The electroweak mixing angle in unified gauge theories—the renormalisation effects

A general analysis of the renormalisation corrections to the unification results for the coupling constants of strong and electroweak interactions is attempted. In particular, the effects of introducing an energy scale intermediate between the unification energy and the low-energy regions are studied and found to be important. This analysis is applied to unification schemes of both kinds, namely, unification at superhigh energies, and unification at accessible energies.     

General electroweak mixing models and extended gauge theories

General electroweak mixing schemes containing an arbitrary number of weak bosons and agreeing with the predictions of the standard model for neutrino scattering and polarized electron scattering in the low energy approximation are analyzed using the propagator matrix formalism. The mean charged boson mass is bounded while the mean neutral boson mass is unbounded and determined as a function of the mean charged one. Under a special assumption on the electroweak mixing parameters the four fermion interaction agrees with the one of extended gauge theories and the mean boson masses agree with the boson masses of the standard model.     

Group-Theoretic analysis of the mixing angle in the electroweak gauge group

In this paper we provide strong mathematical support for the idea that the experimentally measured magnitude 1 –M _W ² /M _Z ² associated with sin² _W in the standard model of electroweak interactions cannot be simultaneously identified with the squared quotient of the electric charge by theSU(2) charge,e ²/g ². In fact, the natural, mathematical requirement that the Weinberg rotation between the gauge fields associated with the third component of the weak isospin (T ₃) and the hypercharge (Y) proceeds from aglobal Lie-group homomorphism of theSU(2) U(l) _Y gauge group in some locally isomorphic Lie group [which then proves to beU(2)], and not from alocal (Lie algebra)isomorphism, imposes strong restrictions so as to fix the single valuee ²/g ²=1/2. The two definitions of sin² _W can only be identified in the asymptotic limit corresponding to an earlier stage of the universe before the spontaneous symmetry breaking had taken place.     

Corrections to the mixing angle in the grand unified theories

The contribution of the two-loop corrections to the mixing angle and the lifetime of the proton are studied in grand unified theories. The corrections to the mixing angle are small giving numerical values that are uncomfortably smaller than the average Weinberg angle. When the mixing angle is chosen closest to the measured value, then the predicted lifetime is in the range of the experimental bounds.     

γ-W0 mixing as an alternative to unified weak-electromagnetic gauge theories

A very simple model based on γ-W⁰ mixing (but not on spontaneously broken gauge symmetry) is shown to reproduce all the quantitative predictions of the standard (Weinberg-Salam) model as far as low-energy phenomenology is concerned. The model predicts intermediate-boson mass relations weaker than Weinberg's. Weinberg's mass formulae, however, can be accommodated in our model provided the γ-W⁰ mixing parameter and the coupling constants (e and g) satisfy a special relation. This relation also guarantees asymptotic SU(2) × U(1), decent high-energy behavior in processes like $\text{v +}\text{v}\text{→}\text{W}{}^{\mathrm{+}}\text{+}\text{W}{}^{\mathrm{?}}$ and the SU(2) × U(1) value for the W^± boson gyromagnetic ratio. Speculative attempts are made to understand the sign and magnitude of “sin² θ_W”.     

Monopole charges in unified gauge theories

Monopole charges, being global quantities, depend on the gauge group of a theory, which in turn is determined by the representations of all its fields. For example, chromodynamics in its present form when combined with electrodynamics has as its gauge group not SU(3) × U(1) but a “smaller” group U(3). The specification of monopole charges for a theory can thus be quite intricate. We report here the result of an investigation in several current gauge theories. Of particular interest is the possible existence in some theories of monopoles carrying multiplicative charges. As a by-product, we clarify some earlier assertions in the literature which seem to us incorrect.     

Sum rules fore + e − annihilation in general electroweak mixing models and extended gauge theories

We derive a lower bound for the integrated cross section for production of weak neutral bosons ine ⁺ e ^? annihilation within general electroweak theories containing an arbitrary number of weak bosons. The low energy parameterC of such theories is expressed in terms of integrals over the weighted cross section fore ⁺ e ^? annihilation at high energies.     

Renormalization group constraints on unified gauge theories

Renormalization group constraints on the behavior of Yukawa and scalar quartic couplings in unified gauge theories are examined. Yukawa couplings are generally asymptotically free whenever the gauge couplings are, but scalar quartic couplings can be asymptotically free only for simple scalar multiplets in large groups with large fermion content. The infrared behavior of Yukawa and scalar quartic couplings implied by the renormalization group equations has interesting and phenomenologically useful consequences: infrared fixed points (or quasifixed points) lead to bounds on masses of fermions and scalars, while scalar quartic couplings can be driven out of the domain of positivity of the classical potential, with possible implications for patterns of symmetry breaking.     

Asymptotic freedom constraints on grand unified gauge theories

Constraints on the fermion and Higgs scalar content of grand unified gauge theories, imposed by the requirement of asymptotic freedom for the gauge couplings, are derived for models which have fermion representations with only color singlets and color triplets. The constraintn _f ≦16 on the numbern _f of flavors of color triplet quarks in pure QCD is removed. Definitive limits are placed on the representation content of theories based on the exceptional groups.     

