Dynamical supersymmetry breaking and gauge anomalies

Some aspects of supersymmetric gauge theories and discussed. It is shown that dynamical supersymmetry breaking does not occur in supersymmetric QED in higher dimensions. The cancellation of both local (perturbative) and global (non-perturbative) gauge anomalies are also discussed in supersymmetric gauge theories. We argue that there is no dynamical supersymmetry breaking in higher dimensions in any supersymmetric gauge theories free of gauge anomalies. It is also shown that for supersymmetric gauge theories in higher dimensions with a compact connected simple gauge group, when the local anomaly-free condition is satisfied, there can be at most a possibleZ ₂ global gauge anomaly in extended supersymmetricSO(10) (or spin (10)) gauge theories inD=10 dimensions containing additional Weyl fermions in a spinor representation ofSO(10) (or spin (10)). In four dimensions with local anomaly-free condition satisfied, the only possible global gauge anomalies in supersymmetric gauge theories areZ ₂ global gauge anomalies for extended supersymmetricSP(2N) (N=rank) gauge theories containing additional Weyl fermions in a representation ofSP(2N) with an odd 2nd-order Dynkin index.     

Flavour and superunification

We consider the possibility of a non-trivial embedding of $\text{(}\text{10}\text{+}\text{5}\text{)}\text{SU}\text{(5)}$ families of spin if$\text{1}\text{2}$ left-handed fermions in a combination of irreducible massless supermultiplets of N extended supersymmetry. We demand the whole spectrum of spin $\text{1}\text{2}$ states to be anomaly free with respect to SU(N). This turns out to be a necessary condition for the absence of anomalies at the SU(5) level. We find two classes of models, with spin $\text{1}\text{2}$ fermions in SU(N) representations associated to one- and two-column Young tableaux, respectively, in which each irreducible massless multiplet occurs at most once. These two classes of models lead to a nontrivial family generation due to supersymmetry. For N = 8 extended supersymmetry, they give at most three and five families, respectively. The first class of models is more natural in the way it excludes SU(5) exotics. The same analysis is extended to the massless multiplets that can be obtained from bilinear composite fields of the (preonic) elementary fields of N extended supergravity. We prove that the generation of families requires the repetition of massless multiplets and that ($\text{10 +}\text{5}$) SU(5) families can only be generated in pairs. General properties of multilinear composite operators of the preonic fields are given and the rôle of massive representations to classify towers of operators with definite spin is pointed out.     

Radial and Regge excitations in unified,grand unified and subconstituent models

Necessary group theoretic conditions for all elementary gauge bosons and fermions of an arbitrary renormalizable gauge theory to lie on Regge trajectories are reviewed. It is then argued that in properly unified gauge theories all particles of a given spin lie on Regge trajectories. This then implied that a properly unified gauge theory has no local U(1) factor groups, and no massive fermion singlets. A consideration of the general pattern of Regge and radial recurrences to be expected in quantum field theories suggests that the presence or absence of spin $\text{3}\text{2}$ quarks and/or leptons in the TeV region will provide crucial clues to enable one to distinguish between various classes of unified, grand unified, and subconstituent models. The correct interpretation of such excited fermions will require correlation with the Higgs boson mass and possible radial and Regge excitations of the weak vector bosons.     

The topological meaning of non-abelian anomalies

We show how the non-abelian anomaly for gauge fields coupled to Weyl fermions in 2n dimensions is related to the non-trivial topology of gauge orbit space. The form of the anomaly and its normalization are shown to follow from a familiar index theorem for a certain (2n + 2)-dimensional Dirac operator. We are thus able to recover and give topological meaning to a variety of results concerning anomalies in 4- and higher-dimensional theories.     

Axial and conformal anomalies for arbitrary spin in gravity and supergravity

We present a general algorithm for the construction of gravitational axial and conformal anomalies for fields of arbitrary spin. A variety of models is then displayed in which one or both of the anomalies vanish by cancellation. Our results are compared (and in the spin $\text{3}\text{2}$ case, contrasted) with previous calcultions.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

Anomaly constraints on nonlinear sigma models

Anomalies in nonlinear sigma models can sometimes be cancelled by local counterterms. We show that these counterterms have a simple topological interpretation, and that the requirements for anomaly cancellation can be easily understood in terms of 't Hooft's anomaly matching conditions. We exhibit the anomaly cancellation on homogeneous spaces $\text{G}\text{H}$ and on general riemannian manifolds $\text{M}$. We include external gauge fields on the manifolds and derive the generalized anomaly-cancellation conditions. Finally, we discuss the implications of this work for superstring theories.     

