4. Deep Inelastic Scattering and Partons Or: Fundamental - 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 4. Deep Inelastic Scattering and Partons Or: Fundamental Constituents at Last References: [HM 9; PRSZR 7.2, 8.1/4-5; HG 6.8-10] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.4.0

(a) Inelastic Scattering → Deep Inelastic Scattering DIS Breit/Brick-Wall Frame: no energy transfer E − E ′ = 0 ; momentum transfer maximal � p ′ Breit = − � p Breit . 1 ⇒ Dissipate energy and momentum into small volume λ 3 � Probe wave length λ Breit ∼ Q 2 = Breit . Now Q 2 � ( 3 . 5GeV ) 2 ∼ ( 0 . 07fm ) 2 ≫ r − 2 N : Energy cannot dissipate into whole N in ∆ t ∼ λ c [Tho] = ⇒ Shoot hole into N , breakup dominates. Deep Inelastic Scattering DIS N ( e ± , e ′ ) X : inclusive, i.e. all outgoing summed.  Lorentz-Invariant: ν = p · q M = E lab − E ′ lab > 0 energy transfer in lab Now characterised by   2 independent variables Invariant mass-squared W 2 = p ′ 2 = M 2 + 2 p · q + q 2 = M 2 + q 2 ( 1 − x )  out of ( θ , q 2 = − Q 2 ,  ⇑  Bjørken- x = − q 2 2 p · q = Q 2 E ′ lab , ν , W , x ) 2 M ν ∈ [ 0;1 ] ; elastic scattering: x = 1 Dimension-less structure functions F 1 , 2 ( x , Q 2 ) parametrise most general elmag. hadron ME. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.4.1

(b) Experimental Evidence E , Q 2 ր : Resonances broaden & disappear into continuum for W ≥ 2 . 5 GeV total depends only weakly on Q 2 at fixed W ≫ M = ⇒ elastic on point constituents. Mott (elastic point) [HG 6.18] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.4.2

Structure Functions F 1 , 2 are Q 2 -Independent at Fixed Bjørken- x � � 2 α � 2 � F 2 ( Q 2 , x ) � d 2 σ � + 2 F 1 ( Q 2 , x ) E ′ 2 cos 2 θ tan 2 θ � = (I.7.6) � lab d Ω d E ′ q 2 ν 2 M 2 F 1 , 2 dimensionless, Q 2 , ν → ∞ but x = Q 2 fixed: F 1 , 2 cannot dep. on Q 2 , only on dimensionless x . 2 M ν SLAC data proton 2 GeV 2 < Q 2 < 30 GeV 2 world data proton, x = 0 . 225 [HG 6.20] Data: no new scale (e.g. mass, constituent radius)! back-scattering events = ⇒ F 1 � = 0 : fermions. [Tho 8.11] ⇒ Interpretation: Virtual photon absorbed by charged, massless spin- 1 = 2 point -constituents: P ARTONS . Idea: Bjørken 1967; name: Feynman 1969; soon identified with Gell-Mann’s “quarks” of isospin. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.4.3

(c) Sequence of Events in the Parton Model [HM 9, PRSZ] Scaling: independence of Q 2 at fixed x . Not a sign of QCD, but only that no new scale in nucleon: point-constituents. Scale-Breaking as sign of “small” interactions between constituents → QCD’s DGLAP-WW (Part III) ⇒ DIS is elastic scattering on P ARTONS : charged, m = 0 spin- 1 = 2 point -constituents. Problem: Partons not in detector − → Confinement hypothesis (later). = ⇒ Assume that collision proceeds in two well-separated stages: struck parton (1) Parton Scattering: Timescale in Breit frame: partons γ t parton ≈ ∆ x c ≈ λ ≈ 1 Q ≪ 0 . 05fm for Q 2 ≫ ( 4GeV ) 2 . rearrange c partons in into nucleon = ⇒ Photon interacts with one parton near-instantaneously, hadrons takes snapshot of parton configuration, frozen in time. spectator partons (2) Hadronisation: final-state interactions rearrange partons into hadron fragments, covert collision energy into new particles (inelastic!). typ. hadron mass ∼ 1GeV ≈ 0 . 2fm 1 Much larger timescale t hadronisation ≈ c ≫ t parton . = ⇒ Describe Scattering and Hadronisation independently of each other, no interference. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.4.4

