Borel Plane Research Articles

On the resurgent approach to Écalle–Voronin's invariants

Given a holomorphic germ at the origin of C with a simple parabolic fixed point, the Fatou coordinates have a common asymptotic expansion whose formal Borel transform is resurgent. We show how to use Écalle's alien operators to study the singularities in the Borel plane and relate them to the horn maps, providing each of Écalle–Voronin's invariants in the form of a convergent numerical series. The proofs are original and self-contained, with ordinary Borel summability as the only prerequisite.

Nonperturbative Ambiguities and the Reality of Resurgent Transseries

In a wide range of quantum theoretical settings—from quantum mechanics to quantum field theory, from gauge theory to string theory—singularities in the complex Borel plane, usually associated to instantons or renormalons, render perturbation theory ill-defined as they give rise to nonperturbative ambiguities. These ambiguities are associated to choices of an integration contour in the resummation of perturbation theory, along (singular) Stokes directions in the complex Borel plane (rendering perturbative expansions non-Borel summable along any Stokes line). More recently, it has been shown that the proper framework to address these issues is that of resurgent analysis and transseries. In this context, the cancelation of all nonperturbative ambiguities is shown to be a consequence of choosing the transseries median resummation as the appropriate family of unambiguous real solutions along the coupling-constant real axis. While the median resummation is easily implemented for one-parameter transseries, once one considers more general multi-parameter transseries the procedure becomes highly dependent upon properly understanding Stokes transitions in the complex Borel plane. In particular, all Stokes coefficients must now be known in order to explicitly implement multi-parameter median resummations. In the cases where quantum-theoretical physical observables are described by resurgent functions and transseries, the methods described herein show how one may cancel nonperturbative ambiguities, and define these observables nonperturbatively starting out from perturbation theory. Along the way, structural results concerning resurgent transseries are also obtained.

QCD sum rules on the complex Borel plane

Borel transformed QCD sum rules conventionally use a real valued parameter (the Borel mass) for specifying the exponential weight over which hadronic spectral functions are averaged. In this paper, it is shown that the Borel mass can be generalized to have complex values and that new classes of sum rules can be derived from the resulting averages over the spectral functions. The real and imaginary parts of these novel sum rules turn out to have damped oscillating kernels and potentially contain a larger amount of information on the hadronic spectrum than the real valued QCD sum rules. As a first practical test, we have formulated the complex Borel sum rules for the phi meson channel and have analyzed them using the maximum entropy method, by which we can extract the most probable spectral function from the sum rules without strong assumptions on its functional form. As a result, it is demonstrated that, compared to earlier studies, the complex valued sum rules allow us to extract the spectral function with a significantly improved resolution and thus to study more detailed structures of the hadronic spectrum than previously possible.

Nonlinear analysis with resurgent functions

Ecalle’s resurgent functions appear naturally as Borel transforms of divergent series like Stirling series, formal solutions of differential equations like Euler series, or formal series associated with many other problems in Analysis and dynamical systems. Resurgence means a certain property of analytic continuation in the Borel plane, whose stability under con- volution (the Borel counterpart of multiplication of formal series) is not obvious. Following the analytic continuation of the convolution of several resurgent functions is indeed a delicate question, but this must be done in an explicit quan- titative way so as to make possible nonlinear resurgent calculus (e.g. to check that resurgent functions are stable under composition or under substitution into a convergent series). This can be done by representing the analytic continuation of the convolution product as the integral of a holomorphic n-form on a singular n-simplex obtained as a suitable explicit deformation of the standard n-simplex. The theory of currents is convenient to deal with such integrals of holomorphic forms, because it allows to content oneself with little regularity: the deformations we use are only Lipschitz continuous, because they are built from the flow of non-autonomous Lipschitz vector fields.

Perturbative expansion of the plaquette toO(α35)in four-dimensional SU(3) gauge theory

Using numerical stochastic perturbation theory, we determine the first 35 infinite volume coefficients of the perturbative expansion in powers of the strong coupling constant $\alpha$ of the plaquette in SU(3) gluodynamics. These coefficients are obtained in lattice regularization with the standard Wilson gauge action. The on-set of the dominance of the dimension four renormalon associated to the gluon condensate is clearly observed. We determine the normalization of the corresponding singularity in the Borel plane and convert this into the $\overline{\mathrm{MS}}$ scheme. We also comment on the impact of the renormalon on non-perturbative determinations of the gluon condensate.

