Instantons in Gauge Theories

The following sections are included:

• PREFACE

• CONTENTS

Since the early seventies, non-Abelian gauge theories have played a major role in the theory of fundamental interactions. With their roots in quantum electrodynamics (QED) they inherited some features of the latter, such as gauge invariance, for instance. However, their similarity to QED is not the reason why they are so cherished…

Instantons have many faces. Historically they were found as topologically nontrivial solutions of the duality equations of the Euclidean Yang–Mills theory with finite action [1]…

It is shown that infrared phenomena in the gauge theories are guided by certain classical solutions of the Yang-Mills equations. The existence of such solutions can lead to a finite correlation length which stops infrared catastrophe. In the present paper we deal only with theories with a compact but abelian gauge group. In this case the problems of correlation length and charge confinement are completely solved.

We find regular solutions of the four dimensional euclidean Yang-Mills equations. The solutions minimize locally the action integrals which is finite in this case. The topological nature of the solutions is discussed.

We propose a description of the vacuum in Yang-Mills theory and arrive at a physical interpretation of the pseudoparticle solution and the attendant violation of symmetries. The existence of topologically inequivalent classical gauge fields gives rise to a family of quantum mechanical vacua, parametrized by a CP-nonconserving angle. The requirement of vacuum stability against gauge transformations renders the vacua chirally noninvariant.

The finite action Euclidean solutions of gauge theories are shown to indicate the existence of tunneling between topologically distinct vacuum configurations. Diagonalization of the Hamiltonian then leads to a continuum of vacua. The construction and properties of these vacua are analyzed. In non-abelian theories of the strong interactions one finds spontaneous symmetry breaking of axial baryon number without the generation of a Goldstone boson, a mechanism for chiral SU(N) symmetry breaking and a possible source of T violation.

A systematic study is made of the relevant degrees of freedom and the dynamics of quantum chromodynamics (QCD). We find that the dynamical properties of QCD are, to a large extent, a consequence of the structure of the vacuum arising from the tunneling between degenerate, classically stable, vacuums, and that the relevant degrees of freedom can be taken to be the Euclidean path histories that can be used to calculate the tunneling in the semiclassical approximation. This nonperturbative vacuum structure appears well suited to the major features of QCD, i.e., the dimensional transmutation that determines the size of the hadrons and the strong-interaction coupling constant, the source of dynamical chiral symmetry breaking, and the mechanism responsible for quark confinement.

The classical Yang–Mills action is invariant with respect to a large class of transformations: (i) 15-parametric conformal group (4 translations, 4 special conformal transformations, 6 Lorentz rotations and 1 dilatation); (ii) global rotations in the color space (isospace) described by 3 parameters in the case of G = SU(2). Some of these symmetries are broken for every given instanton solution. For instance, since the BPST instanton is a localized four-dimensional configuration, it has a center, and for any given position of the center, the translational invariance is of course absent. The symmetries of the action are restored only if one considers the family of all possible instanton solutions as a whole. Then, for any solution with the given center, there is another one, with the center shifted to a different point in the four-dimensional space…

A detailed quantitative calculation is carried out of the tunneling process described by the Belavin-Polyakov-Schwarz-Tyupkin field configuration. A certain chiral symmetry is violated as a consequence of the Adler-Bell-Jackiw anomaly. The collective motions of the pseudoparticle and all contributions from single loops of scalar, spinor, and vector fields are taken into account. The result is an effective interaction Lagrangian for the spinors.

In the transition towards collective coordinates, page 3442, we inserted a factor [Eq. (9.4)] for each collective coordinate because these have to be normalized with a Gaussian integral. However, the relevant Gaussian integrals here are all of the type

The conformal transformation properties of the recently discovered pseudoparticle solution to a pure Yang-Mills theory are studied. It is shown that the solution is invariant under an O(5) subgroup of conformal transformations. A formalism is developed which renders this invariance explicit and which allows a very compact group-theoretical analysis of the propagation of fermions in the field of the pseudoparticle.

We comment on introduction of the instanton collective coordinates in the theories with the spontaneously broken gauge symmetry. In distinction with the pure gauge theories some gauge-invariant quantities acquire explicit dependence on the collective coordinates associated with the gauge field orientation in the gauge group, which looks rather paradoxically. This dependence, however, is absolutely necessary for preserving the Lorentz invariance of the instanton-induced amplitudes.

