Non-perturbative Power Corrections Research Articles

Energy correlators on tracks: resummation and non-perturbative effects

Energy correlators measured inside high-energy jets at hadron colliders have recently been demonstrated to provide a new window into both perturbative and non-perturbative Quantum Chromodynamics. A number of the most interesting features of these correlators, namely their universal scaling behavior and the ability to image the confinement transition, require precise angular resolution, necessitating the use of tracking information in experimental measurements. Theoretically, tracking information can be incorporated into the energy correlators using track functions, which are non-perturbative functions describing the fragmentation of quarks and gluons into charged hadrons. In this paper, we apply our recently developed track function formalism to energy correlators, and study in detail the interplay of track functions with perturbative resummation and non-perturbative power corrections. We provide resummed results for the energy correlators at collinear next-to-leading-logarithmic accuracy and compare with parton shower Monte Carlo simulations. For the two-point correlator the use of tracking has a minimal effect throughout the entire distribution, but it has a significant effect for higher point correlators. Our results are crucial for the theoretical interpretation of recent experimental measurements of the energy-energy correlators.

Renormalons in the energy-energy correlator

The energy-energy correlator (EEC) is an observable of wide interest for collider physics and Standard Model measurements, due to both its simple theoretical description in terms of the energy-momentum tensor and its novel features for experimental studies. Significant progress has been made in both applications and higher-order perturbative predictions for the EEC. Here, we analyze the nature of the asymptotic perturbative series for the EEC by determining its analytic form in Borel space under the bubble-sum approximation. This result provides information on the leading and subleading nonperturbative power corrections through renormalon poles. We improve the perturbative convergence of the overline{textrm{MS}} series for the EEC by removing its leading renormalon using an MSR scheme, which is independent of the bubble-sum approximation. Using the leading MSR scheme power correction determined by fits to thrust, we find good agreement with EEC OPAL data already at mathcal{O} ( {alpha}_s^2 ).

The catchment area of groomed jets at NNLL

Groomed jet observables have a dynamical catchment area which plays a key role in determining the leading nonperturbative power corrections and the impact of the underlying event. Based on field-theoretic arguments, certain moments of the groomed jet radius Rg capture the entirety of the kinematic and grooming parameter dependence of these effects. These moments can be computed perturbatively in the soft drop operator expansion region where these corrections are small, but yet significant to be relevant for precision physics. A precise determination of these moments is thus crucial to faithfully isolate the universal contributions of hadronization and the underlying event. Building on a previously developed effective field theory framework for the doubly differential soft drop groomed jet mass and groomed jet radius measurement, we present here a calculation of these moments at next-to-next-to-leading-logarithmic (NNLL) accuracy including matching into the plain jet mass region. We compare our predictions for these moments against parton-shower Monte Carlo simulations and find good agreement. These results have applications for precision physics with soft drop jet mass such as determination of the strong coupling constant and the top quark mass and for improving hadronization models.

Dispersive analysis of neutral meson mixing

We analyze the neutral meson mixing by directly solving the dispersion relation obeyed by the mass and width differences of the two meson mass eigenstates. We solve for the parameters $x$ and $y$, proportional to the mass and width differences in the charm mixing, respectively, taking the box-diagram contributions to $x(s)$ and $y(s)$ at large mass squared $s$ of a fictitious $D$ meson as inputs. The SU(3) symmetry breaking is introduced through physical thresholds of different $D$ meson decay channels for $y(s)$. These threshold-dependent effects, acting like nonperturbative power corrections in QCD sum rules, stabilize the solutions of $y(s=m_D^2)$ with the $D$ meson mass $m_D$. We then calculate $x(s)$ through the dispersive integration of $y(s)$, and show that our predictions $x(m_D^2)\approx 0.21\%$ and $y(m_D^2)\approx 0.52\%$ are close to the data in both $CP$-conserving and $CP$-violating cases. It is observed that the channel containing di-kaon states provides the major source of SU(3) breaking, which enhances $x(m_D^2)$ and $y(m_D^2)$ by four orders of magnitude relative to the perturbative results. We also predict the coefficient ratio $q/p$ involved in the charm mixing with $|q/p|-1\approx 2\times 10^{-4}$ and $Arg(q/p)\approx 6\times 10^{-3}$ degrees, which can be scrutinized by precise future measurements. The formalism is extended to studies of the $B_{s(d)}$ meson mixing and the kaon mixing, and the small deviations of the obtained width differences from the perturbative inputs explain why the above mixing can be understood via short-distance dynamics. We claim that the puzzling charm mixing is attributed to the strong Glashow-Iliopoulos-Maiani suppression on perturbative contributions, instead of to breakdown of the quark-hadron duality, which occurs only at 15\% level.

