Renormalon Singularities Research Articles

H→γγ to all orders in αs in the large- β0 limit of QCD

In this note, we present the result for the QCD corrections to the decay of the Higgs boson into two photons in the large-$\beta_0$ limit of QCD, providing the first two terms in the heavy-top expansion. From our results, one can easily read off the exact leading-$n_f$ QCD contributions in analytic form to all orders in the strong coupling, $\alpha_s$, where $n_f$ is the number of massless quarks, and identify the leading renormalon singularities. We give explicit results for the leading-$n_f$ coefficients at 6 and 7 loops and use the large-$\beta_0$ result to speculate about the size of yet unknown (but small) higher-order contributions to the QCD series.

Resonant resurgent asymptotics from quantum field theory

We perform an all-order resurgence analysis of a quantum field theory renormalon that contributes to an anomalous dimension in six-dimensional scalar ϕ3 theory and is governed by a third-order nonlinear differential equation. We augment the factorially divergent perturbative expansion associated to the renormalon by asymptotic expansions to all instanton orders, in a conjectured and well-tested formula. A distinctive feature of this renormalon singularity is the appearance of logarithmic terms, starting at second-instanton order in the trans-series. To highlight this and to illustrate our methods, we also analyze the trans-series for a closely related second-order nonlinear differential equation that exhibits a similarly resonant structure but lacks logarithmic contributions.

Small-momentum expansion of heavy-quark correlators in the large-\u03b20 limit and \u03b1s extractions

We calculate the small-momentum expansion of vector, axial-vector, scalar, and pseudo-scalar heavy-quark current correlators in the large-β0 limit of QCD, extending the analysis of Grozin and Sturm beyond the vector current. Our results are used to study the higher-order behaviour of dimensionless ratios of vector and pseudo-scalar moments used for the precise extraction of the strong coupling, αs, from relativistic quarkonium sum rules and lattice data, respectively. We show that these ratios benefit from a partial cancellation of the leading renormalon singularities. Our results can guide the design of combinations of moments with improved perturbative behaviour.

Renormalons beyond the Borel plane

The renormalon singularities are a known source of the divergent behavior of asymptotic perturbative series from field theoretical models. These singularities live in the Borel plane and are responsible for ambiguities in the physical plane. We propose that field theories can have renormalons beyond the usual first Borel plane. We show an example with a scalar field theory where, considering a chain of cat's eyes diagrams, the model develops a Gevrey-3 asymptotic series.

Towards the QED beta function and renormalons at 1/Nf2 and 1/Nf3

We determine the $1/N_f^2$ and $1/N_f^3$ contributions to the QED beta function stemming from the closed set of nested diagrams. At order $1/N_f^2$ we discover a new logarithmic branch-cut closer to the origin when compared to the $1/N_f$ results. The same singularity location appears at $1/N_f^3$, and these correspond to a UV renormalon singularity in the finite part of the photon two-point function.

A new renormalon in two dimensions

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 ground state energy of the Lieb-Liniger model is a non-relativistic manifestation of this phenomenon. We also provide en passant a detailed perturbative calculation of the Lieb-Liniger energy up to two-loops, and we check that it agrees with the prediction of the Bethe ansatz.

Renormalons in integrated spectral function moments and αs extractions

Precise extractions of $\alpha_s$ from $\tau\to {\rm (hadrons)}+\nu_\tau$ 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-$\beta_0$ limit and in full QCD. In large-$\beta_0$, 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 moments whose perturbative series are unstable, while those with good perturbative behaviour benefit from partial cancellations of renormalon singularities. The conclusions can be extended to QCD through a convenient scheme transformation to the $C$-scheme together with the use of a modified Borel transform which make the results particularly simple; the leading IR singularity becomes a simple pole, as in large-$\beta_0$. Finally, for the moments that display good perturbative behaviour, we discuss an optimized truncation based on renormalisation scheme (or scale) variation. Our results allow for a deeper understanding of the perturbative behaviour of integrated spectral function moments and provide theoretical support for low-$Q^2$ $\alpha_s$ determinations.

