We present a formulation to give renormalon-free predictions consistently with fixed order perturbative results. The formulation has a similarity to Lee's method in that the renormalon-free part consists of two parts: one is given by a series expansion which does not contain renormalons and the other is the real part of the Borel integral of a singular Borel transform. The main novel aspect is to evaluate the latter object using a dispersion relation technique, which was possible only in the large-$β_0$ approximation. Here, we introduce an "ambiguity function," which is defined by inverse Mellin transform of the singular Borel transform. With the ambiguity function, we can rewrite the Borel integral in an alternative formula, which allows us to obtain the real part using analytic techniques similarly to the case of the large-$β_0$ approximation. We also present detailed studies of renormalization group properties of the formulation. As an example, we apply our formulation to the fixed-order result of the static QCD potential, whose closest renormalon is already visible.
We compute the static self-energy of SU(3) gauge theory in four spacetime dimensions to order α^{20} in the strong coupling constant. We employ lattice regularization to enable a numerical simulation within the framework of stochastic perturbation theory. We find perfect agreement with the factorial growth of high order coefficients predicted by the conjectured renormalon picture based on the operator product expansion.
Recently evidence has been found that the perturbative QCD potential agrees well with phenomenological potentials and lattice computations of the QCD potential. We review the present status of the perturbative QCD potential and theoretical backgrounds. We also report our recent analysis which shows analytically, on the basis of renormalon dominance picture, that the perturbative QCD potential quickly ``converges'' to a Coulomb-plus-linear form. The Coulomb-plus-linear potential can be computed systematically as we include more terms of the perturbative series; up to three-loop running (our cur…
The paper is motivated by observation of a kind of branes in the vacuum state of the lattice SU(2) gluodynamics. The branes represent two-dimensional vortices whose total area scales in physical units while the non-Abelian action diverges in the ultraviolet. We consider the question whether effects of the branes can be accommodated into the continuum theory. We demonstrate that at least in case of the gluon condensate (plaquette action) and of the heavy quark potential the contribution of the branes corresponds to the ultraviolet renormalon. Thus, the vortices might represent a non-perturbativ…
We study the operator product expansion of the plaquette (gluon condensate) and the self-energy of an infinitely heavy quark. We first compute their perturbative expansions to order $α^{35}$ and $α^{20}$, respectively, in the lattice scheme. In both cases we reach the asymptotic regime where the renormalon behavior sets in. Subtracting the perturbative series, we obtain the leading non-perturbative corrections of their respective operator product expansions. In the first case we obtain the gluon condensate and in the second the binding energy of the heavy quark in the infinite mass limit. The …
The exact mass gap of the O(N) Gross-Neveu model is known, for arbitrary $N$, from non-perturbative methods. However, a "naive" perturbative expansion of the pole mass exhibits an infinite set of infrared renormalons at order 1/N, formally similar to the QCD heavy quark pole mass renormalons, potentially leading to large ${\cal O}(Λ)$ perturbative ambiguities. We examine the precise vanishing mechanism of such infrared renormalons, which avoids this (only apparent)contradiction, and operates without need of (Borel) summation contour prescription, usually preventing unambiguous separation of pe…
We present a nonrelativistic one-particle quantum mechanics whose perturbative S-matrix exhibits a renormalon divergence that we explicitely compute. The potential of our model is the sum of the 2d Dirac $δ$-potential -- known to require renormalization -- and a 1d Dirac $δ$-potential tilted at an angle. We argue that renormalons are not specific to this example and exist for a much wider class of potentials. The ambiguity in the Borel summation of the perturbative series due to the renormalon pole is resolved by the physical condition of causality through careful consideration of the $iε$ pre…
When an asymptotically non-free theory possesses a mass parameter, the ultraviolet (UV) renormalon gives rise to non-perturbative contributions to dimension-four operators and dimensionless couplings, thus has a similar effect as the instanton. We illustrate this phenomenon in O(N) symmetric massive $λϕ^4$ model in the $1/N$ expansion. This effect of UV renormalon is briefly compared with non-perturbative corrections in the magnetic picture of the Seiberg-Witten theory.
