arXiv subjects

Irinel Caprini

Publications and source records attributed to Irinel Caprini.

At least 19 recordsLinked to original sources

arXiv2024

Power corrections to the modified QCD perturbative series based on conformal mapping of the Borel plane

Modifications of the QCD perturbative expansions by the subtraction of the dominant infrared renormalon have been proposed recently as attempts to solve the long-standing discrepancy between fixed-order and contour-improved perturbation theory for the hadronic $\tau$ decays. In this approach, the modified perturbative series is supplemented by a modified gluon condensate in the operator product expansion of the Adler function. Motivated by these works, we revisited recently a formulation of the QCD perturbation theory, proposed some time ago, which takes into account the renormalons by means of the conformal mapping of the Borel plane. One expects that the modified perturbative series obtained in this framework should be accompanied by modified power-suppressed nonperturbative corrections. However, in the previous studies the focus was on the perturbative series and the question of the possible power corrections has not been considered. In the present paper, we investigate for the first time this problem. Using techniques from the mathematical resurgence theory, we derive the expression of the dominant power correction to the perturbative series obtained by conformal mapping of the Borel plane, and discuss its implications for phenomenological applications.

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arXiv2023

Revisiting the convergence of the perturbative QCD expansions based on conformal mapping of the Borel plane

The difference between fixed-order (FO) and contour-improved (CI) formulations of QCD perturbation theory limits the precision of the strong coupling determined from the hadronic decay of the $\tau$ lepton. Recently, several attempts to understand the mathematical origin of the difference and to solve it by subtracting the dominant infrared renormalon divergence have been made. Motivated by these studies, we review in this paper an improved perturbative QCD expansion, defined some time ago, which also exploits the renormalons by means of a suitable conformal mapping of the Borel plane. In particular, we revisit the convergence of the new expansion, by completing the proof presented in a previous paper and showing that the domain of convergence is larger than stated before. We also check the validity of the convergence conditions for the Adler function and the CI and FO expansions of the $\tau$ hadronic spectral function moments, and compare the approach based on conformal mapping with recent solutions to the CIPT-FOPT discrepancy proposed in the literature.

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arXiv2023

Resurgent representation of the Adler function in the large-$β_0$ approximation of QCD

Using a full resummation of the Adler function in the large-$β_0$ approximation of QCD and a mathematical framework of resurgence suitable for the specific properties of the Borel transform in this particular case, we derive a compact resurgent representation of the QCD Adler function, valid in the whole complex momentum plane. The representation is expressed in terms of the inverse Mellin transform of the Borel function and is analytic in the complex momentum plane, except for cuts along the timelike axis and the Landau region of the spacelike axis. It contains a purely nonperturbative term singular at the origin of the coupling plane, depending on a single real arbitrary constant. We compare the resurgent Adler function derived in this work with a previous determination in a similar framework and use its values in the complex plane for a calculation of the hadronic width of the $τ$ lepton in the Standard Model.

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arXiv2021

Conformal mappings in perturbative QCD

We discuss the method of conformal mappings applied to perturbative QCD. The approach is based on the Borel-Laplace integral regulated with the principal value prescription and the expansion of the Borel transform in powers of the variable which performs the conformal mapping of the cut Borel plane onto the unit disk. We write down the expression of the conformal mapping for the most general location of the singularities of the Borel transform and review the properties of the corresponding expansions of the correlators. Unlike the standard perturbative expansions, which are divergent, the modified expansions have a tamed behaviour at large orders and may even converge under some conditions. On the other hand, the expansion functions exhibit nonperturbative features similar to those of the expanded function. Using these properties, it was suggested recently that the expansions based on the conformal mapping of the Borel plane may provide an alternative to the standard OPE. We briefly review the arguments in favour of this conjecture and discuss the application of the method to the Adler function for massless quarks and the static quark self-energy calculated in lattice QCD.