Second-order QCD corrections to event shape distributions in deep inelastic scattering

We compute the next-to-next-to-leading order (NNLO) QCD corrections to event shape distributions and their mean values in deep inelastic lepton–nucleon scattering. The magnitude and shape of the corrections varies considerably between different variables. The corrections reduce the renormalization and factorization scale uncertainty of the predictions. Using a dispersive model to describe non-perturbative power corrections, we compare the NNLO QCD predictions with data from the H1 and ZEUS experiments. The newly derived corrections improve the theory description of the distributions and of their mean values.

Predictions for energy correlators probing substructure of groomed heavy quark jets

We develop an effective field theory (EFT) framework to perform an analytic calculation for energy correlator observables computed on groomed heavy-quark jets. A soft-drop grooming algorithm is applied to a jet initiated by a massive quark to minimize soft contamination effects such as pile-up and multi-parton interactions. We specifically consider the two-particle energy correlator as an initial application of this EFT framework to compute heavy quark jet substructure. We find that there are different regimes for the event shapes, depending on the size of the measured correlator observable, that require the use of different EFT formulations, in which the quark mass and grooming parameters may be relevant or not. We use the EFT to resum large logarithms in the energy correlator observable in terms of the momentum of a reconstructed heavy hadron to NLL′ accuracy and subsequently match it to a full QCD mathcal{O} (αs) cross section, which we also compute. We compare our predictions to simulations in Pythia for e+e− collisions. We find a good agreement with partonic simulations, as well as hadronic ones with an appropriate shape function used to describe nonperturbative effects and the heavy quark hadron decay turned off. We also predict the scaling behavior for the leading nonperturbative power correction due to hadronization. Consequently, we can give a prediction for the energy correlator distribution at the level of the reconstructed heavy hadron. This work provides a general framework for the analysis of heavy quark jet substructure observables.

Transverse momentum spectra at threshold for groomed heavy quark jets

We present the transverse momentum spectrum for a heavy hadron at threshold in a groomed jet initiated by a heavy quark. The cross section is doubly differential in the energy fraction of an identified heavy hadron in the jet and its transverse momentum measured with respect to the groomed (recoil free) jet axis. The grooming is implemented using a soft-drop grooming algorithm and helps us in mitigating the effects of Non-Global logarithms and pile up. For the particular case of a B meson, we identify two distinct regimes of the transverse momentum spectrum and develop an EFT within the formalisms of Soft Collinear Effective Theory (SCET) and Heavy Quark Effective Theory (HQET) for each of these regions. We show how each region can be matched smoothly into the other to provide a prediction for the perturbative transverse momentum spectrum. The EFT also predicts the scaling behavior of the leading non-perturbative power corrections and implements a simple shape function to account for hadronization. We work in the threshold region where the heavy hadron carries most of the energy of the jet since in this regime, we have a very good discriminating power between heavy quark and gluon initiated jets. We observe that the shape of the spectrum is independent of the energy of the jet over a large range of transverse momentum. We propose that this spectrum can be used as a probe of evolution for heavy quark TMD fragmentation function. At the same time, it can be treated as a jet substructure observable for probing Quark-Gluon Plasma (QGP).

Simplicity from recoil: the three-loop soft function and factorization for the energy-energy correlation

We derive an operator based factorization theorem for the energy-energy correlation (EEC) observable in the back-to-back region, allowing the cross section to be written as a convolution of hard, jet and soft functions. We prove the equivalence of the soft functions for the EEC and color singlet transverse-momentum resummation to all-loop order, and give their analytic result to three-loops. Large logarithms appearing in the perturbative expansion of the EEC can be resummed to all orders using renormalization group evolution. We give analytic results for all required anomalous dimensions to three-loop order, providing the first example of a transverse-momentum (recoil) sensitive e+e− event shape whose anomalous dimensions are known at this order. The EEC can now be computed to next-to-next-to-next-to-leading logarithm matched to next-to-next-to-leading order, making it a prime candidate for precision QCD studies and extractions of the strong coupling constant. We anticipate that our factorization theorem will also be crucial for understanding non-perturbative power corrections for the EEC, and their relationship to those appearing in other observables.

Ghost-gluon coupling, power corrections, andΛMS¯from lattice QCD with a dynamical charm

This paper is a first report on the determination of $\Lambda_{\msbar}$ from lattice simulations with 2+1+1 twisted-mass dynamical flavours {\it via} the computation of the ghost-gluon coupling renormalized in the MOM Taylor scheme. We show this approach allows a very good control of the lattice artefacts and confirm the picture from previous works with quenched and ${\rm N}_f$=2 twisted-mass field configurations which prove the necessity to include non-perturbative power corrections in the description of the running. We provide with an estimate of $\Lambda_{\msbar}$ in very good agreement with experimental results. To our knowledge it is the first calculation with a dynamical charm quark which makes the running up to $\alpha_s(M_Z)$ much safer.