According to standard lore, perturbative series of super-renormalizable theories have only instanton singularities. In this paper we show that two-dimensional scalar theories with a spontaneously broken $O(N)$ symmetry at the classical level, which are super-renormalizable, have an IR renormalon singularity at large $N$. Since perturbative expansions in these theories are made around the "false vacuum" in which the global symmetry is broken, this singularity can be regarded as a manifestation of the non-perturbative absence of Goldstone bosons. We conjecture that the Borel singularity in the g…
The large-order behaviour of QCD is dominated by renormalons. On the other hand renormalons do not occur in conformal theories, such as the one describing the infrared fixed-point of QCD at small beta_0 (the Banks--Zaks limit). Since the fixed-point has a perturbative realization, all-order perturbative relations exist between the conformal coefficients, which are renormalon-free, and the standard perturbative coefficients, which contain renormalons. Therefore, an explicit cancellation of renormalons should occur in these relations. The absence of renormalons in the conformal limit can thus be…
We show that the Borel resummed perturbative static potential at N_f=0 converges well, and is in a remarkable agreement with the quenched lattice calculation at distances 1/r > 660 MeV. This shows that Borel resummation is very good at handling the renormalon in the static potential (and in the pole mass), and allows one to use the pole mass in perturbative calculation of heavy quark physics.
We study resummed perturbative contributions due to a heavy top-quark. These renormalon contributions are evaluated for fermion propagators. Results for the top-quark width are given. Estimates of non-perturbative uncertainties are made on the ρ-parameter using different schemes of dealing with the Landau-pole. For the physical top-quark mass the effects are negligible.
We point out that, contrary to what is believed to hold for QCD, renormalons are genuine in QED; i.e. the ambiguities which come with them do not require cancellation by hypothetical non-perturbative contributions. They are just the ambiguities characteristic of any trivial ---and thus effective--- theory. If QED remained an isolated theory up to an energy close to its triviality scale, these ambiguities would surely hint at new physics. This not being so, the renormalon ambiguities in QED lead to no new physics, not even to non-perturbative contributions within QED itself.
In two lectures, we overview the renormalon and renormalon-related techniques and their phenomenological applications. We begin with a single renormalon chain which is a well defined and systematic way to specify the character of corrections in inverse powers of the total energy to observables directly in Minkowski space. Renormalons demonstrate also presence of nonperturbative contributions. We proceed then to multirenormalon chains and argue that they are in fact not suppressed compared to a single chain. On one hand, this phenomenon might be a mechanism for enhancement of power corrections.…
I discuss various situations in which perturbative expansions are used in Yang-Mills theories with asymptotic freedom and establish the limits of its applicability.
The basis of renormalon calculus is briefly discussed. The method is applied to study QCD predictions for three sum rules of deep-inelastic scattering, namely for the Gross-Llewellyn Smith, Bjorken polarized and unpolarized sum rules. It is shown that the renormalon structures of theses sum rules are closely related. These properties are giving us the hint that theoretical expressions of these three sum rules are similar both in the and non-perturbative sectors. Some phenomenological consequances of the new relations are discussed.
We examine the total energy E_tot(r) = 2 m_pole + V_QCD(r) of the b\bar b system within perturbative QCD up to O(α_S^3). We extend a previous analysis by incorporating effects of the non-zero charm-quark mass in loops. We find that, once the renormalon cancellation is performed, E_tot(r) agrees well with typical phenomenological potentials for heavy quarkonia at distances 0.5 GeV^-1 ~< r ~< 3 GeV^-1. We also examine the perturbative predictions for E_tot(r) of other heavy quarkonium systems. Whenever stable predictions are obtained, they agree with each other up to an r-independent constant.