Infrared renormalon in the supersymmetric $\mathbb{C}P^{N-1}$ model on $\mathbb{R}\times S^1$

Abstract 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 $\Lambda R$, where $\Lambda$ is the dynamical scale and $R$ is the $S^1$ radius, is kept fixed (we set $\Lambda R\ll1$ 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 gluon condensate and obtain the corresponding Borel transform $B(u)$. For $\mathbb{R}\times S^1$, we find that the Borel singularity at $u=2$, which exists in the system on the uncompactified $\mathbb{R}^2$ and corresponds to twice the minimal bion action, disappears. Instead, an unfamiliar renormalon singularity emerges at $u=3/2$ for the compactified space $\mathbb{R}\times S^1$. The semi-classical interpretation of this peculiar singularity is not clear because $u=3/2$ is not dividable by the minimal bion action. It appears that our observation for the system on $\mathbb{R}\times S^1$ prompts reconsideration on the semi-classical bion picture of the infrared renormalon.

Renormalon structure in compactified spacetime

Abstract We point out that the location of renormalon singularities in theory on a circle-compactified spacetime $\mathbb{R}^{d-1} \times S^1$ (with a small radius $R \Lambda \ll 1$) can differ from that on the non-compactified spacetime $\mathbb{R}^d$. We argue this under the following assumptions, which are often realized in large-$N$ theories with twisted boundary conditions: (i) a loop integrand of a renormalon diagram is volume independent, i.e. it is not modified by the compactification, and (ii) the loop momentum variable along the $S^1$ direction is not associated with the twisted boundary conditions and takes the values $n/R$ with integer $n$. We find that the Borel singularity is generally shifted by $-1/2$ in the Borel $u$-plane, where the renormalon ambiguity of $\mathcal{O}(\Lambda^k)$ is changed to $\mathcal{O}(\Lambda^{k-1}/R)$ due to the circle compactification $\mathbb{R}^d \to \mathbb{R}^{d-1} \times S^1$. The result is general for any dimension $d$ and is independent of details of the quantities under consideration. As an example, we study the $\mathbb{C} P^{N-1}$ model on $\mathbb{R} \times S^1$ with $\mathbb{Z}_N$ twisted boundary conditions in the large-$N$ limit.

Renormalization-group improvement in hadronic $τ$ decays in 2018

One of the main sources of theoretical uncertainty in the extraction of the strong coupling from hadronic tau decays stems from the renormalization group improvement of the series. Perturbative series in QCD are divergent but are (most likely) asymptotic expansions. One needs knowledge about higher orders to be able to choose the optimal renormalization-scale setting procedure. Here, we discuss the use of Padé approximants as a model-independent and robust method to extract information about the higher-order terms. We show that in hadronic \tauτ decays the fixed-order expansion, known as fixed-order perturbation theory (FOPT), is the most reliable mainstream method to set the scale. This fully corroborates previous conclusions based on the available knowledge about the leading renormalon singularities of the perturbative series.

Direct Photon Production in Pion-Proton Collisions

In this article we calculate the contribution of the higher-twist Feynman diagrams to the large-pT inclusive single direct photon production cross section in πp collisions in case of the running coupling and frozen coupling approaches within holographic QCD. We determined the structure of infrared renormalon singularities of the higher-twist subprocess cross section. Also compared and analyzed the resummed higher-twist cross sections with the ones obtained in the framework of the frozen coupling approach.

Higher-Twist Mechanism and Inclusive Gluon Production in Pion-Proton Collisions

We calculate the contribution of the higher-twist Feynman diagrams to the large-pT inclusive gluon production cross section in πp collisions incase of the running coupling and frozen coupling approaches within perturbative and holographic QCD. The structure of infrared renormalon singularities of the higher-twist subprocess cross section are obtained and the resummed higher-twist cross sections (Borel sum) with the ones obtained in the framework of the frozen coupling approach and leading-twist cross section are compared and analyzed.

Higher-twist mechanism and inclusive gluon production in pion–proton collisions

We calculate the contribution of the higher-twist Feynman diagrams to the large-[Formula: see text] inclusive gluon production cross-section in [Formula: see text] collisions in case of the running coupling and frozen coupling approaches within perturbative and holographic QCD. The structure of infrared renormalon singularities of the higher-twist subprocess cross-section is obtained and the resummed higher-twist cross-sections (Borel sum) with the ones obtained in the framework of the frozen coupling approach and leading-twist cross-section are compared and analyzed.

