QCD - introduction lagrangian, symmetries, running coupling, Coulomb - PowerPoint PPT Presentation

QCD - introduction lagrangian, symmetries, running coupling, Coulomb gauge

Lagrangian

Quantum Chromodynamics we require a theory which has approximate chiral symmetry has approximate SU(3) flavour symmetry accounts for the parton model has colour and colour confinement is renormalizable

QCD gauge SU (3) c local gauge invariance (QED): φ → φ − ˙ A → A + ∇ Λ Λ impose local gauge symmetry: ψ ( x ) → e − i Λ ( x ) ψ ( x ) and get an interacting field theory: � � ¯ ¯ L = ψγ µ ( ∂ µ + ieA µ ) ψ A µ → A µ + ∂ µ Λ ψγ µ ∂ µ ψ →

Quantum Chromodynamics local gauge invariance (QCD): impose local gauge symmetry: ψ ( x ) a → U ab ψ ( x ) b for invariance of L: � � ¯ ¯ ψ a δ ab γ µ ∂ µ ψ b → δ ab γ µ ∂ µ + ig γ µ ( A µ ) ab � � L = ψ a ψ b A µ → UA µ U † + i gU ∂ µ U † eight gluons F µ ν ∝ [ D µ , D ν ] = ig ( ∂ µ A ν − ∂ ν A µ ) − g 2 [ A µ , A ν ]

QCD n f q f [ i γ µ ( ∂ µ + igA µ ) − m f ] q f − 1 � 2Tr( F µ ν F µ ν ) L QCD = ¯ f F µ ν = ∂ µ A ν − ∂ ν A µ + ig [ A µ , A ν ] λ a A µ = A a µ 2 flavour, colour, Dirac indices [ λ a 2 , λ b 2 ] = if abc λ c 2 Tr( λ a λ b ) = 2 δ ab L θ = θ g 2 64 π 2 F µ ν ˜ F µ ν

Symmetries

symmetries in (classical) field theory ∂ µ j µ = 0 q → f ( q ) j µ d d 3 x j 0 � d � � d 3 x � · � dtQ = j = 0 dt

symmetries in QCD U (1) V � d 3 x ( u † u + d † d ) q → e − i θ q u γ µ u + ¯ Q = j µ = ¯ q γ µ q = ¯ d γ µ d symmetry current charge p → e + ν ‘baryon number conservation’ in full SM need 1-gamma_5, which [violated by EW anomaly] intrduces anomaly, ’t Hooft e ff f L has a prefactor of exp(-2 pi/alpha_2) ~ 10^-70

symmetries in QCD U (1) A m u = m d = 0 � d 3 x ( u † γ 5 u + d † γ 5 d ) q → e − i γ 5 θ q u γ µ γ 5 u + ¯ Q 5 = j µ 5 = ¯ d γ µ γ 5 d symmetry current charge this symmetry does not exist in the quantum theory ∂ µ j µ 5 = 3 α s 8 π F ˜ F

symmetries in QCD scale invariance m u = m d = 0 x → λ x q → λ 3 / 2 q ( λ x ) j µ = x ν Θ µ ν A → λ A ( λ x ) symmetry current ∂ µ j µ = Θ µ µ = 0 this symmetry does not exist in the quantum theory qq + α s 12 π F 2 Θ µ µ = m ¯

symmetries in QCD SU(3) V isospin m u = m d q → e i θ T a µ = ¯ d 3 x ψ † T a j a ψγ µ T a Q a = � F ψ F ψ F q symmetry current charge 1 � b † ( x ) τ + b ( x ) − d † ( x ) τ − d ( x ) � Q + | π − ⟩ = � d 3 x | π − ⟩ = | π 0 ⟩ √ 2 Q + H | π − ⟩ = E π − | π 0 ⟩ ; H | π 0 ⟩ = E π − | π 0 ⟩ H | π − ⟩ = E π − | π − ⟩ ; this symmetry is explicitly broken by quark mass and EW effects