A puzzling combination: Disorder × order

There are several examples indicating that the operator combination “disorder × order” creates anomalous spin and/or statistics. In 2 + 1 dimensions there is a similar anomaly related to the disorder operator carrying Z(N) topological charge believed to have an important role in quark confinement. The conventionally defined total angular momentum has an anomalous term $\text{1}\text{2}\text{σ}{}^3\text{((}\text{1}\text{√3}\text{)λ}{}^8\text{)}$ and can take the values $\text{1}\text{2}\text{(}\text{1}\text{3}\text{)}$ in the presence of “disorder × order” in 2 + 1 dimensional SU(2) (SU(#)) gauge theories. It is argued that the existence of a stable soliton solution or the introduction of Higgs fields are not important for the anomaly.     

Dimensional regularization and the two-loop axial anomaly in abelian,non-abelian and supersymmetric gauge theories

The two-loop corrections to the axial anomaly are calculated for a non-abelian gauge theory with fermions using both conventional and supersymmetric dimensional regularization. In both cases we find results consistent with the Adler-Bardeen theorem if we use non-anticommuting γ⁵ of 't Hooft and Veltman. Expectations (based on the supermultiplet structure of the anomalies) that there exists in N = 1 supersymmetric Yang-Mills theory an axial current J⁵ such that $\text{?·J}{}^5\text{～ β(g)F}\text{F}\text{?}$ are discussed.     

Magnetic moments of composite fermions

There have been a considerable number of papers proposing composite models for leptons and quarks. Recently, Glück and Lipkin have stated that reproducing the observed magnetic moments of these fermions presents a serious difficulty for these composite models. We show for a renormalizable theory that, in contrast to Glück's and Lipkin's nonrelativistic arguments, a deeply bound system (with heavy constituent particle masses m_c) of (total) spin $\text{1}\text{2}$, charge e and mass m has the magnetic moment $\text{(e/2m) [1 +}\text{“usual” (QED + QCD + weak) corrections +O}\text{(m/m}{}_{\mathrm{<mtext>c</mtext>}}\text{)}\text{“new” bindng corrections]}$. Although there remains the considerable dynamical problem of obtaining “light” bound fermions from heavy constituents, there is no separate, additional magnetic moment difficulty.     

Symmetry breaking by instantons in supergravity

The non-zero modes of different spin operators on the background of a self-dual gravitational instanton are all related by global supersymmetry transformations and completely cancel in the one-loop term, which is determined entirely by the zero modes. We derive the number of zero modes of each spin. In an asymptotically locally Euclidean self-dual instanton there are 2τ $\text{spin}\text{-}\text{3}\text{2}$ zero modes and 3τ spin-2 zero modes, where τ is the Hirzebruch signature. Up to 3 of the spin-2 zero modes (depending on boundary conditions) may correspond to global rotations. The $\text{spin}\text{-}\text{3}\text{2}$ zero modes break the U(1) chiral symmetry and give rise to helicity-changing amplitudes. Together with the spin-2 zero modes they determine the trace anomaly or scaling behaviour. We can compare our results with the perturbation theory predictions for the axial vector current and trace anomalies in K3, the unique compact self-dual gravitational instanton, because in this case there are no boundary terms. We obtain agreement.     

Large q expansions for q-state gauge-matter Potts models in lagrangian form

We consider the lagrangian form of a q-state generalization of Ising gauge theories with matter fields in d = 3 and 4 dimensions. The theory is exactly soluble in the limit q → ∞ and corrections are easily calculable in power series in $\text{1}\text{q}{}^{\mathrm{<mtext>1</mtext><mtext>d</mtext>}}$. Extrapolating the series for the free energies and latent heats by the method of Padé approximants, we have constructed the phase diagrams for all values of q. Our results agree well with known results for pure spin systems and, for the case q = 2, with Ising Monte Carlo data.     

Threshold properties of electroproduction and annihilation

The threshold relation for deep-inelastic electroproduction that was suggested by Drell and Yan and by West is not a general property of softened field theories. However, for individual Feynman graphs of certain perturbation-expansion models it is valid in the form of an inequality. This is shown for every graph in φ³ theory, and for certain simple graphs when fields and particles with spin are considered, provided that the fields to which the electromagnetic current couples have the same spin as the target particle. The structure functions F₂(ω) and $\text{F}{}_2\text{(ω)}$ for electroproduction and electron-positron annihilation are both expected to behave like |ω ? 1 |^p at ω = 1, with the same power p for each but not necessarily the same coefficient. The behaviour of $\text{F}{}_2\text{(ω)}$ at ω = 0 is investigated and is found to result in finite multiplicity in the annihilation process at infinite energy.     

