Intr Introduc oduction tion to to Qua Quantum ntum Chr hrom - PowerPoint PPT Presentation

Intr Introduc oduction tion to to Qua Quantum ntum Chr hrom omodyna odynamic ics (QC (QCD) ) Jianwei Qiu Theory Center, Jefferson Lab May 29 – June 15, 2018 L ecture Two

QCD is e QC is everywhe rywhere in our univ in our universe se q What is the role of QCD in the evolution of the universe? q How hadrons are emerged from quarks and gluons? q How does QCD make up the properties of hadrons? Their mass, spin, magnetic moment, … q What is the QCD landscape of nucleon and nuclei? Color Confinement Asymptotic freedom Q (GeV) 200 MeV (1 fm) 2 GeV (1/10 fm) Probing momentum q How do the nuclear force arise from QCD? q ...

Unprecedented Intellectual Challenge! q Facts: No modern detector has been able to see quarks and gluons in isolation! Gluons are dark! q The challenge: How to probe the quark-gluon dynamics, quantify the hadron structure, study the emergence of hadrons, …, if we cannot see quarks and gluons? q Answer to the challenge: Theory advances: QCD factorization – matching the quarks/gluons to hadrons with controllable approximations! Experimental breakthroughs: Jets – Footprints of energetic quarks and gluons Quarks – Need an EM probe to “see” their existence, … Gluons – Varying the probe’s resolution to “see” their effect, … Energy, luminosity and measurement – Unprecedented resolution, event rates, and precision probes, especially EM probes, like one at Jlab, …

The heor oretic tical a l appr pproa oache hes – a s – appr pproxim ximations tions q Perturba turbativ tive QC QCD F Factoriza torization: tion: p e – Approximation at Feynman diagram level 1 ⎛ ⎞ DIS : O QR σ + ⊗ ⎜ ⎟ tot ⎝ ⎠ See Metz’s lectures Probe Approximation Structure Sokhan’s lectures Hard-part Power corrections Parton-distribution Furletova’s lectures q Ef Effectiv tive f fie ield the ld theory (EFT): ory (EFT): See Stewart’s lectures – Approximation at the Lagrangian level Cirigliano’s lectures Soft-collinear effective theory (SCET), Non-relativistic QCD (NRQCD), Heavy quark EFT, chiral EFT(s), … See Stevens’ lectures q Othe Other a r appr pproa oache hes: s: Pastore’s lectures Light-cone perturbation theory, Dyson-Schwinger Equations (DSE), Constituent quark models, AdS/CFT correspondence, … q La Lattic ttice QC QCD: : See Stevens’ lecture – Approximation mainly due to computer power Pastore’s lectures Hadron structure, hadron spectroscopy, nuclear structure, phase shift, …

Physical Observables Cross sections with identified hadrons are non-perturbative! Hadronic scale ~ 1/fm ~ 200 MeV is not a perturbative scale Purely infrared safe quantities Observables without identified hadron(s)

Fully infrare Fully infra red sa d safe fe obse observa rvable bles – I s – I Fully inclusive, without any identified hadron! σ total e + e − → hadrons = σ total e + e − → partons The sim The simple plest obse st observa rvable ble in QC in QCD

e + e - è hadrons inclsusive cross sections q e + e - è hadron total cross section – not a specific hadron! 2 Hadrons σ tot e + e − → hadrons ∝ “n” Partons “m” If there is no quantum interference between partons and hadrons, tot P P P P P ∑ ∑ ∑ ∑ ∑ =1 σ ∝ = = m n m n + − + − + − + − e e hadrons e e n e e m → e e m → → → → → n n m m n Unitarity ∝ ∑ tot P σ + − + − e e partons e e m → → m tot tot Finite in perturbation σ = σ + − + − e e hadrons e e partons → → theory – KLN theorem q e + e - è parton total cross section: Calc lcula ulable le in in pQC pQCD

Infrared Safety of e + e - Total Cross Sections q Optical theorem: 2 q Time-like vacuum polarization: IR safety of IR safety of with q IR safety of : If there were pinched poles in Π (Q 2 ), ² real partons moving away from each other ² cannot be back to form the virtual photon again! Rest frame of the virtual photon