symmetries in QCD SU(3) A m u = m d = 0 a q → e − i θ T a γ 5 q µ = ¯ d 3 x ψ † γ 5 T a ψ � Q = j a ψγ µ γ 5 T a ψ 5 symmetry current charge This symmetry is realised in the Goldstone mode. transform the vacuum: e i θ a Q a 5 | 0 ⟩ = | 0 ⟩ } Wigner mode Q a 5 | 0 ⟩ = 0

symmetries in QCD HFM: spin direction is signalled out (NOT by an external field) it could be any direction, so small fluctutatiopns (or large) do not cost energy -> gapless dispersion relationship, E ~ k or k^2. SU(3) A 5 | 0 ⟩ = | θ ⟩ ̸ = | 0 ⟩ e i θ a Q a } Goldstone mode H | θ ⟩ = H e i θ a Q a 5 | 0 ⟩ = e i θ a Q a 5 H | 0 ⟩ = E 0 | θ ⟩ so there is a continuum of states degenerate with the vacuum Excitations of the vacuum may be interpreted as a particle. In this case fluctuations in theta are massless particles called Goldstone bosons.

symmetries in QCD SU(3) A Goldstone boson quantum numbers: | δθ ⟩ = θ a Q a 5 | 0 ⟩ � d 3 xb † ( x ) T a F d † ( x ) | 0 ⟩ ∼ θ a spin singlet, spatial singlet, flavour octet ⇒ the pion octet

Chiral Symmetry Breaking SU L (2) × SU R (2) × U A (1) × U V (1)

Isospin Invariance equal quark masses ψ → e i θ · τ ψ j a ψγ µ τ a ψ V µ = ¯ � d 3 x ψ † τ a ψ Q a V = [ H, Q a V ] = 0

Isospin Invariance [ H, Q a V ] = 0 H ( Q a V | M ⟩ ) = E M ( Q a V | M ⟩ ) d 3 k � � � k ( τ a ) T d k b † k τ a b k − d † Q a V = (2 π ) 3 1 u ⟩ − d ¯ Q + V | ρ 0 ⟩ = Q + √ 2( | u ¯ d ⟩ ) V 1 2( − | u ¯ d ⟩ − | u ¯ Q + V | ρ 0 ⟩ = √ d ⟩ ) Q + V | ρ 0 ⟩ ∝ | ρ + ⟩

Axial Symmetry ψ → e i γ 5 θ · τ ψ j a ψγ µ γ 5 τ a ψ Aµ = ¯ � Q a d 3 x ψ † γ 5 τ a ψ A = [ H, Q a A ] = 0

Axial Symmetry d 3 k � � � b † k λ 2 λ b k λ − d − k λ 2 λ d † Q a A 1 = (2 π ) 3 c k − k λ Q a A 1 ( | + + ⟩ + | − −⟩ ) = ( | + + ⟩ − | − −⟩ ) Q a A 1 ( | + −⟩ + | − + ⟩ ) = 0 J ( J )( J ) = J ( J +1)( J ) H

Axial Symmetry d 3 k � � � b † k τ a d † Q a − k + d − k τ a b k A 2 = (2 π ) 3 s k pion RPA creation operator! Q a A 2 | M ⟩ = | M π a ⟩

Goldstone’s theorem says nothing about the excited pion spectrum. π ′ (1300) ≈ ρ ′ (1450)

gluons

Quantum Chromodynamics Q: are these peculiar gluons real? There was indirect evidence from deviation from DIS scalang. Direct evidence was achieved at DESY with three jet events. REF:http://cerncourier.com/cws/article/cern/ 39747 A three jet event at DESY. August, 1979 John Ellis, Mary Gaillard, Graham Ross