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arXiv2021

Analyticity and Regge asymptotics in virtual Compton scattering on the nucleon

We test the consistency of the data on the nucleon structure functions with analyticity and the Regge asymptotics of the virtual Compton amplitude. By solving a functional extremal problem, we derive an optimal lower bound on the maximum difference between the exact amplitude and the dominant Reggeon contribution for energies $ν$ above a certain high value $ν_h(Q^2)$. Considering in particular the difference of the amplitudes $T_1^\inel(ν, Q^2)$ for the proton and neutron, we find that the lower bound decreases in an impressive way when $ν_h(Q^2)$ is increased, and represents a very small fraction of the magnitude of the dominant Reggeon. While the method cannot rule out the hypothesis of a fixed Regge pole, the results indicate that the data on the structure function are consistent with an asymptotic behaviour given by leading Reggeon contributions. We also show that the minimum of the lower bound as a function of the subtraction constant $S_1^\inel(Q^2)$ provides a reasonable estimate of this quantity, in a frame similar, but not identical to the Reggeon dominance hypothesis.

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arXiv2020

Conformal mapping of the Borel plane: going beyond perturbative QCD

The power corrections in the Operator Product Expansion (OPE) of QCD correlators can be viewed mathematically as an illustration of the transseries concept, which allows to recover a function from its asymptotic divergent expansion. Alternatively, starting from the divergent behavior of the perturbative QCD encoded in the singularities in the Borel plane, a modified expansion can be defined by means of the conformal mapping of this plane. A comparison of the two approaches concerning their ability to recover nonperturbative properties of the true correlator was not explored up to now. In the present paper, we make a first attempt to investigate this problem. We use for illustration the Adler function and observables expressed as integrals of this function along contours in the complex energy plane. We show that the expansions based on the conformal mapping of the Borel plane go beyond finite-order perturbation theory, containing an infinite number of terms when reexpanded in powers of the coupling. Moreover, the expansion functions exhibit nonperturbative features of the true function, while the expansions have a tamed behavior at large orders and are expected even to be convergent. Using these properties, we argue that there are no mathematical reasons for supplementing the expansions based on the conformal mapping of the Borel plane by additional arbitrary power corrections. Therefore, we make the conjecture that they provide an alternative to the standard OPE in approximating the QCD correlator. This conjecture allows to slightly improve the accuracy of the strong coupling extracted from the hadronic $τ$ decay width. Using the optimal expansions based on conformal mapping and the contour-improved prescription of renormalization-group resummation, we obtain $α_s(m_τ^2)=0.314 \pm 0.006$, which implies $α_s(m_Z^2)=0.1179 \pm 0.0008$.

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arXiv2020

Test of analyticity and unitarity for the pion form-factor data around the $ρ$ resonance

High-statistics data on the $e^+e^-\to π^+π^-$ cross section and the pion vector form factor have been obtained recently by several collaborations. Unfortunately, there are some tensions between different datasets, especially the most precise ones, which have not been resolved so far. Additional independent constraints on the data are therefore of interest. We consider a parametrization-free method of analytic extrapolation proposed recently, which is based on a mixed phase and modulus extremal problem and combines rigorous upper and lower bounds with numerical simulations to account for the statistical distributions of the input and output values. Spacelike data on the form factor and measurements of the modulus in the region $(0.65-0.71)$ GeV are used as input. In previous works, the formalism was applied for extrapolating the form factor to low energies. In the present work, we use it as a stringent and model-independent test of consistency with analyticity and unitarity for the high-statistics data around the $ρ$ resonance. The study reveals some inconsistencies, in particular below the $ρ$ peak the BABAR data are slightly higher than the band of extrapolated values, while above the $ρ$ peak all the data are situated at the lower edge of the band. The implications of the results on the two-pion vacuuum polarization contribution to the anomalous magnetic moment of the muon are briefly discussed.

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arXiv2019

Higher-order perturbative coefficients in QCD from series acceleration by conformal mappings

