Lattice Gauge Theories - An Introduction

Lattice Gauge Theories - An Introduction

Andreas Wipf

Theoretisch-Physikalisches Institut Friedrich-Schiller-University Jena

Bayrischzell, März 2018

1 Why lattice field theories

2 Lattice Gauge Theories

3 Observables in lattice gauge theories

4 Fermions on a Lattice

Why discretize quantum field theories 3 / 41

weakly interacting systems :

subsystems almost independent of each other weakly correglated quantum systems

weakly interacting effective dof (quasi particles) quantum electrodynamics

weak interaction weak field gravity

strong interaction at high energies underlyingGaussian fixed point perturbations theory applicable

strongly interacting systems:

properties explained by strong correlations between subsystems strongly correlated quantum systems

high temperature superconductivity ultra cold atoms in optical lattices spin systems near phase transitions strong field gravity

strong interaction at low energies underlying interacting fixes point

dependent on scale a theory can be weakly or strongly interacting needs non-perturbative methods

various approaches 5 / 41

soluble models low dimensions:

exactly soluble models: Ising-, Schwinger-, Thirring model, . . . high symmetry:

conformal symmetry, supersymmetry, integrable systems, dualities,. . . approximations:

mean field, strong coupling expansion, expansions for high/low temperataure, phenomenological models, . . .

functional methods:

∞-system of coupled Schwinger-Dyson equations functional renormalization group

lattice formulation, ab-initio lattice simulation lattice-QFT⇒particular statistical system powerfulsimulation methodsof statistical physics

global gauge transformations 6 / 41

Gauge theories in continuum

all fundamental theories = gauge theories electrodynamics: abelian U(1) gauge theory electroweak model:SU(2)×U(1) gauge theory strong interaction: SU(3) gauge theory

gravity: gauge theory

matter fieldφ(x)∈ V, global gauge transformφ(x)→Ωφ(x) Ω∈ Ggauge group

invariant scalar product onV: (Ωφ,Ωφ) = (φ, φ) invariant Lagrange density

L(φ, ∂_µφ) = (∂_µφ, ∂_µφ)−V(φ) invariant potentialV(Ωφ) =V(φ)

local gauge transformations 7 / 41

construction oflocally gauge invariant theory

φ(x)−→φ⁰(x) =Ω(x)φ(x), Ω(x)∈ G

∂µφwrong transformation property; need covariant derivative Dµφ=∂µφ−igAµφ, gcoupling constant needs newdynamical fieldAµ∈g(g=Lie algebra) requirement:Dµφtransforms asφdoes=⇒

D_µ⁰ = ΩDµΩ⁻¹⇐⇒A⁰_µ= ΩAµΩ⁻¹− i

g∂µΩΩ⁻¹ field strength

Fµν = i

g[Dµ,Dν] =∂µAν−∂νAµ−ig[Aµ,Aν]∈g

gauge invariant Lagrangian 8 / 41

transforms in adjoint representation

Fµν(x)−→Ω(x)Fµν(x)Ω⁻¹(x)

L Lorentz invariant, parity invariant, gauge invariant ⇒

L=−1

4trF^µνFµν+ Dµφ,D^µφ

−V(φ)

principle of minimal coupling:

begin with globally invariant theory,replace∂µ→Dµ

addYang-Mills term−¹₄trF^µνFµν (cp. electrodynamics) symmetries and particle content→Lagrange density (almost)

Parallel transport 9 / 41

C_yx path fromx toy, parametrizedx(s) parallel transportofφalong path:

0= ˙x^µDµφ⇐⇒ dφ(s)

ds =igAµ(s) ˙x^µ(s)φ(s), φ(s)≡φ x(s) cp. time-dependent Schrödinger equation

letx(0) =x andx(1) =y ⇒ φ(y) =Pexp ig

Z 1

0

ds Aµ(s) ˙x^µ(s)

! φ(x)

parallel transport along path C

U(C_yx,A) =Pexp ig Z

Cyx

A

!

