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 One
The plan for my four lectures q The Goal: To unde o understa stand nd the the str strong inte ong interaction dyna tion dynamic ics in te s in term rms of s of Quantum Qua ntum C Chr hrom omo-dyna o-dynamic ics (QC s (QCD), a ), and nd to pr to prepa pare you f ou for upc or upcom oming le ing lectur tures in this sc s in this school hool q The Plan (approximately): From om the the disc discovery of ry of ha hadr drons to m ons to mode odels ls, a , and to the nd to theory of ory of QC QCD Funda Fundamenta ntals of ls of QC QCD, , How to pr ow to probe obe qua quarks/ s/gluons without be luons without being a ing able le to se to see the them? Factoriza torization, Ev tion, Evolution, a olution, and Ele nd Elementa ntary ha ry hard pr d proc ocesse sses s Hadr dron pr on prope opertie ties (m s (mass ss, spin, …) a , spin, …) and str nd struc uctur tures in QC s in QCD Unique niquene ness of ss of le lepton-ha pton-hadr dron sc on scatte ttering ring From om J JLa Lab1 b12 to the to the Ele Electr tron-Ion C on-Ion Collide ollider (EIC r (EIC) )
New pa w partic ticle les, ne , new ide w ideas, a , and ne nd new the w theorie ories s q Early proliferation of new hadrons – “particle explosion”: … and many more!
New pa w partic ticle les, ne , new ide w ideas, a , and ne nd new the w theorie ories s q Proliferation of new particles – “November Revolution”: November Revolution! … and Quark Model QCD many more! EW H 0 Completion of SM?
New pa w partic ticle les, ne , new ide w ideas, a , and ne nd new the w theorie ories s q Proliferation of new particles – “November Revolution”: November Revolution! … and Quark Model QCD many more! EW X, … H 0 Y, … Completion of SM? Z, … Pentaquark, … How do we make sense of all of these? Another particle explosion?
New pa w partic ticle les, ne , new ide w ideas, a , and ne nd new the w theorie ories s q Early proliferation of new hadrons – “particle explosion”: … and many more! q Nucleons has internal structure! ✓ e ~ ◆ 1933: Proton’s magnetic moment µ p = g p 2 m p Otto Stern g p = 2 . 792847356(23) 6 = 2! Nobel Prize 1943 ✓ e ~ ◆ µ n = � 1 . 913 6 = 0! 2 m p
New pa w partic ticle les, ne , new ide w ideas, a , and ne nd new the w theorie ories s q Early proliferation of new hadrons – “particle explosion”: Electric charge distribution … and many more! Proton EM charge q Nucleons has internal structure! radius! 1960: Elastic e-p scattering Robert Hofstadter Neutron Form factors Nobel Prize 1961
New pa w partic ticle les, ne , new ide w ideas, a , and ne nd new the w theorie ories s q Early proliferation of new particles – “particle explosion”: … and many more! q Nucleons are made of quarks! Pr Proton oton Neutr utron on Murray Gell-Mann Quark Model Nobel Prize, 1969
The naïve Quark Model q Flavor SU(3) – assumption: Physical states for , neglecting any mass difference, are represented by 3-eigenstates of the fund’l rep’n of flavor SU(3) q Generators for the fund’l rep’n of SU(3) – 3x3 matrices: with Gell-Mann matrices q Good quantum numbers to label the states: simultaneously diagonalized Isospin: , Hypercharge: q Basis vectors – Eigenstates:
The naïve Quark Model q Quark states: Spin: ½ Baryon #: B = ⅓ Strangeness: S = Y – B Electric charge: q Antiquark states:
Mesons Quark-antiquark flavor states: q Group theory says: 1 flavor singlet + 8 flavor octet states There are three states with : q Physical meson states (L=0, S=0): ² Octet states: ² Singlet states:
Quantum Numbers q Meson states: ² Spin of pair: ² Spin of mesons: ² Parity: ² Charge conjugation: q L=0 states: ( Y=S ) ( Y=S ) Flavor singlet, spin octet q Color: Flavor octet, spin octet No color was introduced!
Baryons 3 quark states: q Group theory says: ² Flavor: ² Spin: q Physical baryon states: ² Flavor-8 ² Flavor-10 Spin-1/2: Spin-3/2: Δ ++ (uuu), … Pr Proton oton Neutr utron on Violation of Pauli exclusive principle Need another quantum number - color!
