Introduction to Lattice QCD

Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions

Anna Hasenfratz University of Colorado

INT Summer School Aug 6-10 Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Why QCD? Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Why QCD?

SU(3)xSU(2)xU(1) Standard Model describes physics (way too) well up to the TeV scale

The discovery of 125GeV nearly SM Higgs (?) & no SUSY give little constraint on BSM physics

Electroweak precision tests are more important than ever and depend on strong interactions Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Why QCD?

QCD is a prototype model of gauge and fermion ﬁelds. Adding scalars is trivial; SUSY not so much.

Models with diﬀerent gauge groups, fermion representations and fermion numbers are candidates for beyond SM phenomenology - but all these candidates are strongly coupled and have to be studied non-perturbatively. Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Why Lattice?

Strong interactions are Asymptotically free • Conﬁning • Chirally broken • The latter two properties are non-perturbative. Physical quantities are not analytic in the QCD coupling! Lattice regularization is the only method that can describe non-perturbative QCD Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Continuum QCD

Continuum Euclidean action: 1 S[A]= d d x F 2 + ψ¯(x)γ D ψ(x) 4 µν µ µ ! " # F a = ∂ Aa ∂ Aa + gf abc Ab Ac µ,ν µ ν − ν µ µ ν D = ∂ igAa ta µ µ − µ Symmetries: SU(3) gauge symmetry • Lorentz, C,P & T • ψ eiαψ • → In the m =0chiral limit ψ eiγ5αψ • → Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Chiral symmetry breaking

Flavor symmetry: in m =0chiral limit

U(N ) U(N ) = U(1) SU(N ) U(1) SU(N ) f V × f A V × f V × A × f A

U(1)A gets broken by the anomaly SU(Nf )A breaks spontaneously N2 1 massless Goldstone bosons (pions). → f − Fermion mass breaks the chiral symmetry explicitly

m2 m π ∼ q mp = mp0 + cmq Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions All Known Physics Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions All Known Physics Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Lattice action: Scalars

Continuum Euclidean system:

S[φ] Z = φe− D ! 1 1 λ S[φ]= d d x (∂ φ(x))2 + m2φ(x)2 + φ(x)4 2 µ 2 4! ! " # Need to 1. Interpret φ D 2. Regularize$ momentum integrals Lattice discretization can do both. Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Lattice action: Scalars

Discretize:

φ(x) φ , x = na −→ n dx a i −→ n ! %i φ dφ D −→ n n ! & Discrete lattice derivative 1 ∂ φ(x) ∆ φ = (φ φ ) µ −→ µ n a n+ˆµ − n 1 ∆µ∗ φn = (φn φn µˆ) a − − Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Lattice action: Scalars

The scalar lattice action: 1 1 λ S[φ]=a4 (∆ φ )2 + m2φ2 + φ4 2 µ n 2 n 4! n n % " # 4 1 1 2 2 λ 4 = a φ (∆∗ ∆ )φ + m φ + φ −2 n µ µ n 2 n 4! n n % " # Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Homework

1) Show that in the λ the scalar lattice action reduces to the →∞ Ising model with φ = 1. (You will have to rescale the ﬁeld to n ± get 1) ± 2) Generalize the discussion of the scalar model to complex scalar and also to 2-component complex scalar. The latter one is the relevant model for the Standard Model Higgs. Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Lattice action: naive fermions

Discretize:

ψ(x) ψ , x = na −→ n dx a i −→ n ! %i ψ ψ¯ dψ dψ¯ D D −→ n n n ! & The naive fermion lattice action:

4 1 S[ψ]=a ψ¯ γ (∆∗ +∆ )ψ + m ψ¯ ψ 2 n µ µ µ n q n n n % " # Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Lattice action: gauge ﬁelds The most important feature is local gauge symmetry. Gauge transformation:

ψ V ψ , ψ¯ ψ¯ V † V SU(3) n → n n n → n n n ∈ and the role of the gauge ﬁeld is to make derivatives gauge invariant ψ¯ ψ ψ¯ U ψ n n+ˆµ −→ n n,µ n+ˆµ with U SU(3) transforming as n,µ ∈ U V U V n,µ −→ n n,µ n+ˆµ Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Lattice action: gauge ﬁelds

Relation to continuum gauge ﬁeld

iagAb tb A (x) U = e− n,µ µ −→ n,µ Gauge invariant quantities:

