Order Behavior Of Perturbation Theory Research Articles

Resurgence in η-deformed Principal Chiral Models

We study the $SU(2)$ Principal Chiral Model (PCM) in the presence of an integrable $\eta$-deformation. We put the theory on $\mathbb{R}\times S^1$ with twisted boundary conditions and then reduce the circle to obtain an effective quantum mechanics associated with the Whittaker-Hill equation. Using resurgent analysis we study the large order behaviour of perturbation theory and recover the fracton events responsible for IR renormalons. The fractons are modified from the standard PCM due to the presence of this $\eta$-deformation but they are still the constituents of uniton-like solutions in the deformed quantum field theory. We also find novel $SL(2,\mathbb{C})$ saddles, thus strengthening the conjecture that the semi-classical expansion of the path integral gives rise to a resurgent transseries once written as a sum over Lefschetz thimbles living in a complexification of the field space. We conclude by connecting our quantum mechanics to a massive deformation of the $\mathcal{N}=2~$ $4$-d gauge theory with gauge group $SU(2)$ and $N_f=2$.

Valley views: instantons, large order behaviors, and supersymmetry

The elucidation of the properties of the instantons in the topologically trivial sector has been a long-standing puzzle. Here we claim that the properties can be summarized in terms of the geometrical structure in the configuration space, the valley. Evidence for this claim is presented in various ways. The conventional perturbation theory and the non-perturbatioe calculation are unified, and the ambiguity of the Borel transform of the perturbation series is removed. A ‘proof’ of Bogomolny's “trick” is presented, which enables us to go beyond the dilute-gas approximation. The prediction of the large order behavior of perturbation theory is confirmed by explicit calculations, in some cases to the 478th order. A new type of supersymmetry is found as a by-product, and our result is shown to be consistent with the non-renormalization theorem. The prediction of the energy levels is confirmed with numerical solutions of the Schrödinger equation.

Sphalerons and large order behaviour of perturbation theory in lower dimension

Sphalerons - unstable static solutions of classical field equations in ( d + 1)-dimensional spacetime - may be viewed as euclidean solutions in d dimensions. We discuss their role in the large order asymptotics of perturbation theory. Specifically, we calculate their contribution to the large order behaviour of the ground state energy in a quantum mechanical model. When the number of negative modes is odd, the single sphaleron contribution dominates, while this contribution vanishes when the number of negative modes is even. These results are confirmed by numerical calculations.

Asymptotics of the cut discontinuity and large order behaviour from the instanton singularity: The case of lattice Schr�dinger operators with exponential disorder

We discuss the relation between the singularity structure of the Borel transform, the asymptotics of the cut discontinuity and the large order behaviour of perturbation theory. In an explicit example — a tight binding model with exponential disorder — we show how to obtain the first instanton singularity from a cluster expansion in the Borel variable, and as an application we determine the exact decay of the density of states asE → ∞. The method opens some perspectives for similar problems arising from different models of Mathematical Physics.

Calorimetric study on the electronic structure in up-us system

Until recent years field theories were only studied from standard perturbation series. These are equivalent to small fluctuation expansions around the classical ground state configurations in which the fields are space-time independent. It has been gradually realized that this procedure may miss some crucial physical features of the theory. In particular the absence of free quarks cannot be explained in quantum chromodynamics from this perturbative viewpoint. One is led to deal with the non-linearities in an essential way. This issue aims at covering the recent advances in this direction typically since 1975 and the issue 23C of Physics Reports.One way of going beyond perturbation theory is to make use of non-perturbative classical solutions which are in general very difficult to obtain. In Yang-Mills theories remarkable advances have been made in the search for self-dual solutions. They are described in the lectures (I, II) by E. Corrigan and R. Stora et al. The quantum meaning of these solutions and the technical problems raised by quantizing fluctuations around them are discussed by J.L. Gervais (III) and D. Gross (IV). The most recent topics of this subject deal with imaginary time solutions, the so-called instantons, which describe quantum tunneling in the semi-classical approximation. Other field configurations such as merons, are also discussed in relation with quark confinement. An attempt to build a general picture of strong interactions on these grounds is displayed in (IV).Recent advances concerning the solitons which are real time, classical Minkowski solutions, are discussed by D. Olive (V). These solutions describe generalized Dirac magnetic monopoles in non-Abelian gauge theories. The resulting model contains confined magnetic charges and unconfined electric charges. If the magnetic and elastic characters can be interchanged by a dual transformation, as discussed by F. Englert et al. in (VI), this would provide a model for quark confinement. This duality transformation has been spelled out in (IX) by C. Itzykson who proves the existence of a confinement phase transition in a lattice model with Z2 gauge symmetry.Another non-linear feature of Yang-Mills theory is that no single gauge condition can be implemented over large field fluctuations. This is discussed in (VII) by Sciuto and also in (III).Another mechanism of confinement is proposed by McCoy and Wu (VIII) in which the propagator has a cut instead of a pole, as it does happen in the two-dimensional Ising model.The lectures of J. Zinn-Justin and G. Parisi, (X) and (XI) deal with the asymptotic estimation of the large order behaviour of perturbation theory, which one can obtain by semi-classical methods. The use of these methods is either to provide a way of improving practical perturbative calculations, or to characterize possible ambiguities due to vacuum tunneling. Additional problems raised in renormalizable theories are discussed in (XI).Another non-perturbative approach is to examine the large N limit of an SU(N) gauge theory. At leading order, only planar Feynman diagrams contribute. They are discussed in (XII) by E. Brézin. There are very subtle questions about the corresponding exact summation in two dimensions which are studied by T.T. Wu in (XV).In two-dimensional field theories exact solutions for the S-matrix have been recently discovered. They are discussed by Karowski in (XIII). In connection with this problem, Lüscher (XIV) has shown the existence of non-linear conserved charges which ensure the absence of particle production. These models are particularly interesting since the mass spectrum cannot be obtained from coupling constant expansion.Finally several related topics in statistical physics are discussed. D. Nelson (XVI) reviews the beautiful properties of the two-dimensional X-Y model which is an ideal example of mechanisms invoked in hadronic field theories. A. Luther (XVII) discussed an attempt to extend to a higher number of dimensions the bosonization of fermion theories which is so powerful in two dimensions. G. Toulouse (XVIII) describes modern ideas in spin glass phase transitions, a possible testing ground for gauge field theories besides its obvious intrinsic interest.