PERTURBATIVE EXPANSION OF τ HADRONIC SPECTRAL FUNCTION MOMENTS

In the extraction of αs from hadronic τ decay data several moments of the spectral functions have been employed. Furthermore, different renormalization group improvement (RGI) frameworks have been advocated, leading to conflicting values of αs. Recently, we performed a systematic study of the perturbative behavior of these moments in the context of the two main-stream RGI frameworks: Fixed Order Perturbation Theory (FOPT) and Contour Improved Perturbation Theory (CIPT). The yet unknown higher order coefficients of the perturbative series were modelled using the available knowledge of the renormalon singularities of the QCD Adler function. We were able to show that within these RGI frameworks some of the commonly employed moments should be avoided due to their poor perturbative behavior. Furthermore, under reasonable assumptions about the higher order behavior of the perturbative series FOPT provides the preferred RGI framework.

Higher-twist effects in the direct photon production and the role of infrared renormalons

We calculate the contribution of the higher-twist effects to the large-${p}_{T}$ inclusive single direct photon production cross section in $\ensuremath{\pi}p$ collisions in the running coupling and frozen coupling approaches within holographic QCD. We obtain the structure of infrared renormalon singularities of the higher-twist subprocess cross section. For a full analysis we compare the resummed higher-twist cross sections with the ones obtained in the framework of the frozen coupling approach and leading-twist cross section.

Perturbative expansion of τ hadronic spectral function moments and α s extractions

Various moments of the hadronic spectral functions have been employed in the determination of the strong coupling α s from tau decays. In this work we study the behaviour of their perturbative series under different assumptions for the large-order behaviour of the Adler function, extending previous work on the tau hadronic width. We find that the moments can be divided into a small number of classes, whose characteristics depend only on generic features of the moment weight function and Adler function series. Some moments that are commonly employed in α s analyses from τ decays should be avoided because of their perturbative instability. This conclusion is corroborated by a simplified α s extraction from individual moments. Furthermore, under reasonable assumptions for the higher-order behaviour of the perturbative series, fixed-order perturbation theory (FOPT) provides the preferred framework for the renormalization group improvement of all moments that show good perturbative behaviour. Finally, we provide further evidence for the plausibility of the description of the Adler function in terms of a small number of leading renormalon singularities.

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.

Meson wave functions from holographic QCD and the role of infrared renormalons in proton–proton collisions

We calculate the contribution of the higher-twist Feynman diagrams to the large-pT inclusive pion production cross section in proton–proton collisions in the cases of the running and frozen coupling approaches within holographic QCD. The structure of the infrared renormalon singularities of the higher-twist subprocess cross section and its resummed expression are found. We compare the resummed higher-twist cross sections with the ones obtained in the framework of the frozen coupling approximation and the leading-twist cross section. We discuss the phenomenological consequences of possible higher-twist contributions to the pion production in proton–proton collisions within holographic QCD.

MESON PRODUCTION IN PROTON–PROTON COLLISIONS IN THE NAIVE NONABELIANIZATION APPROXIMATION AND THE ROLE OF INFRARED RENORMALONS

We calculate the "naive nonabelianization" (NNA) contributions of the higher-twist Feynman diagrams to the large-pT inclusive pion production cross-section in proton–proton collisions in the case of the running coupling and frozen coupling approaches. We compare the resummed "naive nonabelianization" higher-twist cross-sections with the ones obtained in the framework of the frozen coupling approach and leading-twist cross-section. The structure of infrared renormalon singularities of the higher-twist subprocess cross-section and it's resummed expression are found. We discuss the phenomenological consequences of possible higher-twist contributions to the pion production in proton–proton collisions in within NNA.

THE MESON PRODUCTION IN PROTON–PROTON COLLISIONS IN NEXT-TO-LEADING ORDER AND INFRARED RENORMALONS

In this paper, we investigate the next-to-leading order contribution of the higher-twist Feynman diagrams to the large-pT inclusive pion production cross-section in proton–proton collisions and present the general formulae for the higher-twist differential cross-sections in the case of the running coupling and frozen coupling approaches. We compared the resummed next-to-leading order higher-twist cross-sections with the ones obtained in the framework of the frozen coupling approach and leading-twist cross-section. The structure of infrared renormalon singularities of the higher-twist subprocess cross-section and its resummed expression (the Borel sum) are found. It is shown that the resummed result depends on the choice of the meson wave functions used in the calculations. We discuss the phenomenological consequences of possible higher-twist contributions to the meson production in proton–proton collisions in next-to-leading order at RHIC.