Running Coupling

running coupling Khriplovich Yad. F. 10 , 409 (69) 12 − 1 − 2 3 N c 3 N c 3 N f µdg ( µ ) 4 π = − β 0 (4 π ) 2 g 3 ( µ ) α s ( µ 2 ) = (11 − 2 dµ 3 n f ) ln µ 2 / Λ 2 QCD

the running coupling ... UV stable fixed point α = 0 ¯ IR stable fixed point α = 1 , α → ∞ ¯ makes the IR limit stable... 2 f(x) 0 1.5 1 ! 0.5 r → ∞ 0 r → 0 -0.5 0 0.5 1 1.5 2 "

Walking Technicolour The Conformal Window 1.2 1 alpha* may be small enough for pert to be always valid! 0.8 For N f < N AF 0.6 α � β 0.4 0.2 0 -0.2 -0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 α In the infrared limit ; there is no scale dependence. α ( µ ) → α � Thus the theory is conformal and chiral symmetry breaking and confinement are lost.

Banks -Zaks conformal window (of interest for walking TC models) the theory flows to this conformal pt in the IR For QCD this happens when 16.5 > Nf > 8.05 or more generally, since we really need the full form of beta.

Walking Technicolour The Conformal Window Banks and Zaks, NPB196, 189 (82) 0.4 N f = 16.5 For N c f < N f < N AF 0.2 � 0 � the conformal window N f =8 -0.2 -0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 g

Walking Technicolour The Conformal Window For walking TC we want to sit just below the conformal window.

running coupling

running coupling QED & charge screening - + + - - + - r → 0 r → ∞

running coupling QCD & charge antiscreening r b r r gr r b r g r r r

Properties: Asymptotic Freedom QCD and anti-screening Frank Wilczek (1951-)

Coulomb Gauge

Why Coulomb Gauge? • Hamiltonian approach is similar to CQM • all degrees of freedom are physical, no constraints need be imposed • [degree of freedom counting is important for T>0] • T>0 chooses a special frame anyway • no spurious retardation effects • is renormalizable (Zwanziger) • is ideal for the bound state problem • very good for examining gluodynamics

Derivation minimal coupling: gauge group: gauge (Faraday) tensor:

Coulomb Gauge define the chromoelectric field: E i = F i 0 A ia � � A 0 a + gf abc A 0 b A ic E ia = � ˙ define the chromomagnetic field: B i = − 1 2 � ijk F jk A a + 1 2 gf abc � A b � � B a = � � � � A c

Coulomb Gauge Equations of motion: ∂ L ∂ L ∂ β ∂ ( ∂ β A a α ) = ∂ A a α ∂ β F a βα = gj a α + gf abc F b α µ A cµ Gauss’s Law ( ): E a + gf abc � E c = g � a � · � A b · � ( q ) Introduce the adjoint covariant derivative D ab = � ab � � gf abc � � A c D ab · � � E b = g � a ( q )

Coulomb Gauge D ab · � � E b = g � a ( q ) resolve Gauss’s Law: � · � � · � � · � E = � � A = 0 B = 0 E tr � � � E tr = 0 define the full colour charge density: ( q ) + f abc � � a = � q tr · � E b A c Use this in Gauss’s Law to get: � ( � D ab · � ) � = g � a

Coulomb Gauge � ( � D ab · � ) � = g � a Solve for ϕ : g � a = � � a notice that this is a “formal” solution � · � D We have two expressions for the divergence of E � · � E a = �� · D ab A 0 b = �� 2 � a 1 1 A 0 b = g � b ( �� 2 ) � · � � · � D D

Coulomb Gauge H = 1 2( E 2 + B 2 ) = 1 2( E 2 tr � φ � 2 φ + B 2 ) { H c = 1 � d 3 xd 3 y ρ a ( x ) K ab ( x, y ; A ) ρ b ( y ) 2 g g ( �� 2 ) K ab ( x, y ; A ) = � x, a | | y, b � � · � � · � D D