Lattice Quantum Chromodynamics for Pedestrians

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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)

∼ Rexp(−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:

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:

Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization

Wilson gauge action

Wilson gauge action: 2 S [U]= Re Tr 1 − U (n) G g2 µν n∈Λ µ<ν X X

...is a sum over all plaquettes For small lattice spacing a the leading term of the expansion gives

2 a4 S [U]= Re Tr 1−U (n) = Tr F (n)2 +O(a2) G g2 µν 2 g2 µν n∈Λ µ<ν n∈Λ µ,ν X X X X This is called the “naive” continuum limit... More complicated gauge actions involve longer closed loops.

Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization

Homework Prove that up to O(a2) the Wilson gauge action reproduces the continuum action. Do not forget, that the gauge ﬁeld variables do not commute! For this proof you need the Baker-Campbell-Hausdorff formula:

1 exp(A) exp(B)= exp A + B + [A, B] + ... . 2

Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization

Lattice QCD path integral

Lattice path integral for expectation values (n-point functions):

1 O (t) O (0) = D[ψ, ψ] D[U] e−SF [ψ,ψ,U]−SG[U] O [ψ, ψ, U] O [ψ, ψ, U] 2 1 Z 2 1 Z

Partition function:

Z = D[ψ, ψ] D[U] e−SF [ψ,ψ,U]−SG[U] . Z

Christian B. Lang Lattice QCD for Pedestrians Continuum QCD QuantumChromoDynamics Quantization Pure gauge theory Lattice QCD LQCD quantization

Key points The Euclidean lattice formulation has site variables (quarks) and link variables (gaugefields). The gaugefield variables are in the group (not in the algebra). The Wilson gauge field action is a sum over plaquettes. Quantization amounts to summing over all gauge- and fermion-configurations. The naive continuum limit (a → 0 in the action) reproduces the continuum expressions. The formulation is explicitly gauge invariant.

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Monte Carlo integration

We want to compute expectation values like 1 hOi = D[U] e−SG[U] O[U] with Z = D[U] e−SG[U] . Z Z Z Monte Carlo simulation = approximation

1 N 1 hOi ≈ e−S[Un] O[U ] ≈ O[U ] , N n N n n=1 Un with X probabilityX ∝ e−S[Un]

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Markov chain of conﬁgurations:

U0 −→ U1 −→ U2 −→ ... ′ ′ ′ P(Un = U |Un−1 = U)= T (U ← U) ≡ T (U |U) ′ ′ 0 ≤ T (U |U) ≤ 1 , U′ T (U |U)= 1

Balance condition: P

! T (U′|U) P(U) = T (U|U′) P(U′) XU XU ensures:

T T T T P(0) → P(1) → P(2) → ... → P (= equilibrium distribution)

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Detailed balance :

T (U′|U) P(U)= T (U|U′) P(U′)

is a sufﬁcient condition.

Metropolis algorithm Step 1: Choose some candidate configuration U′ according to ′ some a priori selection probability T0(U |U), where U = Un−1. Step 2: Accept the candidate configuration U′ as the new configuration Un with the acceptance probability

′ ′ T0(U|U ) exp − S[U ] T (U′|U)= min 1, . A ′ T0(U |U) exp − S[U] ! Step 3: Repeat these steps from the beginning.

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Example: Ising model

Spin conﬁguration: spins sn ∈ {−1, 1}

S[s]= β n,µ snsn+ˆµ n+y^

′ P n−x^ n n+x^ sn = −sn,old n−y^ ′ ′ TA(s |s)= min 1, exp − (S[s ] − S[s]) i.e. accept if ′ exp − (S[s ] − S[s]) > random number ∈ (0, 1) Note: ∆S is local due to ′ − ′ − || S[s ] S[s] = [β(sn+1ˆ + sn−1ˆ + sn+2ˆ + sn−2ˆ)](sn sn)

Test/update each spin once for all spins = one

sweep β

Measure observables after each sweep

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Examples for Ising spin conﬁgurations (cf. http://thy.phy.bnl.gov/www/xtoys.html, executable/xising)

hot (disordered) cold (ordered)

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Metropolis for Wilson’s gauge action