Using an Effective Charges Method to extract [formula omitted] from event shape moments in [formula omitted] annihilation

We use an Effective Charges (ECH) method to extract ΛMS¯, and hence αs(Mz), from event shape moments in e+e− annihilation. We compare these results with ones obtained using standard MS¯ perturbation theory. The ECH method at NLO is found to perform better than standard MS¯ perturbation theory when applied to means of event shape observables. For example, when we apply the NLO ECH method to 〈1−T〉 we get αs(Mz)=0.1193±0.0003. However ECH at NNLO is found to work less well than ECH at NLO, and the ECH method also fails to describe data for higher moments of event shapes. We attempt to explain this by considering the ECH β-function as an asymptotic series. We also examine the effect of adding two different models for non-perturbative power corrections to the perturbative approximation given by the ECH method and MS¯ perturbation theory. Whilst only small power corrections are required when using ECH at NLO, it is found that these models are insufficient to counteract the undesirable behaviour of ECH at NNLO.

Hadronisation effects in event-shape moments

We study the moments of hadronic event shapes in $e^+e^-$ annihilation within the context of next-to-next-to-leading order (NNLO) perturbative QCD predictions combined with non-perturbative power corrections in the dispersive model. This model is extended to match upon the NNLO perturbative prediction. The resulting theoretical expression has been compared to experimental data from JADE and OPAL, and a new value for $\alpha_s(M_Z)$ has been determined, as well as of the average coupling $\alpha_0$ in the non-perturbative region below $\mu_I=2$ GeV within the dispersive model: \alpha_s(M_Z)&=0.1153\pm0.0017(\mathrm{exp})\pm0.0023(\mathrm{th}),\alpha_0&=0.5132\pm0.0115(\mathrm{exp})\pm0.0381(\mathrm{th}), The precision of the $\alpha_s(M_Z)$ value has been improved in comparison to the previously available next-to-leading order analysis. We observe that the resulting power corrections are considerably larger than those estimated from hadronization models in multi-purpose event generator programs.

On the renormalization of theories of a scalar chiral superfield

An exact renormalization group for theories of a scalar chiral superfield is formulated, directly in four dimensional Euclidean space. By constructing a projector which isolates the superpotential from the full Wilsonian effective action, it is shown that the nonperturbative nonrenormalization theorem follows, quite simply, from the flow equation. Next, it is argued that there do not exist any physically acceptable non-trivial fixed points. Finally, the Wess-Zumino model is considered, as a low energy effective theory. Following an evaluation of the one and two loop β-function coefficients, to illustrate the ease of use of the formalism, it is shown that the β-function in the massless case does not receive any nonperturbative power corrections.

NNLO moments of event shapes ine+e−annihilation

We compute the next-to-next-to-leading order (NNLO) QCD corrections to the first five moments of six event shape variables related to three-particle final states in electron-positron annihilation; the thrust, the heavy jet mass, the C-parameter, the wide and total jet broadenings and the three-to-two-jet transition parameter in the Durham algorithm Y3. The NNLO corrections to the first moment are moderate for all event shapes, while the renormalisation scale dependence of the theoretical prediction is substantially reduced compared to the previously existing NLO results. From a comparison with data from JADE and OPAL, we observe that the energy dependence of the moments of the wide jet broadening and Y3 can be largely explained without any non-perturbative power corrections, while the other observables exhibit a clear need for power-like contributions at low centre-of-mass energy.

A resummable β-function for massless QED

Within the set of schemes defined by generalized, manifestly gauge invariant exact renormalization groups for QED, it is argued that the β-function in the four-dimensional massless theory cannot possess any nonperturbative power corrections. Consequently, the perturbative expression for the β-function must be resummable. This argument cannot be extended to flows of the other couplings or to the anomalous dimension of the fermions and so perturbation theory does not define a unique trajectory in the critical surface of the Gaussian fixed point. Thus, resummability of the β-function is not inconsistent with the expectation that a non-trivial fixed point does not exist.

