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Lattice Quantum Chromodynamics for Pedestrians

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Lattice Quantum Chromodynamics for Pedestrians QuantumChromoDynamics Pure gauge theory Lettuce Field Theory An introduction Christian B. Lang Christian B. Lang Lattice QCD for Pedestrians QuantumChromoDynamics Pure gauge theory Lattice Quantum Field Theory An introduction Christian B. Lang Christian B. Lang Lattice QCD for Pedestrians QuantumChromoDynamics Pure gauge theory Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization 1 QuantumChromoDynamics Continuum QCD Quantization Lattice QCD LQCD quantization 2 Pure gauge theory Monte Carlo integration Confinement at strong coupling How to set the scale? The three limits of LQCD Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Continuum QCD “Declaration of QCD” We assume that QCD is the quantum field theory of quarks and gluons, defined by a Lagrangian and action of the form 1 L = Tr F F + ψ (D/ + m ) ψ 2 g2 µν µν f f f Xf S = d 4x L Z This theory can be solved from first principles, with minimal number of input parameters (bare quark masses and a scale fixing parameter). Hadron properties should be computable from QCD. Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Euclidean space-time: xµ real, x4 = i t. Quarks ψf : flavor f = u, d, s, c, t, b; color c = 1, 2, 3 Gluons Aµ: Fµν = ∂µAν − ∂ν Aµ + i [Aµ, Aν ] Parameters (couplings): g, mu, md , ms, mc, mt , mb (note: bare parameters, scale dependence!) Homework Mental gymnastics: Imagine what changes, if SU(3) → U(1) e.g., Photons: Aµ: Fµν = ∂µAν − ∂ν Aµ Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Gauge invariance A local transformation with an SU(3) matrix Ω(x): ψ(x) → ψ′(x)=Ω(x)ψ(x) ′ ψ(x) → ψ (x)= ψ(x)Ω(x)† ′ † † Aµ(x) → Aµ(x)=Ω(x)Aµ(x)Ω(x) + i ∂µΩ(x) Ω(x) leaves the action invariant ′ S[ψ′, ψ , A′]= S[ψ, ψ, A] (1) - Ω ∈ SU(3) → Ω(x)†Ω(x)= 1 - Cf. rotation invariance, e.g., ψ(x)Ω(x)†Ω(x)ψ(x)= ψ(x)ψ(x) - Action and observable (asymptotic) states have to be color singlets! - This is the SU(3)color gauge invariance! Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Quantization Quantization via Feynman path integral: CN (t) = hN(t)N(0)i ∝ [DA Dψ Dψ] e−S(A,ψ,ψ) N(t) N(0) ∼ expR (−EN t) Observables (like N) are built from A, ψ and ψ Gluons: bosonic variables are commuting: A(x)A(y)= A(y)A(x) Quarks: Grassmann variables are anti-commuting: ψ(x)ψ(y)= −ψ(y)ψ(x) Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Path integral = integral over all field configurations = integral over all field values at all space-time points! How to sum over all ∞∞ field configurations? We need - Regularization and renormalization: Momentum cut-off, dimensional regularization, lattice - Approximation: Perturbation theory or non-perturbative methods Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Perturbation theory Expansion in g leads to Feynman graphs However: g is small only at very large energy/momentum transfer (asymptotic freedom; Gross, Politzer, Wilczek) g2(µ) 1 αs(µ)= ∼ 4 π β0 log(µ/Λ) Λ depends on the regularization scheme. → Parton picture Running coupling (Della Morte et al., arXiv:hep-lat/0209023) Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Problems that cannot be attacked with perturbation theory: Chiral symmetry breaking Explicit: Non-zero quark masses Spontaneous: The pion is a Goldstone boson Confinement and the low energy properties of hadrons Hadron masses Low energy parameters (decay constants, current quark masses, LEC of Chiral Perturbation Theory) Form factors, matrix elements, structure functions We need non-perturbative methods! Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Lattice QCD Kenneth Wilson suggested 1974 to regularize QCD by introducing a 4-d (Euclidean) space-time lattice. Gauge field variables Uµ(x) ∈ SU(3) (3x3 complex, unitary matrices on each link) Quark field variables ψ(x), ψ(x) (f ) (ψα,c(x) are color 3-vectors, Dirac 4-spinors, nf vectors and Grassmann variables, on each lattice site) Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization How to discretize a gauge theory? Continuum: Free fermion action (Dirac operator): 0 4 SF [ψ, ψ]= d x ψ(x) γµ∂µ + m ψ(x) Z Derivative: 1 ∂ ψ(x) → ψ(n +ˆµ) − ψ(n − µˆ) µ 2a Lattice action: 4 ψ(n +ˆµ) − ψ(n − µˆ) S0 [ψ, ψ]= a4 ψ(n) γ + m ψ(n) F µ 2a n∈ ! XΛ µX=1 Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Gauge invariance is invariance under gauge transformations: ψ(n) → ψ′(n) = Ω(n) ψ(n) ′ ψ(n) → ψ (n) = ψ(n) Ω(n)† Note: each Ω(n) ∈ SU(3) is a 3x3 matrix; it may be different for each site (gauge symmetry = local symmetry)! Gauge invariance of lattice action? ′ ψ(n)ψ(n) → ψ (n) ψ′(n)= ψ(n) Ω(n)† Ω(n) ψ(n)= ψ(n)ψ(n) ′ ψ(n)ψ(n +ˆµ) → ψ (n) ψ′(n +ˆµ)= ψ(n) Ω(n)† Ω(n +ˆµ) ψ(n +ˆµ) Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization We need to replace the term by a new form with a new field variable: ψ(n) Uµ(n) ψ(n +ˆµ) The new field is the gauge (transporter) field Uµ(n) ∈ SU(3) with the transformation ′ † Uµ(n) → Uµ(n)=Ω(n) Uµ(n) Ω(n +ˆµ) Then ′ ′ ′ ψ (n) Uµ(n) ψ (n +ˆµ) † ′ = ψ(n) Ω(n) Uµ(n) Ω(n +ˆµ) ψ(n +ˆµ) † † = ψ(n) Ω(n) Ω(n) Uµ(n) Ω(n +ˆµ) Ω(n +ˆµ) ψ(n +ˆµ) = ψ(n) Uµ(n) ψ(n +ˆµ) Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Lattice gauge field: oriented link variable † U−µ(n) ≡ Uµ(n − µˆ) ✛ ✲ n − µˆ n n n +ˆµ ✉ ✉✉† ✉ U−µ(n) ≡ Uµ(n − µˆ) Uµ(n) Compare with the continuum gauge transporter G(x, y)= P exp i A · ds ZCxy ! along a link from x = n to y = n +ˆµ: G(n, n +ˆµ) = exp (i a Aµ(n)) 2 = Uµ(n)= 1 + i a Aµ(n)+ O(a ) Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Fermion action 4 U (n)ψ(n+ˆµ) − U (n)ψ(n−µˆ) S [ψ, ψ, U]= a4 ψ(n) γ µ −µ +mψ(n) F µ 2a n∈ 1 XΛ Xµ= n n+µ Ψ Ψ µ “Link term”: (n)Uµ (n) (n+ ) Homework Prove that up to O(a) the corresponding interaction term is like for the continuum action. Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Can we build action terms just with gauge field variables? Any term like that will do: L[U]= Tr Uµ(n) " # (n,µY)∈L (L is a closed loop of links.) Why? Consider the smallest loop: a plaquette! n +ˆν Uµ(n +ˆν) n +ˆµ +ˆν t✲✦❛ t Uµν (n) = Uµ(n) Uν (n +ˆµ) Uν (n) ✬✩Uν (n +ˆµ) ☞✻▲ ✻ ☞✻▲ ×U−µ(n +ˆµ +ˆν) U−ν (n +ˆν) ✫✪ † † t ✲✦❛ t = Uµ(n) Uν (n +ˆµ) Uµ(n +ˆν) Uν (n) n n +ˆµ Uµ(n) Is it gauge invariant? Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Gauge transformation {Ω(n)} of the plaquette term: ′ ′ ′ † ′ † Tr Uµ(n) Uν (n +ˆµ) Uµ(n +ˆν) Uν (n) = h † i † Tr Ω(n)Uµ(n) Ω(n +ˆµ) Ω(n +ˆµ) Uν (n +ˆµ) Ω(n +ˆν +ˆµ) h † † † † ×Ω(n +ˆν +ˆµ) Uµ(n +ˆν) Ω(n +ˆν) Ω(n +ˆν)Uν (n) Ω(n) = † † i Tr Uµ(n) Uν (n +ˆµ) Uµ(n +ˆν) Uν (n) h i (note: (ABC)† = C† B† A†) Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Gauge transformation {Ω(n)} of the plaquette term: ′ ′ ′ † ′ † Tr Uµ(n) Uν (n +ˆµ) Uµ(n +ˆν) Uν (n) = h † i † Tr Ω(n)Uµ(n) Ω(n +ˆµ) Ω(n +ˆµ) Uν (n +ˆµ) Ω(n +ˆν +ˆµ) h † † † † ×Ω(n +ˆν +ˆµ) Uµ(n +ˆν) Ω(n +ˆν) Ω(n +ˆν)Uν (n) Ω(n) = † † i Tr Uµ(n) Uν (n +ˆµ) Uµ(n +ˆν) Uν (n) h i (note: (ABC)† = C† B† A†) Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization Gauge transformation {Ω(n)} of the plaquette term: ′ ′ ′ † ′ † Tr Uµ(n) Uν (n +ˆµ) Uµ(n +ˆν) Uν (n) = h † i † Tr Ω(n)Uµ(n) Ω(n +ˆµ) Ω(n +ˆµ) Uν (n +ˆµ) Ω(n +ˆν +ˆµ) h † † † † ×Ω(n +ˆν +ˆµ) Uµ(n +ˆν) Ω(n +ˆν) Ω(n +ˆν)Uν (n) Ω(n) = † † i Tr Uµ(n) Uν (n +ˆµ) Uµ(n +ˆν) Uν (n) h i (note: (ABC)† = C† B† A†) Christian B.
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