Lowest order (LO) perturbative calculation p 1 k 1 q Lowest order Feynman diagram: q Invariant amplitude square: k 2 p 2 1 1 Tr 2 4 2 | M | e e N p p µ ν ⎡ ⎤ = γ ⋅ γ γ ⋅ γ ⎣ ⎦ + − → Q c 2 1 2 2 e e QQ s 2 2 s ( p p ) = + 1 2 ( ) ( ) Tr k m k m ⎡ ⎤ × γ ⋅ + γ γ ⋅ − γ 1 Q 2 Q ⎣ ⎦ µ ν 2 t ( p k ) = − 1 1 2 4 2 2 2 2 2 2 = e e N ( m t ) ( m u ) 2 m s ⎡ ⎤ − + − + 2 u ( p k ) = − ⎣ ⎦ Q c Q Q Q 2 s 2 1 q Lowest order cross section: d σ 1 + − e e Q Q 2 2 → | M | whe re s Q Threshold constraint = = + − → 2 e e QQ d t 16 s π 2 2 2 m 4 m 2 4 ⎡ ⎤ πα Q Q (0) 2 e N em 1 1 ∑ ∑ σ = σ = + − ⎢ ⎥ 2 + − → Q c e e Q Q 3 s s s ⎢ ⎥ Q Q ⎣ ⎦ One of the best tests for the number of colors

Next-to-leading order (NLO) contribution q Real Feynman diagram: E 2 p q . with x i i i 1,2,3 = = = i s s / 2 + crossing ⎛ ⎞ 2 p . q ∑ ⎜ ⎟ i ⎝ ⎠ i x 2 ∑ = = i s i 2 1 x x x 1 cos , cycl . ( ) ( ) − = − θ 1 2 3 23 q Contribution to the cross section: IR as x3 → 0 d 2 2 CO as θ 13 → 0 σ 1 x x α + + − → e e QQg s C 1 2 = θ 23 → 0 F dx dx 2 1 x 1 x ( )( ) σ π − − 0 1 2 1 2 Div ivergent a nt as x s x i → 1 Need the d the vir virtua tual c l contrib ontribution a ution and a nd a r regula gulator! tor!

How does dimensional regularization work? q Complex n -dimensional space: Im(n) (1) Start from here: (2) Calculate IRS quantities UV renormalization here Theory cannot be renormalized! a renormalized theory (3) Take ε è 0 for IRS quantities only 4 6 Re(n) UV-finite, IR divergent UV-finite, IR-finite

Dimensional regularization for both IR and CO q NLO with a dimensional regulator: 2 ε 2 ⎡ 1 ⎤ ( ) 4 4 1 3 19 Γ − ε ⎛ ⎞ α π µ ⎛ ⎞ ⎡ ⎤ (1) (0) ² Real: s σ = σ + + ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ 3, 2, ε ε 2 2 3 Q 1 3 2 4 ( ) π Γ − ε ε ε ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ ² Virtual: 2 ε ⎡ 1 1 ⎤ 2 ( ) ( ) 2 4 4 1 3 Γ − ε Γ + ε ⎛ ⎞ ⎡ ⎤ α π µ π ⎛ ⎞ (1) (0) s 4 σ = σ − − + − ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ 2, 2, 2 2 ε ε 3 Q 1 2 2 2 ( ) π Γ − ε ε ε ⎝ ⎠ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ ⎣ ⎦ α ⎡ ⎤ (1) (1) (0) ² NLO: s O No ε dependence! ( ) σ + σ = σ + ε ⎢ ⎥ 3, 2, 2 ε ε π ⎣ ⎦ α ⎡ ⎤ ² Total: tot (0) ( 1) (1) ( 2 ) (0 ) ( 2 ) O 1 s O σ = σ + σ + σ + α = σ + + α 2 3, 2 , s 2 ⎢ ⎥ s ε ε π ⎣ ⎦ σ tot is Infrared Safe! σ tot is independent of the choice of IR and CO regularization Go be Go beyond the ond the inc inclusiv lusive tota total c l cross se oss section? tion?

Hadronic cross section in e+e- collision q Normalized hadronic cross section: : R e + e − ( s ) ≡ σ e + e − → hadrons ( s ) N c = 3 σ e + e − → µ + µ − ( s )  1 + α s ( s ) �  1 + α s ( s ) � X e 2 + O ( α 2 2 + ... s ( s )) ≈ N c q π π q = u,d,s " ! r # 1 + 2 m 2 1 − 4 m 2 X e 2 q q + N c + O ( α s ( s )) q s s q = c,...

Fully infra Fully infrare red sa d safe fe obse observa rvable bles - II s - II No identified hadron, but, with phase space constraints σ Jets e + e − → hadrons = σ Jets e + e − → partons Jets – “tra ts – “trace” or “footprint” of ” or “footprint” of pa partons rtons Thrust distribution in e + e - collisions etc.

Jets – trace of partons q Jets – “total” cross-section with a limited phase-space Not any specific hadron! q Q: will IR cancellation be completed? ² Leading partons are moving away from each other E 2 δ ² Soft gluon interactions should θ not change the direction of an energetic parton → a “jet” δ Z -axis ε √ s= δ ’ – “trace” of a parton q Many Jet algorithms E 1 Sterman-Weinberg Jet