ψ¯nγµUn,µψn+µ,φnUn,µφn+µ,

Un,µ for any closed loop

&C The gauged lattice action could be

S[ψ] Tr( U + hc)+a3(ψ¯ γ U ψ + h.c. + amψ¯ ψ ) ∼ n,µ n µ n,µ n+µ n n n ' ( % &C Is it that simple? Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Lattice action

The simplest gauge action is the plaquette (Wilson gauge)

β S[ψ]= Tr(U + U† ) 6 ! ! n p % % 3 ¯ ¯ ¯ +a (ψnγµUn,µψn+µ + ψnγµUn† µ,µψn µ + amψnψn) − − but we could take any other closed loop (or combination of loops). Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Lattice action

Does it reproduce at least the naive a 0 continuum limit? → Expand: 2 aAµ(n) a 2 U = e− =1 aA (n)+ A (n) + ... n,µ − µ 2 µ leads to 2 a Gµν U = e− , G = F + (a) ! µν µν O so 4 2 6 Tr(U + U† )=2Tr1 + a Tr(F ) + (a ) ! ! µν O i.e. correct continuum form if β =2N/g 2. Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Homework

3) Prove the relations on the previous slide. Be careful: how many terms are there in the Tr(U + U† )? What is the trace of the ni ! ! SU(3) generators? ) 4) Can you derive similar expressions for the 1x2 and other small loops? These terms show up in improved actions, like the Symanzik gauge action. Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Observables The expectation value of any operator is given

1 Sg [U] = U e− , 'O( Z D n,µO n,µ ! & Sg [U] Z = U e− D n,µ n,µ ! & Operators of physical quantities are called observables. Expectation value of a non-gauge invariant operator vanishes. Numerical simulations: create conﬁgurations with probability

Sg [U] e− , and calculate expectation values. Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Wilson loops & the static potential The Wilson loop: closed gauge loop An RxT Wilson loop describes a quark-antiquark pair propagating at distance R for time T

V (R)T W (R, T ) e− ∼ where V (R) is the static potential In a conﬁning theory e V (R)=c + + σR R Coulomb + linear terms (σ is the string tension) Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Glueballs

Glueballs are pure gauge bound states Plaquettes can create glueball states (diﬀerent combinations are taken to describe diﬀerent quantum numbers). If they are separated at distance T

m T (0) #(T ) e− G '! ! (∼ where mG is the glueball mass. Glueballs are notoriously diﬃcult to calculate. (Tricks, tricks and more tricks are needed.) Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Strong coupling expansions

Before powerful computers there was strong coupling expansion ... At small β

Sg [U] β/6 Tr(U +U† ) β e− = e− p ! ! = 1 Tr(U + U† )+... − 6 ! ! ! p & " # 1 β (U) = U (U) 1 Tr(U + U† )+... 'O ( Z D n,µO − 6 ! ! n,µ p ! & & " # Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Strong coupling expansions

Which terms survive U?WhereU and U or three U’s meet D † $ dUTr(UV1)Tr(U†V2)=Tr(V1V2) ! (OK, this should really be done with group characters.) Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Wilson loop at strong coupling 1 β (U) = U (U) 1 Tr(U + U† )+... 'O ( Z D n,µO − 6 ! ! n,µ p ! & & " # Cover every link with an Need RxT plaquettes to cover it all opposite directional one: W (R, T ) βRT eT (Rlog(β)) ∼ ∼ The potential

V (R)= Rlog(β) − linear in R. The strong coupling gauge model is conﬁning! Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Wilson loop at strong coupling Next order : plaquette sticking out: β4, multiplicity 4*RxT, etc

The string tension:( u = β/6 ) σ = ln(u)+4u4 + 12u5 10u6 ... − − Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Glueballs in strong coupling

Lowest order : connect the two plaquettes with a tube:

β 4T T 4ln(β/6) ( ) e− × 'O( ∼ 6 ∼ giving m = 4lnu 3u ... G − ± The rest depends on the glue ball quantum numbers. Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Message from strong coupling expansion

The strong coupling pure gauge system is Conﬁning • Has massive glueballs • Meson spectrum shows chiral symmetry breaking (but that’s • not exact) Is there any use for the strong coupling expansion in the are of supercomputers? Introduction QCD fundamentals The lattice action Strong coupling Lattice fermions Homework

3) Calculate the next order term to the strong coupling expansion of the potential. (It is known to order 14) 4) Calculate the glueball mass in next order strong coupling expansion. Take two plaquettes, parallel to each other for simplicity. 5) Feeling ambitious? Calculate the potential between two Polyakov lines at ﬁnite temperature in the strong coupling. References: Montvay&Munster has extensive discussion about the strong coupling expansion.