DETERMINATION OF THE STRONG COUPLING FROM HADRONIC TAU DECAYS USING RENORMALIZATION GROUP SUMMED PERTURBATION THEORY

We determine the strong coupling constant αs from the τ hadronic width using a renormalization group summed (RGS) expansion of the QCD Adler function. The main theoretical uncertainty in the extraction of αs is due to the manner in which renormalization group invariance is implemented, and the as yet uncalculated higher order terms in the QCD perturbative series. We show that new expansion exhibits good renormalization group improvement and the behavior of the series is similar to that of the standard CIPT expansion. The value of the strong coupling in [Formula: see text] scheme obtained with the RGS expansion is [Formula: see text]. The convergence properties of the new expansion can be improved by Borel transformation and analytic continuation in the Borel plane. This is discussed elsewhere in these issues.

STRONG COUPLING FROM THE TAU HADRONIC WIDTH BY NON-POWER QCD PERTURBATION THEORY

Starting from the divergent character of the perturbative expansions in QCD and using the technique of series acceleration by the conformal mappings of the Borel plane, I define a novel, non-power perturbative expansion for the Adler function, which simultaneously implements renormalisation-group summation and has a tamed large-order behaviour. The new expansion functions, which replace the standard powers of the coupling, are singular at the origin of the coupling plane and have divergent perturbative expansions, resembling the expanded function itself. Confronting the new perturbative expansions with the standard ones on specific models investigated recently in the literature, I show that they approximate in an impressive way the exact Adler function and the spectral function moments. Applied to the τ hadronic width, the contour-improved and the renormalisation-group summed non-power expansions in the [Formula: see text] scheme lead to the prediction [Formula: see text], which translates to [Formula: see text].

Perturbative expansion of the QCD Adler function improved by renormalization-group summation and analytic continuation in the Borel plane

We examine the large-order behaviour of a recently proposed renormalization-group-improved expansion of the Adler function in perturbative QCD, which sums in an analytically closed form the leading logarithms accessible from renormalization-group invariance. The expansion is first written as aneffective series in powers of the one-loop coupling, and its leading singularities in the Borel plane are shown to be identical to those of the standard "contour-improved" expansion. Applying the technique of conformal mappings for the analytic continuation in the Borel plane, we define a class of improved expansions, which implement both the renormalization-group invariance and the knowledge about the large-order behaviour of the series. Detailed numerical studies of specific models for the Adler function indicate that the new expansions have remarkable convergence properties up to high orders. Using these expansions for the determination of the strong coupling from the the hadronic width of the $\tau$ lepton we obtain, with a conservative estimate of the uncertainty due to the nonperturbative corrections, $\alpha_s(M_\tau^2)= 0.3189^{+ 0.0173}_{-0.0151}$, which translates to $\alpha_s(M_Z^2)= 0.1184^{+0.0021}_{-0.0018}$.

Semiclassical Realization of Infrared Renormalons

Perturbation series in quantum field theory (QFT) are generally divergent asymptotic series which are also typically not Borel resummable in the sense that the resummed series is ambiguous. The ambiguity is associated with singularities in the Borel plane on the positive real axis. In quantum mechanics there are cases in which the ambiguity that arises in perturbation theory cancels against a similarly ambiguous contribution from instanton-anti-instanton events. In asymptotically free gauge theories, this mechanism does not suffice because perturbation theory develops ambiguities associated with singularities in the Borel plane which are closer to the origin by a factor of about N (the rank of the gauge group) compared to the singularities realized by instanton events. These are called IR renormalon poles, and on R^{4} they do not possess any known semiclassical realization. By using continuity on R(3) × S(1), and by generalizing the works of Bogomolny and Zinn-Justin to QFT, we identify saddle point field configurations, e.g., bion-antibion events, corresponding to singularities in the Borel plane which are of order N times closer to the origin than the four-dimensional instanton-anti-instanton singularities in the Borel plane. We conjecture that these are the leading singularities in the Borel plane and that they are the incarnation of the elusive renormalons in the weak coupling regime.