A treatment of the gauge zero modes about an instanton in a singular gauge places them on the same footing as all other zero modes and simplifies the calculation of the collective-coordinate part of the instanton determinant. This determinant is calculated first for the gauge group SU(3) and then for general SU(N). The answers differ from previously published results: For SU(3), the reason for this difference is trivial [the inclusion of certain factors of whose absence from 't Hooft's original SU(2) calculation was recently discovered] but the effects on quantum-chromodynamic calculations may be important; for large N, the reasons are more involved, but the usual conclusion that instantons are absent in the planar limit is unaffected.

The fact that SU(2) allows inequivalent embeddings inside larger groups is shown to have some consequences for the interaction of instantons.

As has been explained in Sec. I all finite action field configurations are classified according to their topological charge Q. If Q is fixed the minimum of the action in the given class is achieved on self-dual or anti-self-dual fields. So far we have discussed mainly the classical BPST instanton corresponding to Q = ±1. It is quite natural to ask how far one can go along the same lines in the case of |Q| > 1…

I present some exact solutions of the Polyakov-Belavin-Schwartz-Tyupkin equation F_μv = F_μv for an SU(2) gauge theory in Euclidean space. My solutions describe a system with an arbitrary number of pseudoparticles, with arbitrary scale parameters and arbitrary separations, arranged along a line. The action for an n-pseudoparticle solution is precisely n times the action for a single pseudopartlcle.

The known Euclidean Yang-Mills pseudoparticle solutions with Pontryagin index n are parametrized by 5n constants describing the size and location of each pseudoparticle. By insisting on conformal covariance of the solutions, we show that more general solutions exist—they are parametrized by 5n + 4 constants. We further demonstrate that the additional degrees of freedom are not gauge artifacts and correspond to a new degeneracy of pseudoparticle configurations.

A complete construction, involving only linear algebra, is given for all self-dual euclidean Yang–Mills fields.

The recent work of Atiyah, Hitchin, Drinfeld, and Manin is used to discuss self-dual Yang-Mills solutions for the compact gauge groups O(n), SU(n), and Sp(n). It is shown that the resulting solutions contain the correct number of parameters for all values of the topological charge. Although explicit construction of a general self-dual field requires the solution of a finite-dimensional, nonlinear matrix equation, we show that for widely separated instantons this equation can be solved perturbatively, providing a systematic expansion about the dilute-gas limit and a physical interpretation of the independent parameters in this limit. Further, closed-form expressions can be obtained for the general SU(2) solutions with topological charge 2 or 3. Finally, explicit isospin-½ and isospin-1 propagators are derived for a massless scalar field in the presence of the general self-dual SU(2) solution.

We show that at least 8n−3 parameters are required to specify an n-pseudoparticle solution in Euclidean SU(2) Yang-Mills theory.

The number of parameters entering a Euclidean Yang-Mills solution with topological charge k is determined for a theory constructed from an arbitrary Lie group G. It is shown that his number is precisely that required to specify the position, scale, and relative group orientation of k independent solutions each with minimum topological charge 1. Such minimal single-pseudoparticle solutions can be obtained by embedding the familiar SU₂ pseudoparticle of Belavin et al. into the general Lie group.

The calculation of propagators in the arbitrary self-dual background corresponding to k instantons of a generic form was started by Christ et al. (see paper in Sec. III). These authors considered the SU(2) gauge group and the scalar field of isospin ½ and 1; they followed the method of Ref. [2]. The analysis has been extended to arbitrary gauge groups and different representations of the scalar field in Ref. [1].…

The Green's functions for massless spinor and vector particles propagating in a self-dual but otherwise arbitrary non-Abelian gauge field are shown to be completely determined by the corrresponding Green's functions of scalar particles. Simple, explicit algebraic expressions are constructed for the scalar Green's functions of isospin-½ and isospin-1 particles in the self-dual field of a configuration of n pseudoparticles described by 5n arbitrary parameters.

The structure of the Green function for a scalar field, transforming under the adjoint representation of a gauge group, in the background field of an arbitrary self-dual instanton field, is considered. It is shown that it can be written in the same elegant form as the Green function for the fundamental vector representation of the group, given the introduction of a certain conformally invariant matrix. The same method solves the more general problem of finding the Green function for a field transforming under the direct product of two groups. Simple expressions for the solutions of the corresponding massless Dirac equation are obtained. The results on the products of representations may be applied, iteratively, to construct the Green function and massless solutions of the Dirac equation for any higher representation of the group.