The present calculations in perturbative QCD reach the order $α_s^4$ for several correlators calculated to five loops, and the huge computational difficulties make unlikely the full six-loop calculation in the near future. This situation has practical consequences, in particular the treatment of the higher orders of the perturbation series for the current-current correlator of light quarks is one of the main sources of errors in the extraction of the strong coupling from hadronic $τ$ decays. Several approximate estimates of the next coefficients of the corresponding Adler function have been proposed, using various arguments. In the present paper we exploit the analytic structure of the Adler function in the Borel plane, which allows the definition of an improved perturbative expansion in powers of a conformal variable which maps the cut Borel plane onto the unit disk. The new expansions converge in a larger domain of the Borel plane and, when reexpanded in powers of the strong coupling, yield definite values for the higher perturbative coefficients. We apply the method to the Adler function in the $\bar{\rm MS}$ scheme and to a suitable weighted integral of this function in the complex $s$ plane, chosen such as to avoid model-dependent assumptions on analyticity. Our results $c_{5,1}=287 \pm 40$, $c_{6,1}=2948 \pm 208$ and $c_{7,1}=(1.89 \pm 0.75)\times 10^4$, for the six, seven and eigth-loop coefficients, respectively, agree with a recent determination from Padé approximants applied to the perturbative expansion of the hadronic $τ$ decay rate.

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arXiv2018

Pion electromagnetic form factor at high precision with implications to $a_μ^{ππ}$ and the onset of perturbative QCD

We extend recently developed methods used for determining the electromagnetic charge radius and $a_μ^{ππ}$ to obtain a determination of the electromagnetic form factor of the pion, $F_π^V(t)$, in several significant kinematical regions, using a parametrization-free formalism based on analyticity and unitarity, and with the inclusion of precise inputs from both timelike and spacelike regions. On the unitarity cut, below the first inelastic threshold we use the precisely known phase of the form factor, known from $ππ$ elastic scattering via the Fermi-Watson theorem, and above the inelastic threshold a conservative integral condition on the modulus. We also use as input the experimental values of the modulus at several energies in the elastic region, where the data from $e^+e^-\to π^+π^-$ and $τ$ hadronic decays are mutually consistent, as well as the most recent measurements at spacelike momenta. The experimental uncertainties are implemented by Monte Carlo simulations. At spacelike values $Q^2=-t>0$ near the origin, our predictions are consistent and significantly more precise than the recent QCD lattice calculations. The determinations at larger $Q^2$ confirm the late onset of perturbative QCD for exclusive quantities. From the predictions of $|F_π^V(t)|^2$ on the timelike axis below 0.63 GeV, we obtain the hadronic vacuum polarization (HPV) contribution to the muon anomaly, $a_μ^{ππ}|_{\leq 0.63\gev} = (132.97\pm 0.70)\times 10^{-10}$, using input from both $e^+e^-$ annihilation and $τ$ decay, and $a_μ^{ππ}|_{\leq 0.63 \gev} = (132.91\pm 0.76)\times 10^{-10}$ using only $e^+e^-$ input. Our determinations can be readily extended to obtain such contributions in any interval of interest lying between $2 m_π$ and 0.63 GeV.

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arXiv2018

Renormalization-scheme variation of a QCD perturbation expansion with tamed large-order behavior

The renormalization-scheme and scale dependence of the truncated QCD perturbative expansions is one of the main sources of theoretical error of the standard model predictions, especially at intermediate energies. Recently, a class of renormalization schemes, parametrized by a single real number $C$, has been defined and investigated in the frame of the standard perturbation expansions in powers of the coupling. In the present paper we investigate the $C$-scheme variation of a Borel-improved QCD perturbation series, which implements information about the large-order divergent character of perturbation theory by means of an optimal conformal mapping of the Borel plane. In the new expansions, the powers of the strong coupling are replaced by a set of expansion functions with properties which resemble those of the expanded correlators, having in particular a singular behavior at the origin of the complex coupling plane. On the other hand, the new expansions have a tamed increase at high orders, as demonstrated by previous studies in the $\overline{\text{MS}}$ renormalization scheme. Using as examples the Adler function and the hadronic decay width of the $τ$ lepton, we investigate the properties of the Borel-improved expansions in the $C$-scheme, in comparison with the standard expansions in the $C$-scheme and the expansions in $\overline{\text{MS}}$. The variation with the renormalization scale and the prescription for the choice of an optimal value of the parameter $C$ are discussed. The good large-order behavior of the Borel-improved expansions is proved also in the $C$-scheme, which is a further argument in favor of using them in applications of perturbative QCD at intermediate energies.

