Introduction to Lattice QCD Lecture 2

Lattice fermions Wilson fermions Staggered fermions Continuum limit

Anna Hasenfratz University of Colorado

INT Summer School Aug 6-10 Lattice fermions Wilson fermions Staggered fermions Continuum limit Fourier transform Standard deﬁnitions: π/a 4 4 ipna d p ipna f (p)=a e− f , f = e f (p). n n (2π)4 n π/a ! "− f (p) is periodic within the Brillouin zone 2π f (p)=f (p + m ), m Z, a µ µ ∈ i.e. the lattice momentum is restricted to p π/a =Λ . | µ|≤ cutoﬀ Notation: k = ap dimensionless momentum; π/a d4p p = π/a (2π)4 Ausefulformula− # # eink =(2π)4δ(k) n !µ Lattice fermions Wilson fermions Staggered fermions Continuum limit Fourier transform

The derivatives

4 2 apµ φ (∆ ∆∗ )φ = sin ( )φ( p)φ(p) n µ µ n − a2 2 − n p ! " 1 i ψ¯ (∆ +∆∗ )ψ = sin(ap )ψ¯( p)ψ(p) 2 n µ µ n a µ − n p ! " Notation: pˆ = 2 sin apµ = p + (a2) µ a 2 µ O Lattice fermions Wilson fermions Staggered fermions Continuum limit Free scalars - again

The free scalar lattice action: 1 1 S[ψ]= a4 (∆ φ )2 + m2φ2 2 µ n 2 n n ! $ % a4 2 ap = (ˆp2 + m2)φ(p)φ( p), pˆ = sin( µ ) 2 − µ a 2 "p pˆ p substitution relates continuum & lattice. The continuum → and lattice descriptions agree for small p. Lattice fermions Wilson fermions Staggered fermions Continuum limit Fermion doubling

The naive fermion lattice action:

4 ¯ ¯ S[ψ]= a ψnγµ(∆µ∗ +∆µ)ψn + mψnψn n,µ ! & ' i = a4 ψ¯( p)[ sin(p a)γ + m]ψ(p) − a µ µ "p Now the continuum to lattice replacement is p sin(p a)/a µ → µ very diﬀerent at the edge of the Brillouin zones! Naive lattice fermions describe 16 continuum species. Lattice fermions Wilson fermions Staggered fermions Continuum limit Homework Derive the energy-momentum relation both for scalar and naive fermions. That’s the reliable way to understand the particle content of the systems. 1) the 2-point function is

e iap(n m) φ φ = − − % n m& pˆ2 + m2 "p

2) Integrate out p0. This is a complex contour integral, so ﬁnd the poles ω(p)= ip . ± 0 3) How many poles are there? At what momenta? 4) Now repeat for the fermions. Lattice fermions Wilson fermions Staggered fermions Continuum limit Homework - Spectral representation

e iap(n m) φ φ = − − % n m& pˆ2 + m2 "p

Lattice fermions Wilson fermions Staggered fermions Continuum limit Fermion doubling Nilsen-Ninomiya no-go theorem: It is not possible to construct a lattice fermion action that is Ultra local • Chirally symmetric • Has the correct continuum limit • Undoubled • Solutions: Wilson fermions - break chiral symmetry • Staggered fermion - brake taste symmetry • Ginsparg-Wilson fermions - extend chiral symmetry; local but • not ultra local actions (still universal). Twisted mass fermions • Other variants • Lattice fermions Wilson fermions Staggered fermions Continuum limit Wilson fermions Modify the fermion action so the modes at p π become heavy: µ ∼ 4 SW = a ψ¯n(DW + m)ψn n ! 1 D = γ (∆∗ +∆ ) ar ∆∗ ∆ W 2 µ µ µ − µ µ & ' The Wilson term is quadratic in ∆µ like the scalar kinetic term. It (a) irrelevant operator • O Lifts the doublers • Breaks chiral symmetry even in the chiral limit (it’s a mass • term) Lattice fermions Wilson fermions Staggered fermions Continuum limit Wilson fermions - hopping form The Wilson action after rescaling ψ √2κψ, κ = 1 → 2ma+8r 4 SW = a ψ¯nψn κ(ψ¯n(r γµ)Un,µψn+µ + ψ¯n(r + γµ)Un,µψn µ) − − − n ! & ' κ is the ”hopping parameter”, r =0standard choice

Kappa

T=0 Critical surface κcr (β) Deviation from 1/8 is the K c measure of chiral symmetry breaking.

continuum limit

0 beta infinity Lattice fermions Wilson fermions Staggered fermions Continuum limit Hopping parameter expansion

Hopping parameter expansion: expand in terms of κ (valid for small κ or large mass)

( Sg ψψ¯ ) Z = [Uψψ¯ ]e − − (1 κψ¯ (r γ )U ψ D − n − µ n,µ n+µ " (link κψ¯n(r + γµ)Un,µψn µ)) − − Only terms with 1-1 ψ¯ and ψ contribute leads to closed gauge → loops. Lattice fermions Wilson fermions Staggered fermions Continuum limit Eﬀective gauge action

Sg l Z = [U]e− κ c ReTr U D l closed loops " ! (C Re-exponentiate to ﬁnd the eﬀective gauge action from the fermions

S = 16κ4N (3 ReTrU )+... (r = 1) eﬀ f − ! p ! Fermions always introduce an eﬀective positive gauge coupling - true for other formulations, large κ,etc. Lattice fermions Wilson fermions Staggered fermions Continuum limit Staggered fermions

Simple idea: distribute the 4 components of the Dirac spinor to diﬀerent lattice sites. Counting: 1 component per site x 16 fold doubling = 4 species or tastes Lattice fermions Wilson fermions Staggered fermions Continuum limit Staggered fermions Formalism: Unitary transformation ψ Ω ψ , Ω = γn0 γn1 γn2 γn3 makes n → n n# n 0 1 2 3 the naive fermion action spin-diagonal:

1 ¯ S = ψn# αµ(n)[Un,µψn+ˆµ U† ψn µˆ]+m ψn# ψn# 2a − n µ,µˆ − n − n ! ! α (n)=( 1)n0+ +nµ 1 is a phase factor, 1 µ − ··· − ± Drop 3 of the four components of ψ χ and we have a 4-taste → staggered action 1 S = χ¯nαµ(n)[Un,µχn+ˆµ U† χn µˆ]+m χnχn 2a − n µ,µˆ − n − n ! ! Lattice fermions Wilson fermions Staggered fermions Continuum limit Staggered fermions

How do we recover a 4-component Dirac spinor? 1 Ψαa = Ωαaχ ,η=0or 1 n 8 η 2n+η µ η ! i.e. collect them from a hypercube. (a is taste, α is spinor ) Problems? the different elements of a Dirac spinor ”see” different gauge fields, leads to taste breaking. The coarser the gauge fields, the worst it is; Lattice fermions Wilson fermions Staggered fermions Continuum limit Staggered fermions Some staggered symmetries: Translation: • χ(n) ξ (n)χ(n +ˆµ), χ¯(n) ξ (n)¯χ(n +ˆµ), → µ → µ nν Un,µ Un+ˆµ,µ, (ξµ(n)=( 1) ν>µ ) → − !

Remnant U(1): • iθ χ e± χ , n = even or odd n → n protects the quark mass but gives only 1 pion per 4 tastes Lattice fermions Wilson fermions Staggered fermions Continuum limit Staggered fermions

Rooting: In order to reduce the 4 tastes to 1 the 4th root of the staggered determinant is taken in simulations. The resulting lattice action is non-local but there is growing evidence that in the continuum limit it approaches the correct universality class.