Lagrangian, Symmetries, Running Coupling, Coulomb Gauge Lagrangian Quantum Chromodynamics We Require a Theory Which Has Approximate Chiral Symmetry
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 = F µ F˜ L 64 2 µ
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
i ¯ 3 q e q jµ =¯q µq =¯u µu + d µd Q = d x (u†u + d†d) symmetry current charge
p e+ ‘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
i 5 ¯ 3 q e q jµ5 =¯u µ 5u + d µ 5d Q5 = d x (u† 5u + d† 5d) symmetry current charge
this symmetry does not exist in the quantum theory
3 µj = s F F˜ µ5 8 symmetries in QCD scale invariance mu = md =0 x x q 3/2q( x) µ A A( x) jµ = x symmetry current
µ µ jµ = µ =0
this symmetry does not exist in the quantum theory
µ = mqq¯ + s F 2 µ 12 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 ¯ =0 IR stable fixed point ¯ =1, ⇥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 0.2 0 -0.2 -0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 In the infrared limit (µ) ; 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 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 = A a + gfabcA b A c 2 Coulomb Gauge Equations of motion: L = L ( Aa ) Aa a a abc b cµ F = gj + gf F µA Gauss’s Law ( ): E a + gfabcA b E c = g a · · (q) Introduce the adjoint covariant derivative D ab = ab gfabcA c D ab E b = g a · (q) Coulomb Gauge D ab E b = g a · (q) resolve Gauss’s Law: E = E E tr =0 A =0 B =0 tr · · · define the full colour charge density: a = q + f abcE b A c (q) tr · Use this in Gauss’s Law to get: (D ab ) = g a · Coulomb Gauge (D ab ) = g a · Solve for ϕ: a g a = notice that this is a “formal” solution D · We have two expressions for the divergence of E E a = DabA0b = 2 a · · 1 1 A0b = ( 2) g b D D · · Coulomb Gauge 1 1 H = (E2 + B2)= (E2 2 + B2) 2 2 tr { 1 H = d3xd3y a(x)K (x, y; A) b(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 ( + B B ) 2 J J· · Inner product: 1/2 1/2 = DA H H | J J J Coulomb Gauge Schwinger, Kriplovich Christ and Lee 3 H = d x †(x)( i + m) (x) q · 3 1 H =tr d x ( + B B ) YM J ·J · 3 H = g d x †(x) A (x) qg · 1 3 3 1 a b H = d xd y (x)K (x, y; A) (y) C 2 J ab J a a abc b c (x)= †(x)T (x)+f (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. · 3 g 2 Zwanziger FA[g] = tr d x (A ) g A = gAg † g g† • The absolute minimum of F[g] fixes one field configuration on the gauge orbit. The set of such minima is the Fundamental Modular Region. • The Faddeev Popov operator is positive definite in the FMR The Gribov Ambiguity Coulomb gauge and the Gribov problem det( D) = 0 ∇ · .A1 A a =0 · The Gribov Ambiguity Coulomb gauge and the Gribov problem det( D) = 0 ∇ · Fundamental modular region Gribov region A a =0 · The Gribov Ambiguity Coulomb gauge and the Gribov problem Fundamental modular region det( D) = 0 Gribov region ∇ · FMR is convex GR contains the FMR identify boundary FMR contains A=0 configurations physics lies at the intersection of the FMR and GR Zwanziger; van Baal Feynman Rules a a i Feynman Rules a k1 i1 k2 i2 k i n+1 n b Coulomb Gauge 1 1 g g H = dx E2 + B2 dxdyf abcEb(x)Ac(x) xa 2 yd f def Ee(y)Af (y) 2 −2 ⟨ | D ∇ D | ⟩ ! ! ∇ · ∇ · " # K(x y; A) − an instantaneous potential that depends on the gauge potential K generates the beta function 15.0 K is renormalization group invariant 10.0 ) 0 5.0 (2r QQ (r) - V 0.0 K is an upper limit to the Wilson loop potential QQ ( V 0 r -5.0 -10.0 K is infrared enhanced at the Gribov boundary 0.0 2.0 4.0 6.0 8.0 10.0 r/r0 Szczepaniak & Swanson Coulomb Gauge Khriplovich Yad. F. 10, 409 (69) running coupling 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 QCD - lattice gauge theory lattice intro, Monte Carlo, SU(2)-Higgs Lattice Gauge Theory — Review Wegner (1971): gauged Z(2) spin model Wilson (1974): lattice gauge theory Creutz (1980): numerical lattice gauge theory F. Wegner, J. Math. Phys. 12, 2259 (1971) K. Wilson, PRD10, 2445 (1974) M. Creutz, PRD21, 2308 (1980) extra - in SE from d/dt^2 - D^2 -.