Arbitrary Order In Perturbation Theory Research Articles

Evolution of structure functions in momentum space

We formulate the momentum-space Dokshitzer–Gribov–Lipatov-Altarelli–Parisi (DGLAP) evolution equations for structure functions measurable in deeply inelastic scattering. We construct a six-dimensional basis of structure functions that allows for a full three flavor structure and thereby provides a way to calculate perturbative predictions for physical cross sections directly without unobservable parton distribution functions (PDFs) and without the associated scheme dependence. We derive the DGLAP equations to first non-zero order in strong coupling αs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _\ extrm{s}$$\\end{document}, but the approach can be pursued to arbitrary order in perturbation theory. We also numerically check our equations against the conventional PDF formulation.

A canonical Hamiltonian for open quantum systems

If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to be the one whose dissipator is minimal. We show that the canonical Hamiltonian thus defined is equivalent to the Hamiltonian initially defined by Lindblad, and that it is uniquely specified by requiring the dissipator’s jump operators to be traceless, extending a uniqueness result known previously in the special case of Markovian master equations. For a system weakly coupled to its environment, we give a recursive formula for computing the canonical effective Hamiltonian to arbitrary orders in perturbation theory, which we can think of as a perturbative scheme for renormalizing the system’s bare Hamiltonian.

Perturbative structure of two- and four-point functions of color charge in a non-Gaussian small-x action

We compute the perturbative expansion of the two- and four-point functions of color charges in the Color Glass Condensate framework considering the quartic correction to the McLerran–Venugopalan (MV) model of Gaussian color charge fluctuations. Expressions for these correlators in the perturbative expansion for small and large non-Gaussian color charge fluctuations are derived for arbitrary orders in perturbation theory. We explicitly show that the perturbative series does not converge at higher orders as expected. We apply the Borel–Padé resummation method to our problem to construct a convergent series. It is shown that the fully non-perturbative solution can be described by the Borel–Padé approximants constructed from the first few terms of the perturbative series for small non-Gaussian fluctuations.

Infrared problem in perturbative quantum field theory

We propose a mathematically rigorous construction of the scattering matrix and the interacting fields in models of relativistic perturbative quantum field theory with massless fields and long-range interactions. We consider quantum electrodynamics and a certain model of interacting scalar fields in which the standard definition of the scattering matrix is not applicable because of the infrared problem. We modify the Bogoliubov construction using the ideas of Dollard, Kulish and Faddeev. Our modified scattering matrix and modified interacting fields are constructed with the use of the adiabatic limit which is expected to exist in arbitrary order of perturbation theory. In the paper, we prove this assertion in the case of the first- and the second-order corrections to the modified scattering matrix and the first-order corrections to the modified interacting fields. We study the physical properties of our construction. We conclude that the electrons and positrons are always surrounded by irremovable clouds of photons. Moreover, the physical energy-momentum operators do not coincide with the standard ones and their joint spectrum does not contain the mass hyperboloid.

Derivation of the Redfield Quantum Master Equation and Corrections to It by the Bogoliubov Method

Following the ideas N. N. Bogoliubov used to derive the classical and quantum nonlinear kinetic equations, we derive the Redfield quantum linear master equation, which is widely used in the theory of open quantum systems. This method allows one to calculate the joint state of a system and a reservoir at every instant of time, as well as to obtain quantum master equations as autonomous differential equations in an arbitrary order of perturbation theory. We prove that under certain conditions the expressions for the corrections of all orders are well defined. We also discuss the question of whether the quantum dynamics described by the derived quantum master equations is Markovian.

