Absence of even-integer ζ\zeta-function values in Euclidean physical quantities in QCD

At order $\alpha_s^4$ in perturbative quantum chromodynamics, even-integer $\zeta$-function values are present in Euclidean physical correlation functions like the scalar quark correlation function or the scalar gluonium correlator. We demonstrate that these contributions cancel when the perturbative expansion is expressed in terms of the so-called $C$-scheme coupling $\hat\alpha_s$ which has recently been introduced in Ref. [1]. It is furthermore conjectured that a $\zeta_4$ term should arise in the Adler function at order $\alpha_s^5$ in the $\overline{\rm MS}$-scheme, and that this term is expected to disappear in the $C$-scheme as well.Comment: 5 pages; 2 refs added, version published in Phys. Lett.

Trans-series from condensates

The Shifman-Vainshtein-Zakharov (SVZ) sum rules provide a method to obtain trans-series expansions in many quantum field theories, in which exponentially small corrections are calculated by combining the operator product expansion with the assumption of vacuum condensates. In some solvable models, exact expressions for trans-series can be obtained from non-perturbative results, and this makes it possible to test the SVZ method by comparing its predictions to these exact trans-series. In this paper we perform such a precision test in the example of the fermion self-energy in the Gross-Neveu model. Its exact trans-series expansion can be extracted from the large $N$ solution, at the first non-trivial order in $1/N$. It is given by an infinite series of exponentially small corrections involving factorially divergent power series in the 't Hooft parameter. We show that the first two corrections are associated to two-quark and four-quark condensates, and we reproduce the corresponding power series exactly, and at all loops, by using the SVZ method. In addition, the numerical values of the condensates can be extracted from the exact result, up to order $1/N$.Comment: 42 page

Large N instantons from topological strings

The $1/N$ expansion of matrix models is asymptotic, and it requires non-perturbative corrections due to large $N$ instantons. Explicit expressions for large $N$ instanton amplitudes are known in the case of Hermitian matrix models with one cut, but not in the multi-cut case. We show that the recent exact results on topological string instanton amplitudes provide the non-perturbative contributions of large $N$ instantons in generic multi-cut, Hermitian matrix models. We present a detailed test in the case of the cubic matrix model by considering the asymptotics of its $1/N$ expansion, which we obtain at relatively high genus for a generic two-cut background. These results can be extended to certain non-conventional matrix models which admit a topological string theory description. As an application, we determine the large $N$ instanton corrections for the free energy of ABJM theory on the three-sphere, which correspond to D-brane instanton corrections in superstring theory. We also illustrate the applications of topological string instantons in a more mathematical setting by considering orbifold Gromov--Witten invariants. By focusing on the example of $\mathbb{C}^3/\mathbb{Z}_3$, we show that they grow doubly-factorially with the genus and we obtain and test explicit asymptotic formulae for them.Comment: 31 page

Instantons, renormalons and the theta angle in integrable sigma models

Some sigma models which admit a theta angle are integrable at both $\vartheta=0$ and $\vartheta=\pi$. This includes the well-known $O(3)$ sigma model and two families of coset sigma models studied by Fendley. We consider the ground state energy of these models in the presence of a magnetic field, which can be computed with the Bethe ansatz. We obtain explicit results for its non-perturbative corrections and we study the effect of the theta angle on them. We show that imaginary, exponentially small corrections due to renormalons remain unchanged, while instanton corrections change sign, as expected. We find in addition corrections due to renormalons which also change sign as we turn on the theta angle. Based on these results we present an explicit non-perturbative formula for the topological susceptibility of the $O(3)$ sigma model in the presence of a magnetic field, in the weak coupling limit.Comment: version 2: 45 pages, 8 figures, minor corrections in text and equation

On the structure of trans-series in quantum field theory

Many observables in quantum field theory can be expressed in terms of trans-series, in which one adds to the perturbative series a typically infinite sum of exponentially small corrections, due to instantons or to renormalons. Even after Borel resummation of the series in the coupling constant, one has to sum this infinite series of small exponential corrections. It has been argued that this leads to a new divergence, which is sometimes called the divergence of the OPE. We show that, in some interesting examples in quantum field theory, the series of small exponential corrections is convergent, order by order in the coupling constant. In particular, we give numerical evidence for this convergence property in the case of the free energy of integrable asymptotically free theories, which has been intensively studied recently in the framework of resurgence. Our results indicate that, in these examples, the Borel resummed trans-series leads to a well defined function, and there are no further divergences.Comment: 31 pages, 7 figure

Asymptotic expansions, resurgence and large order behaviour of quantum chromodynamics