> -(d/ Lattice Gauge Theory — Review dtau^2 + D^2) Euclidean field theory with a spacetime regulator x0 ix4 S iS E iS[ ] S [ ] D e D e E E E (xµ) (anµ) n D d µ nµ n µ Maps quantum field theory to a statistical mechanics problem Lattice Gauge Theory — Review Complex scalar field: 4 2 2 SE = d x [( µ )†( µ )+m † + 0( † ) ] 0 2 ˆ† ˆ + ( ˆ† ˆ 1) ( ˆ† ˆ + ˆ† ˆ ) n n n n n n+µ n+µ n n µ a ˆ 2 1 2 = = 0 = 2 2 2d + a m0 umklapp (over turn) is when the sum of two momneta in a BZ results in one Lattice Gauge Theory — Review outside of teh BZ (and then gets mapped back into it) lattice symmetries discrete translation: momentum conservation up to 2 /a This is ok since the IR EFT only deals with p 1/a hypercubic symmetry: a subgroup of O(4). This is potentially a problem, but all hypercubic operators that are also not O(4) symmetric are irrelevant ➙ no problem for the IR EFT, i.e., O(4) symmetry is recovered as an accidental symmetry. 4 ex. hypercubic, not O(4), symmetric; dim 6: † µ µ Lattice Gauge Theory — Review Gauge Fields Consider the gauge covariant laplacian 2 2 n+µ n µ 2 †D †[ iA + A ] µ µ a n discrete derivative ruins the gauge transformation Lattice Gauge Theory — Review Gauge Fields one could press ahead and obtain gauge invariance up to O(a5) but terms like A2 and AµAµAµAµ are induced in the IR effective field theory — which is a disaster: the first is dimension 2; the second is not O(4) invariant; and tuning is required to eliminate these. we should build in gauge invariance from the start! Lattice Gauge Theory — Review Gauge Fields Think of the covariant derivative as a connection in an internal space. Then it is natural to regard the gauge field as a link variable. ig x+µ A (z)dzµ U(x, µ) P e x µ R Lattice Gauge Theory — Review Gauge Fields Gauge transformation: (x) †(x) (x) U(x, µ) †(x)U(x, µ) (x + µ) Thus the gauge covariant derivative is 2 1 2 a (D ) [2 †(x) (x) †(x) U(x, µ) (x+µ) †(x+µ) U †(x, µ) (x)] 2 µ 2 x,µ Lattice Gauge Theory — Review Gauge Fields Closed loops of link variables form gauge invariant objects The most local pure gauge object is a plaquette ˆ µ (x) µˆ x Lattice Gauge Theory — Review Gauge Fields a4 (x)=1 ia2F a T a F a F b T aT b + ... µ µ 2 µ µ 1 S = (1 tr (x)) N µ x,µ> 2N = 2 g0 Lattice Gauge Theory — Review Fermions Represented as Grassmann fields, which must be explicitly integrated S S DU D ¯ D e F = DU det(D / [U]) e E (evaluating this determinant is extremely expensive!) iS[A]+i d4x ¯(i/ m gA/) +i ¯ +i ¯ D ¯ D DAµe R R R 1 iS[A] i ¯(i/ m gA/) = DA det(i/ m gA/)e µ R Lattice Gauge Theory — Review Fermions Attempts to simulate directly lead to the fermion sign problem. (cf. flipping rows or columns in the determinant) Lattice Gauge Theory — Review these zeroes are IN the Brillouin zone! Fermions fermion doubler problem µ 4 µ ¯ D a ¯(x) U(x, µ) (x + µ) U †(x µ, µ) (x µ) /(2a) µ x,µ NB WIlson fermionds break chiral symmetry (hence multiplicative mass 1 µ 1 renormalization) and recovering it in the S (p)=m + i sin(ap ) IR requires careful tuning a µ µ Zeros at pµ =(0, 0, 0, 0) pµ =(0, /a, 0, 0) … i.e., there are 2 4 ‘species’ of fermions. Lattice Gauge Theory — Review Fermions massless fermion problem (removing the doublers necessarily breaks chiral symmetry) chiral fermion problem Weyl spinor => 8 LH + 8 RH Weyl spinors. (chiral gauge theories cannot be placed on the lattice) Lattice Gauge Theory — Review called the “Haar measure” — is gauge invariant and integrates over the gauge manifold Gauge Fields — the measure S Z = DUe E DU = dU(x, µ) x,µ U(1) SU(2) i (x,µ) i a /2 U(x, µ)=e U(x, µ)=e · = a + i a 0 · d 1 dU = dU = (1 a2)d4a 2 Lattice Gauge Theory — Review Gauge Fields — the measure dU = detg 1/2 d | | g is the metric on the group manifold gAB = tr[U † ( AU) U †( BU)] Lattice Gauge Theory — Review Gauge Fields — the Wilson loop Work in axial gauge: Aˆ(x)=0,U(x, tˆ)=1 T t R 1 S tr (R, T ) = Z DU †(R, T ) U(0 R, T ) (0,T) †(R, 0) U(0 R, 0) (0, 0)e E · Make the spectral decomposition 2 E (R)T tr (R, T ) = M(0) n e n | | | n V (R)T tr (R, T ) e T R Lattice Gauge Theory — Review Gauge Fields — the Wilson loop Evaluate in the strong coupling (small beta) limit T R dU U(x, µ)=0 1 dU(x, µ)[U(x, µ)] [U †(y, )] = ij k N xy µ i jk RT 2N log(g2/2N)RT tr (R, T ) = =e 0 g2 0 This is an area law — colour confinement! Lattice Gauge Theory — Review Note that by the same argument (compact) QED is also confining, but it is separated from the continuum by a first order phase transition (to the massless photon phase). There is a line of phase transitions in 4d SU(N), but it ends, and the strong coupling and weak coupling regimes are smoothly connected. Thus SU(N) gauge theory is confining. string tension U(1) 4d Lattice Gauge Theory Monte Carlo evaluation of the integral 1 1 S [A] = DA [A, M ] det(M[A])e E O Z µO important shift to Euclidean space! 