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arXiv2018

Model-independent constraint on the pion scalar form factor and light quark masses

We investigate the pion scalar form factor in the Meiman-Okubo framework, implementing the phase below the inelastic $K\bar K$ threshold, where it is known from the $ππ$ scalar isoscalar phase shift $δ_0^0$ by Watson theorem. State-of-the-art knowledge of the perturbative QCD expansion of the scalar correlator and the phase shift $δ_0^0$ is used as input. No assumptions about the phase above the inelastic threshold or the possible zeros of the form factor in the complex plane are necessary. We obtain a model-independent constraint relating the sum of the light quark masses to the slope and the curvature of the pion scalar form factor at the origin. The recent lattice results for the light quark masses and the pion scalar radius are found to satisfy this constraint. We obtain also a strong correlation between the pion scalar radius and the curvature of the form factor, with rather high values predicted for the curvature.

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arXiv2017

Hyperasymptotics and quark-hadron duality violations in QCD

We investigate the origin of the quark-hadron duality-violating terms in the expansion of the QCD two-point vector correlation function at large energies in the complex $q^2$ plane. Starting from the dispersive representation for the associated polarization, the analytic continuation of the operator product expansion from the Euclidean to the Minkowski region is performed by means of a generalized Borel-Laplace transform, borrowing techniques from hyperasymptotics. We establish a connection between singularities in the Borel plane and quark-hadron duality violating contributions. Starting with the assumption that for QCD at $N_c=\infty$ the spectrum approaches a Regge trajectory at large energy, we obtain an expression for quark-hadron duality violations at large, but finite $N_c$.

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arXiv2017

Perturbative Expansions in QCD Improved by Conformal Mappings of the Borel Plane

Perturbation expansions appear to be divergent series in many physically interesting situations, including in quantum field theories like quantum electrodynamics (QED) and quantum chromodynamics (QCD), where the perturbative coefficients exhibit a factorial growth at large orders. While this feature has little impact on physical predictions in QED, it can have nontrivial consequences in applications of perturbative QCD at moderate energies. In particular, it affects the theoretical error in the extraction of the strong coupling $α_s$ from hadronic $τ$ decays, despite progress of perturbative calculations available at present to four loops. We discuss a new type of perturbative expansion for QCD correlators, which uses instead of the standard powers of the coupling a new set of expansion functions. These functions are defined by means of an optimal conformal mapping of the Borel complex plane, which implements the known features of the high-order divergence in terms of the lowest Borel-plane singularities. The properties of the expansion functions resemble those of the expanded correlators, by exhibiting in particular the singular behaviour of the correlators at $α_s=0$. We prove the good convergence properties of the new expansions on mathematical models that simulate the physical polarization function for light quarks and its derivative (the Adler function), in various prescriptions of renormalization-group summation.

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arXiv2017

Electromagnetic charge radius of the pion at high precision

We present a determination of the pion charge radius from high precision data on the pion vector form factor from both timelike and spacelike regions, using a novel formalism based on analyticity and unitarity. At low energies, instead of the poorly known modulus of the form factor, we use its phase, known with high accuracy from Roy equations for $ππ$ elastic scattering via the Fermi-Watson theorem. We use also the values of the modulus at several higher timelike energies, where the data from $e^+e^-$-annihilation and $τ$-decay are mutually consistent, as well as the most recent measurements at spacelike momenta. The experimental uncertainties are implemented by Monte-Carlo simulations. The results, which do not rely on a specific parametrization, are optimal for the given input information and do not depend on the unknown phase of the form factor above the first inelastic threshold. Our prediction for the charge radius of the pion is $r_π=(0.657 \pm 0.003) \fm $, which amounts to an increase in precision by a factor of about 2.7 compared to the PDG average.

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arXiv2017

Model-independent constraints on hadronic form factors with above-threshold poles

Model-independent constraints on hadronic form factors, in particular those describing exclusive semileptonic decays, can be derived from the knowledge of field correlators calculated in perturbative QCD, using analyticity and unitarity. The location of poles corresponding to below-threshold resonances, i.e., stable states that cannot decay into a pair of hadrons from the crossed channel of the form factor, must be known a priori, and their effect, accounted for through the use of Blaschke factors, is to reduce the strength of the constraints in the semileptonic region. By contrast, above-threshold resonances appear as poles on unphysical Riemann sheets, and their presence does not affect the original model-independent constraints. We discuss the possibility that the above-threshold poles can provide indirect information on the form factors on the first Riemann sheet, either through information from their residues or by constraining the discontinuity function. The bounds on form factors can be improved by imposing, in an exact way, the additional information in the extremal problem. The semileptonic $K\to π\ell ν$ and $D\to π\ellν$ decays are considered as illustrations.

