Bare Yukawa Coupling Research Articles

Summary of Results inN = 1 SupersymmetricSU(2) Gauge Theories

We summarize some results in 4d, N = 1 supersymmetric SU(2) gauge theories: the exact effective superpotentials, the vacuum structure, and the exact effective Abelian couplings for arbitrary bare masses and Yukawa couplings.

From effective Lagrangians, to chiral bags, to Skyrmions with the large-Nc renormalization group.

We explicitly relate effective meson-baryon Lagrangian models, chiral bags, and Skyrmions in the following way. First, effective Lagrangians are constructed in a manner consistent with an underlying large-${\mathit{N}}_{\mathit{c}}$ QCD. An infinite set of graphs dress the bare Yukawa couplings at leading order in 1/${\mathit{N}}_{\mathit{c}}$, and are summed using semiclassical techniques. What emerges is a picture of the large-${\mathit{N}}_{\mathit{c}}$ baryon reminiscent of the chiral bag: hedgehog pions for r\ensuremath{\ge}${\mathrm{\ensuremath{\Lambda}}}^{\mathrm{\ensuremath{-}}1}$ patched onto bare nucleon degrees of freedom for r\ensuremath{\le}${\mathrm{\ensuremath{\Lambda}}}^{\mathrm{\ensuremath{-}}1}$, where the ``bag radius'' ${\mathrm{\ensuremath{\Lambda}}}^{\mathrm{\ensuremath{-}}1}$ is the UV cutoff on the graphs. Next, a novel renormalization group (RG) is derived in which the bare Yukawa couplings, baryon masses, and hyperfine baryon mass splittings run with \ensuremath{\Lambda}. Finally, this RG flow is shown to act as a filter on the renormalized Lagrangian parameters: when they are fine-tuned to obey Skyrme-model relations the continuum limit \ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty} exists and is, in fact, a Skyrme model; otherwise there is no continuum limit.

Strong-coupling expansions for chiral models of electroweak symmetry breaking

We consider chiral U( N) × U( N) models with fermions in the limit of infinitely large local bare Yukawa coupling. When the scalar field is subject to non-linear constraint, phase transitions in these models are seen to be identical to those in the corresponding purely bosonic ones. Relaxing the non-linear constraint, we compute the seventh-order strong-coupling series for the susceptibility in these models and analyze them numerically for the U(2) × U(2) case. We find that in four dimensions the approach to the phase transition follows to a good accuracy the mean-field critical behavior, indicating the absence of non-trivial fixed points at strong coupling and being consistent with the first-order nature of the transition. In three dimensions, the strongly-coupled bosonic U(2) × U(2) model (without gauge fields) has a first-order transition strong enough to accommodate electroweak baryogenesis only for a narrow region of the bare parameter space.

Numerical simulations of an SU(2)L ⊗ SU(2)R-symmetric Higgs-Yukawa model on the QUADRICS Q16

We report on our work on the SU(2)_L x SU(2)_R symmetric Higgs Yukawa Model with mirror fermion action. Our model describes a fermion Higgs system in the limit of vanishing gauge coupling. Setting the bare Yukawa coupling of the mirror fermions G_\chi to zero, we want to determine the triviality bounds on the renormalized Yukawa coupling of the fermions G_{R\psi} and the scalar self-coupling g_R on 8^3x16 and 16^3x32 lattices.

Solvability, consistency, and the renormalization group in large-Nc models of hadrons.

We establish the following fundamentals about Lagrangian models of meson-baryon interactions in the large-$N_{c}$ limit: 1. Calculating the leading-order contribution to $1$-meson/$2$-baryon Green's functions in the $1/N_c$ expansion involves summing an infinite class of divergent Feynman diagrams. So long as the bare Lagrangian properly obeys all large-$N_c$ selection rules, this all-loops resummation is accomplished exactly by solving coupled classical field equations with a short-distance cutoff. 2. The only effect of the resummation is to renormalize the bare Yukawa couplings, baryon masses and hyperfine baryon mass splittings of the model. 3. In the process, the large-$N_{c}$ renormalization group flow of these bare parameters is completely determined. We conjecture that variants of the Skyrme model emerge as UV fixed points of such flows.

