p ) q ν ¯ 2 2 M (I.7.5C) � �� � (most general elast. S = 1 2 ) � �� � Dirac: modify point-form Pauli: anomalous mag. term F 1 ( 0 ) = Z , F 2 ( 0 ) = κ � q µ q ν � � �� � e ± inelastic inclusive W µν = F 1 ( q 2 , x ) + F 2 ( q 2 , x ) p µ − p · q p ν − p · q − g µν q 2 q µ q 2 q ν q 2 M 2 ν M (most general inel. hadronic) (I.7.6W) PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University I.7.5

Preview: Electron Scattering on Hadronic Systems see II.1-3,5 [Martin fig. 2.4] [DFHMS fig. 7.1] [HG 6.11] nucleus quasi-elast. inelast. on quarks levels on bound N on nucleon excited in nucleus PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University I.7.6

Next: II. Phenomena Shapes of Nuclei, Bethe-Weizsäcker Mass Formula Familiarise yourself with: [PRSZR 5.4, 2.3, 3.1/3; HG 6.3/4, (14.5), 16.1; cursorily PRSZR 18, 19] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University I.7.7