Cuts for two-body decays at colliders

Fixed-order perturbative calculations of fiducial cross sections for two-body decay processes at colliders show disturbing sensitivity to unphysically low momentum scales and, in the case of H → γγ in gluon fusion, poor convergence. Such problems have their origins in an interplay between the behaviour of standard experimental cuts at small transverse momenta (pt) and logarithmic perturbative contributions. We illustrate how this interplay leads to a factorially divergent structure in the perturbative series that sets in already from the first orders. We propose simple modifications of fiducial cuts to eliminate their key incriminating characteristic, a linear dependence of the acceptance on the Higgs or Z-boson pt, replacing it with quadratic dependence. This brings major improvements in the behaviour of the perturbative expansion. More elaborate cuts can achieve an acceptance that is independent of the Higgs pt at low pt, with a variety of consequent advantages.

Resurgence and 1/N Expansion in Integrable Field Theories

In theories with renormalons the perturbative series is factorially divergent even after restricting to a given order in 1/N, making the 1/N expansion a natural testing ground for the theory of resurgence. We study in detail the interplay between resurgent properties and the 1/N expansion in various integrable field theories with renormalons. We focus on the free energy in the presence of a chemical potential coupled to a conserved charge, which can be computed exactly with the thermodynamic Bethe ansatz (TBA). In some examples, like the first 1/N correction to the free energy in the non-linear sigma model, the terms in the 1/N expansion can be fully decoded in terms of a resurgent trans-series in the coupling constant. In the principal chiral field we find a new, explicit solution for the large N free energy which can be written as the median resummation of a trans-series with infinitely many, analytically computable IR renormalon corrections. However, in other examples, like the Gross-Neveu model, each term in the 1/N expansion includes non-perturbative corrections which can not be predicted by a resurgent analysis of the corresponding perturbative series. We also study the properties of the series in 1/N. In the Gross-Neveu model, where this is convergent, we analytically continue the series beyond its radius of convergence and show how the continuation matches with known dualities with sine-Gordon theories.

Critical behavior of the 2d scalar theory: resumming the N8LO perturbative mass gap

We apply the optimized perturbation theory (OPT) to resum the perturbative series describing the mass gap of the bidimensional ϕ4 theory in the ℤ2 symmetric phase. Already at NLO (one loop) the method is capable of generating a quite reasonable non-perturbative result for the critical coupling. At order-g7 we obtain gc = 2.779(25) which compares very well with the state of the art N8LO result, gc = 2.807(34). As a novelty we investigate the supercritical region showing that it contains some useful complimentary information that can be used in extrapolations to arbitrarily high orders.

Stochastic computation of g − 2 in QED

We perform a numerical computation of the anomalous magnetic moment (g − 2) of the electron in QED by using the stochastic perturbation theory. Formulating QED on the lattice, we develop a method to calculate the coefficients of the perturbative series of g − 2 without the use of the Feynman diagrams. We demonstrate the feasibility of the method by performing a computation up to the α3 order and compare with the known results. This program provides us with a totally independent check of the results obtained by the Feynman diagrams and will be useful for the estimations of not-yet-calculated higher order values. This work provides an example of the application of the numerical stochastic perturbation theory to physical quantities, for which the external states have to be taken on-shell.

On the perturbative expansion of exact bi-local correlators in JT gravity

We study the perturbative series associated to bi-local correlators in Jackiw-Teitelboim (JT) gravity, for positive weight λ of the matter CFT operators. Starting from the known exact expression, derived by CFT and gauge theoretical methods, we reproduce the Schwarzian semiclassical expansion beyond leading order. The computation is done for arbitrary temperature and finite boundary distances, in the case of disk and trumpet topologies. A formula presenting the perturbative result (for λ ∈ ℕ/2) at any given order in terms of generalized Apostol-Bernoulli polynomials is also obtained. The limit of zero temperature is then considered, obtaining a compact expression that allows to discuss the asymptotic behaviour of the perturbative series. Finally we highlight the possibility to express the exact result as particular combinations of Mordell integrals.

Tree weights and renormalization in QFT

In this chapter we define specific tree weights which appear natural when considering a certain approach to non-perturbative renormalization in QFT, namely the constructive renormalization. Several examples of such tree weights are explicitly given in Appendix A. A fundamental step in QFT is to compute the logarithm of functional integrals used to define the partition function of a given model This comes from a fundamental theorem of enumerative combinatorics, stating the logarithm counts the connected objects. The main advantage of the perturbative expansion of a QFT into a sum of Feynman amplitudes is to perform this computation explicitly: the logarithm of the functional integral is the sum of Feynman amplitudes restricted to connected graphs. The main disadvantage is that the perturbative series indexed by Feynman graphs typically diverges.

Large order behaviour of perturbation theory

In quantum field theory (QFT), the main analytic tool to calculate physical quantities is the perturbative expansion. Following, Dyson's intuitive argument, the divergence of perturbative series was demonstrated in some models of quantum mechanics (QM) with polynomial potentials, using the Schrödinger equation. Later, it was proposed to study the problem within a path integral formulation. A systematic method in field theory was proposed by Lipatov, using the field integral representation of the φ4 4 field theory and instantons. It can be shown that the ground-state energy of the quartic anharmonic oscillator is analytic in a cut-plane. The imaginary part of the energy on the cut is related to barrier penetration. The behaviour of the perturbative coefficients at large orders is related to the behaviour of the imaginary part for small and negative coupling and can be obtained by instanton methods. The method has been generalized to the class of potentials for which (in general complex) instanton contributions have been calculated. The same method can be readily applied to boson field theories, while the extension to field theories involving fermions, like Quantum QED, requires additional considerations. The general conclusion is that, in QFT, all perturbative series, expanded in terms of a loop-expansion parameter, are divergent series.

Understanding Phase Transitions in Supersymmetric Quantum Electrodynamics With Resurgence Theory

Using resurgence theory to describe phase transitions in quantum field theory shows that information on non-perturbative effects like phase transitions can be obtained from a perturbative series expansion.

Hyperasymptotic approximation to the plaquette and determination of the gluon condensate

We give the hyperasymptotic expansion of the plaquette with a precision that includes the terminant associated to the leading renormalon. Subleading effects are also considered. The perturbative series is regulated using the principal value prescription for its Borel integral. We use this analysis to give a determination of the gluon condensate in SU(3) pure gluodynamics that is independent of the scale and renormalization scheme used for the coupling constant: $$ {\left\langle {G}^2\right\rangle}_{\mathrm{PV}}\left({n}_f=0\right)=3.15(18){r}_0^{-4} $$ G 2 PV n f = 0 = 3.15 18 r 0 − 4 .