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PHYS 6610: Graduate Nuclear and Particle Physics I 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 2021 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 2021 H. W. Grießhammer, INS, George Washington University II.4.0 (a) Inelastic Scattering ! Deep Inelastic Scattering DIS 0 0 Breit/Brick-Wall Frame: no energy transfer E −E = 0; momentum transfer maximal~pBreit = −~pBreit. 1 3 Probe wave length lBreit ∼ p =) Dissipate energy and momentum into small volume lBreit. Q2 2 2 −2 −2 Now Q & (3:5GeV) ∼ (0:07fm) rN : l Energy cannot dissipate into whole N in Dt ∼ c [Tho] =) Shoot hole into N, breakup dominates. Deep Inelastic Scattering DIS N(e±;e0)X: inclusive, i.e. all outgoing summed. 2 p · q 0 Now characterised by Lorentz-Invariant: n = = E − E > 0 energy transfer in lab M lab lab 6 2 independent variables6 2 02 2 2 2 2 6 Invariant mass-squared W = p = M + 2p · q + q = M + q (1 − x) out of (q;q2 = −Q2; 6 4 −q2 = Q2 * E0 ;n;W;x) Bjørken-x = 2 [0;1]: inelasticity (elastic scattering: x = 1) lab 2p · q = 2Mn 2 Dimension-less structure functions F1;2(x;Q ) parametrise most general elmag. hadron ME. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.4.1 (b) Experimental Evidence E;Q2 %: Resonances broaden & disappear into continuum for W ≥ 2:5 GeV total depends only weakly on Q2 at fixed W M =) elastic on point constituents. Mott (elastic point) [HG 6.18] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.4.2 2 Structure Functions F1;2 are Q -Independent at Fixed Bjørken-x 2 2 2 2 d s 2a 02 2 q F2(Q ;x) 2F1(Q ;x) 2 q (I.7.6) 0 = 2 E cos + tan dW dE lab Q 2 n M 2 Q2 F dimensionless, Q2;n ! 1 but x = fixed: F cannot dep. on Q2, only on dimensionless x. 1;2 2Mn 1;2 SLAC data proton 2 GeV2 < Q2 < 30 GeV2 world data proton, x = 0:225 [HG 6.20] Data: no new scale (e.g. mass, constituent radius)! back-scattering events =) F 6= 0: fermions. 1 [Tho 8.11] 1 =) Interpretation: Virtual photon absorbed by charged, massless spin- 2 point-constituents: PARTONS. Idea: Bjørken 1967; name: Feynman 1969; soon identified with Gell-Mann’s “quarks” of isospin. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.4.3 (c) Sequence of Events in the Parton Model [HM 9, PRSZ] Scaling: independence of Q2 at fixed x. Not a sign ofQCD, 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) 1 =) DIS is elastic scattering onP ARTONS: charged, m ≈ 0 spin- 2 point-constituents. Problem: Partons not in detector −!Confinement hypothesis (later). =) Assume that collision proceeds in two well-separated stages: (1) Parton Scattering: Timescale in Breit frame: struck parton partons Dx 1 fm γ t ≈ ≈ l ≈ 0:05 for Q2 (4GeV)2. rearrange parton c Q c into =) Photon interacts with one parton near-instantaneously, hadrons takes snapshot of parton configuration, frozen in time. partons in nucleon spectator partons (2) Hadronisation: final-state interactions rearrange partons into hadron fragments, covert collision energy into new particles (inelastic!). 1 fm Much larger timescale thadronisation ≈ ≈ 0:2 tparton. typ. hadron mass∼ 1GeV c =) Describe Scattering and Hadronisation independently of each other, no interference. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 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 ? m Problem: Transverse momenta~pq of partons sum to zero, but cannot simply infer them from p ! parton momentum parton momentum 2 Solution: Use W;Q ! 