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arXiv2017

Towards tests of quark-hadron duality with functional analysis and spectral function data

The presence of terms that violate quark-hadron duality in the expansion of QCD Green's functions is a generally accepted fact. Recently, a new approach was proposed for the study of duality violations (DVs), which exploits the existence of a rigorous lower bound on the functional distance, measured in a certain norm, between a "true" correlator and its approximant calculated theoretically along a contour in the complex energy plane. In the present paper we pursue the investigation of functional-analysis based tests towards their application to real spectral function data. We derive a closed analytic expression for the minimal functional distance based on the general weighted $L^2$ norm and discuss its relation with the distance measured in $L^\infty$ norm. Using fake data sets obtained from a realistic toy model in which we allow for covariances inspired from the publicly available ALEPH spectral functions, we obtain by Monte Carlo simulations the statistical distribution of the strength parameter that measures the magnitude of the DV term added to the usual operator product expansion (OPE). The results show that, if the region with large errors near the end-point of the spectrum in $τ$ decays is excluded, the functional-analysis based tests using either $L^2$ or $L^\infty$ norms are able to detect, in a statistically significant way, the presence of DVs in realistic spectral function pseudodata.

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arXiv2016

Precise determination of the low-energy hadronic contribution to the muon $g-2$ from analyticity and unitarity: An improved analysis

The two-pion low-energy contribution to the anomalous magnetic moment of the muon, $a_μ\equiv(g-2)_μ/2$, expres sed as an integral over the modulus squared of the pion electromagnetic form fac tor, brings a relatively large contribution to the theoretical error, since the low accuracy of experimental measurements in this region is amplified by the drastic increase of the integration kernel. We derive stringent constraints on the two-pion contribution by exploiting analyticity and unitarity of the pion electromagnetic form factor. To avoid the poor knowledge of the modulus of this function, we use instead its phase, known with high precision in the elastic region from Roy equations for pion-pion scattering via the Fermi-Watson theorem. Above the inelastic threshold we adopt a conservative integral condition on the modulus, determined from data and perturbative QCD. Additional high precision data on the modulus in the range $0.65-0.71$ GeV, obtained from $e^+e^-$ annihilation and $τ$-decay experiments, are used to improve the predictions on the modulus at lower energies by means of a parametrization-free analytic extrapolation. The results are optimal for a given input and do not depend on the unknown phase of the form factor above the inelastic threshold. The present work improves a previous analysis based on the same technique, including more experimental data and employing better statistical tools for their treatment. We obtain for the contribution to $a_μ$ from below 0.63 GeV the value $(133.258 \pm 0.723)\times 10^{-10}$, which amounts to a reduction of the theoretical error by about $6 \times 10^{-11}$.

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arXiv2016

Constraints on the virtual Compton scattering on the nucleon in a new dispersive formalism

The dispersive representation of the virtual Compton forward scattering amplitude has been recently reexamined in connection with the evaluation of the Cottingham formula for the proton-neutron electromagnetic mass difference. The most difficult part of the analysis is related to one of the invariant amplitudes, denoted as $T_1(ν, Q^2)$, which requires a subtraction in the standard dispersion relation with respect to the energy $ν$ at fixed photon momentum squared $q^2=-Q^2$. We propose an alternative dispersive framework, which implements analyticity and unitarity by combining the Cauchy integral relation at low and moderate energies with the modulus representation of the amplitude at high energies. Using techniques of functional analysis, we derive a necessary and sufficient condition for the consistency with analyticity of the subtraction function $S_1(Q^2)=T_1(0, Q^2)$, the electron-proton cross sections measured at low and moderate energies and the Regge model assumed to be valid at high energies. From this condition we obtain model-independent constraints on the subtraction function, confronting them with the available information on nucleon magnetic polarizabilities and results reported recently in the literature. The formalism can be used also for testing the existence of a fixed pole at $J=0$ in the angular momentum plane, but more accurate data are necessary for a definite answer.

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