Wolfgang Bock

Phase diagram of a lattice SU(2) ⊗ SU(2) scalar-fermion model with naive and Wilson fermions

Abstract We present the phase structure of the chiral SU(2) L ⊗ SU(2) R scalar-fermion model on the lattice with on-site Yukawa coupling y and Wilson-Yukawa coupling w for positive y and w . The hopping parameter κ of the four-component scalar field of fixed length is both positive and negative. From the different behaviour of several observables ferromagnetic, paramagnetic and antiferromagnetic phases can be distinguished. They split into different regions or phases with small and large y + 4 w . A similar structure is also found in the quenched approximation. In addition, in the unquenched case a ferrimagnetic phase is found at negative κ around y + 4 w ≈ √2. We discuss fermion masses in various regions and point out the possibilities of decoupling the unwanted fermion doublers in the continuum limit in analogy to the Wilson mechanism.

Decoupling Doublers of Chiral Lattice Fermions in a Quenched Fermion Higgs Model

Abstract We study in the quenched approximation the fermion-scalar sector of a lattice version of the standard model in the broken symmetry phase using a Wilson-Yukawa coupling to deal with the fermion doubling. For relatively strong but fixed Wilson-Yukawa coupling the fermion doublers can be easily decoupled as the critical point is approached by giving them masses of the order of the cutoff.

Unquenched investigation of fermion masses in a chiral fermion theory on the lattice

Abstract Using dynamical fermions we study the decoupling of the fermion doublers of chiral lattice fermions in the broken symmetry phase of a scalar-fermion model. The model uses a chiral invariant Wilson term, called a Wilson-Yukawa term with coupling strength w , in addition to the usual Yukawa coupling y . We find qualitative agreement with our results obtained before in the quenched approximation. First in the w = 0 case we establish a strong Yukawa coupling region. Then for relatively large but fixed w we confirm that the fermion doublers can be easily decoupled by giving them masses of the order of the cutoff as the symmetry restoring transition is approached.

Fermion-Higgs model with reduced staggered fermions

Abstract We introduce a lattice fermion-Higgs model with one component ‘reduced staggered’ fermions. In order to use the fermion field as efficiently as possible we couple the two staggered flavors to the O(4) Higgs field leading to a model with only one SU(2) doublet in the scaling region. The number of fermions is doubled in a numerical investigation of the model with the hybrid Monte Carlo algorithm. We present results for the phase diagram, particle masses and renormalized couplings on lattices ranging in size from 6 3 24 to 16 3 24.

Search for an upper bound of the renormalized Yukawa coupling in a lattice fermion-Higgs model ☆

Abstract We study the scaling laws for the fermion mass and the scalar field expectation value in the weak coupling region of the broken phase of a lattice regularized chiral-invariant SU(2)L ⊗ SU(2)R fermion-Higgs model with bare Yukawa coupling y and Wilson-Yukawa coupling w. In particular we concentrate on the region in the vicinity of the line A, which is the line of maximal values of y + 4w on the critical surface containing the gaussian fixed point. We have not found any indication for the existence of a nontrivial fixed point on that line or anywhere else in the weak coupling region. The renormalized Yukawa coupling yR as a function of the fermionic correlation length appears to be bounded from above. The upper bound obtained from the numerical data at w = 0 is compatible with the perturbative unitarity bound. Furthermore, in the weak coupling region, including the line A, it is not possible to choose w such that the unwanted fermion doublers would be removed from the physical particle pectrum.

Non-perturbative study of the Yukawa coupling in an SU(2)⊗SU(2) model with quenched naive lattice fermions

Abstract We initiate a non-perturbative study of Yukawa coupling in an SU(2) L ⊗SU(2) R invariant scalar-fermion lattice model. As a first step we calculate fermion masses in the broken phase using naive fermions in the quenched approximation. At small Yukawa coupling, taking into account appreciable finite-size effects, our results are consistent with perturbation theory. At large bare Yukawa coupling we find the fermion mass and the renormalized Yukawa coupling growing with increasing scalar correlation length, indicating a non-perturbative behaviour.

Fermion masses at strong Wilson-Yukawa coupling in the symmetric phase☆

Abstract The fermion mass spectrum is studied in the strong-coupling symmetric (PMS) phase of a scalar-fermion model derived from a lattice proposal of the standard model with Wilson-Yukuwa coupling. We find that the fermions are massive and argue for the following interpretation of our numerical results in this PMS phase. The lowest lying fermion spectrum in this phase consists only of a Dirac fermion which is a singlet with respect to the SU(2) L group (the “neutral” fermion). We give evidence that the SU(2) L doublet fermion (the “charged” fermion), in terms of which the model is originally formulated and which is present in the weak-coupling symmetric phase, is absent in the PMS phase. The neutral fermion is only weakly coupled to the scalar particles and appears to become non-interacting in the scaling region. If this interpretation is correct, the Wilson-Yukawa approach does not lead to a satisfactory lattice formulation of the standard model.

Can the couplings in the fermion-Higgs sector of the standard model be strong?

Abstract We present results for the renormalized quartic self-coupling λ R and the Yukawa coupling y R in a lattice fermion-Higgs model with two SU(2) L doublets, mostly for large values of the bare couplings. One-component (“reduced”) staggered fermions are used in a numerical simulation with the Hybrid Monte Carlo algorithm. The fermion and Higgs masses and the renormalized scalar field expectation value are computed on L 3 24 lattices where L ranges from 6 to 16. In the scaling region these quantities are found to have a 1 L 2 dependence, which is used to determine their values in the infinite-volume limit. We then calculate the y R and λ R from their tree-level definitions in terms of the masses and renormalized scalar field expectation value, extrapolated to infinite volume. The scalar field propagators can be described momenta up to the cut-off by one-fermion-loop renormalized perturbation theory and the results for λ R and y R come out to be close to the tree-level unitarity bounds. There are no signs that are in contradiction with the triviality of the Yukawa and quartic self-coupling.

Spontaneous symmetry breaking on the lattice generated by Yukawa interaction

We study by numerical simulation a lattice Yukawa model with naive fermions at intermediate values of the Yukawa coupling constant y when the nearest-neighbour coupling κ of the scalar field Φ is very weakly ferromagnetic (κ ≈ 0) or even antiferromagnetic (κ < 0) and the non-vanishing value of 〈Φ〉 is generated by the Yukawa interaction. The renormalized Yukawa coupling yR achieves here its maximal value and this y-region is thus of particular importance for lattice investigations of strong Yukawa interaction. However, here the scalar field propagators have a very complex structure caused by fermion loop corrections and by the proximity of phases with antiferromagnetic properties. We develop methods for analyzing these propagators and for extracting the physical observables. We find that going into the negative κ region, the scalar field renormalization constant becomes small and yR does not see, to exceed the unitarity bound, making the existence of a non-trivial fixed point in the investigated Yukawa model quite unlikely.

The eigenvalue spectra in Z(2) ⊗ Z(2) and SU(2) ⊗ SU(2) fermion-Higgs models

We present an analysis of the eigenvalue spectra of the Dirac operator M and related operators M′ and M†M in an SU(2) ⊗ SU(2) and a Z(2) ⊗ Z(2) fermion-Higgs model in the quenched approximation. We especially study the spectra in the symmetric and broken symmetry phases along the crossover line which was recently discovered in these models in numerical simulations. It turns out that in the symmetric phases of both models zero modes emerge along the crossover line. In the case of the Z(2) model the zero modes follow the crossover line into the broken symmetry phase whereas in the case of the SU(2) model they appear to stop at the phase transition line between the symmetric and the broken phases.