∈ G, A=Aµdx^µ

gauge invariant variables 10 / 41

pathsC_yx andC_zy can be composed:C_zy ◦ C_yx =C_zx U(C_zy ◦ C_yx,A) =U(C_zy,A)U(C_yx,A) exists (useless?) nonabelianStokes theorem

gauge transformation

U(C_yx,A⁰) = Ω(y)U(C_yx,A) Ω⁻¹(x) fromx toy parallel transportet field

U(C_yx)φ(x) transforms as φ(y) gauge invariant objects(over-complete)

trU(C_xx) holonomies φ(y),U(C_yx)φ(x)

scalar products

Lattice field theories (Euclidean) 11 / 41

field theory in continuous spacetimeR^d ill-defined (UV-divergences) spacetime continuum→discretize spacetime

e.g.hypercubic latticeΛwith lattice constanta

lattice sites, lattice links, lattice plaquettes, lattice cubes, . . . minimal momentump=2π/a

⇒theory regularized in UV

matter fieldφ(x)→φ_x, x ∈Λlattice field

derivative→difference oparatoror lattice derivative, e.g.

(∂µφ)_x = 1 a

φ(x +aeµ)−φ(x) gauge theory:covariant lattice derivative

(Dµφ)_x ≡ 1 a

φ_x+aeµ−Ux,µφ_x

gauge invariant matter action 12 / 41

U_x,µparallel transporter fromx tox+aeµ

lattice action for matter field (a=1) Smatter=X

x,µ

Dµφ_x,Dµφ_x +X

x

V(φ_x)

=−2<X

x,µ

φ_x+eµUx,µφ_x +X

x

2d φ_x, φ_x

+V(φ_x

IR cutoff

finite latticeΛ =Z⁴→Nt×N³ needed in simulations

extrapolate toN→ ∞ classical spin system

nearest neighbour interaction Smatterreal, positive

locally gauge invariant

x¹ x² t

1

N 1

N

gauge invariant gauge action 13 / 41

new dynamical compact fieldU_x,µ∈ G

parallel transporter along link fromx tox+aeµ

replaces dynamical noncompact fieldAµ(x)∈g relation via parallel transport

A

smooth on scale a ⇒

Ux,µ≈e^{ig aAµ(x)} =1+ig aAµ(x) +. . . covariant derivative

(Dµφ)_x = 1 a

φ_x+aeµ− 1+ig aAµ(x) +. . . φ_x

= (∂_µφ)_x −ig(Aµφ)_x +O(a) there areO(a²)improved lattice derivative

plaquette variable 14 / 41

x x+eµ

x+eµ+eν

x+eν

plaquette p

transporterUµν(x) around plaquette

paralell transport around plaquette p∼(x, µ, ν)

U_p=U_x+eν,−eνU_{x+eµ+eν,−eµ}U_x+eµ,eνU_x,µ Baker-Hausdorff formula

U_x,µ≈e^iagAµ(x), a1

⇒Up=e^{ia2gFµν(x)+O(a3)} transforms homogeneously

U_p(x)→Ω(x)U_p(x)Ω⁻¹(x)

U_p+U_p^†≈2·1−a⁴g²F_µν² (x) +O(a⁶)

Yang-Mills theory on lattice 15 / 41

lattice action for gauge field configurationU={U_x,µ} SW(U) = 1

g²N X

p

tr

1−1 2

Up+U_p^†

(Wilson) .

in particular forG=SU(2) SW= 1

2g² X

p

tr(1−Up)

improved lattice action (Symanzik)

SYM−SSy=O(a²)

⇒faster convergence to continuum limita→0

functional integral 16 / 41 functional integral over lattice gauge fields{U_x,µ}={U_`}

Z

DAµ(x)−→^? Z Y

(x,µ)

dUx,µ= Z Y

`

dU`, `: link actionand measuremust be gauge invariant

recallUx,µ→Ω_x+eµUx,µΩ⁻¹_x

gauge invariance⇒dUx,µleft- and right-invariant (normalized) Haar measure expectation values in pure lattice gauge theory

hOiˆ = 1 Z

Z Y

`

dU`O(U)e^−SW(U)

partition function

Z = Z Y

`

dU`e^−SW(U)

invariant integration 17 / 41

considerirreducible representationsU→ R(U)of compactG, dim=d_R Peter-Weyl theorem:The functions{R(U)^ab}form an orthogonal basis on L₂(dU), and

R^ab,R^0cd

≡ Z

R¯^ab(U)R^0cd(U)dU= δ_RR⁰ dR

δ_acδ_bd,

Lemma:The charactersχ_R(U) =trR(U)form a ON-basis of invariant functions, f(U) =f(ΩUΩ⁻¹)in L₂(dU), such that χ_R, χ_R⁰

