Peter Hasenfratz

Perfect lattice action for asymptotically free theories

Abstract There exist lattice actions which give cut-off independent physical predictions even on coarse grained lattices. Rotation symmetry is restored, the spectrum becomes exact and, in addition, the classical equations have scale invariant instanton solutions. This perfect action can be made short ranged. It can be determined by combining analytical calculations with numerical simulations on small lattices. We illustrate the method and the benefits on the d = 2 non-linear σ-model.

The connection between the Λ parameters of lattice and continuum QCD

Abstract By calculating the two- and three-point functions at the one-loop level in weak-coupling lattice QCD we have found the numerical relation between the Λ parameter of the lattice theory and that of the continuum theory: Λ ( α =1 ) MOM = 83.5 Λ lattice for SU(3), Λ ( α =1 ) MOM = 57.5 Λ lattice for SU(2). This relation sets the scale for all lattice calculations in QCD.

Lattice QCD without tuning, mixing and current renormalization

Abstract The classically perfect action of QCD requires no tuning to get the pion massless in the broken phase: the critical bare mass mqc is zero. Neither the vector nor the flavour non-singlet axial vector currents need renormalization. Further, there is no mixing between 4-fermion operators in different chiral representations. The order parameter of chiral symmetry requires, however, a subtraction which is given here explicitly. These results are based on the fact that the fixed point action satisfies the Ginsparg-Wilson remnant chiral symmetry condition. On chiral symmetry related questions any other local solution of this condition will produce similar results.

Chemical potential on the lattice

The naive way of introducing chemical potential on the lattice leads to quadratic divergences even for free fermions. Starting from the analogy between the chemical potential and the fourth component of an abelian gauge field, a simple solution is proposed. For Wilson fermions it leads to a trivial modification of the hopping parameter of quarks propagating along the imaginary time direction.

The quark bag model

Abstract The quark bag model is reviewed here with particular emphasis on spectroscopic applications and the discussion of exotic objects as baryonium, gluonium, and the quark phase of matter. The physical vacuum is pictured in the model as a two-phase medium. In normal phase of the vacuum, outside hadrons, the propagation of quark and gluon fields is forbidden. When small bubbles in a second phase are created in the medium of the normal phase with a characteristic size of one fermi, the hadron constituent fields may propagate inside the bubbles in normal manner. The bubble (bag) is stabilized against the pressure of the confined hadron constituent fields by vacuum pressure and surface tension. Inside the bag the colored quarks and gluons are governed by the equations of quantum chromodynamics.

Goldstone boson related finite size effects in field theory and critical phenomena with O(N) symmetry

Abstract Chiral perturbation theory provides a systematic large volume expansion in powers of 1/ L d −2 , where L = V 1/ d is the size of the system and d > 2 is the dimension. Different observables, including the transversal and longitudinal two-point functions, are calculated up to order (1/ L d −2 ) 2 . In the scaling region the results can be used to control the finite-size effects in numerical simulations.

Renormalization group study of scalar field theories

An approximate RG equation is derived and studied in scalar quantum field theories in d dimensions. The approximation allows for an infinite number of different couplings in the potential, but excludes interactions containing derivatives. The resulting non-linear partial differential equation can be studied by simple means. Both the gaussian and the non-gaussian fixed points are discribed qualitatively correctly by the equation. The RG flows n d = 4 and the problem of defining an “effective” field theory are discussed in detail.

Prospects for perfect actions

Abstract The fixed-point (FP) action in QCD, although it is local and determined by classical equations, is difficult to parametrize well and is expensive to simulate. But the stake is high: the FP action has scale invariant instanton solutions, has no topological artifacts, satisfies the index theorem on the lattice, does not allow exceptional configurations, requires no tuning to get the pion massless and is expected to reduce the cut-off effects significantly. An overview is given including a discussion on tests in Yang-Mills theory, QCD and d = 2 spin and gauge models.

The exact mass gap of the O(3) and O(4) non-linear σ-models in d = 2☆

Abstract By comparing the Bethe Ansatz solution with perturbation theory we obtain the exact results m = 8 e Λ MS and m = ( 32 π e ) 1 2 Λ MS for the O(3) and O(4) non-linear σ-models, respectively. Here Λ MS is the renormalization group invariant scale in the MS scheme.

The scales of euclidean and hamiltonian lattice QCD

Abstract Our earlier result on ΛF.g.MOM/ΛEucl.latt. is confirmed by recalculating this ratio using the background field method. The relation between the scales of hamiltonian and euclidean SU(N) lattice gauge theory is also determined. We obtained Λ latt H Λ latt E =0.968 e −0·5495 N 2 = 0.91 N=3 0.84 N=2 . It is in strong disagreement with the numbers previously used in the literature. It is argued that the strong coupling expansions for the string tension should be carefully reanalyzed.