En les teories quàntiques de camps, les prediccions numèriques d’observables físics només es poden calcular amb expansions pertorbatives en potències de les constants d'acoblament, els paràmetres que determinen la força de les interaccions entre camps. Mentre que l’èxit predictiu de la teoria quàntica de camps no es pot negar, aquests càlculs pertorbatius estan plens de divergències. D’una banda, els coeficients de l’expansió pertorbativa es calculen a partir d’integrals de loops que són divergents la majoria de les vegades. Algunes d’aquestes divergències estan associades a termes no físics que es poden sostreure. En altres casos, s’aplica un procés de renormalització per cancelar aquestes divergències, però això suposa l'elecció d’un conveni teòric (escala i esquema) de la qual els observables físics no poden dependre. D’altra banda, un cop les integrals de loops han sigut renormalitzades, l’expansió resultant encara suma a una resposta infinita per tots els valors no nuls de la constant d'acoblament. Això succeeix perquè els coeficients de l’expansió creixent factorialment amb l’ordre. Tot i així, aquestes expansions es poden entendre com expansions asimptòtiques, que codifiquen el comportament de l’observable en el límit quan la constant d'acoblament s’acosta a zero, i l’observable es pot aproximar truncant l’expansió a un terme òptim. Aquest segon tipus de divergència no está limitat, de fet, a la teoria quàntica de camps, sinó que pot apareixer en diferents contextos de les matemàtiques i la física: per exemple, en expansions pertorbatives dels valors propis de l’energia d’un sistema de la mecànica quàntica, o com a solucions formals d’una equació diferencial. A la part I d’aquesta tesi, l’objecte principal d’estudi és la constant d'acoblament forta i les expansions pertorbatives d’observables físics a la quàntica chromodinàmica. Primer, discitum breument com les divergències de loops d’un gluó propagant-se a l’espai amb correccions quàntiques poden ser absorbides dins de la constant d'acoblament forta durant el procés de renormalització. Aquest procés, no obstant, implica el cost d’introduir dependències en l’escala i l’esquema dins la constant d'acoblament, per tant, aquesta no és un observable físic de la teoria. Això motiva una redefinició de la constant d'acoblament tal que la seva dependència en l’esquema es redueix a un sol paràmetre. Després utilitzem aquesta redefinició de la constant d'acoblament en anàlisis fenomenològics d’observables físics associats a dispersions electró-positró, i a la desintegració del Higgs i del tau en hadrons. Demostrem que eleccions apropiades d’aquest paràmetre d’esquema pot donar lloc a millores substancials de les prediccions pertorbatives d’aquests observables. A la part II, discutim les divergències d’expansions asimptòtiques en el context d’integrals de camí. Convencionalment, el mètode de la sumació de Borel asigna una resposta finita a les expansions divergents. Tot i així, la suma de Borel podria no contenir tota la informació d’una funció, perquè a aquesta li poden faltar correccions exponencialment petites. Llavors considerem una petita variació de la sumació de Borel, on una transformada de Borel generalitzada (una transformada de Laplace inversa) és seguida d’una transformada de Laplace direccionals. Aquestes eines ens permet donar, potser, millors respostes a problemes típics de la sumació de Borel, com la pèrdua de les correccions exponencials i les ambigüitats de la sumació de Borel. A més, definim ressurgència com una connexió entre la discontinuïtat d’una funció i els coeficients de la seva expansió asimptòtica. A partir d’aquesta definició, podem reduir el problema de ressurgència a un problema de correccions exponencials perdudes en les expansions asimptòtiques i podem relacionar diferents formes d’entendre la ressurgència que es troben a la literatura.For realistic quantum field theories, numerical predictions of physical observables can only be calculated from perturbative expansions in powers of the couplings, the parameters that determine the strength of the field interactions. While the predictive success of quantum field theory is undeniable, these perturbative computations are plagued with divergences. On one hand, the coefficients of the perturbative expansion are computed from loop integrals that are divergent most of the times. Some of these divergences are associated with unphysical terms that can be subtracted. In other cases, a renormalisation procedure is applied to cancel these divergences, but this entails a choice of theoretical conventions (scale and scheme) which physical observables cannot depend on. On the other hand, once the loop integrals have been renormalised, the resulting expansion still sums to an infinite answer for all non-vanishing values of the coupling. This is due to the fact that the coefficients of the expansion grow factorially with the order. Still, these expansions can be understood as asymptotic expansions, which encode the limiting behaviour of the observable for small coupling, and whose truncation to an optimal term yields numerical approximations of the observable. This second kind of divergence is in fact not limited to quantum field theories, but it may arise in different contexts of mathematics and physics: for instance, in perturbative approximations to the energy eigenvalues of a quantum mechanic system, or in formal solutions to differential equations. In part I of this dissertation, the main object of study is the strong coupling constant and the perturbative expansions of physical observables in quantum chromodynamics. First, we briefly discuss how the loop divergence of the quantum corrected gluon propagator can be absorbed inside the strong coupling constant during the renormalisation. This process, however, comes at the cost of introducing scale and scheme dependences into the coupling, therefore it is not a physical observable of the theory. This motivates a coupling redefinition whose scheme dependence is reduced to a single parameter. We then use this coupling redefinition in phenomenological analysis of physical observables associated to electron-positron scattering, and to Higgs and tau decays into hadrons. We demonstrate that appropriate choices of this scheme parameter can lead to substantial improvements in perturbative predictions of these observables. In part II, we discuss the divergence of asymptotic expansions in the context of path integrals. Conventionally, the method of Borel summation assigns a finite answer to the divergent expansion. Still, the Borel sum might not encode the full information of a function, because it misses exponentially small corrections. We then consider a slight variation of the conventional Borel summation, in which a generalised Borel transform (an inverse Laplace transform) is followed by a directional Laplace transform. These tools allow us to give perhaps better answers to typical problems in Borel summation: missing exponential corrections and ambiguities in the Borel summation. In addition, we define resurgence as a connection between the discontinuity of a function and the coefficients of its asymptotic expansion. From this definition, we can reduce resurgence to the problem of missing exponential corrections in asymptotic expansions and correlate different approaches to resurgence found in the literature