Soft spin 32 fermions require gravity and supersymmetry

If massless fermions of spin $\text{3}\text{2}$ have non-vanishing low-energy couplings, the fermions must have massless partners of spin 2, and all particles to which the fermions couple must display supersymmetry.     

Supersymmetry anomalies and some aspects of renormalization

We show how the $\text{L}$-matrix elements avoid the problem of supersymmetry breaking by the gauge fixing and ghost terms for renormalization in the Wess-Zumino gauge. Possible origins of supersymmetry anomalies are discussed. Gauge and gravitational anomalies induce a supersymmetry anomaly which has two distinct terms, one of which is gauge invariant. We give the expression for the noninvariant term for 2n-dimensional spacetime and for the invariant part in four dimensions. This anomaly, although cohomologically nontrivial, is still consistent with result that in superspace no supersymmetry anomaly is generated.     

Anomalies and odd dimensions

The parity-violating effective action for theories of fermions coupled to external gauge and gravitational fields in odd dimensions is computed exactly. This action is then used to compute gauge and gravitational anomalies in even dimensions. This derivation of the anomalies elucidates the relation of covariant to consistent anomalies as well as the relation between the Abelian anomaly and the non-Abelian anomaly in two lower dimensions.     

Absorption and dispersion of electromagnetic eigenwaves of electron-positron plasma in a strong magnetic field: H. Pérez Rojas and A. E. Shabad. Instituto de Investigación Técnica Fundamental,Havana, Cuba

Quantum theories of N-component Fermi fields are formulated in terms of overcomplete sets of bilocal pseudo-spin operators in analogy with the Holstein-Primakoff theory of ordinary spin systems. Classical constrained Hamiltonian systems arise in the large N limit revealing the semi-classical nature of the large N approximation. Such systems are briefly analyzed in the context of the Gross-Neveu $\text{(}\text{ψ}\text{ψ)}{}^2$ model, where the dynamics of c-number (rather than Grassman) fields governs the semi-classical limit.     

Path-integral formulation of chiral invariant fermion models in two dimensions

We study the Thirring and chiral-invariant Gross-Neveu (CGN) models using the functional integral method. By introducing an auxiliary vector field we disclose a relation with two-dimensional gauge theories coupled to fermions and then extend a technique based on a chiral change in the functional variables to study purely fermionic models.We obtain the exact Klaiber solution for the massless Thirring model (for spin $\text{1}\text{2}$) in a very simple way and we then extend our technique to investigate the CGN model. We show the factorization of a free fermionic part at the level of Green functions on very general grounds. We then impose certain restrictions on the behavior of the fields — which render our treatment exact only in the zero winding number sector, but allow the computation of the U(1) part of the CGN Green functions exactly, showing, in particular, its complete decoupling from the color part and the almost long-range order behaviour in the infrared region.In our approach, the non-triviality of the jacobian arising from the chiral transformation — directly related to the topological density and the axial anomaly — appears to be crucial for the functional integral treatment of these models.     

The possibility of new fermions with ΔI = 0 mass

In the Glashow-Weinberg-Salam model the fermions have $\text{ΔI =}\text{1}\text{2}$ masses from the breaking of the weak SU(2) gauge symmetry. In many enlarged models, such as those from grand unified and/or supersymmetric theories, there are additional fermions with undetermined ΔI = 0 (SU(2) invariant) masses. We study these new fermions. They induce flavour changing neutral currents. We show that the mixing angles of ΔI = 0 fermions of mass order M with normal $\text{ΔI =}\text{1}\text{2}$ fermions of mass order m ? M are order η or η², where η = m/M. If M > 150 δ, δ being a model-dependent mixing parameter of order a normal fermion mass, the amplitudes of all FCNC processes are below the experimental sizes and limits. Thus for δ?0.1 GeV, M can be as low as 20 GeV, close to the present experimental lower bound. δ is fixed, and less than 0.1 GeV for all relevant cases, if we assume the mass hierarchy of the known fermions is not the result of a particular choice of ΔI = 0 mass parameters. If produced ΔI = 0 mass fermions will be noticeable by the mass degeneracy within their isospin multiplets. There will be an enhanced ratio of neutral to charged decays over the normal fermions. Standard GUT predictions are changed little.