The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects

Abstract We study the dynamics of four dimensional gauge theories with adjoint fermions for all gauge groups, both in perturbation theory and non-perturbatively, by using circle compactification with periodic boundary conditions for the fermions. There are new gauge phenomena. We show that, to all orders in perturbation theory, many gauge groups are Higgsed by the gauge holonomy around the circle to a product of both abelian and nonabelian gauge group factors. Non-perturbatively there are monopole-instantons with fermion zero modes and two types of monopole-anti-monopole molecules, called bions. One type are magnetic bions which carry net magnetic charge and induce a mass gap for gauge fluctuations. Another type are neutral bions which are magnetically neutral, and their understanding requires a generalization of multi-instanton techniques in quantum mechanics — which we refer to as the Bogomolny-Zinn-Justin (BZJ) prescription — to compactified field theory. The BZJ prescription applied to bion-anti-bion topological molecules predicts a singularity on the positive real axis of the Borel plane (i.e., a divergence from summing large orders in peturbation theory) which is of order N times closer to the origin than the leading 4-d BPST instanton-anti-instanton singularity, where N is the rank of the gauge group. The position of the bion-anti-bion singularity is thus qualitatively similar to that of the 4-d IR renormalon singularity, and we conjecture that they are continuously related as the compactification radius is changed. By making use of transseries and Écalle’s resurgence theory we argue that a non-perturbative continuum definition of a class of field theories which admit semi-classical expansions may be possible.

Expansion functions in perturbative QCD and the determination ofαs(Mτ2)

The conventional series in powers of the coupling in perturbative QCD have zero radius of convergence and fail to reproduce the singularity of the QCD correlators like the Adler function at $\alpha_s=0$. Using the technique of conformal mapping of the Borel plane, combined with the "softening" of the leading singularities, we define a set of new expansion functions that resemble the expanded correlator and share the same singularity at zero coupling. Several different conformal mappings and different ways of implementing the known nature of the first branch-points of the Adler function in the Borel plane are investigated, in both the contour-improved (CI) and fixed-order (FO) versions of renormalization group resummation. We prove the remarkable convergence properties of a set of new CI expansions and use them for a determination of the strong coupling from the hadronic $\tau$ decay width. By taking the average upon this set, with a conservative treatment of the errors, we obtain $\alpha_s(M_\tau^2)= 0.3195^{+ 0.0189}_{- 0.0138}$.

Determination of [formula omitted]: a conformal mapping approach

We discuss a new class of expansions in perturbative QCD, based on the technique of conformal mappings of the Borel plane, and apply them for the determination of $\alpha_s$ from the hadronic decays of the $\tau$ lepton. Using the expansion up to fifth order in the $\bar{\rm MS}$ scheme, the method leads to the prediction $\alpha_s(M_\tau^2)= 0.320\pm 0.011$.

Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems

A precise description of the singularities of the Borel transform of solutions of a level-one linear differential system is deduced from a proof of the summable-resurgence of the solutions by the perturbative method of J. Écalle. Then we compare the meromorphic classification (Stokes phenomenon) from the viewpoint of the Stokes cocycle and the viewpoint of alien derivatives. We make explicit the Stokes–Ramis matrices as functions of the connection constants in the Borel plane and we develop two examples. No assumption of genericity is made.

Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c = 1 Matrix Models

We address the nonperturbative structure of topological strings and c=1 matrix models, focusing on understanding the nature of instanton effects alongside with exploring their relation to the large-order behavior of the 1/N expansion. We consider the Gaussian, Penner and Chern-Simons matrix models, together with their holographic duals, the c=1 minimal string at self-dual radius and topological string theory on the resolved conifold. We employ Borel analysis to obtain the exact all-loop multi-instanton corrections to the free energies of the aforementioned models, and show that the leading poles in the Borel plane control the large-order behavior of perturbation theory. We understand the nonperturbative effects in terms of the Schwinger effect and provide a semiclassical picture in terms of eigenvalue tunneling between critical points of the multi-sheeted matrix model effective potentials. In particular, we relate instantons to Stokes phenomena via a hyperasymptotic analysis, providing a smoothing of the nonperturbative ambiguity. Our predictions for the multi-instanton expansions are confirmed within the trans-series set-up, which in the double-scaling limit describes nonperturbative corrections to the Toda equation. Finally, we provide a spacetime realization of our nonperturbative corrections in terms of toric D-brane instantons which, in the double-scaling limit, precisely match D-instanton contributions to c=1 minimal strings.

Complexified path integrals and the phases of quantum field theory

The path integral by which quantum field theories are defined is a particular solution of a set of functional differential equations arising from the Schwinger action principle. In fact these equations have a multitude of additional solutions which are described by integrals over a complexified path. We discuss properties of the additional solutions which, although generally disregarded, may be physical with known examples including spontaneous symmetry breaking and theta vacua. We show that a consideration of the full set of solutions yields a description of phase transitions in quantum field theories which complements the usual description in terms of the accumulation of Lee–Yang zeroes. In particular we argue that non-analyticity due to the accumulation of Lee–Yang zeros is related to Stokes phenomena and the collapse of the solution set in various limits including but not restricted to, the thermodynamic limit. A precise demonstration of this relation is given in terms of a zero dimensional model. Finally, for zero dimensional polynomial actions, we prove that Borel resummation of perturbative expansions, with several choices of singularity avoiding contours in the complex Borel plane, yield inequivalent solutions of the action principle equations.

