On the asymptotic transition to complexity in quantum chromodynamics

Quantum chromodynamics (QCD) is a renormalizable gauge theory that successfully describes the fundamental interaction of quarks and gluons. The rich dynamical content of QCD is manifest, for example, in the spectroscopy of complex hadrons or the emergence of quark–gluon plasma. There is a fair amount of uncertainty regarding the behavior of perturbative QCD in the infrared and far ultraviolet regions. Our work explores these two domains of QCD using non-linear dynamics and complexity theory. We find that local bifurcations of the renormalization flow destabilize asymptotic freedom and induce a steady transition to chaos in the far ultraviolet limit. We also conjecture that, in the infrared region, dissipative non-linearity of the renormalization flow supplies a natural mechanism for confinement.     

Renormalization group approach to function approximation and to improving subsequent approximations

We establish a relation between bijective functions and renormalization group transformations and find their renormalization group invariants. For these functions, taking into account that they are globally one-to-one, we propose several improved approximations (compared with the power series expansion) based on this relation. We propose using the obtained approximations to improve the subsequent approximations of physical quantities obtained, in particular, by one of the main calculation techniques in theoretical physics, i.e., by perturbation theory. We illustrate the effectiveness of the renormalization group approximation with several examples: renormalization group approximations of several analytic functions and calculation of the nonharmonic oscillator ground-state energy. We also generalize our approach to the case of set maps, both continuous and discrete.     

Perturbation Theory and the Analytic Approach in the Context of the Inclusive τ-Lepton Decay

We perform a comparative analysis of different forms of pertubative expansions in spacelike and timelike regions. In the context of the inclusive -lepton decay, we compare the results obtained using the standard perturbation theory and the Shirkov–Solovtsov analytic approach, which modifies the perturbative expansions such that the new approximations reflect basic principles of the theory, such as renormalization invariance, spectrality, and causality. We show the advantages and self-consistency of the analytic approach in describing the -lepton decay.     

An Ansatz for the asymptotics of hypergeometric multisums

Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the asymptotic expansion of such sequences is to find a recurrence satisfied by them, convert it into a differential equation satisfied by their generating series, and analyze the singularities in the complex plane. We propose a shortcut by constructing directly from the structure of the hypergeometric term a finite set, for which we conjecture (and in some cases prove) that it contains all the singularities of the generating series. Our construction of this finite set is given by the solution set of a balanced system of polynomial equations of a rather special form, reminiscent of the Bethe ansatz. The finite set can also be identified with the set of critical values of a potential function, as well as with the evaluation of elements of an additive K-theory group by a regulator function. We give a proof of our conjecture in some special cases, and we illustrate our results with numerous examples.     

Renormalization group, causality, and nonpower perturbation expansion in QFT

The structure of the QFT expansion is studied in the framework of a new “invariant analytic” version of the perturbative QCD. Here, an invariant coupling constant α(Q ² /Λ ² ) = β ₁ α_s(Q ² )/(4π) becomes a Q ² -analytic invariant function α _an (Q²/Λ ² ) ≡A(x), which, by construction, is free of ghost singularities because it incorporates some nonperturbative structures. In the framework of the “analyticized” perturbation theory, an expansion for an observable F, instead of powers of the analytic invariant charge A(x), may contain specific functions A_n(x)=[aⁿ(x)] _an , the “nth power of a(x) analyticized as a whole.” Functions A _n>2(x) for small Q² ≤Λ ² oscillate, which results in weak loop and scheme dependences. Because of the analyticity requirement, the perturbation series for F(x) becomes an asymptotic expansion à la Erdélyi using a nonpower set {A _n (x)}. The probable ambiguities of the invariant analyticization procedure and the possible inconsistency of some of its versions with the renormalization group structure are also discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 55–66, April, 1999.     

Analytic Invariant Charge in QCD with Suppression of Nonperturbative Contributions at Large <Emphasis Type="Italic">Q</Emphasis><Superscript>2</Superscript>

Based on the analytic invariant charge obtained from the results of the standard perturbation theory up to the four-loop approximation, we construct a “synthetic” model of the invariant charge in quantum chromodynamics. In the suggested model, the perturbative discontinuity on the timelike semiaxis in the complex Q² plane is preserved, and nonperturbative contributions not only cancel nonphysical perturbation theory singularities in the infrared region but also rapidly decrease in the ultraviolet region. On one hand, the effective coupling function in this model is enhanced at zero (the dual superconductivity property of the quantum chromodynamics vacuum); on the other hand, a dynamical gluon mass appears. In our approach, fixing the parameter corresponding to the string tension parameter and normalizing (for example, at the point M_τ) entirely fix the synthetic invariant-charge model. The dynamical gluon mass m_g is then fixed and is stable as the number of loops of the original perturbative approximation increases. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 221–239, November, 2005.     