A certain pattern of divergence of perturbative expansions in quantum field theories, related to their small and large momentum behaviour, is known as renormalons. We review formal and phenomenological aspects of renormalon divergence. We first summarize what is known about ultraviolet and infrared renormalons from an analysis of Feynman diagrams. Because infrared renormalons probe large distances, they are closely connected with non-perturbative power corrections in asymptotically free theories such as QCD. We discuss this aspect of the renormalon phenomenon in various contexts, and in partic…
We propose a clear definition of the gluon condensate within the large-$β_0$ approximation as an attempt toward a systematic argument on the gluon condensate. We define the gluon condensate such that it is free from a renormalon uncertainty, consistent with the renormalization scale independence of each term of the operator product expansion (OPE), and an identical object irrespective of observables. The renormalon uncertainty of $\mathcal{O}(Λ^4)$, which renders the gluon condensate ambiguous, is separated from a perturbative calculation by using a recently suggested analytic formulation. The…
A Borel resummation scheme of subtracting the perturbative contribution from the average plaquette is proposed using the bilocal expansion of Borel transform. It is shown that the remnant of the average plaquette, after subtraction of the perturbative contribution, scales as a dim-4 condensate. A critical review of the existing procedure of renormalon subtraction is presented.
We embed in a generalized Borel procedure the notion of renormalization and renormalons. While there are several efforts in literature to have a semi-classical understanding of the renormalons, here we argue that this is not the fundamental issue and show how to deal with the problem. We find that the effective Lagrangians describing the effects of renormalons are non-local in space but local in time. The quark-antiquark potential in QCD with an infinite number of fermions is also analyzed. The connection between the analyzable functions, the Callan-Symanzyk equation and the renormalons, provi…
I illustrate the role of infrared renormalons in computing inclusive B-decay spectra. I explain the relation between the leading ambiguity in the definition of Sudakov form factor \sim \exp(NΛ/M) and that of the pole mass, and show how these ambiguities cancel out between the perturbative and non-perturbative components of the b-quark distribution in the meson.
We analyze inclusive $τ$ decay using a modified version of the $a$ expansion, a non-perturbative technique in which the effective coupling is analytic in the infrared region. The modification involves renormalization group improvement of the integrand in a spectral representation for the D-function prior to implementing the $a$ expansion. The advantage of this approach is that it enables us to monitor the structure of the induced power corrections and to ensure that these are consistent with the operator product expansion. Numerically the method agrees well with experiment: the comparison is m…
Using numerical stochastic perturbation theory, we determine the first 35 infinite volume coefficients of the perturbative expansion in powers of the strong coupling constant $α$ 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 im…
In the leading order of the large-$N$ approximation, we study the renormalon ambiguity in the gluon (or, more appropriately, photon) condensate in the 2D supersymmetric $\mathbb{C}P^{N-1}$ model on~$\mathbb{R}\times S^1$ with the $\mathbb{Z}_N$ twisted boundary conditions. In our large~$N$ limit, the combination $ΛR$, where $Λ$ is the dynamical scale and $R$~is the $S^1$ radius, is kept fixed (we set $ΛR\ll1$ so that the perturbative expansion with respect to the coupling constant at the mass scale~$1/R$ is meaningful). We extract the perturbative part from the large-$N$ expression of the gluo…
Perturbative calculations of the static QCD potential have the $u=3/2$ renormalon uncertainty. In the multipole expansion performed within pNRQCD, this uncertainty at LO is known to get canceled against the ultrasoft correction at NLO. To investigate the net contribution remaining after this renormalon cancellation, we propose a formulation to separate the ultrasoft correction into renormalon uncertainties and a renormalon independent part. We focus on very short distances $Λ_{\rm QCD} r \lesssim 0.1$ and investigate the ultrasoft correction based on its perturbative evaluation in the large-$β…
Precise extractions of $α_s$ from $τ\to {\rm (hadrons)}+ν_τ$ and from $e^+e^-\to {\rm (hadrons)}$ below the charm threshold rely on finite energy sum rules (FESRs) where the experimental side is given by integrated spectral function moments. Here we study the renormalons that appear in the Borel transform of polynomial moments in the large-$β_0$ limit and in full QCD. In large-$β_0$, we establish a direct connection between the renormalons and the perturbative behaviour of moments often employed in the literature. The leading IR singularity is particularly prominent and is behind the fate of m…