ABJM quantum spectral curve at twist 1: algorithmic perturbative solution

We present an algorithmic perturbative solution of ABJM quantum spectral curve at twist 1 in sl (2) sector for arbitrary spin values, which can be applied to, in principle, arbitrary order of perturbation theory. We determined the class of functions- products of rational functions in spectral parameter with sums of Baxter polynomials and Hurwitz functions - closed under elementary operations, such as shifts and partial fractions, as well as differentiation. It turns out, that this class of functions is also sufficient for finding solutions of inhomogeneous Baxter equations involved. For the latter purpose we present recursive construction of the dictionary for the solutions of Baxter equations for given inhomogeneous parts. As an application of the proposed method we present the computation of anomalous dimensions of twist 1 operators at six loop order. There is still a room for improvements of the proposed algorithm related to the simplifications of arising sums. The advanced techniques for their reduction to the basis of generalized harmonic sums will be the subject of subsequent paper. We expect this method to be generalizable to higher twists as well as to other theories, such as mathcal{N} = 4 SYM.

All-orders behaviour and renormalons in top-mass observables

We study a simplified model of top production and decay, consisting in a virtual vector boson W* decaying into a massive-massless t- overline{b} quark-antiquark pair. The top has a finite width and further decays into a stable vector boson W and a b quark. We then consider the emission or the virtual exchange of one gluon, with all possible light-quark loop insertions. These are the dominant diagrams in the limit of an infinite number of light flavours. We devise a procedure to compute this process fully, by analytic and numerical methods, and for any infrared-safe final-state observables. We examine the results at arbitrary orders in perturbation theory, and assess the factorial growth associated with renormalons. We look for renormalon effects leading to corrections of order ΛQCD, that we dub “linear” renormalons, in the inclusive cross section (with and without selection cuts), in the mass of the reconstructed-top system, and in the average energy of the final-state W boson, considering both the pole and the overline{mathrm{MS}} scheme for the top mass. We find that the total cross section without cuts, if expressed in terms of the overline{mathrm{MS}} mass, does not exhibit linear renormalons, but, as soon as selection cuts are introduced, jets-related linear renormalons arise in any mass scheme. In addition, we show that the reconstructed mass is affected by linear renormalons in any scheme and that the average energy of the W boson (that we consider as a simplified example of leptonic observable), in any mass scheme, has a renormalon in the narrow-width limit, that is however screened at large orders for finite top widths, provided the top mass is in the overline{mathrm{MS}} scheme.

Generalized ladder operators for the perturbed harmonic oscillator

In this paper, we construct corrections to the raising and lowering (i.e. ladder) operators for a quantum harmonic oscillator subjected to a polynomial type perturbation of any degree and to any order in perturbation theory. We apply our formalism to a couple of examples, namely q and p4 perturbations, and obtain the explicit form of those operators. We also compute the expectation values of position and momentum for the above perturbations. This construction is essential for defining coherent and squeezed states for the perturbed oscillator. Furthermore, this is the first time that corrections to ladder operators for a harmonic oscillator with a generic perturbation and to an arbitrary order of perturbation theory have been constructed.

On the effective field theory of intersecting D3-branes

We study the effective field theory of two intersecting D3-branes with one common dimension along the lines recently proposed in ref. [1]. We introduce a systematic way of deriving the classical effective action to arbitrary orders in perturbation theory. Using a proper renormalization prescription to handle logarithmic divergencies arising at all orders in the perturbation series, we recover the first order renormalization group equation of ref. [1] plus an infinite set of higher order equations. We show the consistency of the higher order equations with the first order one and hence interpret the first order result as an exact RG flow equation in the classical theory.

Wilsonian effective action of superstring theory

By integrating out the heavy fields in type II or heterotic string field theory one can construct the effective action for the light fields. This effective theory inherits all the algebraic structures of the parent theory and the effective action automatically satisfies the Batalin-Vilkovisky quantum master equation. This theory is manifestly ultraviolet finite, has only light fields as its explicit degrees of freedom, and the Feynman diagrams of this theory reproduce the exact scattering amplitudes of light states in string theory to any arbitrary order in perturbation theory. Furthermore in this theory the degrees of freedom of light fields above certain energy scale are also implicitly integrated out. This energy scale is determined by a particular parameter labelling a family of equivalent actions, and can be made arbitrarily low, leading to the interpretation of the effective action as the Wilsonian effective action.