Measure-category properties of Borel plane sets and Borel functions of two variables

Let Open image in new window stand for the Fubini-type product of σ-ideals Open image in new window . We consider mixed measure-category σ-ideals Open image in new window and Open image in new window (called the Mendez σ-ideals), and study some features of their structure. We show that Open image in new window and Open image in new window are not invariant with respect to nonzero rotations. Using Fremlin’s results, we describe nice Borel bases of Open image in new window and Open image in new window . The rest of the paper is devoted to uniform versions of the Nikodym theorem and the Lusin theorem for Borel functions of two variables.

Deep inelastic and dipole scattering on finite length hot [formula omitted] SYM matter

Deep inelastic scattering of R -currents and the scattering of a small dipole on finite length hot N = 4 SYM matter are discussed. In each case we find the scale when scattering becomes strong is determined by a saturation momentum Q s 2 ∼ L T 3 / x where L is the length of the matter. For R -currents we analyze the operator product expansion. For infinite length matter the series generated by the OPE is not Borel summable but we are able to determine the exponential part of the tunneling amplitude determining F 2 when Q 2 Q s 2 ≫ 1 from the position of the singularity closest to the origin on the real axis of the Borel plane. In finite length matter the OPE series is not convergent but it is Borel summable. When a small dipole, and the string connecting the ends of the dipole, pass through hot matter there is an induced motion of the string in the 5th dimension. When T 4 L cosh η , with the η the rapidity of the string, is large enough the string would normally break into several parts after leaving the medium, however, this cannot happen in the classical approximation in which we work.

MIXED GEVREY ASYMPTOTICS

In this paper, we illustrate with two examples that the computation of Stokes multipliers in problems in which the asymptotic expansions are not of Gevrey order one, is much more complicated and that the singularities in the corresponding Borel plane can be essential singularities.

Infrared freezing of Euclidean QCD observables

We consider the leading one-chain term in a skeleton expansion for QCD observables and show that for energies ${Q}^{2}>{\ensuremath{\Lambda}}^{2}$, where ${Q}^{2}={\ensuremath{\Lambda}}^{2}$ is the Landau pole of the coupling, the skeleton expansion result is equivalent to the standard Borel integral representation, with ambiguities related to infrared (IR) renormalons. For ${Q}^{2}<{\ensuremath{\Lambda}}^{2}$ the skeleton expansion result is equivalent to a previously proposed modified Borel representation where the ambiguities are connected with ultraviolet (UV) renormalons. We investigate the ${Q}^{2}$-dependence of the perturbative corrections to the Adler-$D$ function, the GLS sum rule and the polarized and unpolarized Bjorken sum rules. In all these cases the one-chain result changes sign in the vicinity of ${Q}^{2}={\ensuremath{\Lambda}}^{2}$, and then exhibits freezing behavior, vanishing at ${Q}^{2}=0$. Finiteness at ${Q}^{2}={\ensuremath{\Lambda}}^{2}$ implies specific relations between the residues of IR and UV renormalon singularities in the Borel plane. These relations, only one of which has previously been noted (though it remained unexplained), are shown to follow from the continuity of the characteristic function in the skeleton expansion. By considering the compensation of nonperturbative and perturbative ambiguities we are led to a result for the ${Q}^{2}$-dependence of these observables at all ${Q}^{2}$, in which there is a single undetermined nonperturbative parameter, and which involves the skeleton expansion characteristic function. The observables freeze to zero in the infrared. We briefly consider the freezing behavior of the Minkowskian ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ ratio.

Renormalons and analytic properties of the β function

The presence or absense of renormalon singularities in the Borel plane is shown to be determined by the analytic properties of the Gell-Mann - Low function \beta(g) and some other functions. A constructive criterion for the absense of singularities consists in the proper behavior of the \beta function and its Borel image B(z) at infinity, \beta(g)\sim g^\alpha and B(z)\sim z^\alpha with \alpha\le 1. This criterion is probably fulfilled for the \phi^4 theory, QED and QCD, but is violated in the O(n)-symmetric sigma model with n\to\infty.