The gauge action has locally the form

β S[U (n)′] = Re Tr 6 1 − U (n)′ A µ loc N µ 6 with A = i=1 Pi P i.e., sum over “staples” Pi (3 orthogonal directions!) We need the change of the action:

β ∆S = S[U (n)′] − S[U (n)] = − Re Tr U (n)′ − U (n) A µ loc µ loc N µ µ h i Note: A is not affected by the change of Uµ(n)! Updating one link requires only information from the bordering plaquettes (staples)

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Action

Let’s compute:

Start with some (e.g. hot or cold) conﬁguration. Update each link of the conﬁguration = one sweep. Run for many equilibrating sweeps. Continue sweeping and measure observables every k sweeps (k depends on the Gauge configurations autocorrelation time).

Measure observables

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Conﬁnement at strong coupling

Consider the observable “Wilson loop”:

r Wilson loop W [U]= Tr Uµ(n) " # (n,µY)∈Cr×t where Cr×t is a closed loop around a r × t rectangle. t Its expectation value deﬁnes a “static potential” V (r) at spatial separation r q(r) q(r) between a quark and an anti-quark: t r propagation of qq state −tV (r) −t ∆E hWLi ∝ e 1 + O(e ) q(0) q(0) = e−nt aV (r) 1 + O(e−nt a∆E ) Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Strong coupling expansion

1 β hWC i = ′ D[U] exp Re Tr[UP ] Tr Ul Z 3 ! " # Z XP Yl∈C 1 β † = ′ D[U] exp Tr[UP ]+ Tr[UP] Tr Ul . Z 6 ! " # Z XP Yl∈C In the (group) integral only terms with oppositely oriented links survive.

SU(3) dUUab = 0

RSU(3) dUUabUcd = 0 † 1 RSU(3) dUUab(U )cd = 3 δad δbc † RSU(3) dUUab(U )ba = 3

R Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Thus β nA hW i∝ = enA log(β/18) = enr nt log(β/18) C 18

hWCi∝ exp − a nt V (r) 1 → V (r = a n )= σ r with σ = − log(β/18) 6= 0 r a2 (Wilson conﬁnement criterion)

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Measuring the static potential

Alternative observable: Polyakov loop r Polyakov loop NT −1 P(m~ )= Tr U (m~ , j)  4  j 0 Y=  

hP(m~ )P(~n)†i∝ e−NT aV (r) 1 + O(e−NT a∆E ) “Static” potential: potential between “static” charges (not well-deﬁned for dynamical sea quarks → string breaking)

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

How to set the scale?

β = 5.95 β = 6.20 0.9 a V(an) 0.8 1.0

0.7 0.8 0.6

0.6 0.5

0.4 0.4 2 4 6 8 2 4 6 8 10 12 14 16 n n Example for evaluation of the static potential: linearly rising!

The lattice spacing is not yet ﬁxed. We need a physical scale parameter to ﬁx it.

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

(R. Sommer radius) 1 Experiment (analysed with non-rel. Schrödinger equation) gives for 2 the force the dimensionless combination r0 F(r0)= 1.65 at r0 = 0.5 fm. 2 We then ﬁnd the position x in the lattice potential, where that vakue is observed; thus x = r0/a or a = r0/x 3 Compare with lattice potential:

dV (r) d B B F(r)= = A + + σ r = − + σ dr dr r r 2 4 B and σ a2 determined from a ﬁt to the lattice potential. This has to be done at each value of the gauge coupling β = 6/g2 and gives a(β).

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Scaling according to renormalization group (running coupling constant)

β − 1 1 2 2β2 1 2 a(g)= β g 0 exp − 1 + O(g ) Λ 0 2β g2 L 0 is conﬁrmed by the lattice results.

One ﬁnds a(g → 0) → 0!

In full LGT (including the quark sea) one can take measured hadron masses to set the scale.

asymptotic freedom confinement perturbation theory g=infinite g=0

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

3 β = 6.0 β = 6.2 2 β = 6.4 Cornell Supports the string picture: 1

0 0 )] r 0 -1 [V(r)-V(r -2

-3

-4 0.5 1 1.5 2 2.5 3 r/r0 The static potential derived from Wilson loops shows correct scaling (Bali, Phys. Rep. 343 (2001) 1)

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Alternative methods to set the scale

One needs Nf + 1 parameters: α(g), mu (g), md (g), ms(g), mc (g), mb(g), mt (g) (all scale-variant!). Thus one needs Nf masses (mπ, mK , mD,...) and one scale parameter. The scale should be measurable in experiment and in LQCD, statistically reliable, and weakly dependent on the quark masses. (cf. recent review of Rainer Sommer at LATTICE2013: http://www.lattice2013.uni-mainz.de/72_ENG_HTML.php):

Use some hadron mass parameter (e.g., mΩ or fπ or fK ); needs extrapolation to or work at the physical point.