1 = [A, M ] O O A { } S [A] det(M[A])e E A S [A] { } DAµ det(M[A]) e E warnings: autocorrelation, critical slowing down, determinant Lattice Gauge Theory compute a hadron mass... C(t)= 0 †(t) (0) 0 = ¯ 5 (x, 0) |S S | S iHt iHt C(t)= 0 e †(0)e (0) 0 | S S | iHt iHt C(t)= 0 e †(0)e n n (0) 0 | S | |S | n i(E E )t 2 C(t)= e n 0 n (0) 0 | |S | | n C(t) m = log eff C(t + 1) Lattice Gauge Theory one seeks ‘plateaus’ warnings: how are these defined?, close levels meff t meff t Lattice Gauge Theory renormalise • (for m=0) select the coupling, g • work in units of the lattice spacing, a • compute a physical quantity, such as mN. Set a such that mN = 0.94 GeV. • This gives a(g) for one point. Repeat to trace out a(g), or g(a). Attempt to take limit a->0 and V->infinity. Lattice Gauge Theory more warnings: • signal/noise: good interpolators! • finite density • light cone correlations • systems with many scales (Ds) • statistics/operators/correlators • highly excited states • unstable states warnings: how is a plateau defined? closely spaced levels? umklapp (over turn) is when the sum of two momneta in a BZ results in one Lattice Gauge Theory — Review outside of teh BZ (and then gets mapped back into it) continuum limit The lattice spacing has been scaled out of the problem. It is recovered upon renormalisation. cf. mˆ ( )=a( )mphys (a)[g0(a)] Or can extract ratios as (a) There should be no phase transition if one wants to obtain strong coupling continuum physics. The system correlation length should be large wrt the lattice spacing /a 1 amgap 1 Lattice Gauge Theory — Review lattices C Hoelbling, PoS (LATTICE2010) 011, arXiv:1102.0410. Lattice Gauge Theory — Review unquenching C. T. H. Davies et al., PRL 92, 022001, (2004) (input: m ,m ,m m , (m + m )/2,m) K (2S) (1S) u d s SU(2)-Higgs the importance of importance sampling! V_n (R=1) is rapidly driven to zero! Monte Carlo n/2 Vn = n ( 2 + 1) importance sampling (x) g(x) I = f(x) I = (x) 1 1 (x) I = g(ˆxi)+O xˆ N i N (x) detailed balance guarantees the fiixed SU(2)-Higgs point exists: P(U -> U’) C(U) = P(U’ -> U) C(U’) Monte Carlo To “throw darts” one generates a Markov chain of field configurations: U(x, µ), (x) U(x, µ), (x) ... { }1 { }2 P P DU (U U )=1 (& memoryless stochastic process) P [U]= DU (U U) [U ] (a fixed point probability density exists) C P C exp( S[U]) [U]= C DU exp( S[U ]) ergodicity b is the su(2) four-vector representing B SU(2)-Higgs (actually, not normalized) Monte Carlo — scalar field Form B(x)= [U(x, µ) (x + µ)+U †(x µ) (x µ)] µ Then S = [( (x) b(x))2 + ( (x)2 1)2 b2(x)] x thus local actions are important! heat bath S ( ) 4 Seek a new configuration via dP ( ) e E d P Metropolis Propose a new configuration; accept it if the action is lowered; otherwise accept it conditionally with probability min(1, exp[V ( ) V ( )]) SU(2)-Higgs Monte Carlo — gauge field heat bath Consider a single link SU xµ = tr U(x, µ)W †(x, µ) | 2 2 W (x, µ)= (x) †(x + µ)+ U(x )U(x + ,µ),U†(x + µ, )+ =µ U †(x , )U(x ,µ),U†(x + µ, ) =µ SU(2)-Higgs Monte Carlo — gauge field heat bath And update: W = rWˆ tr(UW†) ˆ dP (U) e 2 dU W SU(2) r tr(U) dP (UWˆ ) e 2 dU dP (a ) 1 a2e ra0 (1 a2) da d a µ 0 0 U U(a)Wˆ SU(2)-Higgs Monte Carlo — autocorrelation O(t + )O(t) ( )= t O2(t) t (t and ! are “algorithmic time”) O = ¯( x, 1/(2am)) ¯( x, 0) (3d phi4 theory) SU(2)-Higgs Results need Higgs in the fundamental for no phase trans E. Fradkin and S. Shenker, PRD19, 3682 (1979) F. Knechtli, hep-lat/9910044