Renormalization in classical mechanics and many body quantum field theory

We attempt to give apedagogical introduction to perturbative renormalization. Our approach is to first describe, following Linstedt and Poincaré, the renormalization of formal perturbation expansions for quasi-periodic orbits in Hamiltonian mechanics. We then discuss, following [FT1, FT2], the renormalization of the formal ground state energy density of a many Fermion system. The construction of formal quasi-periodic orbits is carried out in detail to provide a relatively simple model for the considerably more involved, and perhaps less familiar, perturbative analysis of a field theory. As we shall see, quasi-periodic orbits and many Fermion systems have a number of important features in common. In particular, as Poincaré observed in the classical case and [FT1, FT2] pointed out in the latter, the formal expansions considered here both contain divergent subseries. Dedicated to Professor Shmuel Agmon Research supported in part by the Natural Sciences and Engineering Research Council of Canada.     

Analytic continuation of Dirichlet-Neumann operators

Summary. The analytic dependence of Dirichlet-Neumann operators (DNO) with respect to variations of their domain of definition has been successfully used to devise diverse computational strategies for their estimation. These strategies have historically proven very competitive when dealing with small deviations from exactly solvable geometries, as in this case the perturbation series of the DNO can be easily and recursively evaluated. In this paper we introduce a scheme for the enhancement of the domain of applicability of these approaches that is based on techniques of analytic continuation. We show that, in fact, DNO depend analytically on variations of arbitrary smooth domains. In particular, this implies that they generally remain analytic beyond the disk of convergence of their power series representations about a canonical separable geometry. And this, in turn, guarantees that alternative summation mechanisms, such as Padé approximation, can be effectively used to numerically access this extended domain of analyticity. Our method of proof is motivated by our recent development of stable recursions for the coefficients of the perturbation series. Here, we again utilize this recursion as we compare and contrast the performance of our new algorithms with that of previously advanced perturbative methods. The numerical results clearly demonstrate the beneficial effect of incorporating analytic continuation procedures into boundary perturbation methods. Moreover, the results also establish the superior accuracy and applicability of our new approach which, as we show, allows for precise calculations corresponding to very large perturbations of a basic geometry. Received October 10, 2000 / Revised version received January 21, 2002 / Published online June 17, 2002     

On Gevrey solutions of threefold singular nonlinear partial differential equations

We study Gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain. We deal with both Fuchsian and irregular singularities, and allow the presence of a singular perturbation parameter. By means of the Borel–Laplace summation method, we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their Gevrey asymptotic expansion in the perturbation parameter. This result rests on the Malgrange–Sibuya theorem, and it requires to prove that the difference between two neighboring solutions is exponentially small, what in this case involves an asymptotic estimate for a particular Dirichlet-like series.     

Singularities of Jacobi series on C2 and the Poisson process equation

A classical theorem of Gabor Szego relates the singularities of real zonal harmonic expansions with those of associated analytic functions of a single complex variable. Zeev Nehari developed the counterpart for Legendre series on the C-plane by generalizing Szego's theorem. This paper function theretically identifies the singularities of analytic symmetric Jacobi series on C² with those of analytic functions on the C-plane. One feature is that information about the singularities of solutions of Solomon Bochner's Poisson process equation flow from the expansion coefficients. Others are that the Szego and Nehari theorems appear on characteristic subspaces. And, that this PDE, unlike those normally encountered in function theory, is hyperbolic in the real domain.     

Smooth Feshbach map and operator-theoretic renormalization group methods

A new variant of the isospectral Feshbach map defined on operators in Hilbert space is presented. It is constructed with the help of a smooth partition of unity, instead of projections, and is therefore called smooth Feshbach map. It is an effective tool in spectral and singular perturbation theory. As an illustration of its power, a novel operator-theoretic renormalization group method is described and applied to analyze a general class of Hamiltonians on Fock space. The main advantage of the new renormalization group method over its predecessors is its technical simplicity, which it owes to the use of the smooth Feshbach map.     

Influence of hydrodynamic fluctuations on the phase transition in the E and F models of critical dynamics

We use the renormalization group method to study the E model of critical dynamics in the presence of velocity fluctuations arising in accordance with the stochastic Navier-Stokes equation. Using the Martin-Siggia-Rose theorem, we obtain a field theory model that allows a perturbative renormalization group analysis. By direct power counting and an analysis of ultraviolet divergences, we show that the model is multiplicatively renormalizable, and we use a two-parameter expansion in ∈ and δ to calculate the renormalization constants. Here, ∈ is the deviation from the critical dimension four, and δ is the deviation from the Kolmogorov regime. We present the results of the one-loop approximation and part of the fixedpoint structure. We briefly discuss the possible effect of velocity fluctuations on the arge-scale behavior of the model.     