The fermion sector is an important part of both QCD and the Standard Model. Although the very existence of instantons is due to the nontrivial topology in the space of the gauge fields, introducing the massless fermion fields in the Yang–Mills Lagrangian and considering them in the instanton background leads to drastic consequences — this observation was first made by 't Hooft in his pioneering paper [1]. The point is that instantons make explicit the nonconservation of certain fermion quantum numbers associated with the triangle anomaly in the axial-vector current [2]…

We formulate the Euclidean Yang-Mills gauge theory for isospin in terms of multispinors of SU(2)×SU(2)[=0(4)]×SU(2). The Dirac equation for fermions with arbitrary isospin interacting with the self-dual, conformally covariant Yang-Mills field is analyzed and completely solved for the isovector case. The relevance for this problem of the Atiyah-Singer index theory and its relation to the anomalous divergence of the axial-vector current are also explained.

In models of fermions coupled to gauge fields certain current-conservation laws are violated by Bell-Jackiw anomalies. In perturbation theory the total charge corresponding to such currents seems to be still conserved, but here it is shown that nonperturbative effects can give rise to interactions that violate the charge conservation. One consequence is baryon and lepton number nonconservation in V−A gauge theories with charm. Another is the nonvanishing mass squared of the η.

A new restriction on fermion quantum numbers in gauge theories is derived. For instance, it is shown that an SU(2) gauge theory with an odd number of left-handed fermion doublets (and no other representations) is mathematically inconsistent.

In numerous applications one often encounters situations in which the field configuration considered is not the exact solution of the equations of motion. Historically, one of the first examples of this type is an analog ensemble of the I/A “atoms” introduced in Ref. [1], the so-called dilute instanton gas (I stands for the instanton and A for the anti-instanton). Each given pseudoparticle in the ensemble is affected by the presence of others, and it is quite obvious that the topologically neutral gas cannot, strictly speaking, correspond to a minimum of the action and, hence, is not the exact solution. In other words the action of n pseudoparticles in the I/A ensemble S_n ≠ (8πn/g²); instead,

A simple method is presented for doing systematic constrained instanton calculations in models such as Ф⁴ or Higgs theories where the presence of a mass term prevents the existence of a classical solution. As an application, instanton estimates of the large-order behavior of the perturbation series in massive Ф⁴₄ theory are derived. (These estimates agree with those of Frishman and Yankielowicz.)

The effects of large-scale vacuum fluctuations on instantons of a small size ρ are considered. We find the instanton density with account of the gluon vacuum condensate. It is shown that the instanton gas approximation fails at unexpectedly small ρ. The instanton interaction with the gluon condensate already becomes strong at ρ≈(1.1GeV)⁻¹, while the absolute bound for the use of quasiclassical methods lies at ρ≈(0.5GeV)⁻¹.

The collective-coordinate method for the quasizero modes is suggested. Quasizero mode means that a direction in functional space exists where the action varies slowly. As in the case exact zero modes the corresponding integration is nongaussian and should be performed exactly. The method is illustrated by calculating the instanton-anti-instanton interaction in double-well quantum mechanics.

Supersymmetric (SUSY) gauge theories are potentially one of the most important fields of application of instantons. The reason why instantons may be so important is the existence of the so-called flat directions or valleys in a large class of models with matter fields. In these models the vacuum states are degenerate along certain directions in the space of fields, and the corresponding potential energy is zero classically and to any finite order in perturbation theory. The degeneracy may or may not be lifted by nonperturbative effects. In many cases instantons result in a remarkable dynamic effect: color and/or SUSY spontaneous breaking in the weak coupling regime [1–7] where all approximations made are under full theoretic control and instantons are protected from the infrared disaster by induced masses of the gauge fields (for a review see Refs. [8, 9])…

There is no Hermitean supersymmetry in Euclidean four-space. The simplest supersymmetry has complex four-component spinorial parameters. We give its algebraic structure and the automorphisms of the algebra, as well as a representation in terms of fields and an invariant Lagrangian. The results are relevant to the counting and the construction of the solutions of the many-instanton problem.

We show that the eigenvalue equations for the fluctuation of scalars, fermions and gluon around any classical self-dual solution of the Yang–Mills theory have the same spectrum of non-zero eigenvalues. In the case of a supersymmetric Yang–Mills theory this implies that the one loop correction around any self dual instanton is just given by a counting of the zero modes of the gluon, fermion and ghost.

Instantons in the simplest supersymmetric Yang–Mills theory are considered. We introduce bosonic and fermionic collective coordinates and study how they change under the supersymmetry transformations. The instanton measure is shown to be explicitly invariant under the transformations. We discuss the relation between quantum anomalies and the functional form of the instanton measure.