OPE and sum rules for correlators of pseudoscalar and axial currents

We study the operator product expansion (OPE) and quark-hadron duality for two-and three-point correlators of the axial (A) and pseudoscalar (P) currents of the light quarks. In the chiral limit, these correlators are often dominated by nonperturbative power corrections leading to subtleties of quark-hadron duality relations and of the extraction of properties of light pseudoscalars. For the two-point correlators, we show the sum rule for 〈PP〉 to be sensitive to the excited light pseudoscalar. For the three-point correlators, we derive the Ward identities which provide the normalization of the pion electromagnetic form factor at zero momentum transfer. For large momentum transfer, we demonstrate the way the correct behavior of the pion form factor in agreement with perturbative QCD emerges from condensate terms in the OPE for the 〈PV P〉 and 〈AV P〉 correlators. The local-duality sum rule for 〈AV A〉 is shown to lead to the pion form factor with the required properties for all values of the momentum transfer.

Running coupling and power corrections in non-linear evolution at the high-energy limit

A main feature of high-energy scattering in QCD is saturation in the number density of gluons. This phenomenon is described by non-linear evolution equations, JIMWLK and BK, which have been derived at leading logarithmic accuracy. In this paper we generalize this framework to include running coupling corrections to the evolution kernel. We develop a dispersive representation of the dressed gluon propagator in the background of Weiszäcker–Williams fields and use it to compute O ( β 0 n − 1 α s n ) corrections to the kernel to all orders in perturbation theory. The resummed kernels present infrared-renormalon ambiguities, which are indicative of the form and importance of non-perturbative power corrections. We investigate numerically the effect of the newly computed perturbative corrections as well as the power corrections on the evolution. We present numerical results for the evolution rate as a function of the saturation scale. We find that running-coupling corrections are necessary for making quantitative predictions even when the saturation scale is very large; at present energies also the power corrections are important.

Event shapes in deep inelastic scattering at HERA

Mean values and differential distributions of event-shape variables have been studied in neutral current deep inelastic scattering using an integrated luminosity of 82.2 pb −1 collected with the ZEUS detector at HERA. The kinematic range is 80 < Q 2 < 20 480 GeV 2 and 0.0024 < x < 0.6 , where Q 2 is the virtuality of the exchanged boson and x is the Bjorken variable. The data are compared with a model based on a combination of next-to-leading-order QCD calculations with next-to-leading-logarithm corrections and the Dokshitzer–Webber non-perturbative power corrections. The power-correction method provides a reasonable description of the data for all event-shape variables studied. Nevertheless, the lack of consistency of the determination of α s and of the non-perturbative parameter of the model, α 0 ¯ , suggests the importance of higher-order processes that are not yet included in the model.

Quark-hadron duality and hadron properties from correlators of pseudoscalar and axial currents

We study the operator product expansion (OPE) and quark-hadron duality for 2- and 3-point correlators of the axial ($A$) and pseudoscalar ($P$) currents of the light quarks. In the chiral limit these correlators are often dominated by nonperturbative power corrections leading to subtleties of quark-hadron duality relations and of the extraction of properties of light pseudoscalars. For the 2-point correlators, we show the sum rule for $⟨PP⟩$ to be sensitive to the excited light pseudoscalar. For the 3-point correlators, we derive the Ward identities which provide the normalization of the pion electromagnetic form factor at zero momentum transfer. For large momentum transfer, we demonstrate the way the correct behavior of the pion form factor in agreement with perturbative QCD emerges from condensate terms in the OPE for the $⟨PVP⟩$ and $⟨AVP⟩$ correlators. The local-duality sum rule for $⟨AVA⟩$ is shown to lead to the pion form factor with the required properties for all values of the momentum transfer.

Non-perturbative power corrections to ghost and gluon propagators

We study the dominant non-perturbative power corrections to the ghost and gluon propagators in Landau gauge pure Yang-Mills theory using OPE and lattice simulations. The leading order Wilson coefficients are proven to be the same for both propagators. The ratio of the ghost and gluon propagators is thus free from this dominant power correction. Indeed, a purely perturbative fit of this ratio gives smaller value ($\simeq 270$MeV) of $\Lambda_{\ms}$ than the one obtained from the propagators separately($\simeq 320$MeV). This argues in favour of significant non-perturbative $\sim 1/q^2$ power corrections in the ghost and gluon propagators. We check the self-consistency of the method.

A new determination of the electromagnetic nucleon form factors from QCD Sum Rules

We obtain the electromagnetic form factors of the nucleon, in the space-like region, using three-point function Finite Energy QCD Sum Rules. The QCD calculation is performed to leading order in perturbation theory in the chiral limit, and also to leading order in the non-perturbative power corrections. For the Dirac form factor, F 1 ( q 2 ) , we get a very good agreement with the data for both the proton and the neutron, in the currently accessible experimental region of momentum transfers. Unfortunately this is not the case, though, for the Pauli form factor F 2 ( q 2 ) , which has a soft q 2 -dependence proportional to the quark condensate 0 | q ¯ q | 0 > .