Associativity of the operator product expansion

We consider a recursive scheme for defining the coefficients in the operator product expansion (OPE) of an arbitrary number of composite operators in the context of perturbative, Euclidean quantum field theory in four dimensions. Our iterative scheme is consistent with previous definitions of OPE coefficients via the flow equation method or methods based on Feynman diagrams. It allows us to prove that a strong version of the “associativity” condition holds for the OPE to arbitrary orders in perturbation theory. Such a condition was previously proposed in an axiomatic setting in Hollands [SIGMA 5, 090 (2009)] and has interesting conceptual consequences: (1) One can characterise perturbations of quantum field theories abstractly in a sort of “Hochschild-like” cohomology setting, (2) one can prove a “coherence theorem” analogous to that in an ordinary algebra: the OPE coefficients for a product of two composite operators uniquely determine those for n composite operators. We concretely prove our main results for the Euclidean φ44 quantum field theory, covering also the massless case. Our methods are rather general, however, and would also apply to other, more involved, theories such as Yang-Mills theories.

Webs and posets

The non-Abelian exponentiation theorem has recently been generalised to correlators of multiple Wilson line operators. The perturbative expansions of these correlators exponentiate in terms of sets of diagrams called webs, which together give rise to colour factors corresponding to connected graphs. The colour and kinematic degrees of freedom of individual diagrams in a web are entangled by mixing matrices of purely combinatorial origin. In this paper we relate the combinatorial study of these matrices to properties of partially ordered sets (posets), and hence obtain explicit solutions for certain families of web-mixing matrix, at arbitrary order in perturbation theory. We also provide a general expression for the rank of a general class of mixing matrices, which governs the number of independent colour factors arising from such webs. Finally, we use the poset language to examine a previously conjectured sum rule for the columns of web-mixing matrices which governs the cancellation of the leading subdivergences between diagrams in the web. Our results, when combined with parallel developments in the evaluation of kinematic integrals, offer new insights into the all-order structure of infrared singularities in non-Abelian gauge theories.

Correlators, Feynman Diagrams, and Quantum No-hair in deSitter Spacetime

We provide a parametric representation for position-space Feynman integrals of a massive, self-interacting scalar field in deSitter spacetime, for an arbitrary graph. The expression is given as a multiple contour integral over a kernel whose structure is determined by the set of all trees (or forests) within the graph, and it belongs to a class of generalized hypergeometric functions. We argue from this representation that connected deSitter n-point vacuum correlation functions have exponential decay for large proper time-separation, and also decay for large spatial separation, to arbitrary orders in perturbation theory. Our results may be viewed as an analog of the so-called cosmic-no-hair theorem in the context of a quantized test scalar field. This work has significant overlap with a paper by Marolf and Morrison, which is being released simultaneously.

Massless Interacting Scalar Quantum Fields in deSitter Spacetime

We generalize key aspects of arXiv:1010.5367 (and also arXiv:1010.5327) to the case of {\em massless} $\lambda \phi^{2n}$ quantum field theory on deSitter spacetime. As in that paper, our key objective is to derive a suitable "Mellin-Barnes-type" representation of deSitter correlation functions in a deSitter-invariant state, which holds to arbitrary orders in perturbation theory, and which incorporates renormalization. The representation is suitable for the study of large distance/time properties of correlation functions. It is arrived at via an analytic continuation from the corresponding objects on the sphere, and, as in the massive case, relies on the use of graph-polynomials and their properties, as well as other tools. However, the perturbation expansion is organized somewhat differently in the massless case, due to the well-known subtleties associated with the "zero-mode" of the quantum field. In particular, the correlation functions do not possess a well-defined limit as the self-coupling constant of the field goes to zero, reflecting the well-known non-existence of a deSitter invariant state in the free massless scalar theory. We establish that generic correlation functions cannot grow more than polynomially in proper time for large time-like separations of the points. Our results thus leave open the possibility of quantum induced IR-instabilities of deSitter spacetime on very large time-scales.