(d) Relating Elastic Parton Scattering & Nucleon DIS What is the Bjørken- x ? – The Infinite Momentum Frame p ⊥ q of partons sum to zero, but cannot simply infer them from p µ ! Problem: Transverse momenta � parton momentum parton momentum Solution: Use W , Q 2 → ∞ to boost q µ along N -momentum axis � p N momentum N momentum Lorentz boost into Infinite Momentum Frame IMF . against q µ boost Transverse parton momenta unchanged, but longitudinal now p � → γ p � p ⊥ − q ≫ M , | � q | . q p ⊥ = ⇒ Transverse motion time-dilated: Hadronisation indeed much slower: rearranging by � q → 0 . IMF is also a Breit/Brick-wall frame : = ⇒ Parton carries momentum fraction 0 ≤ ξ ≤ 1 of total nucleon momentum. = ( ξ p + q ) 2 = Assume Elastic Scattering on Parton: ( ξ p ) 2 ! ⇒ 2 ξ p · q + q 2 = 0 ; ( ξ p ) 2 cancels. ⇒ ξ = − q 2 2 p · q = Q 2 Bjørken- x � = fraction of hadron momentum which is carried by = 2 M ν = x : parton struck by photon in Infinite Momentum Frame IMF . PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.4.5

Compare Elastic and Inelastic Cross Sections more careful: [HM ex. 9.3ff; CL] Idea: DIS = incoherent superposition of elastic scattering on individual partons in IMF. = ⇒ Relate Inelastic to Elastic Cross Section e µ → e µ on Point-Fermion in lab frame: � � 2 α � 2 � F 2 ( Q 2 , x ) � d 2 σ � + 2 F 1 ( Q 2 , x ) cos 2 θ tan 2 θ � 2 E ′ 2 = inelastic (I.7.6): � d Ω d E ′ q 2 ν M 2 inel � � 2 � � � � E ′ 1 − q 2 elastic (I.7.4): d σ Z α cos 2 θ 2 M 2 tan 2 θ E with E ′ = � = � 2 E sin 2 θ 1 + E d Ω 2 E 2 M ( 1 − cos θ ) el 2 q 2 = ( k − k ′ ) 2 = − 2 k · k ′ = − 2 EE ′ ( 1 − cos θ ) = − 4 EE ′ sin 2 θ use 2 � � 2 Z α E ′ � 2 � � � E ′ d 2 σ 1 − q 2 cos 2 θ 2 M 2 tan 2 θ E δ [ E ′ − � = ⇒ = M ( 1 − cos θ )] � d Ω d E ′ q 2 1 + E 2 E 2 el PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.4.6

Compare Elastic and Inelastic Cross Sections more careful: [HM ex. 9.3ff; CL] Idea: DIS = incoherent superposition of elastic scattering on individual partons in IMF. = ⇒ Relate Inelastic to Elastic Cross Section e µ → e µ on Point-Fermion in lab frame: � � 2 α � 2 � F 2 ( Q 2 , x ) � d 2 σ � + 2 F 1 ( Q 2 , x ) cos 2 θ tan 2 θ � 2 E ′ 2 = inelastic (I.7.6): � d Ω d E ′ q 2 ν M 2 inel � � 2 Z α E ′ � 2 � � � E ′ d 2 σ 1 − q 2 cos 2 θ 2 M 2 tan 2 θ E δ [ E ′ − � = M ( 1 − cos θ )] elastic (I.7.4): � d Ω d E ′ 1 + E q 2 2 E 2 el � � use E ′ M ( 1 − cos θ )] = E ′ + EE ′ E 1 + E E δ [ E ′ − δ [ E ′ − E M ( 1 − cos θ ) M ( 1 − cos θ ) ] 1 + E E � �� � � �� � � �� � = − ν = − q 2 / ( 2 M ) = 1 by δ -distribution = δ [ ν + q 2 ν δ [ 1 + q 2 q 2 2 M ] = 1 2 M ν ] = 1 2 p · q ] = 1 ν δ [ 1 + ν δ [ 1 − x ] expected for elastic scattering � � � 2 Z α � 2 � 1 � � d 2 σ q 2 q 2 cos 2 θ 2 M 2 ν tan 2 θ � 2 E ′ 2 δ [ 1 + = ⇒ = ν − 2 p · q ] � d Ω d E ′ q 2 2 el PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.4.6

Compare Elastic and Inelastic Cross Sections more careful: [HM ex. 9.3ff; CL] Idea: DIS = incoherent superposition of elastic scattering on individual partons in IMF. = ⇒ Relate Inelastic to Elastic Cross Section e µ → e µ on Point-Fermion in lab frame: � � 2 α � 2 � F 2 ( Q 2 , x ) � d 2 σ � + 2 F 1 ( Q 2 , x ) cos 2 θ tan 2 θ � 2 E ′ 2 = inelastic (I.7.6): � d Ω d E ′ q 2 ν M 2 inel � � 2 Z α � 2 � 1 � � d 2 σ q 2 q 2 cos 2 θ 2 M 2 ν tan 2 θ � 2 E ′ 2 = ν − δ [ 1 + 2 p · q ] elastic (I.7.4): � d Ω d E ′ q 2 2 = − x el � �� � + q 2 F 2 ( Q 2 , x ) = Z 2 δ [ 1 = ⇒ Elastic on 1 point-fermion with momentum p depends only on x : 2 p · q ] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.4.6