Introduction to Quantum Chromodynamics (QCD)

Introduction to Quantum Chromodynamics (QCD) Jianwei Qiu August 16 19, 018 Four Lectures The 3 rd WHEPS, August 16-4, 018, Weihai, Shandong

q The Goal: The plan for my four lectures To understand the strong interaction dynamics, and hadron structure, in terms of Quantum Chromo-dynamics (QCD) q The Plan (approximately): From hadrons to partons, the quarks and gluons in QCD Fundamentals of QCD, Factorization, Evolution, and Elementary hard processes Two Lectures Hadron structures and properties in QCD Parton distribution functions (PDFs), Transverse momentum dependent PDFs (TMDs), Generalized PDFs (GPDs), and Multi-parton correlation functions Two lectures

New particles, new ideas, and new theories q Early proliferation of new hadrons particle explosion : and many more!

New particles, new ideas, and new theories q Proliferation of new particles November Revolution : November Revolution! Quark and Model many more! EW QCD H 0 Completion of SM?

New particles, new ideas, and new theories q Proliferation of new particles November Revolution : November Revolution! Quark and Model many more! EW QCD How do we make sense of all of these? H 0 X, Y, Completion of SM? Z, Pentaquark, Another particle explosion?

New particles, new ideas, and new theories q Early proliferation of new hadrons particle explosion : and many more! q Nucleons has internal structure! 1933: Proton s magnetic moment Otto Stern Nobel Prize 1943 e~ µ p = g p m p g p =.79847356(3) 6=! e~ µ n = 1.913 6= 0! m p

New particles, new ideas, and new theories q Early proliferation of new hadrons particle explosion : and many more! q Nucleons has internal structure! 1960: Elastic e-p scattering Electric charge distribution Proton EM charge radius! Robert Hofstadter Nobel Prize 1961 Form factors Neutron

New particles, new ideas, and new theories q Early proliferation of new particles particle explosion : and many more! q Nucleons are made of quarks! Proton Neutron 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: Isospin:, Hypercharge: simultaneously diagonalized 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 q Group theory says: flavor states: 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 Spin-1/: ² Flavor-10 Spin-3/: Proton Neutron Δ ++ (uuu), 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: Recall: Antisymmetric 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 = 1 3 [4µ d µ u ] µ u µ d /3 1/3 = µn µ p Exp = 0.68497945(58)

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 Discovery of spin ½ quarks, and partonic structure! What holds the quarks together? ² Localized probe: Q = (p p 0 ) 1fm ² Two variables: 1 Q 1fm Q =4EE 0 sin ( /) x B = Q m N = E E 0 The birth of QCD (1973) Quark Model + Yang-Mill gauge theory Nobel Prize, 1990

Quantum Chromo-dynamics (QCD) q Fields: = A quantum field theory of quarks and gluons = Quark fields: spin-½ Dirac fermion (like electron) Color triplet: Flavor: q QCD Lagrangian density: Gluon fields: spin-1 vector field (like photon) Color octet: q QED force to hold atoms together: L QED (,A)= X f [(i@µ ea µ ) µ m f ] f 1 4 [@ µa @ A µ ] f QCD is much richer in dynamics than QED Gluons are dark, but, interact with themselves, NO free quarks and gluons

q Gauge Invariance: Gauge property of QCD 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

q Propagators: Feynman rules in QCD Quark: Gluon: i ab k i ab k apple i k m ij g µ + k µk k 1 1 for a covariant gauge apple g µ + k µn + n µ k k n for a light-cone gauge n A(x) =0 with n =0 Ghost:: i ab k

Feynman rules in QCD

q Scattering amplitude: Renormalization, why need? = + + +... = E i PS I E I E i 1 E I E 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!

Physics of renormalization q UV divergence due to high mass states, not observed = - + Low mass state q Combine the high mass states with LO LO: + = High mass states Renormalized coupling NLO: - +... No UV divergence! q Renormalization = re-parameterization of the expansion parameter in perturbation theory

Renormalization 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!

q Interaction strength: QCD Asymptotic Freedom μ and μ 1 not independent Discovery of QCD Asymptotic Freedom Collider phenomenology Controllable perturbative QCD calculations Nobel Prize, 004

q Running quark mass: Effective Quark Mass Quark mass depend on the renormalization scale! q QCD running quark mass: q Choice of renormalization scale: q Light quark mass: for small logarithms in the perturbative coefficients 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 and CO divergences are generic problems of a massless perturbation theory

Infrared Safety q Infrared safety: Infrared safe = κ > 0 Asymptotic freedom is useful only for quantities that are infrared safe

Foundation of perturbative QCD q Renormalization QCD is renormalizable Nobel Prize, 1999 t Hooft, Veltman q Asymptotic freedom weaker interaction at a shorter distance Nobel Prize, 004 Gross, Politzer, Welczek q Infrared safety and factorization calculable short distance dynamics pqcd factorization connect the partons to physical cross sections J. J. Sakurai Prize, 003 Mueller, Sterman Look for infrared safe and factorizable observables!