Properly separating and subtracting renormalons in the framework of the operator product expansion (OPE) is a way to realize high precision computation of QCD effects in high energy physics. We propose a new method (FTRS method), which enables to subtract multiple renormalons simultaneously from a general observable. It utilizes a property of Fourier transform, and the leading Wilson coefficient is written in a one-parameter integral form whose integrand has suppressed (or vanishing) renormalons. The renormalon subtraction scheme coincides with the usual principal-value prescription at large o…
Perturbative expansions of QCD observables in powers of $α_s$ are believed to be asymptotic and non-Borel summable due to the existence of singularities in the Borel plane (renormalons). This fact is connected with the factorization of scales (which is inherent to QCD and asymptotic freedom) and jeopardizes the convergence of the perturbative expansion and the accurate determination of power-suppressed corrections. This problem is more acute for physical systems composed by one or more heavy quarks. In lattice regulations, it reflects on the appearance of power-like divergences in the inverse …
We study an effect of the off-shellness of the quark and antiquark inside a heavy quarkonium system on IR renormalons contained in the perturbative computations of the quarkonium energy levels. We demonstrate that, when the off-shellness p_Q^2 -m_Q^2 \sim alpha_S^2 m_Q^2 is larger than m_Q Lambda_QCD, renormalons in the energy levels as calculated in perturbative QCD are suppressed by a factor Lambda_QCD/alpha_S^2 m_Q as compared to those in 2 m_pole + V_QCD(r). In this case the residual O(Lambda_QCD^4) renormalon has the same dimension as that of the leading gluon-condensate contribution.
We report on the following consequences of the relations between non-singlet contributions to $g_1^N$ and $F_1^N$ structure functions found within infrared renormalon model: the discovery of new next-to-leading order inequalities between non-singlet polarized and unpolarized parton densities, and the existence of the effects of similarity between non-perturbative and perturbative contributions to the unpolarized and polarized Bjorken sum rules.
The Borel summation technique of infrared renormalons is applied to the lattice determination of heavy quark mass. With Borel summation a physical heavy quark pole mass and binding energy of a heavy-light meson can be defined in a rigorous and calculable manner. A notable feature of the Borel summation, compared to the usual perturbative cancellation of IR renormalons, is an automatic scale separation. The two approaches of handling renormalon divergence are compared in the B-meson as well as in an (imaginary) heavy-light meson with a mass much larger than the inverse of the lattice spacing.
In these lectures we give a concise introduction to the ideas of renormalon calculus in QED and QCD. We focus in particular on the example of the Adler D function of vacuum polarization, and on relations between perturbative renormalon ambiguities and corresponding non-perturbative Operator Product Expansion (OPE) ambiguities. Recent work on infrared freezing of Euclidean observables is also discussed.
We calculate the leading $1/N_f$ perturbative contributions to the polarized nonsinglet structure functions $g_3$ and $g_5$ to all orders in $α_s$. The contributions from the first renormalon pole are determined. It is a measure for the ambiguity of the perturbative calculation and is assumed to dominate the power corrections. The corrections $Δg_3$ and $Δg_5$ are given as functions of the Bjorken variable $x$ and turn out to be negligable. The anomalous dimensions of the leading twist operators are obtained in the next-to-leading order.
We propose a simple and easy-to-implement scheme for a renormalon-free gluon condensate (GC) matrix element, which is analogous to implementations of short-distance heavy-quark mass renormalization schemes existing in the literature already for a long time. Because the scheme is based on a perturbative subtraction at the level of the matrix element, with a freely adaptable infrared factorization scale, it can be implemented with little effort for any observable where the GC is relevant. The scheme depends on the renormalon norm of the GC which has to be supplemented independently. We apply the…
We review recent studies of the operator product expansion of the plaquette and of the associated determination of the gluon condensate. One first needs the perturbative expansion to orders high enough to reach the asymptotic regime where the renormalon behavior sets in. The divergent perturbative series is formally regulated using the principal value prescription for its Borel integral. Subtracting the perturbative series truncated at the minimal term, we obtain the leading non-perturbative correction of the operator product expansion, i.e. the gluon condensate, with superasymptotic accuracy.…
I illustrate a recent work on the large-order behaviour of the perturbative expansion (and the related power-suppressed ambiguities) arising from infrared renormalons, in the context of top mass measurements in open-top production processes.
