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 III. Descriptions 2. Perturbative QCD Or: Why we Believe References: [PRSZR 8.1-3, 14; HM 2.15, 10.3-9, 11.4/6-7; Tho 10.7/8; Ryd 3, end of 9.6; HG 12.3; PS 16.7; Per 6.5; lots more. . . ] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.0
(a) An Ideal World: QCD With Small Coupling Constant (b) From Colours to Potentials (c)Running Coupling & Asymptotic Freedom QED: [Ryd, end of 9.6] QCD: [PS 16.7, Per 6.5] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.1
Running Coupling in QCD: Now Known to O ( α 4 s ) � = 3 -Loop SU ( N c ) Gauge Theory at LO (1-loop) 4 π : α s ( q 2 ) = N f quark flavours with m 2 q < q 2 (for m q = 0 ) [ 11 N c − 2 N f ] ln ( q 2 / Λ 2 QCD ) [Gross, Politzer/Wilczek, ’t Hooft 1973] Today calculated up to & including O ( α 3 s ) relative to LO: horrific diagrams, beautifully agrees with data. = ⇒ QCD has only one parameter. Data: α s ( M z ) = 0 . 1181 ± 0 . 0013 or Λ QCD ≈ 250 MeV . charm threshold ← bottom threshold ↓ [PDG 2015] • perturbative renormalisation procedure • gauge group is SU c ( N c ) , and N c = 3 This Confirms: • flavour N f = ( uds )+( c )+( b ) increases like R -factor with √ s PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.2
The Low- q 2 Regime: Infrared Slavery 4 π α s ( q 2 ) = QCD ) + O ( α 3 s ) [ 11 N c − 2 N f ] ln ( q 2 / Λ 2 Naïvely apply running = ⇒ [Deur/. . . Phys. Lett. B665 (2008) 349] α s > 1 at some √ s ≈ 1GeV = ⇒ Perturbation theory breaks down at low s . = ⇒ Must resort to non-perturbative methods! Infrared Slavery offers chance of confinement. Is typical size of charge-smearing set by 1 Λ QCD ≈ 250MeV ≈ 1fm ? = ⇒ Hadron size, confinement? PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.3
(d) Quarkonia and Perturbative QCD = QED N 2 c − 1 for α s ( q 2 ) ≪ 1 = q at large s = q 2 . ⇒ Test on positronium-like q ¯ LO QCD ˆ Positronium: H-atom with reduced mass µ = m e → m e 2 positronium e + e − quarkonium q ¯ q − α α s − 4 pot. V ( r ) r 3 r γ glue � 4 � 2 µ q − α 2 µ e − 3 α s binding E n 2 n 2 2 n 2 Should work best for heaviest system: Bottomonium b ¯ b 1 − 1 ⇒ If truly Coulombic, then E 1 − E 2 = 27 2 2 = = . 2 2 − 1 1 E 2 − E 3 5 3 2 = ⇒ Long-range part not really Coulombic! = ⇒ Add phenom. QCD String Potential V ( r ) = − 4 α s 3 r + σ r String constant σ ≈ 1GeV fm ≈ 10 5 N fm by fit to spectra, universal in b ¯ b , c ¯ c ,... . PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.4
(e) QCD-Inspired, Phenomenological Potentials How Non-Relativistic are Quarkonia? � E kin ∼ E bind Typical velocities: v typ = systemmass � α 2 1 Positronium: v typ = 2 n 2 ∼ α = 137 ≪ 1 = ⇒ very non-relativistic. � 1GeV Bottomonium: v typ ≈ 10GeV ≈ 0 . 3 = ⇒ Relativistic effects will be large: � p 2 p 4 p 2 = m q + � 2 m q − � m 2 q + � q + ... – kin. energy 8 m 3 = ⇒ Lamb shift lowers 1S state. – Hypefine Splitting: Positronium spin-spin int. like for mag. dipoles H HFS = 2 π σ 2 δ ( 3 ) ( α � σ 1 · � � r ) 3 m 2 e H HFS = 2 π 4 α s σ 2 δ ( 3 ) ( 3 � σ 1 · � → Quarkonium: chromo-magnetic interaction between spins � r ) 3 m 2 q ∂ V ( r ) 1 � L · � – Fine Structure: H FS = S splits P -wave states with same J but different L , S . 2 m 2 r ∂ r 1 8 m 2 � ∇ 2 V ( r ) ∝ δ ( 3 ) ( – Darwin Term/Zitterbewegung H Darwin = � r ) in Coulombic potential. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.5