=δ_RR⁰ identities

orthogonality: R^ab, χ_R0

= χ_R0,R^ab

=δ_RR0

d_R δab

gluing:

Z

dΩχ_R(UΩ⁻¹)χ_R0(ΩV) =δ_RR0

d_R χ_R(UV) cutting:

Z

dΩχ_R(ΩUΩ⁻¹V) = 1

d_Rχ_R(U)χ_R(V) decomposition of unity: X

R

d_Rχ_R(U) =δ(1,U)

the curse of dimensionality 18 / 41

functional integral on finited-dimensional lattice dVdim(G)−dimensional integral SU(2) gauge theory, moderate hyper-cubic 16⁴-lattice⇒

786 432−dimensional integral cannot be calculated numerically!

stochastic methods

generate many configurations distributed according toe^−action method of important sampling

Monte Carlo (MC) algorithms (Metropolis, Heat bath, . . . ) with ferminions: expensive (hybrid MC + . . . )

observables in pure lattice gauge theories 19 / 41

only gauge invariant observables (Elitzur theorem) traces ofparallel transporters along loops

W[C] =tr(U`<sub>n</sub>· · ·U`₁), C=`<sub>n</sub>◦ · · · ◦`₁ Wilson loops W[R,T]rectangular loop, edge lengthsR,T

static energy of a staticqq-pair¯ separated byR V_q¯q(R) =− lim

T→∞

1

T loghW[R,T]i string tension and Lüscher term

Vq¯q(R)∼σR+ const.− c

R +o R⁻¹

T

R

static quarks in representationR 20 / 41 confinement⇒σ >0

⇒ W ∼exp(−σRT)area law (strong coupling) only colorless (gauge invariant) states are seen

Click here

−6

−4

−2 0 2 4 6 8

0.0 0.5 1.0 1.5 2.0 2.5

V˜R/µ

µR

R= 7 R= 14 R= 27 R= 64 R= 77 R= 77^′ R= 182 R= 189

linear potentialsfor static quarks in different G2representations

string-breaking for static charges in adjoint of SU(N) or for G₂ 21 / 41

dynamical quarks meson, diquarkqq¯ → 2 mesons, diquarks

charges in adjoint or G2

energy scale = 2 mglueball

decay products: glue-lumps

−2

−1 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6

V˜R/µ

µR

R= 7, β= 30, N= 48³ R= 14, β= 30, N= 48³ R= 7, β= 20, N= 32³ R= 14, β= 20, N= 32³

some observables in pure gauge theories 22 / 41

confinement:⇒only colorless (gauge invariant) states are seen QCD:confinement at low temperature, no gluons

glueballs= colourless bound states of gluons

state by acting withinterpolating operatoron vacuum

|ψ(τ)i= ˆO(τ)|0i, O(τ) =ˆ e^τHˆO(0)e^−τHˆ

two-point function

G_E(τ) =h0|TO(τ) ˆˆ O(0)|0i=X

n

|h0|O|ni|ˆ ²e^−Enτ

asymptotically large Euclidean time

GE(τ)−→ |h0|O|0i|ˆ ²+|h0|O|1i|ˆ ²e^−E1τ

1+O e^{−τ(E2−E1)}

excited state withh0|O|1i 6=ˆ 0→asymptotics O|0iˆ and|1ishould have same quantum numbers

parity, angular momentum (cubic group), charge conjugation, . . . glueballs:Ocombination of paralles transporters

masses of glueballsin MeV MC-simulation of Chen et al.

J^PC 0⁺⁺ 2⁺⁺ 0⁻⁺ 1⁺⁻ 2⁻⁺ 3⁺⁻

mG[MeV] 1710 2390 2560 2980 3940 3600

J^PC 3⁺⁺ 1⁻⁻ 2⁻⁻ 3⁻⁻ 2⁺⁻ 0⁺⁻

mG[MeV] 3670 3830 4010 4200 4230 4780

gauge theories at finite temperature 24 / 41

partition function:β-periodic gauge fields Z(β) =

I Y

(x,µ)

dU_x,µe^−SW(U)

space tim

e

P^x

Z ⇒thermodynamic potentials T <Tc :confinement→glueballs T >Tc :deconfinement→gluon plasma phase diagram, order of transition(s) order parameter: Polyakov loopPx

