# Lagrangian, Symmetries, Running Coupling, Coulomb Gauge Lagrangian Quantum Chromodynamics We Require a Theory Which Has Approximate Chiral Symmetry

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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 SUc (3) local gauge invariance (QED): A A + Λ φ φ Λ˙ → ∇ → − impose local gauge symmetry: iΛ(x) ψ(x) e− ψ(x) → and get an interacting ﬁeld theory: = ψ¯γµ∂ ψ ψ¯γµ(∂ + ieA )ψ A A + ∂ Λ L µ → µ µ µ → µ µ ! ! Quantum Chromodynamics local gauge invariance (QCD): impose local gauge symmetry: ψ(x) U ψ(x) a → ab b for invariance of L: = ψ¯ δabγµ∂ ψ ψ¯ δabγµ∂ + igγµ(A ) ψ L a µ b → a µ µ ab b ! ! i " # Aµ UAµU † + U∂µU † → g eight gluons F [D , D ] = ig(∂ A ∂ A ) g2[A , A ] µν ∝ µ ν µ ν − ν µ − µ ν QCD 1 n f 1 = q¯ [iγ (∂µ + igAµ) m ]q Tr(F F µν ) LQCD f µ − f f − 2 µν !f Fµν = ∂µAν ∂ν Aµ + ig[Aµ, Aν ] a − a λ Aµ = Aµ 2 ﬂavour, colour, Dirac indices λa λb λc [ , ] = if abc 2 2 2 Tr(λaλb) = 2δab (1) g2 = θ F µν F˜ Lθ 64π2 µν Symmetries symmetries in (classical) ﬁeld theory µ q f(q) jµ ∂ j =0 → µ d d d3xj Q = d3x j =0 dt 0 ≡ dt ∇ · symmetries in QCD U(1)V iθ ¯ 3 q e− q jµ =¯qγµq =¯uγµu + dγµd Q = d x (u†u + d†d) → symmetry current charge p e+ν ‘baryon number conservation’ → in full SM need 1-gamma_5, which intrduces anomaly, ’t Hooft eﬀf L has a [violated by EW anomaly] prefactor of exp(-2 pi/alpha_2) ~ 10^-70 symmetries in QCD U(1)A mu = md =0 iγ5θ ¯ 3 q e− q jµ5 =¯uγµγ5u + dγµγ5d Q5 = d x (u†γ5u + d†γ5d) → symmetry current charge this symmetry does not exist in the quantum theory 3α ∂µj = s F F˜ µ5 8π symmetries in QCD scale invariance mu = md =0 x λx q →λ3/2q(λx) → µν A λA(λx) jµ = xν Θ → symmetry current µ µ ∂ jµ = Θµ =0 this symmetry does not exist in the quantum theory α Θµ = mqq¯ + s F 2 µ 12π symmetries in QCD SU(3)V mu = md isospin iθT a a a a 3 a q e F q j = ψ¯γ T ψ Q = d xψ†T ψ → µ µ F F ! symmetry current charge + 1 3 + 0 Q π− = d x b†(x)τ b(x) d†(x)τ −d(x) π− = π | ⟩ √2 − | ⟩ | ⟩ + 0 0 0 H π− = E !π− ; " Q H π− = E π ; H #π = E π | ⟩ π− | ⟩ | ⟩ π− | ⟩ | ⟩ π− | ⟩ this symmetry is explicitly broken by quark mass and EW effects symmetries in QCD SU(3)A mu = md =0 a iθT γ5 a a a 3 a q e− q j = ψ¯γµγ5T ψ Q = d xψ†γ5T ψ → µ 5 symmetry current !charge This symmetry is realised in the Goldstone mode. transform the vacuum: iθaQa e 5 0 = 0 | ⟩ | ⟩ } Wigner mode Qa 0 = 0 5| ⟩ 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 iθaQa e 5 0 = θ = 0 | ⟩ | ⟩ ̸ | ⟩ } Goldstone mode iθaQa iθaQa H θ = He 5 0 = e 5 H 0 = E θ | ⟩ | ⟩ | ⟩ 0| ⟩ 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 = θaQa 0 | ⟩ 5| ⟩ a 3 a θ d xb†(x)T d†(x) 0 ∼ F | ⟩ ! spin singlet, spatial singlet, ﬂavour octet ⇒ the pion octet Chiral Symmetry Breaking SU (2) SU (2) U (1) U (1) L × R × A × V Isospin Invariance iθ·τ ψ → e ψ equal quark masses a = ¯ a jVµ ψγµτ ψ Qa = d3xψ†τ aψ V ! a [H, QV ]=0 Isospin Invariance