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

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 + 4 w 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 y R 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.

Scalar sector of an SU(2)L ⊗ SU(2)R Yukawa model

At intermediate values of the bare Yukawa coupling y , where the SSB is caused by the Yukawa interaction, the scalar propagators show very complex structures which, however, we can explain by the vicinity of phases with antiferromagnetic ordering and by the feedback of the fermion loops. We develop methods for extracting the wave function renormalization constant Z π and the scalar mass m σ . We present first results on the influence of fermions on m σ . Our results on lattices from 6 3 16 up to 10 3 16 allow a preliminary discussion of the finite-size effects.

Properties of the hypercharge sector of the standard model on the lattice

Using a hopping parameter expansion, we demonstrate the decoupling of fermion doubler modes in the hypercharge U(1) Y lattice gauge theory. We also show analytically that it is possible to tune the bare Yukawa coupling to a value such that the fundamental physical fermion am f=0 (as is necessary in order to obtain a finite physical mass m f in the continuum limit), and we calculate this value. The appropriate special case of these results is compared with numerical simulations of the frozen-gauge limit of the U(1) Y theory, and good agreement is found. The role of the Yukawa coupling in a chiral lattice gauge theory is contrasted with that in a nonchiral Yukawa model.

A U(1) L ⊗ U(1) R symmetric Yukawa model in the phase with spontaneously broken symmetry

The lattice regularized U(1) L ⊗ U(1) R symmetric scalar-fermion model with a mirror pair of fermion fields is investigated in the phase with spontaneously broken symmetry. The reflection positivity condition, ensuring unitarity in Minkowski space, is proven in this lattice formulation in a wide range of bare parameters. Numerical Monte Carlo calculations with dynamical fermions are performed on 4 3 × 8, 4 3 × 16 and 6 3 × 16 lattices at moderately strong bare Yukawa couplings and at very small and infinite bare quartic scalar coupling. It is shown that not only the lattice fermion doublers but also the mirror fermion can be made heavy, and therefore the light fermion spectrum is “chiral” as in the standard model. The “vacuum stability” bound on the Higgs-boson mass is discussed in lattice perturbation theory and non-perturbatively. It implies that for heavy mirror fermions the Higgs-boson mass has to be close to the non-perturbative upper limit.

Fermion propagator in lattice chiral gauge theories: Analytic results

Using a hopping parameter expansion, we demonstrate the decoupling of fermion doubler modes in lattice chiral gauge theories, where both the right-handed and left-handed fermions may be non-singlets. The decoupling of the doublers for the case of a single right-handed fermion was previously first shown by Aoki, Lee, and Xue. We also show analytically that it is possible to tune the bare Yukawa coupling to a value such that the fundamental physical fermion mass is zero in lattice units.

U(1) L ⊗ U(1) R symmetric yukawa model in the symmetric phase

The lattice regularized U(1) L ⊗ U(1) R symmetric scalar fermion model with explicit mirror fermions is investigated in the phase with unbroken symmetry. As a first part of a non-perturbative investigation, in the present work numerical Monte Carlo calculations with dynamical fermions are performed on 4 3 × 8, 6 3 × 12 and 8 3 × 16 lattices near the expected perturbative gaussian fixed point, in order to study the cutoff-dependent upper limit on the renormalized scalar quartic- and Yukawa-couplings. The bare Yukawa coupling of the mirror fermion is fixed at zero, which is near the region of parameter space where in the broken phase decoupling of the mirror fermion is expected. Lattice perturbation theory and fermion hopping parameter expansion is exploited for supporting the numerical simulations.

Results from a strong-coupling expansion for a lattice Yukawa model with a real scalar field.

Results are presented from a strong bare Yukawa coupling expansion for a lattice Yukawa theory with a real scalar field. It is found the effective action involves competing interactions, consistent with the existence of a ferrimagnetic phase at intermediate Yukawa coupling $y$. We also give evidence that the (bosonic) continuum theory defined at the ferromagnetic-paramagnetic phase boundary at large $y$ is free.