Polyakov loop 25 / 41

center symmetry:

non-periodic gauge trafo by center trafo

order parameter:

Polyakov loopPx

Px =tr ^Nt

Y

x0=1

U_(x0,x),0

SU(3): center=Z3

broken belowTc

restored aboveTc

−0.5 −0.4 −0.3 −0.2−0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5

−0.4

−0.3

−0.2

−0.1 0

0.1 0.2

0.3 0.4

0.5

0 2000 4000 6000

Im

Re

histogram of Polyakov loop

expected phase diagram of QCD

fermions 27 / 41

fermions on the lattice

functional approach:ψα(x)anticommuting

{ψα(x), ψβ(y)}={ψ¯α(x),ψ¯β(y)}={ψα(x),ψ¯β(y)}=0 fermionic integration = multi-dimensionalGrassmann integral

Z

DψDψ¯· · · ≡ Z

Y

x

Y

α

dψα(x)dψ¯α(x). . .

expectation value of observableAˆ h0|A|0iˆ = 1

ZF

Z

DψDψ¯A( ¯ψ, ψ)e^{−SF(ψ,ψ)¯} partition function

ZF= Z

DψDψ¯e^−SF

bilinear classical actionSFfor the fermion field SF=

Z

d^dx L(ψ,ψ),¯ L= ¯ψ(x)Dψ(x) Grassmann integration→determinant of fermion operator

ZF= Z

DψDψ¯exp

− Z

d^dxψ(x¯ )Dψ(x)

=detD corresponding formula forcomplex scalars

ZB= Z

DφDφ¯exp

− Z

d^dxφ(x¯ )Aφ(x)

= 1

detA Majorana fermions(susy)

ZF= Z

Dψexp

− Z

d^dxψ(x)Dψ(x)

=Paff(D)

fermion determinant 29 / 41

expectation valuesin full lattice gauge theory h(U)i= 1

Z Z

O(U)dµ(U), dµ(U) =det(D)e^−S[U]DU, Z = Z

dµ(U) subtle: first order Dirac operator on lattice

on finite latticeD(huge) matrix

stochastic methods applicable if det(D)e^−S[U]>0 usually:Disγ₅-hermitean

γ₅Dγ5=D^†

eigenvalues come in complex conjugated pairs⇒determinant real P(λ)≡det(λ−D) =detγ₅(λ−D)γ5=det λ−D^†

=P^∗(λ^∗)

problems with fermions 30 / 41

λroot⇒λ^∗root, real, not necessarily positive sign problemif detDchanges sign

example

D=∂/+m+O γ₅hermitean⇐⇒∂µ=−∂_µ^†, O=O^†, [γ₅,O] =0 natural choice

∂˚µf

(x) = 1

2(f(x +eµ)−f(x−eµ)) gauge theories:chiral symmetry for massless fermions

e^iαγ5De^iαγ5=D⇔ {γ₅,D}=0 naive Dirac operator

D=γ^µ˚∂_µ+m γ₅-hermitean, chirally symmetric form=0

doublers on lattice withN sites: fermion Green function hx

1

∂˚+m

0i= 1 N

N

X

n=1

e^ipnx

m+i˚p_n, ˚p_n=sinp_n, p_n= 2πn N Green function on the lattice with 40 sites.

x m=0.2

x m=0.1

fermion Green function on one-dimensional lattice withN=40

dispsersion relations 32 / 41

p

0 π

˚p² ˆp² p²

dispersion relations for−∂² p²:continuum relation

˚p²from∂˚

ˆp²from nearest neighbor Laplacian (∼Wilson operator)

( ˆ∆f)(x) =X

µ(f(x +eµ)−2f(x) +f(x −eµ))

˚∂µ ⇒chiral andγ₅−hermitean∂/ doublersin spectrum

Wilson operator 33 / 41

Theorem (Nielsen-Ninomiya)

exists no translational invariant D fulfilling

1 locality: D(x−y).e^−γ|x−y|,

2 continuum limit:lima→0D(p) =˜ P

µγ^µpµ,

3 no doublers:D(p)˜ is invertible if p6=0,

4 chirality:{γ₅,D}=0.

