Derivative expansion for the effective action of chiral gauge fermions. The abnormal parity component

Explicit exact formulas are presented, for the leading order term in a strict chiral covariant derivative expansion, for the abnormal parity component of the effective action of two- and four-dimensional Dirac fermions in the presence of scalar, pseudo-scalar, vector and axial vector background fields. The formulas hold for completely general internal symmetry groups and general configurations. In particular, the scalar and pseudo-scalar fields need not be on the chiral circle. Received: 31 January 2001 / Published online: 12 April 2001     

The effective action for chiral fermions

We present an exact formula for the imaginary part of the effective action for chiral fermions in complex representations of the gauge group G, in the presence of an arbitrary external gauge field configuration. This result provides a unified treatment of both local and global anomalies and a precise characterization of some of the special properties of chiral fermions.     

Direct construction of the effective action of chiral gauge fermions in the anomalous sector

The anomaly implies an obstruction to a fully chiral covariant calculation of the effective action in the abnormal-parity sector of chiral theories. The standard approach then is to reconstruct the anomalous effective action from its covariant current. In this work, we use a recently introduced formulation which allows one to directly construct the non-trivial chiral invariant part of the effective action within a fully covariant formalism. To this end we develop an appropriate version of Chan’s approach to carry out the calculation within the derivative expansion. The result to four derivatives, i.e., to leading order in two and four dimensions and next-to-leading order in two dimensions, is explicitly worked out. Fairly compact expressions are found for these terms.     

The soliton of the effective chiral action in a heat-kernel expansion

It is shown that the heat-kernel expansion (rather than the gradient expansion) of the effective chiral action of the quark loop yields in next-to-leading (i.e., fourth) order an effective low-energy lagrangian which possesses stable soliton solutions. The baryon number-one soliton solution of this effective chiral lagrangian is found numerically.     

The small-volume expansion of gauge theories coupled to massless fermions

The small-volume expansion of the low-lying glueball states for SU(2) and SU(3) gauge theory, coupled to massless fermions with periodic and antiperiodic boundary conditions, is determined. For SU(3) with periodic boundary conditions the vacuum is eightfold degenerate and breaks part of the cubic group spontaneously. In all cases the scalar-to-tensor mass ratio m_A1⁺⁺/m_E⁺⁺ is 1.1 to 1.3 as in the pure-gauge case. We also discuss chiral symmetry.     

The soliton of the effective chiral action

The baryon-number-1 soliton solution of the effective chiral meson action of QCD is found numerically. For this purpose the underlying fermion determinant (quark loop) is calculated in a non-perturbative way by diagonalizing the corresponding Dirac hamiltonian. The vacuum energy is defined within a static version of the proper-time regularization. Topological soliton solutions are found for cut-offs below some critical value, above which only topologically trivial solutions exist.     

On the chiral limit in lattice gauge theories with Wilson fermions

The chiral limit κ ? κ _c (β) in lattice gauge theories with Wilson fermions and problems related to near-to-zero (’exceptional’) eigenvalues of the fermionic matrix are studied. For this purpose we employ compact lattice QED in the confinement phase. A new estimator $\tilde m_\pi$ for the calculation of the pseudoscalar mass m _π is proposed which does not suffer from ’divergent’ contributions at κ ? κ _c (β)We conclude that the main contribution to the pion mass comes from larger modes, and ’exceptional’ eigenvalues play no physical role. The behaviour of the subtracted chiral condensate $\left\langle {\bar \psi \psi } \right\rangle _{subt}$ near κ _c (β) is determined. We observe a comparatively large value of $\left\langle {\bar \psi \psi } \right\rangle _{subt} \cdot Z_P^{ - 1}$ , which could be interpreted as a possible effect of the quenched approximation.     

The effective chiral Lagrangian from dimension-six parity and time-reversal violation

We classify the parity- and time-reversal-violating operators involving quark and gluon fields that have effective dimension six: the quark electric dipole moment, the quark and gluon chromo-electric dipole moments, and four four-quark operators. We construct the effective chiral Lagrangian with hadronic and electromagnetic interactions that originate from them, which serves as the basis for calculations of low-energy observables. The form of the effective interactions depends on the chiral properties of these operators. We develop a power-counting scheme and calculate within this scheme, as an example, the parity- and time-reversal-violating pion–nucleon form factor. We also discuss the electric dipole moments of the nucleon and light nuclei.     

Derivative expansions of the non-equilibrium effective action

New techniques for evaluating the closed time path effective action for non-equilibrium quantum fields are presented. A derivative expansion is performed using a proper time kernel. Applications relevant to the scalar field theory of thermal inflation are discussed and dissipation terms resummed. The effective action of the electromagnetic field is also considered. In this case the leading term can be related to the conductivity of a plasma and has a simple interpretation in terms of the classical Drude theory of conductivity.     

The anomalous effective action in three-dimensional gauge theories

Summary We analyse the effective action for gauge fields in odd dimensions, obtained by integrating out the fermions in the Feynman path integral. In particular, we discuss the generation of a Chern-Simons term by massless fermions minimally coupled to an Abelian gauge field. We review two methods of revealing the presence of a Chern-Simons term in the effective action: first, as the consequence of a nontrivial holonomy of the fermionic ground state, then as the result of the generation of an anomalous imaginary part of the effective action. We derive the most general form of the anomalous effective action at the lowest nontrivial order of a derivative expansion in time. We discuss the implications of our analysis for the theory of the fractional quantum Hall effect as well as for the quantization of anomalous theories. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.     

Topological fluctuations and breaking of chiral symmetry in gauge theories involving massless fermions

Using the method of Euclidean functional integrals, we show the occurrence of a certain types of chiral symmetry breaking as a result of local fluctuations in the winding number. We mainly restrict our discussion to QED₂ in order to have an independent check of our methods from the known solution of this model. We do not use instantons (pseudo-particles), and we also avoid functional integration over fields with global winding (Pontryagin) number different from zero.     

Finiteness of the Coulomb gauge QCD perturbative effective action

At 2-loop order in the Coulomb gauge, individual Feynman graphs contributing to the effective action have energy divergences. It is proved that these cancel in suitable combinations of graphs. This has previously been shown only for transverse external fields. The calculation results in a generalization of the Christ–Lee term which was inserted into the Hamiltonian.     

Improved effective action for light quarks beyond the chiral limit

We propose an improvement of the Diakonov–Petrov effective action on the basis of the Lee–Bardeen results for the quark determinant in the instanton field. This improved effective action provides a proper account of the current quark masses, which is particularly important for strange quarks. This action is successfully tested by calculations of the quark condensate, the masses of the pseudoscalar meson octet and axial-anomaly low-energy theorems. Received: 19 October 1998 / Revised version: 17 January 1999 / Published online: 20 May 1999     

Summing the derivative expansion of the effective action

The derivative expansion of the effective action is a perturbative development in derivatives of the fields. The expansion breaks down when some of the derivatives are too large. We show how to sum exactly the first and second derivatives and treat perturbatively derivatives higher than second.     

On the Yang-Mills effective action with instanton gauge field

The calculation of the determinant for the second order covariant derivative operator is analyzed in the space R⁴ and for Yang-Mills instanton field configurations. The problems inherent to the ultraviolet ζ-function renormalization method and infrared divergencies of this operator are reviewed. A method for estimating the asymptotic coefficients of its determinant has been discussed and the failure of the general conformal invariance induced by any regularization technique has also been considered. A particular solution valid for a Yang-Mills multi instanton configuration, valuable in order to get the non conformally invariant piece of the general solution, is the main result obtained here. We give it as an integral equation, namely in a semi-explicit form.