Static force (see above): r0 or r1 (similar deﬁnition; not optimal on coarse lattices). (See next page for quark mass dependence!)

Gradient ﬂow: t0 or w0

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Slide from Rainer Sommer’s talk at Lattice 2013 Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD Gradient ﬂow: behaviour of plaquette expectation value under differential (smearing) equation in ’time’ τ for the gauge links: dV (τ) ∂S (V ) = − P V (τ) with V (0)= U . dτ dV

Then ﬁnd the dimensionless values of t0 (Lüscher,) or w0 (BMW) such that 2 d 2 2 τ hE(τ)i|τ=t0 = 0.3 or τ hτ E(τ)i|τ=w = 0.3 dτ 0 where E is the plaquette variable. Fig. from M. Lüscher, JHEP 1008, 071 (2010)

M. Lüscher, JHEP 1008, 071 (2010); see also Lüscher’s talk at LATTICE2013 BMW, JHEP 1209, 010 (2012) : w0 a = w0,phys = 0.1755(18)(04)fm.

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

The three limits of LQCD

Continuum limit: a(g, m) → 0 (g → 0) Lattice artifacts should become small → Improvement programme

Thermodynamic limit: L → ∞ (L · a = const.) Hadron physics in a box of a few fm → Finite volume effects can be utilized

Chiral limit: m → m0 (Mπ → Mπ,exp) Physical u, d quark masses are small → We want to understand chiral symmetry breaking

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Continuum limit

The same physical image represented on lattices of linear extent 5, 10, 15, 20, 30 (in units of a) corresponding to lattice spacings a of 30 mm, 15 mm, 7.5 mm, and 5mm.

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Continuum limit

The same physical image represented on lattices of linear extent 5, 10, 15, 20, 30 (in units of a) corresponding to lattice spacings a of 30 mm, 15 mm, 7.5 mm, and 5mm.

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Continuum limit

The same physical image represented on lattices of linear extent 5, 10, 15, 20, 30 (in units of a) corresponding to lattice spacings a of 30 mm, 15 mm, 7.5 mm, and 5mm.

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Continuum limit

The same physical image represented on lattices of linear extent 5, 10, 15, 20, 30 (in units of a) corresponding to lattice spacings a of 30 mm, 15 mm, 7.5 mm, and 5mm.

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Continuum limit

The same physical image represented on lattices of linear extent 5, 10, 15, 20, 30 (in units of a) corresponding to lattice spacings a of 30 mm, 15 mm, 7.5 mm, and 5mm.

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Continuum limit

The same physical image represented on lattices of linear extent 5, 10, 15, 20, 30 (in units of a) corresponding to lattice spacings a of 30 mm, 15 mm, 7.5 mm, and 5mm.

Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Confinement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD Key points QCD can be formulated on a Euclidean space-time lattice Quantization amounts to summing over all gauge configuration; this can be approximated by Monte Carlo sums Strong coupling confinement can be proven, towards weaker coupling MC calculations show non-vanishing string tension and asymptotic freedom The lattice spacing “runs”: a(β) The three limits of LQCD: continuum, thermodynamic, chiral.

Gluonic vacuum ﬂuctuations, movie c Leinweber et al., www.physics.adelaide.edu.au/cssm/research) click to start Christian B. Lang Lattice QCD for Pedestrians Monte Carlo integration QuantumChromoDynamics Conﬁnement at strong coupling Pure gauge theory How to set the scale? The three limits of LQCD

Homework

Are the operators ψnψn, ψnUn,µψn+ˆµ, and ψnψn+ˆµ gauge invariant? Which gauge invariant operators can you construct out of site-fermions and link-gaugeﬁelds? What particle states could they couple to? What would be a glueball state?

Christian B. Lang Lattice QCD for Pedestrians