Phenomenological Potentials: Constituent Quark Model – Take perturbative QCD results for colour factors etc. – Fit string constant σ , quark (constituent) mass m q , α s . – Non-relativistic potential with some retardation effects: HFS, FS (LS coupling), Darwin, Lamb,. . . Results Bottom : m b ≈ 5GeV , α s ( ϒ ) ≈ 0 . 2 , σ ϒ ≈ 1 GeV fm Charm : m c ≈ 1 . 5GeV , α s ( J / ψ ) ≈ 0 . 25 , σ J / ψ ≈ 1 GeV fm – Constituent quark masses of b and c slightly larger than their QCD (current quark) masses: small “dressing” . – QCD string constant same for b and c : universal – Charmonium less Coulombic; more relativistic; more sensitive to QCD string. – Confirms perturbative colour factors. = ⇒ SU ( N c = 3 ) . – But usually HFS somewhat small, LS somewhat big. Neglects many relativistic radiative/retardation effects. [PRSZR] QCD-inspired Constituent Quark Model was important to boost confidence in QCD. – Now we need to go beyond and do “true” QCD! PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.6
(f) QCD for Quarkonium Decay Perturbative QCD needs α s ( q 2 → ∞ ) ≪ 1 . = ⇒ Focus again on lowest quarkonium states. Kinematics forbids strong decay: M ϒ ≈ 2 M b − E ϒ bind < 2 M B ( B / ¯ B -meson: ¯ b / b + light quark, e.g. b ¯ u ) bind < 2 m eff bind − E ϒ ⇒ E B = u eff. mass of light quark in B meson is large. = ⇒ Bottomonium & Charmonium only decay by q ¯ q annihilation into gluons or photons. Parity determines gluon/photon number (HW). Translate positronium: charge Z q , N c = 3 colours. | Ψ ( 0 ) | 2 probability of q ¯ q at same place ( S -wave). Γ [ 1 1 S 0 → γγ ] = 3 4 π ( Z 2 q α ) 2 | Ψ ( 0 ) | 2 m 2 q 4 π α 2 2 s Γ [ 1 1 S 0 → gg ] = | Ψ ( 0 ) | 2 m 2 3 ���� q colour factor Z 4 q α 2 ⇒ Ratio Γ [ q ¯ q → γγ ] q → gg ] = 9 q ) [ 1 + O ( α s ) QCD corrections ] independent of | Ψ ( 0 ) | 2 and m q . = α 2 Γ [ q ¯ s ( q ¯ 2 Experimental signal: gg hadronises into 2 hadron jets over longer timescale (factorisation assumption). PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.9
Determining α s ( q 2 ) in Quarkonia Bottomonium: γγ decay not yet seen. α 2 Charmonium: Γ [ η c → γγ ] Γ [ η c → gg ] = 8 s ( J / ψ ) = [ 3 . 1 ± 1 . 2 ] × 10 − 4 = ⇒ α s ( J / ψ ) = 0 . 25 ± 0 . 05 α 2 9 J / ψ and ϒ are 3 S 1 states: = ⇒ Only decay into odd number of gauge bosons (parity, see HW). q lepton virtual γ q ¯ lepton ∝ ( Z q α ) 2 Γ [ γ + 2jets ] ∝ ( Z q α ) 2 α 3 = α = α s ⇒ Γ [ leptons ] Γ [ leptons ] Γ [ 3jets ] s = Γ [ γ + 2jets ] ∝ ; ; α 3 Z 2 q αα 2 α 2 Z 2 q αα 2 Z 2 Γ [ 3jets ] q α s s s s Include QCD corrections to high orders. Lots of experimental information, many b ¯ b states & decays not yet seen. α s ( ϒ ) = 0 . 163 ± 0 . 016 α s ( J / ψ ) = 0 . 25 ± 0 . 05 But only one datum on plot. = ⇒ Can do even better. [PDG 2015] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.10
(g) Perturbative QCD Corrections in e + e − Annihilation q LO: 2-jet event virtual γ q ¯ Leading QCD correction: 3-jet event ( ) � � 1 + α s ( q 2 ) R = N c ∑ Z 2 q π q [Mar 5.12] Includes m q � = 0 corrections of QCD. [PDG 2012 46.7] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.11
2 & 3 Jet Events: Evidence of Gluons at Large √ s PETRA 1979 2 jets ≃ α s ( s ) < 1 for large √ s . 3 jets If third jet, its total charge is often zero . Ratio [PRSZR] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.12
Angular Distribution of 3- and 4-Jet Events from QCD PETRA at DESY [Per 6.9] [Tho 10.19] 3-Jet Events: angular distribution tests gluon spin: J P = 1 − . You could calculate this with what we learned. 4-Jet Events: test ggg vertex ⇐ ⇒ local SU ( 3 ) gauge symmetry . PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.13
(h) QCD in Proton-Antiproton Processes CERN, ongoing Interactions and colour factors in p ¯ p Rutherford’s gold foil data QCD data [Per 6.4] [Per 6.5] parton ) 2 → 0 ⇒ s parton ≫ t parton = ( k parton − k ′ Consider scattering on partons under small angles = α 2 s ( q 2 ) ⇒ Rutherford-like d σ 9 = d Ω ≈ + corrections from 3-gluon vertex 0 sin 4 θ 8 4 E 2 ���� 2 colour This Confirms: • Short-distance potential ∝ 1 . • Gluon massless J − = 1 − particle. • Colour factors: SU ( N c = 3 ) . r PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.2.14
Recommend
More recommend