a [H, QV ]=0 a a H(QV |M⟩)=EM (QV |M⟩) 3 a d k † a − † a T QV = 3 bkτ bk dk(τ ) dk ! (2π) " # + 0 + 1 QV ρ = QV ( uu¯ dd¯ ) | ⟩ √2 | ⟩− ⟩ + 0 1 QV ρ = ( ud¯ ud¯ ) | ⟩ √2 −| ⟩−| ⟩ + 0 + QV |ρ ⟩∝|ρ ⟩ Axial Symmetry ψ → eiγ5θ·τ ψ a = ¯ a jAµ ψγµγ5τ ψ a 3 † a Q = d xψ γ5τ ψ A ! a [H, QA]=0 Axial Symmetry 3 a d k † † QA1 = 3 ck bkλ2λbkλ d kλ2λd kλ (2π) − − − ! " # Qa ( + + + ) = ( + + ) A1 | ⟩ | − −⟩ | ⟩ − | − −⟩ Qa ( + + + ) = 0 A1 | −⟩ | − ⟩ (J)(J) (J+1)(J) JH = J Axial Symmetry 3 a d k a a † † QA2 = 3 sk bkτ d k + d kτ bk (2π) − − ! " # pion RPA creation operator! Qa M = Mπa A2| ⟩ | ⟩ 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 N 1 N 2 N 3 c − 3 c − 3 f 1 dg(µ) β0 3 2 4π µ = g (µ) αs(µ ) = (1) dµ −(4π)2 (11 2 n ) ln µ2/Λ2 − 3 f 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 AF 0.6 For Nf <N 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 α(µ) α ; there is no scale dependence. → 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 c AF Nf = 16.5 For Nf <Nf <N 0.2 0 the conformal window -0.2 Nf=8 -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 r 0 →∞ → running coupling QCD & charge antiscreening r b r r r gr b r r gr 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: Ei = F i0 Eia = A˙ ia A0a + gfabcA0bAic − −∇ define the chromomagnetic field: 1 Bi = ijkF −2 jk 1 B a = Aa + gfabcAb Ac ∇× 2 × Coulomb Gauge Equations of motion: ∂ ∂ ∂β L = L ∂(∂βAaα) ∂Aaα β a a abc b cµ ∂ Fβα = gjα + gf FαµA Gauss’s Law ( ): E a + gfabcAb E c = gρa ∇ · · (q) Introduce the adjoint covariant derivative D ab = δab gfabcAc ∇− D ab E b = gρa · (q) Coulomb Gauge D ab E b = gρa · (q) resolve Gauss’s Law: E = E φ Etr =0 A =0 B =0 tr −∇ ∇ · ∇ · ∇ · define the full colour charge density: ρa = ρq + f abcE b Ac (q) tr · Use this in Gauss’s Law to get: (D ab )φ = gρa − · ∇ Coulomb Gauge (D ab )φ = gρa − · ∇ Solve for ϕ: a g a φ = ρ notice that this is a “formal” solution − D ∇ · We have two expressions for the divergence of E E a = DabA0b = 2φa ∇ · −∇ · −∇ 1 1 A0b = ( 2) gρb D −∇ D ∇ · ∇ · Coulomb Gauge 1 1 H = (E2 + B2)= (E2 φ 2φ + B2) 2 2 tr − ∇ { 1 H = d3xd3y ρa(x)K (x, y; A)ρb(y) c 2 ab g 2 g Kab(x, y; A)= x, a ( ) y, b | D −∇ D | ∇ · ∇ · Coulomb Gauge A complication due to the curved gauge manifold 1 1/4 ij 1/2 1/4 Hˆ = g− pˆ g g pˆ g− 2m i j g = metric =det( D ) Faddeev-Popov determinant J ∇ · 1 3 1 H = d x ( − Π Π + B B ) 2 J J· · Inner product: 1/2 1/2 Ψ Φ = DA Ψ∗Φ H H − | J → J J Coulomb Gauge Schwinger, Kriplovich Christ and Lee 3 H = d x ψ†(x)( iα + βm)ψ(x) q − · ∇ 3 1 H =tr d x ( − Π Π + B B ) YM J ·J · 3 H = g d x ψ†(x)α Aψ(x) qg − · 1 3 3 1 a b H = d xd y − ρ (x)K (x, y; A) ρ (y) C 2 J ab J a a abc b c ρ (x)=ψ†(x)T ψ(x)+f Π (x) A (x) · g g Kab(x, y; A)= x, a ( 2) y, b | D −∇ D | ∇ · ∇ · The Gribov Ambiguity • A =0 does not uniquely specify the gauge field.

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