Color q Minimum requirements: ² Quark needs to carry at least 3 different colors ² Color part of the 3-quarks’ wave function needs to antisymmetric q SU(3) color: Antisymmetric Recall: color singlet state: q Baryon wave function: Antisymmetric Symmetric Symmetric Symmetric Antisymmetric
A complete example: Proton q Wave function – the state: q Normalization: q Charge: q Spin: q Magnetic moment: ✓ µ n ◆ µ n = 1 µ u ≈ 2 / 3 = − 0 . 68497945(58) 3[4 µ d − µ u ] − 1 / 3 = − 2 µ d µ p Exp
How to “see” substructure of a nucleon? q Modern Rutherford experiment – Deep Inelastic Scattering: SLAC 1968: e ( p ) + h ( P ) → e 0 ( p 0 ) + X ² Localized probe: Q 2 = � ( p � p 0 ) 2 � 1 fm � 2 1 Q ⌧ 1 fm ² Two variables: Q 2 = 4 EE 0 sin 2 ( θ / 2) Q 2 x B = 2 m N ν ν = E − E 0 Discovery of spin ½ quarks, and partonic structure! What holds the quarks together? Nobel Prize, 1990 The birth of QCD (1973) – Quark Model + Yang-Mill gauge theory
Quantum Chromo-dynamics (QCD) = A quantum field theory of quarks and gluons = q Fields: Quark fields: spin- ½ Dirac fermion (like electron) Color triplet: Flavor: Gluon fields: spin-1 vector field (like photon) Color octet : q QCD Lagrangian density: q QED – force to hold atoms together: f [( i ∂ µ − eA µ ) γ µ − m f ] ψ f − 1 X 4 [ ∂ µ A ν − ∂ ν A µ ] 2 L QED ( φ , A ) = ψ f QCD is much richer in dynamics than QED Gluons are dark, but, interact with themselves, NO free quarks and gluons
Gauge property of QCD q Gauge Invariance: where q Color matrices: Generators for the fundamental representation of SU3 color q Gauge Fixing: Allow us to define the gauge field propagator: with the Feynman gauge
Ghost in QCD q Ghost: Ghost so that the optical theorem (hence the unitarity) can be respected
Feynman rules in QCD q Propagators: i Quark: γ · k − m δ ij ✓ ◆� i δ ab − g µ ν + k µ k ν 1 − 1 Gluon: k 2 k 2 λ for a covariant gauge � i δ ab − g µ ν + k µ n ν + n µ k ν k 2 k · n for a light-cone gauge n · A ( x ) = 0 with n 2 = 0 i δ ab Ghost:: k 2
Fe Feynm ynman rule n rules in QC s in QCD
Renorm normaliza lization, why ne tion, why need? d? q Scattering amplitude: = + + + ... E i E i E I 1 ⎛ ⎞ = PS ... + ... + ⇒ ∞ ∫ ⎜ ⎟ E E I − ⎝ ⎠ i I UV divergence: result of a “sum” over states of high masses Uncertainty principle: High mass states = “Local” interactions No experiment has an infinite resolution!
Physic Physics of re s of renorm normaliza lization tion q UV divergence due to “high mass” states, not observed = + - “High mass” states “Low mass” state q Combine the “high mass” states with LO Renormalized = + LO: coupling + ... No UV divergence! - NLO: q Renormalization = re-parameterization of the expansion parameter in perturbation theory
Renorm normaliza lization Group tion Group q Physical quantity should not depend on renormalization scale μ renormalization group equation: q Running coupling constant: q QCD β function: q QCD running coupling constant: Asymptotic freedom!
QCD Asymptotic Freedom q Interaction strength: μ 2 and μ 1 not independent Discovery of QCD Asymptotic Freedom Collider phenomenology Nobel Prize, 2004 – Controllable perturbative QCD calculations
Effe Effective tive Qua Quark rk Ma Mass ss q Ru2nning quark mass: Quark mass depend on the renormalization scale! q QCD running quark mass: q Choice of renormalization scale: for small logarithms in the perturbative coefficients q Light quark mass: QCD perturbation theory (Q>> Λ QCD ) is effectively a massless theory
Infrared and collinear divergences q Consider a general diagram: for a massless theory ² Singularity Infrared (IR) divergence ² Collinear (CO) divergence IR IR a and C nd CO div O divergenc nces a s are g gene neric ric pr prob oble lems s of of a a massle ssless ss pe perturba turbation the tion theory ory
Infra Infrare red Sa d Safe fety ty q Infrared safety: Infrared safe = κ > 0 Asymptotic freedom is useful only for quantities that are infrared safe
Recommend
More recommend