Critical behavior of polar modes in lead phosphate near the ferroelastic phase transition

Until recent years field theories were only studied from standard perturbation series. These are equivalent to small fluctuation expansions around the classical ground state configurations in which the fields are space-time independent. It has been gradually realized that this procedure may miss some crucial physical features of the theory. In particular the absence of free quarks cannot be explained in quantum chromodynamics from this perturbative viewpoint. One is led to deal with the non-linearities in an essential way. This issue aims at covering the recent advances in this direction typically since 1975 and the issue 23C of Physics Reports.One way of going beyond perturbation theory is to make use of non-perturbative classical solutions which are in general very difficult to obtain. In Yang-Mills theories remarkable advances have been made in the search for self-dual solutions. They are described in the lectures (I, II) by E. Corrigan and R. Stora et al. The quantum meaning of these solutions and the technical problems raised by quantizing fluctuations around them are discussed by J.L. Gervais (III) and D. Gross (IV). The most recent topics of this subject deal with imaginary time solutions, the so-called instantons, which describe quantum tunneling in the semi-classical approximation. Other field configurations such as merons, are also discussed in relation with quark confinement. An attempt to build a general picture of strong interactions on these grounds is displayed in (IV).Recent advances concerning the solitons which are real time, classical Minkowski solutions, are discussed by D. Olive (V). These solutions describe generalized Dirac magnetic monopoles in non-Abelian gauge theories. The resulting model contains confined magnetic charges and unconfined electric charges. If the magnetic and elastic characters can be interchanged by a dual transformation, as discussed by F. Englert et al. in (VI), this would provide a model for quark confinement. This duality transformation has been spelled out in (IX) by C. Itzykson who proves the existence of a confinement phase transition in a lattice model with Z2 gauge symmetry.Another non-linear feature of Yang-Mills theory is that no single gauge condition can be implemented over large field fluctuations. This is discussed in (VII) by Sciuto and also in (III).Another mechanism of confinement is proposed by McCoy and Wu (VIII) in which the propagator has a cut instead of a pole, as it does happen in the two-dimensional Ising model.The lectures of J. Zinn-Justin and G. Parisi, (X) and (XI) deal with the asymptotic estimation of the large order behaviour of perturbation theory, which one can obtain by semi-classical methods. The use of these methods is either to provide a way of improving practical perturbative calculations, or to characterize possible ambiguities due to vacuum tunneling. Additional problems raised in renormalizable theories are discussed in (XI).Another non-perturbative approach is to examine the large N limit of an SU(N) gauge theory. At leading order, only planar Feynman diagrams contribute. They are discussed in (XII) by E. Brézin. There are very subtle questions about the corresponding exact summation in two dimensions which are studied by T.T. Wu in (XV).In two-dimensional field theories exact solutions for the S-matrix have been recently discovered. They are discussed by Karowski in (XIII). In connection with this problem, Lüscher (XIV) has shown the existence of non-linear conserved charges which ensure the absence of particle production. These models are particularly interesting since the mass spectrum cannot be obtained from coupling constant expansion.Finally several related topics in statistical physics are discussed. D. Nelson (XVI) reviews the beautiful properties of the two-dimensional X-Y model which is an ideal example of mechanisms invoked in hadronic field theories. A. Luther (XVII) discussed an attempt to extend to a higher number of dimensions the bosonization of fermion theories which is so powerful in two dimensions. G. Toulouse (XVIII) describes modern ideas in spin glass phase transitions, a possible testing ground for gauge field theories besides its obvious intrinsic interest.

The resurgence of instantons in string theory

Nonperturbative eects in string theory are usually associated to D{branes. In many cases it can be explicitly shown that D{brane instantons control the large{order behavior of string perturbation theory, leading to the well{known (2g)! growth of the genus expansion. This paper presents a detailed treatment of nonperturbative solutions in string theory, and their relation to the large{order behavior of perturbation theory, making use of transseries and resurgent analysis. These are powerful techniques addressing general nonperturbative contributions within non{linear systems, which are developed at length herein as they apply to string theory. The cases of topological strings, the Painlev e I equation describing 2d quantum gravity, and the quartic matrix model, are explicitly addressed. These results generalize to minimal strings and general matrix models. It is shown that, in order to completely understand string theory at a fully nonperturbative level, new sectors are required beyond the standard D{brane sector.