nicetopologicalproof

give up chiral invariance: Wilson fermions S_w=Snaive− r

2 X

x

ψ¯_xa∆ψˆ _x =X

x

ψ¯_xD_wψ_x ,

Wilson operator

Dw=γ^µ∂˚µ−ar 2

∆ˆ

spectrum of free massive Wilson operator 34 / 41

√d

continuums

ℜλ_p

m+2rd m+2r

ℑλ_p

m spectrum

result 1

γ₅hermitian {γ₅,D} 6=0

complex eigenvalues in thermodynamic limit (r =1) staggered fermions, Ginsparg-Wilson fermions

lattice action

SF=X

x

ψ¯_x(Dwψ)_x

gauge invariance:first parallel transport and then compare(r =1) (Dw)_xy = (m+d)δ_xy

−1 2

X

µ

(1+γ^µ)Uy,−µδ_x,y−eµ+ (1−γ^µ)Uy,µδ_x,y+eµ

rescaling (Wilson)

ψ→ 1

√m+d ψ gauge invariant action

Sw=X

x

ψ¯_xψ_x−κX

x,µ

ψ¯_x−eµ(1+γ^µ)Ux,−µψ_x + ¯ψ_x+eµ(1−γ^µ)Ux,µψ_x

hopping parameterκ= (2m+2d)⁻¹

full lattice field theory 36 / 41

latticefunctional integrals Z =

Z Y

`

dU`

Y

x

dψ_xdψ¯_x e^{−Sg(U)−SF(ψ,ψ)¯}

= Z Y

`

dU` det(D[U])e^−Sg(U)

= Z Y

`

dU`sign(detD) (detM)^1/2e^−Sg(U)

M=D^†D⇒detM ≥0.

try stochastic method with

dµ(U) = (detM)^1/2e^−Sg(U)DU

pseudo fermions 37 / 41

expectation values

hO[U]i=

Rdµ(U)sign(detD)O(U) Rdµ(U)sign(detD) problem withre-weighing: sign(detD)may average to zero fermion determinant: method ofpseudofermion fields

(detM)^1/2= Z Y

p

Dφ^†_pDφ_pe^−SPF, SPF=

NPF

X

p=1

φ_p,M^−qφ_p

q·N_PF =1/2. If detD>0⇒ Z =

Z Y

`

dU`DφDφ^∗e^{−Sg(U)−SPF(U,φ,φ†)} HMC algorithm: force given by gradient of non-localS_g+S_PF

observables depending on femion fields 38 / 41

rHMC dynamicsM^−q→rational approximation M^−q≈α₀+

NR

X

r=1

α_r M+β_r

fermion correlators:SF quadratic inψ⇒Wick contraction e.g. interpolating operator for pion

Oπ(t) =X

x

ψ(t¯ ,x)τ γ₅ψ(t,x)

Wick-contraction

h0|O^†_π(t)Oπ(0)|0i= 1 Z

Z Y

`

dU`GFGF det(D[U])e^−Sg(U)

∼amplitude·e^−mπt

⇒masses of bound states: mesons, baryons, glueballs, . . .

mesons (baryon number 0)

Name O T J P C

π uγ¯ ₅d SASS 0 - + η ¯uγ5u SASS 0 - +

a ud¯ SASS 0 + +

f ¯uu SASS 0 + +

ρ uγ¯ µd SSSA 1 - + ω uγ¯ µu SSSA 1 - + b uγ¯ ₅γ_µd SSSA 1 + + h ¯uγ₅γµu SSSA 1 + +

increase overlap with vacuum: smearing of sources and sinks diagonalization of correlation matrix

G₂masses of mesons, diquarks, baryons 40 / 41

heavy ensemble light ensemble

Wellegehausen, Maas, Smekal, AW (2013)

summary, . . . 41 / 41

simulations: stochastic, linear algebra, programming

works for QCD atT =0 andT >0 (fermionsβ-anti-periodic) but:fermions difficult and expensive

thermodynamic and continuum extrapolations:N→ ∞anda→0 realistic quark masses achieved

problem:finite baryon density, det(D)complex

⇒conventional MC does not work

simulations forsupersymmetric YM theories lattice breaks supersymmetry

some results of mass spectrum ofN =1 SYM new result onN = (2,2)andN = (8,8)

relevant for AdS/CFT (Gregory-Laflamme instability) books: Montvay-Münster, Rothe, Lang-Gattringer, AW, . . .