Simulation of dynamical fermions in lattice QCD

Péter Petreczky Physics Department and RIKEN-BNL

INT, February 10, 2008 Strong interactions and QCD

Structure and Interaction of Hadrons

Quantum Chromo Dynamics (QCD)

SU(3) non-Ablelian gauhe theory

gluon self interactions asymptotic freedom confinement : Quarks and gluons cannot exist as free particles Color singlet states

Mesons Baryons Lattice Formuation of QCD

evolution operator in imaginary time

Integral over functions integral with very large (but finite) dimension ( > 1000) Lattice Monte-Carlo Methods

fermion determinant

Costs : Quarks and gluon fields on a lattice

fermion doubling ! 16 d.o.f ! Wilson fermions

Wilson (1975)

chiral symmetry is broken even in the massless case !

additive mass renormalization

Wilson Dirac operator is not bounded from below

difficulties in numerical simulations Hadron properties , spectral functions Staggered fermions

Kogut, Susskid (1975)

different flavors, spin componets sit in different corners of the Brillouin zone or in hypercube (3+1)-d: 4-flavor theory rooting trick

EoS and phase diagram of QCD (2+1) d: 2-flavor theory, interesting for graphene

(1+1) d: 1-flavor theory (no doubling) Chiral fermions on the lattice ?

We would like the following properties for the lattice Dirac operator:

Nielsen-Ninomiya no-go theorem : conditions one 1-4 cannot be satisfied simultaneously Nielsen, Ninomiya (1981)

Wilson fermion formulation gives up 4) Staggered fermion formulation gives up 3) Ginsparg-Wilson fermions

Ginsparg, Wilson (1982)

• anti-commutation properties are recovered in the continuum limit (a->0) • the r.h.s. of the Ginsparg-Wilson relation is zero for the solutions

mildest way to break the chiral symmetry on the lattice : physical consequences of the chiral symmetry are mantained ( e.g. soft pion theorem etc. ) Generalized chiral symmetry and topology

GW relation Luescher (1998)

flavor singlet transformation :

Luescher (1998)

Hasenfratz, Laliena, Niedermeyer (1998)

for flavor non-singlet transformation no anomaly ! Constructing chiral fermion action I

Overlap fermions :

Neuberger (1998)

using it can be shown that

GW relation with R=1/2 Constructing chiral fermion action II

Domain wall fermions : introduce the fictitious 5 th dimension of extent :

Shamir (1993)

Extensively used in mumerical simulations : ( see P. Boyle, 2007 for review ) Better staggered fermions ?

Fermion formulations gave up conditions 3. or 4. what about condition 2., i.e. cubic symmetry ?

one possibility :

Wilczek (1987)

2-flavors instead of 4 ! Another possibility is to use the analogy with graphene, Creutz (2007) gauge the theory is straightforward: use plaquette action ! because of the symmetry no additional parameters are needed in the gauge action

as in the staggered fermion formulation different flavors sit in the different corners of the Brillouin zone Simulating lattice fermions

because of the determinant no local updates are possible (e.g. Metropolis)

pseudo-fermion fields :

introduce (Gaussian) conjugate momenta

Hybrid Monte-Carlo algorithm : update the fields and conjugate momenta with discretized classical equation of motion accept the new configuration with probability