Random walk approximation in a chiral Yukawa-model and global symmetries

The fermion propagator is investigated in a chiral Yukawa-model with explicit mirror fermions applying the random walk approximation to the hopping parameter expansion. It is shown that the globalSU(2)L⊗SU(2)R symmetry breaking due to the mass splitting within fermion doublets does influence the critical behaviour of the fermion spectrum in the continuum limit. In particular, in the case of a mirror pair of split doublets, whereSU(2)L⊕SU(2)R is broken toSU(2)L, no evidence is found for a dynamical spectrum doubling at infinitely strong bare Yukawa-couplings, in contrast to the case with degenerate doublets andSU(2)L⊗SU(2)R symmetry.

Study of different lattice formulations of a Yukawa model with a real scalar field

We determine the phase structure and properties of a 4D lattice model with a real scalar field interacting with staggered fermions via a local Yukawa interaction. For large bare Yukawa coupling, the phase diagram differs significantly from that for the model with a hypercubic Yukawa interaction; in particular, we find a new symmetric phase in which fermions are massive. We comment on implications for extracting regularization scheme-independent continuum physics from lattice scalar-fermion models.

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

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.

Study of the φ4-model with Yukawa coupling and dynamical fermions

We study the one-component φ 4-model with Yukawa coupling and dynamical fermions. The unbiased Hybrid Monte Carlo algorithm is applied to this model for the first time. Our results show that inside the broken symmetry phase all the measured quantities change continuously when the bare Yukawa coupling Y is increased and the fermion sector behaves approximately like a free fermion with mass m f related to the scalar field vacuum expectation value 〈 φ〉 by m f = Y 〈 φ〉 even for large Y. We do not find evidence for a phase transition induced by large Y inside the broken symmetry phase. We also present a method to simulate the effective potential in the presence of dynamical fermions. For small Y, we demonstrate that the measured effective potential agrees well with lowest order perturbative calculation in Y.

The sigma-model with Wilson lattice fermions

The σ-model with Wilson fermions is considered in one-loop lattice perturbation theory and in the hopping-parameter expansion at large bare couplings. Chiral-symmetry restoration in the large cut-off limit of perturbation theory is only possible if asymmetric counterterms are added to the lattice action. In the hopping-parameter expansion at infinitely large bare Yukawa coupling, dynamical parity doubling of the fermion occurs.

Equivalence of the four-point interaction and the Yukawa interaction. I. A boson model

It is shown that the compositeness conditions by themselves do not ensure the complete equivalence of the Yukawa-type interaction with the corresponding four-point interaction. In our study of the corresponding situation in the context of the Lee model versus the separable-potential model, we find in the limit in which the bare Yukawa coupling constant becomes infinite, the finite-energy wave functions calculated from the two theories coincide. It is shown, however, that although the finite-energy spectra of the two theories are the same, the Yukawa theory has additional spectral contributions at infinity. It is further demonstrated that it is only by removing all contributions from the spectrum at infinity that the two theories can be made essentially identical. It is therefore suggested that the proofs of the equivalence of the four-fermion interaction and the Yukawa-type interaction need to be reexamined.

Equivalence of the four-point interaction and the Yukawa interaction. II. A fermion model

It is shown that the compositeness conditions by themselves do not ensure the complete equivalence of the Yukawa-type interaction with the corresponding four-point interaction. In our study of the corresponding situation in the context of the Lee model versus the separable-potential model, we find in the limit in which the bare Yukawa coupling constant becomes infinite, the finite-energy wave functions calculated from the two theories coincide. It is shown, however, that although the finite-energy spectra of the two theories are the same, the Yukawa theory has additional spectral contributions at infinity. It is further demonstrated that it is only by removing all contributions from the spectrum at infinity that the two theories can be made essentially identical. It is therefore suggested that the proofs of the equivalence of the four-fermion interaction and the Yukawa-type interaction need to be reexamined.