Generalized scaling function at strong coupling

We considered folded spinning string in AdS_5 x S^5 background dual to the Tr(D^S Phi^J) operators of N=4 SYM theory. In the limit S,J-> \infty and l=pi J/\sqrt\lambda\log S fixed we compute the string energy with the 2-loop accuracy in the worldsheet coupling \sqrt\lambda from the asymptotical Bethe ansatz. In the limit l-> 0 the result is finite due to the massive cancelations with terms coming from the conjectured dressing phase. We also managed to compute all leading logarithm terms l^{2m}\log^n l/\lambda^n/2 to an arbitrary order in perturbation theory. In particular for m=1 we reproduced results of Alday and Maldacena computed from a sigma model. The method developed in this paper could be used for a systematic expansion in 1/\sqrt\lambda and also at weak coupling.

Consistency relations for non-Gaussianity

We investigate consistency relations for non-Gaussianity. We provide a model-independentdynamical proof for the consistency relation for three-point correlation functions from theHamiltonian and field redefinition. This relation can be applied to single-field inflation,multi-field inflation and the curvaton scenario. This relation can also be generalized ton-point correlation functions up to arbitrary order in perturbation theory and with arbitrarynumber of loops.

The Operator Product Expansion for Perturbative Quantum Field Theory in Curved Spacetime

We present an algorithm for constructing the Wilson operator product expansion (OPE) for perturbative interacting quantum field theory in general Lorentzian curved spacetimes, to arbitrary orders in perturbation theory. The remainder in this expansion is shown to go to zero at short distances in the sense of expectation values in arbitrary Hadamard states. We also establish a number of general properties of the OPE coefficients: (a) they only depend (locally and covariantly) upon the spacetime metric and coupling constants, (b) they satisfy an associativity property, (c) they satisfy a renormalization group equation, (d) they satisfy a certain microlocal wave front set condition, (e) they possess a “scaling expansion”. The latter means that each OPE coefficient can be written as a sum of terms, each of which is the product of a curvature polynomial at a spacetime point, times a Lorentz invariant Minkowski distribution in the tangent space of that point. The algorithm is illustrated in an example.

Higher Order Asymptotic Optimal Policies for Partial Equilibrium Economies

We apply perturbation theory to solve the optimal control problem of an investor with time-additive power utility over intermediate consumption and final wealth. Under general conditions we show existence of a power series repre- sentation for the prevailing optimal consumption and investment policies around a benchmark model. Each term in the power series is characterized in closed form by a recursive general formula that allows analytical computations up to an (in principle) arbitrary order in perturbation theory.

Percolating granular superconductors.

We investigate diamagnetic fluctuations in percolating granular superconductors. Granular superconductors are known to have a rich phase diagram including normal, superconducting, and spin-glass phases. Focusing on the normal-superconducting and the normal-spin-glass transition at low temperatures, we study the diamagnetic susceptibility chi((1)) and the mean square fluctuations of the total magnetic moment chi((2)) of large clusters. Our work is based on a random Josephson network model that we analyze with the powerful methods of renormalized field theory. We investigate the structural properties of the Feynman diagrams contributing to the renormalization of chi((1)) and chi((2)). This allows us to determine the critical behavior of chi((1)) and chi((2)) to arbitrary order in perturbation theory.

A theorem on the power of supersymmetry in Matrix theory

For the so-called source-probe configuration in Matrix theory, we prove the following theorem concerning the power of supersymmetry (SUSY): let δ be a quantum-corrected effective SUSY transformation operator expandable in powers of the coupling constant g as δ=∑ n⩾0 g 2 n δ ( n) , where δ (0) is of the tree-level form. Then, apart from an overall constant, the SUSY Ward identity δΓ=0 determines the off-shell effective action Γ uniquely to arbitrary order of perturbation theory, provided that the SO(9) symmetry is preserved. Our proof depends only on the properties of the tree-level SUSY transformation laws and does not require the detailed knowledge of quantum corrections.