Within the framework of gauge SUSY theories we discuss correlation functions of the type (W²(x),S²(0)) where S is the chiral matter superfield (in the one-flavor model). SUSY implies that these correlation functions do not depend on coordinates and vanish identically in perturbation theory. We develop a technique for the systematic calculation of instanton effects. It is shown that even in the limit x → 0 the correlation functions at hand are not saturated by small-size instantons with radius ρ ˜ x; a contribution of the same order of magnitude comes from the instantons of characteristic size ρ ˜ l/v (v is the vacuum expectation value of the scalar field, and we concentrate on the models with v > Λ where Λ is the scale parameter fixing the running gauge coupling constant). If v > Λ both types of instantons can be consistently taken into account. The computational formalism proposed is explicitly supersymmetric and uses the language of instanton-associated superfields. We demonstrate, in particular, that one can proceed to a new variable, ρ_inv, which can be naturally considered as a supersymmetric generalization of the instanton radius. Unlike the ordinary radius ρ, this variable is invariant under the SUSY transformations. If one uses ρ_inv instead of ρ the expressions for the instanton contribution can be rewritten in the form saturated by the domain ρ²_inv=0. The cluster decomposition as well as x-independence of the correlation functions considered turn out to be obvious in this formalism.

We discuss in detail the supersymmetric instanton calculus of NSVZ and extend it to chiral matter fields in the adjoint representation. The constant Green functions induced by the instanton of supersymmetric SU(2) gauge theories are calculated systematically for the cases with and without scalar vev's bigger than the scale of the gauge theory and for nonvanishing small masses of chiral fields. One instanton contributions to the Green functions containing four fields without large vevs would disturb clustering; but they are argued to vanish; two-instanton effects then lead to a pattern which quantitatively agrees with factorization and the anomaly relation.

We extend a previously formulated instanton superfield method from SU(2) to SU(N) gauge theories without classical vacuum expectation values of scalar fields. We consider supersymmetric Yang-Mills, supersymmetric QCD, and a class of chiral theories with matter in the fundamental and antisymmetric tensor representation. Via cluster decomposition, our results provide non-trivial consistency checks; in particular they enable us to conclude that for most of the chiral theories the approach without classical scalar VEV's is inconsistent.

A superfield formalism is developed for superinstantons is supersymmetric SU(N) gauge theories.

A superfield formalism is developed for instantons in supersymmetric gauge theories with matter fields in an arbitrary group.

Below the reader will find two review papers devoted to instantons in gauge theories. Many questions which are tacitly assumed or only briefly mentioned in the original publications are discussed here in detail. Along with these two reviews I would like to recommend the book “Solitons and Instantons” by R. Rajaraman (North-Holland, 1987). Although gauge theories per se are discussed in less detail, the book gives a broader introduction to a wide range of questions related to topology, including solitons and monopoles. It also covers several topics not included in this volume, such as instantons in two-dimensional models, instantons and high orders of perturbation theory, etc.

The following sections are included:

• Introduction

• Instantons and bounces in particle mechanics
  • Euclidean functional integrals
  • The double well and instantons
  • Periodic potentials
  • Unstable states and bounces

• The vacuum structure of gauge field theories
  • Old stuff
  • The winding number
  • Many vacua
  • Instantons: generalities
  • Instantons: particulars
  • The evaluation of the determinant and an infrared embarrassment

• The Abelian Higgs model in 1+1 dimensions

• 't Hooft's solution of the U(1) problem
  • The mystery of the missing meson
  • Preliminaries: Euclidean Fermi fields
  • Preliminaries: chiral Ward identities
  • QCD (baby version)
  • QCD (the real thing)
  • Miscellany

• The fate of the false vacuum
  • Unstable vacua
  • The bounce
  • The thin-wall approximation
  • The fate of the false vacuum
  • Determinants and renormalization
  • Unanswered questions

• Appendix 1: How to compute determinants

• Appendix 2: The double well done doubly well

• Appendix 3: Finite action is zero measure

• Appendix 4: Only winding number survives

• Appendix 5: No wrong-chirality solutions

• Notes and references

An attempt is made to present an instanton “calculus” in a relatively simple form. The physical meaning of instantons is explained by the example of the quantum-mechanical problem of energy levels in a two-bumped potential. The nonstandard solution to this problem based on instantons is analyzed, and the reader is acquainted with the main technical elements used in this approach. Instantons in quantum chromodynamics are then considered. The Euclidean formulation of the theory is described. Classical solutions of the field equations (the Belavin-Polyakov-Shvarts-Tyupkin instantons) are obtained explicitly and their properties are studied. The calculation of the instanton density is described and the complete result is given for an arbitrary number of colors. The effects associated with fermion fields are briefly described.