1 to boost qµ along N-momentum axis~p N momentum N momentum Lorentz boost into Infinite Momentum Frame IMF. against q µ k boost k ? Transverse parton momenta unchanged, but longitudinal now pq −! g pq M;j~pq j. ? =) Transverse motion time-dilated: Hadronisation indeed much slower: rearranging by~pq ! 0. IMF is also a Breit/Brick-wall frame: =) Parton carries momentum fraction 0 ≤ x ≤ 1 of total nucleon momentum. ! Assume Elastic Scattering on Parton: (x p)2 = (x p + q)2 =) 2x p · q + q2 = 0; (x p)2 cancels. −q2 Q2 Bjørken-x = fraction of hadron momentum which is carried by =) x = = = x: b 2p · q 2M n parton struck by photon in Infinite Momentum Frame IMF. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.4.5 No, Not That IMF – And Not That IMF Either PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.4.6 2 1 2 2 Q 2 d s(Q ;x = p·q ) Z d s(xp) Sum cross sections, =) 2 = dx q(x) 0 ∑ 0 elastic dW dE all partons q dW dE no QM interference. inel 0 on parton 2 2 2 Q p!xp Q Q p!xp with d[1 − ] −! d[1 − ] = xd [x − ] and M2 ≡ p2 −! (xp)2 = x 2M2 2p · q 2xp · q 2p · q | {z } = d(x − x): parton momentum must match exp. kinematics 1 2 2 2 Z 2 d s(Q ;x) 2a 2 q 02 2 x x Q 2 q 0 = 2 cos E dx ∑ Zq q(x) + 2 2 tan d(x − x) dW dE inel Q 2 all partons q n x 2M n 2 0 | {z } = x=M slaughter d distribution 2 2 2 d s(Q ;x) 2a 2 q 02 2 x 1 2 q 0 = 2 cos E × ∑ Zq q(x) + tan dW dE inel Q 2 all partons q n M 2 Compare Elastic and Inelastic Cross Sections more careful: [HM ex. 9.3ff; CL] Idea: DIS = incoherent superposition of elastic scattering in IMF on individual partons. =) Relate Inelastic to Elastic Cross Section em ! em on Point-Fermion in lab frame: 2 ! 0 2 ds Za 2 q E Q 2 q 0 E elastic (I.7.4): = cos 1 + tan with E = dW 2 q 2 E 2M2 2 E el 2E sin 2 1 + M (1 − cosq) q use −Q2 ≡ q2 = (k − k0)2 = −2k · k0 = −2EE0(1 − cosq) = −4EE0 sin2 m ! 0! 2 e 2 0 2 0 2 d s 2Za E 2 q E Q 2 q 0 E =) = cos 1 + tan d[E − ] E0 Q2 E M2 E dW d el 2 2 2 1 + M (1 − cosq) PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H. W. Grießhammer, INS, George Washington University II.4.7 2 1 2 2 Q 2 d s(Q ;x = p·q ) Z d s(xp) Sum cross sections, =) 2 = dx q(x) 0 ∑ 0 elastic dW dE all partons q dW dE no QM interference. inel 0 on parton 2 2 2 Q p!xp Q Q p!xp with d[1 − ] −! d[1 − ] = xd [x − ] and M2 ≡ p2 −! (xp)2 = x 2M2 2p · q 2xp · q 2p · q | {z } = d(x − x): parton momentum must match exp. kinematics 1 2 2 2 Z 2 d s(Q ;x) 2a 2 q 02 2 x x Q 2 q 0 = 2 cos E dx ∑ Zq q(x) + 2 2 tan d(x − x) dW dE inel Q 2 all partons q n x 2M n 2 0 | {z } = x=M slaughter d distribution 2 2 2 d s(Q ;x) 2a 2 q 02 2 x 1 2 q 0 = 2 cos E × ∑ Zq q(x) + tan dW dE inel Q 2 all partons q n M 2 Compare Elastic and Inelastic Cross Sections more careful: [HM ex. 9.3ff; CL] Idea: DIS = incoherent superposition of elastic scattering in IMF on individual partons. =) Relate Inelastic to Elastic Cross Section em ! em on Point-Fermion in lab frame: 2 0 2 0 2 d s 2Za E 2 q E Q 2 q 0 E elastic (I.7.4): = cos 1 + tan d[E − ] E0 Q2 E M2 E dW d el 2 2 2 1 + M (1 − cosq) E0 E E0 E EE0 use d[E0 − ] = 1 + (1 − cosq) d[E0 − E + (1 − cosq)] E 1 + E (1 − cosq) E M M M | {z } | {z } | {z } 2 = 1 by d -distribution = −n = Q =(2M) Q2 1 Q2 1 Q2 1 = d[n − ] = d[1 − ] = d[1 − ] = d[1 − x] 2M n 2Mn n 2p · q n as expected for elastic scattering X 2 2 2 2 d s 2a 2 q 02 2 1 Q 2 q Q =) 0 = 2 cos E Z + 2 tan d[1 − ] dW dE el Q 2 n 2M n 2 2p · q 2 2 2 2 d s 2a 2 q 02 F2(Q ;x) 2 F1(Q ;x) 2 q inelastic (I.7.6): = cos E + tan d dE0 Q2 2 M 2 = x W inel n z }| { Q2 =) F of elastic on 1 point-fermion with momentum p only dep. on x: F (Q2;x) = Z2 d[1− ] 2 2 2p · q PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2021 H.
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