Exact magnetic monopole solutions in Yang-Mills and unified gauge theories

We study the magnetic monopoles in non-Abelian gauge theories. The exact static, spherically symmetric solutions of the magnetic monopoles in both Yang-Mills and unified gauge theories are obtained. The energyE of the static system is calculable and it is either zero or infinite. The existence of the magnetic monopole solution is a consequence of symmetry rather than dynamics. We propose a new definition of the electromagnetic field tensor, which relates the static solution of gauge fields and the magnetic monopole solution. Experimental implications are discussed.     

The electroweak standard model in the axial gauge

We derive the Feynman rules of the standard model in the axial gauge. After this we prove that the fields and do not correspond to physical particles. As a consequence, these fields cannot appear as incoming or outgoing lines in Feynman graphs. We then calculate the contribution of these fields in the case of a particular decay mode of the top quark.Received: 28 January 2004, Revised: 12 February 2004, Published online: 8 April 2004     

Generalized local gauge symmetry and the Ward-Takahashi identities in unified field theories

We discuss the symmetry basis of unified field theories, i.e., the generalized concept of local gauge symmetry, and its physical implications. The generalized Ward-Takahashi identities and the explicit constraints among renormalization constants are derived by using the path integral in a specific model. These constraints are confirmed at the one-loop level.Work supported in part by the U.S. Energy Research and Development Administration.     

Grand unified theory precursors and nontrivial fixed points in higher-dimensional gauge theories

Within the context of traditional logarithmic grand unification at M(GUT) approximately equal to 10(16) GeV, we show that it is nevertheless possible to observe certain GUT states such as X and Y gauge bosons at lower scales, perhaps even in the TeV range. We refer to such states as "GUT precursors." These states offer an interesting alternative possibility for new physics at the TeV scale, and could be used to directly probe GUT physics even though the scale of gauge coupling unification remains high. Our results also give rise to a Kaluza-Klein realization of nontrivial fixed points in higher-dimensional gauge theories.     

A precise determination of the electroweak mixing angle from semileptonic neutrino scattering

The cross-section ratio of neutral-current and charged-current semileptonic interactions of muon-neutrinos on isoscalar nuclei has been measured with the result:R ^v =0.3093±0.0031 for hadronic energy larger than 4 GeV. From this ratio we determined the electroweak mixing angle sin² θ _W , wherem _c is the charm-quark mass in GeV/c². Comparison with direct measurements ofm _w andm _z determines the radiative shift of the intermediate boson mass Δr=0.077±0.025(exp.)±0.038(syst.), in agreement with the prediction. Assuming the validity of the electroweak standard theory we determined ?=0.990?0.013(m _c ?1.5)±0.009(exp.)±0.003(theor.).     

Comparison of grand unified theories with electroweak and strong coupling constants measured at LEP

Using the renormalization group equations one can evolve the electroweak and strong coupling constants, as measured at LEP, to higher energies in order to test the ideas of grand unified theories, which predict that the three coupling constants become equal at a single unification point. With data from the DELPHI Collaboration we find that in the minimal non-supersymmetric standard model with one Higgs doublet a single unification point is excluded by more than 7 standard deviations. In contrast, the minimal supersymmetric standard model leads to good agreement with a single unification scale of 10^16.0±0.3 GeV. Such a large scale is compatible with the present lower limits on the proton lifetime. The best fit is obtained for a SUSY scale around 1000 GeV and limits are derived as function of the strong coupling constant. The unification point is sensitive to the number of Higgs doublets and only the minimal SUSY model with two Higgs doublets is compatible with GUT unification, if one takes the present limits on the proton lifetime into account.     

Electromagnetic mass shift of pions in unified gauge field theories and the light-cone

The electromagnetic mass shift of pions is discussed in unified gauge field theories of weak and electromagnetic interactions to second order of the electric charge and the pion mass, respectively. We calculate the contributions of the next to leading light-cone singularities (quark mass terms) and it is found that these do not contribute to the divergences in the mass shift provided the masses of the bare proton quark and neutron quark are equal. γ, Z, W, φ exchanges are taken into account. Different bare quark masses in general make the mass shift divergent. In the tadpole contribution, the Born term causes a divergence unless we extrapolate first to q² = 0.     

The role of spurions in Higgs-less electroweak effective theories

Inspired by recent developments of moose models, we reconsider low-energy effective theories of Goldstone bosons, gauge fields and chiral fermions applied to low-energy QCD and to Higgs-less electroweak symmetry breaking. Couplings and the corresponding reduction of symmetry are introduced via constraints enforced by a set of non-propagating covariantly constant spurion fields. Relics of the latter are used as small expansion parameters conjointly with the usual low-energy expansion. Certain couplings can only appear at higher orders of the spurion expansion and, consequently, they become naturally suppressed independently of the idea of dimensional deconstruction. At leading order this leads to a set of generalized Weinberg sum rules and to the suppression of non-standard couplings to fermions in Higgs-less EWSB models with the minimal particle content. Within the latter, higher spurion terms allow for a fermion mass matrix with the standard CKM structure and C P violation. In addition, Majorana masses for neutrinos are possible. Examples of non-minimal models are briefly mentioned.Received: 8 January 2004, Revised: 7 February 2004, Published online: 2 April 2004