Infrared power corrections for the average thrust < T > in e+e- annihilation are analyzed in the framework of renormalon resummation, motivated by analogy with the skeleton expansion in QED and the BLM approach. Performing the ``massive gluon'' renormalon integral a renormalization scheme invariant result is obtained. We find that a major part of the discrepancy between the known next-to-leading order (NLO) calculation and experiment can be explained by resummation of higher order perturbative terms. This fact does not preclude the infrared finite coupling interpretation with a substantial 1/Q…
In a recent work by some of us it was shown that the long-standing discrepancy between the QCD perturbation series for the inclusive hadronic tau decay rate computed in the CIPT and FOPT expansion approaches can be understood from the fact that CIPT has an infrared (IR) sensitivity that it not compatible with the standard form of the operator production expansion (OPE). For concrete IR renormalon models for the QCD Adler function the resulting CIPT-FOPT discrepancy, the asymptotic separation, can be calculated analytically from the Borel representation of the CIPT series expansion. If the know…
As higher order perturbative series are available, it is becoming necessary to include nonperturbative effects in QCD calculations using the OPE. In order to systematically determine nonperturbative effects and to incorporate them into theoretical calculations, the renormalon problem should be resolved. We use a renormalon subtraction method utilizing Fourier transform to determine nonperturbative matrix elements of HQET, $\barΛ$ and $μ_π^2$. This is the first determination performed with subtraction of the $u=1$ renormalon.
In integrable field theories in two dimensions, the Bethe ansatz can be used to compute exactly the ground state energy in the presence of an external field coupled to a conserved charge. We generalize previous results by Volin and we extract analytic results for the perturbative expansion of this observable, up to very high order, in various asymptotically free theories: the non-linear sigma model and its supersymmetric extension, the Gross--Neveu model, and the principal chiral field. We study the large order behavior of these perturbative series and we give strong evidence that, as expected…
We study the free energy of integrable, asymptotically free field theories in two dimensions coupled to a conserved charge. We develop methods to obtain analytic expressions for its trans-series expansion, directly from the Bethe ansatz equations, and we use this result to determine the structure of its Borel singularities. We find a new class of infrared renormalons which does not fit the traditional expectations of renormalon physics proposed long ago by 't Hooft and Parisi. We check the existence of these new singularities with detailed calculations based on the resurgent analysis of the pe…
Precise theoretical predictions are a key ingredient for an accurate determination of the structure of the Langrangian of particle physics, including its free parameters, which summarizes our understanding of the fundamental interactions among particles. Furthermore, due to the absence of clear new-physics signals, precise theoretical calculations are required in order to pin down possible subtle deviations from the Standard Model predictions. The error associated with such calculations must be scrutinized, as non-perturbative power corrections, dubbed infrared renormalons, can limit the ultim…
We study the large order behaviour in perturbation theory of the Bjorken, Ellis-Jaffe and Gross-Llewellyn-Smith sum rules. In particular, we consider their first infrared renormalons, for which we obtain their analytic structure with logarithmic accuracy and also an approximate determination of their normalization constant. Estimates of higher order terms of the perturbative series are given. The Renormalon subtracted scheme is worked out for these observables and compared with experimental data. Overall, good agreement with experiment is found. This allows us to obtain {\hat a}_0 and some hig…
For precise QCD prediction of observables, the ambiguity due to renormalons in perturbative calculations should be appropriately separated from Wilson coefficients in the framework of the operator-product-expansion. Recently, a new method has been developed which utilizes the properties of Fourier transform to separate multiple renormalons simultaneously from the Wilson coefficients. To understand how this method works analytically, we perform a renormalon separation from various toy models with the one-loop beta function. We confirm that each of the results is consistent with the theoretical …
Perturbative expansions for short-distance quantities in QCD are factorially divergent and this deficiency can be turned into a useful tool to investigate nonperturbative corrections. In this work, we use this approach to study the structure of power corrections to parton quasi-distributions and pseudo-distributions which appear in lattice calculations of parton distribution functions. As the main result, we predict the functional dependence of the leading power corrections to quasi(pseudo)-distributions on the Bjorken $x$ variable. We also show that these corrections can be strongly affected …