SU(3) Color Gauge Invariance and the Jaffe–Witten Mass Gap in QCD

We discuss the role of group theory, the theory of distributions, and some theorems of the theory of functions of complex variable in connection with the so-called Jaffe–Witten mass gap in QCD, which is responsible for the large-scale structure of the QCD ground state and hence plays a dominant role in QCD as a theory of strong interactions at low energies. We show how the mass gap may appear without violating the SU(3) color gauge invariance of QCD. The theory of generalized functions (distributions) and the Weierstrass–Sochocki–Casorati theorem are used in order to perform the renormalization of the regularized mass gap.     

Explicit Expressions for Timelike and Spacelike Observables of Quantum Chromodynamics in Analytic Perturbation Theory

We study the possibility of expressing the invariant QCD coupling function (i.e., the effective coupling constant) in an explicit analytic form in two- and three-loop approximations as well as in the case of the Padé-transformed -function. Both the timelike and spacelike domains are investigated. Technical aspects of the Shirkov–Solovtsov analytic perturbation theory are considered. Explicit expressions for the two- and three-loop effective coupling functions in the timelike domain are obtained. In the last case, we apply a new method of expanding functions represented in an arbitrary loop order of perturbation theory in powers of the two-loop function. The comparison with numerical data in the infrared region shows that the obtained explicit expressions for the three-loop functions agree fully with the exact numerical results.     

A Relation between formal and Asymptotic Equivalence for linear Systems Containing A Parameter

We use an elementary method to draw analytic conclusions from divergent formal power series solutions of systems of differential equations containing a parameter and give some applications to the theory of turning points. Our main result shows that a divergent formal series transformation of one system into another in which the coefficients satisfy certain estimates is necessarily the asymptotic expansion of an actual transformation. We use it to show the following. Given a two dimensional system ε^Pdy/dx = A(x,ε)y with A holomorphic at (x₀,0), suppose that x₀ is formally not a turning point in the sense that no singularities appear at x₀ during the standard formal solution procedure with formal fractional power series in ε. Then the formal solution is necessarily a uniform asymptotic representation of a fundamental matrix of the system on a full neighborhood of x₀. (This conclusion is known to fail under weaker hypotheses on A). We also obtain similar but less complete results for higher order systems under more specialized hypotheses.     

Ground States in the Spin Boson Model

We prove that the Hamiltonian of the model describing a spin which is linearly coupled to a field of relativistic and massless bosons, also known as the spin-boson model, admits a ground state for small values of the coupling constant λ. We show that the ground-state energy is an analytic function of λ and that the corresponding ground state can also be chosen to be an analytic function of λ. No infrared regularization is imposed. Our proof is based on a modified version of the BFS operator theoretic renormalization analysis. Moreover, using a positivity argument we prove that the ground state of the spin-boson model is unique. We show that the expansion coefficients of the ground state and the ground-state energy can be calculated using regular analytic perturbation theory.     

Epsilon-expansion in the N-component ϕ 4 model

The formalism of projection Hamiltonians is applied to the N-component O(N)-invariant ϕ⁴ model in the Euclidean and p-adic spaces. We use two versions of the ε-expansion (with ε = 4 − d and with ε = α − 3d/2, where α is the renormalization group parameter) and evaluate the critical indices ν and η up to the second order of the perturbation theory. The results for the (4− d)-expansion then coincide with the known results obtained via the quantum-field renormalization-group methods. Our calculations give evidence that in dimension three, both expansions describe the same non-Gaussian fixed point of the renormalization group. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 365–384, March, 2006.     

短期利率模型中隐含波动率的重构（英文）

This paper concerns an inverse problem of recovering implied volatility in shortterm interest rate model from the market prices of zero-coupon bonds. Based on linearization, an analytic solution, which is given as a power series, is derived for the direct problem.By neglecting high order terms in the power series, an integral equation about the perturbation of volatility is formulated and the Tikhonov regularization method is applied to solve the integral equation. Finally numerical experiments are given and the results show that the method is effective.     

Exponential Renormalization

Moving beyond the classical additive and multiplicative approaches, we present an “exponential” method for perturbative renormalization. Using Dyson’s identity for Green’s functions as well as the link between the Faà di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed. Eventually, we argue that the new method has several attractive features and encompasses the BPHZ method. The latter can be seen as a special case of the new procedure for renormalization scheme maps with the Rota–Baxter property. To our best knowledge, although very natural from group-theoretical and physical points of view, several ideas introduced in the present paper seem to be new (besides the exponential method, let us mention the notions of counter-factors and of order n bare coupling constants).