From Lagrangian to Physical Observables q Theorists: Lagrangian = complete theory q Experimentalists: Cross Section Observables q A road map from Lagrangian to Cross Section: Particles Fields Interactions Symmetries Feynman Rules Lagrangian Green Functions Hard to solve exactly Correlation between fields Observables S-Matrix Cross Sections Solution to the theory = find all correlations among any # of fields + physical vacuum

QCD is everywhere in our universe 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 00 MeV (1 fm) GeV (1/10 fm) Q (GeV) Probing momentum q How do the nuclear force arise from QCD? q...

q Facts: Unprecedented Intellectual Challenge! No modern detector has been able to see quarks and gluons in isolation! q The challenge: Gluons are dark! 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,

Theoretical approaches approximations q Perturbative QCD Factorization: Approximation at Feynman diagram level DIS σ tot : e p 1 + O QR Probe Hard-part q Effective field theory (EFT): Approximation at the Lagrangian level Structure Parton-distribution Approximation Power corrections Soft-collinear effective theory (SCET), Non-relativistic QCD (NRQCD), Heavy quark EFT, chiral EFT(s),, or in terms of hadron d.o.f., q Other approximation or model approaches: Light-cone perturbation theory, Dyson-Schwinger Equations (DSE), Constituent quark models, AdS/CFT correspondence, q Lattice QCD: Approximation mainly due to computer power Hadron structure, hadron spectroscopy, nuclear structure, phase shift,

Physical Observables Cross sections with identified hadrons are non-perturbative! Hadronic scale ~ 1/fm ~ 00 MeV is not a perturbative scale Purely infrared safe quantities Observables with a lot of hadrons, but, without specifically identified hadron(s) Identified hadrons QCD factorization

Fully infrared safe observables total e + e!hadrons Fully inclusive with a lot of hadrons, but, without any specifically identified hadron(s) in terms of particle type and momentum! QCD: total e + e!hadrons = e total + e!partons The simplest observable in QCD! BESIII can measure it with precision

If there is no quantum interference between partons and hadrons, =1 σ + + + + P = P P = P P hadrons tot ee ee n ee m m n ee m m n n n m m n σ + e + e - è hadrons inclsusive cross sections q e + e - è hadron total cross section not a specific hadron! tot e + e!hadrons / P tot ee partons ee + m m σ tot + + ee hadrons q e + e - è parton total cross section: Partons m Hadrons n Unitarity tot = σ Finite in perturbation ee partons theory KLN theorem Calculable in pqcd

Infrared Safety of e + e - Total Cross Sections q Optical theorem: 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 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 q Lowest order Feynman diagram: p 1 k 1 q Invariant amplitude square: M Q 1 1 Tr s 4 µ ν + = ee N ee QQ c γ p γ γ p1γ ( k1 mq) ( k mq) Tr γ γ µ γ γ + ν 4 = eeqn c ( m ) ( ) Q t + mq u + mqs s q Lowest order cross section: dσ + ee QQ 1 = M + ee QQ dt 16π s where s = Q p k s= ( p + p ) 1 t = ( p k ) 1 1 1 u = ( p k ) Threshold constraint σ (0) 4 em eq N πα = σ 1 ee + = QQ c + Q Q 3s m s Q 1 4m s Q One of the best tests for the number of colors

Next-to-leading order (NLO) contribution q Real Feynman diagram: x i = Ei pi. q with i 1,,3 s / = s = i x i pi. q i = = s ( ) = ( θ ) 1 x x x 1 cos, cycl. 1 3 3 q Contribution to the cross section: 1 σ d σ α x x + ee QQg s 1 + = CF 0 dx1dx π 1 x1 1 x ( )( ) + crossing IR as x3 0 CO as θ 13 0 θ 3 0 Divergent as x i 1 Need the virtual contribution and a regulator!

How does dimensional regularization work? q Complex n-dimensional space: Im(n) (1) Start from here: UV renormalization a renormalized theory () Calculate IRS quantities here Theory cannot be renormalized! (3) Take εè 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: ( 1 ε ) ( ε ) ε (1) (0) 4 αs 4πµ Γ 1 3 19 ² Real: σ3, ε = σ, ε 3 π Q 1 3 + + Γ ε ε 4 ² Virtual: ε (1) (0) 4 α ( 1 ) ( 1 s 4πµ Γ ε Γ + ε) 1 3 π σ, ε = σ, ε 4 + 3 π Q ( 1 ε Γ ) ε ε ² NLO: ² Total: α π ( ) (1) (1) (0) s σ3, ε + σ, ε = σ + O ε No ε dependence! tot (0) ( 1) (1) (0 α σ = σ + σ3, ε + σ, ε + O α = σ 1+ + α π σ tot is Infrared Safe! ( ) ) s ( ) s O s σ tot is independent of the choice of IR and CO regularization Go beyond the inclusive total cross section?

Hadronic cross section in e+e- collision q Normalized hadronic cross section: R e+ e (s) e+ e!hadrons(s) e + e!µ + µ (s) X apple N c = 3 N c e q 1+ apple s(s) + O( s(s)) 1+ s(s) q=u,d,s "! r # X +N c e q 1+ m q 4m q 1 + O( s (s)) s s q=c,... +...

Homework (1) 1) Fill in the steps needed to complete the calculation of leading order QCD cross section for total e + e!partons on slide 38: σ (0) 4 em eq N πα = σ 1 ee + = QQ c + Q Q 3s m s Q 1 4m s Q

Backup slides