A superloop approach to supersymmetric gauge theories

A supersymmetric extension of the loop equations of motion in gauge theories is presented. The ansatz used for solving these superloop equations has the form of an average of the superloop product — supersymmetric path-dependent phase factor. The averaging action is determined by the superloop equations themselves. An example of the abelian spinor gauge superfield is considered in some detail.     

Comment to “No‐go result for supersymmetric gauge theories in the causal approach”, by D. R. Grigore and G. Scharf (previous article: Ann. Phys. (Berlin) 17, 864 (2008))

Fulfilling a request of the editor I try to put the aforementioned paper into perspective of relevant results obtained before. In particular I point out that supersymmetry of the S‐operator in cases where it exists is not necessarily in danger. Local couplings reveal however that some composite operators undergo a breaking of their supercovariance, a fact which has been established by Elisabeth Kraus.     

Non-perturbative aspects in supersymmetric gauge theories

We review our present understanding of those non-perturbative features of supersymmetric gauge theories that are believed to determine the properties of their ground states (vacua). A wide variety of theories is discussed in detail: pure Yang-Mills, theories with massive or massless real matter fields, theories with chiral matter, both for SU(N) and for the case of a general compact gauge group (as for instance E₈). Depending on some general features of the theory under consideration, various (perhaps) unexpected phenomena are shown to occur. Among these the breakdown of the (perturbatively established) non-renormalization theorem, the occurrence of runaway vacua in certain limits, the spontaneous dynamical breaking of supersymmetry itself in some chiral theories. Throughout the report we restrict ourselves to the confining picture instanton method, occasionally complementing it with the information coming from chiral and supersymmetric Ward-Takahashi identities. We compare our results with the ones suggested earlier by effective Lagrangian methods and, only briefly, with those obtained by other groups in the Higgs picture.     

Instanton effects in supersymmetric gauge theories

We investigate how in supersymmetric gauge theories non-perturbative effects are able to generate non-trivial vacuum properties otherwise forbidden by perturbative non-renormalization theorems. This conclusion can be reliably drawn since the constancy of certain Green functions — due to supersymmetry (SUSY) — allows one to connect vacuum-dominated large distances with short-distance behaviour which is reliably computed by instanton methods. In all the cases we discuss (without matter, with massive or massless matter in real representations and, finally, with matter in complex representations) instanton calculations imply the occurrence of a variety of condensates. For the pure SUSY gauge theory, a gluino condensate induces the spontaneous breaking of Z_2N. For massive super-quantum chromodynamics (SQCD) we find a peculiar mass dependence of matter condensates whose origin is traced to mass singularities of non-zero mode instanton contributions. These contributions force the massless limit of SQCD to differ from the strictly massless case, in which the spontaneous breaking of chiral symmetries is induced. Inconsistency with an anomaly equation forces either infinite matter condensates or spontaneous SUSY breaking in the massless cases. For non-constant Green functions, instantons are shown to provide new calculable short-distance singularities of an obvious non-perturbative nature.     

Vacuum alignment in supersymmetric gauge theories

The vacuum alignment problem is analyzed in the context of supersymmetric gauge theories with dynamical symmetry breaking. Three cases are distinguished, depending on whether the vacuum expectation value of the weak gauge current superfield in vacuum characterized by the orientation Ω is zero for all Ω, for some Ω, or for no Ω. In the first case, a non-renormalization theorem is proved to all orders in the weak coupling, showing that the usual criterion of minimizing the vacuum energy density is insufficient to determine the alignment, and possible resolutions of the problem are discussed. The second case is similar, except that the possible alignments are resricted to the range of Ω giving a vanishing VEV and the weak gauge group may then be broken non-minimally. In the third case, supersymmetry is itself broken at the tree level by the weak interactions. The supersymmetric generalization of the Schwinger mechanism for dynamical mass generation is described.     

Superpropagators for broken supersymmetric abelian gauge theories

By considering an arbitrary globally supersymmetric abelian gauge theory, the most general shifts on the matter and gauge superfields are performed, the superpropagators are derived and employed for discussing the structure of the terms generated into the effective action.     

Superspace Ward identities in supersymmetric gauge theories

In superspace formulation of supersymmetric gauge theories, gauge invariance requires an infinite set of identities between the infinite set of renormalization constants. Using Ward identities in superspace, the same is derived. These identities at one loop level are also demonstrated.     

Anomaly matching conditions in supersymmetric gauge theories

Sufficient conditions are proven for 't Hooft's consistency conditions to hold at points in the moduli space of supersymmetric gauge theories. Known results for anomaly matching in supersymmetric QCD are rederived as a sample application of the results. The results can be used to show that the anomaly matching conditions hold for s-confining theories.     

A natural framework for the minimal supersymmetric gauge theories

We show that the sequence of Jordan algebras M _inf3 ^sup1 , M _inf3 ^sup2 , M _inf3 ^sup4 , and M _inf3 ^sup8 , whose elements are in the 3×3 Hermitean matrices over , , , and O, respectively, provide an elegant and natural framework in which to describe supersymmetric gauge theories. The four minimal supersymmetric gauge theories are in a one-to-one correspondence with these four Jordan algebras and, hence, with the four division algebras.     

The beta function in supersymmetric gauge theories. Instantons versus traditional approach

Within the background field formalism we discuss vacuum loops in supersymmetric gauge theories. A direct connection is revealed between the absence (or presence) of high-order contributions and infrared regularization. A simple explanation is given why the instanton amplitude is exhausted by one loop whilst in the standard supergraph technique the effective action contains terms of all orders in the coupling constant. We present an exact relation between the Gell-Mann-Low function and the anomalous dimensions of matter superfields stemming from instanton calculus.     

Nonlattice simulation for supersymmetric gauge theories in one dimension

Lattice simulation of supersymmetric gauge theories is not straightforward. In some cases the lack of manifest supersymmetry just necessitates cumbersome fine-tuning, but in the worse cases the chiral and/or Majorana nature of fermions makes it difficult to even formulate an appropriate lattice theory. We propose circumventing all these problems inherent in the lattice approach by adopting a nonlattice approach for one-dimensional supersymmetric gauge theories, which are important in the string or M theory context. In particular, our method can be used to investigate the gauge-gravity duality from first principles, and to simulate M theory based on the matrix theory conjecture.     

Remark on the consistent gauge anomaly in supersymmetric theories

We present a direct field theoretical calculation of the consistent gauge anomaly in the superfield formalism, on the basis of a definition of the effective action through the covariant gauge current. The scheme is conceptually and technically simple and the gauge covariance in intermediate steps reduces calculational labors considerably. The resultant superfield anomaly, being proportional to the anomaly d^abc=trT^a{T^b,T^c}, is minimal without supplementing any counterterms. Our anomaly coincides with the anomaly obtained by Marinković as the solution of the Wess–Zumino consistency condition.     

Patterns of symmetry breaking in supersymmetric gauge theories

We give a group-theoretical analysis of the spontaneous breaking of the gauge group in supersymmetric Yang-Mills theories with unbroken supersymmetry.     

Resurgent analysis of localizable observables in supersymmetric gauge theories

Journal of High Energy Physics - The coset Sp(2, ℝ)/U(1) is parametrized by two real scalar fields. We generalize the formalism of auxiliary tensor (bispinor) fields in U(1) self-dual...