We derive high-precision results for the $\textit{e}$ $\textit{e}$ heavy jet mass (HJM) d$\textit{σ}$/d$\textit{ρ}$ and dihemisphere mass (DHM) d $\textit{σ}$/(d$\textit{s}$ d$\textit{s}$ ) distributions, for $\textit{s}$ ~ $\textit{s}$ , in the dijet region. New results include: i) the N LL resummation for HJM of large logarithms ln ($\textit{ρ}$) at small $\textit{ρ}$ including the exact two-loop non-global hemisphere soft function, the 4-loop cusp anomalous dimension and the 3-loop hard and jet functions, ii) N LL results for DHM with resummation of logarithms ln($\textit{s}$ /$\textit{Q}$ ) when there is no large separation between $\textit{s}$ and $\textit{s}$ , iii) profile functions for HJM to give results simultaneously valid in the peak and tail regions, iv) a complete two-dimensional basis of non-perturbative functions which can be used for double differential observables, that are needed for both HJM and DHM in the peak region, and v) an implementation of renormalon subtractions for large-angle soft radiation to $$ \mathcal{O}\left({\alpha}_s^3\right) $$ together with a resummation of the additional large ln($\textit{Qρ}$/Λ ) logarithms. Here $\textit{Q}$ is the $\textit{e}$ $\textit{e}$ center-of-mass energy. Our resummation results are combined with known fixed-order $$ \mathcal{O}\left({\alpha}_s^3\right) $$ results and we discuss the convergence and remaining perturbative uncertainty in the cross section. We also prove that, at order 1/$\textit{Q}$, the first moment of the HJM distribution involves an additional non-perturbative parameter compared to the power correction that shifts the tail of the spectrum (where 1 ≫ $\textit{ρ}$ ≫ Λ /$\textit{Q}$). This differs from thrust where a single non-perturbative parameter at order 1/$\textit{Q}$ describes both the first moment and the tail, and it disfavors models of power corrections employing a single non-perturbative parameter, such as the low-scale effective coupling model. In this paper we focus only on the dijet region, not the far-tail distribution for $\textit{ρ}$ ≳ 0$\textit{.}$2 beyond which the trijet factorization and resummation become important.

Austria

André Hoang

Spain

Vicent Mateu

United States

Matthew Schwartz

Iain Stewart

Precision $\textit{e}$

We update a previous N LL + $$ \mathcal{O}\left({\alpha}_s^3\right) $$ determination of the strong coupling from a global fit to thrust data by including newly available perturbative ingredients, upgrading the renormalization scales to include a fully canonical scaling region, and implementing the log resummation in a way which ensures the integrated cross section is unaffected by the leading 1/$\textit{Q}$ hadronization power corrections. Detailed discussions are provided concerning the stability of the results under variations of the fit range and the importance of summing up higher-order logarithmic terms for convergence and stability. We show that high-precision results can be achieved even when carrying out a more conservative fit by restricting the dataset to a region which is more clearly dominated by dijet events. This leads to $\textit{α}$ ($\textit{m}$ ) = 0.1136 ± 0$\textit{.}$0012 with $\textit{χ}$ /dof = 0.86, fully compatible with earlier results using a larger fit range. We also demonstrate that a number of additional effects associated to power corrections have a small impact on this fit result, including modifications to the renormalon substraction scheme for dijet power corrections and the inclusion of three-jet power correction models. The fit is also shown to provide very good agreement with data outside the fit range.

Miguel Benitez

Switzerland

Gherardo Vita

On determining $\textit{α}$

We compute the inclusive jet function for boosted heavy quarks to $$ \mathcal{O}\left({\alpha}_s^3\right) $$ . The jet function is defined and calculated in the framework of boosted Heavy-Quark Effective Theory (bHQET). It describes the effect of radiation collimated in narrow jets arising from energetic heavy quarks on observables probing the jet invariant mass $\textit{M}$ in the region where $\textit{M}$ – $\textit{m}$ ≪ $\textit{m}$ , with $\textit{m}$ the heavy quark mass. This kinematic situation is relevant e.g. in boosted top (pair) production at high-energy colliders. We have verified that our result satisfies non-Abelian exponentiation and checked that our calculation reproduces the known cusp and non-cusp anomalous dimensions of the jet function to $$ \mathcal{O}\left({\alpha}_s^3\right) $$ . We also confirmed that the $$ {n}_{\ell}^2{\alpha}_s^3 $$ contribution, where $\textit{n}$ is the number of massless quark flavors, agrees with the prediction from renormalon calculus. Our computation provides the last missing piece to obtain the N LL resummed (self-normalized) thrust distribution used for the calibration of the top quark mass parameter in parton-shower Monte Carlo generators. Our result also contributes to the invariant mass distribution of reconstructed top quarks at N LL , which can be employed for a precise top mass determination at future lepton colliders. As a by-product, we obtain the relation between the pole and short-distance jet-mass schemes at $$ \mathcal{O}\left({\alpha}_s^3\right) $$ . Finally, we estimate the non-logarithmic contribution to the four-loop jet function based on renormalon dominance.

Alberto Clavero

Germany

Robin Brüser

Maximilian Stahlhofen

Three-loop jet function for boosted heavy quarks

Processes involving only massless or massive quarks at tree-level get corrections from massive (lighter, heavier, or equal-mass) secondary quarks starting at two-loop order, generated by a virtual gluon splitting into a massive quark anti-quark pair. One convenient approach to compute such two-loop corrections is starting with the one-loop diagram considering the virtual gluon massive. Carrying out a dispersive integral with a suitable kernel over the gluon mass yields the desired two-loop result. On the other hand, the Mellin-Barnes representation can be used to compute the expansion of Feynman integrals in powers of a small parameter. In this article we show how to combine these two ideas to obtain the corresponding expansions for large and small secondary quark masses to arbitrarily high orders in a straightforward manner. Furthermore, the convergence radius of both expansions can be shown to overlap, being each series rapidly convergent. The advantage of our method is that the Mellin representation is obtained directly for the full matrix element from the same one-loop computation one needs in large-β 0 computations, therefore many existing results can be recycled. With minimal modifications, the strategy can be applied to compute the expansion of the one-loop correction coming from a massive gauge boson. We apply this method to a plethora of examples, in particular those relevant for factorized cross sections involving massless and massive jets, recovering known results and obtaining new ones. Another bonus of our approach is that, postponing the Mellin inversion, one can obtain the small- and large-mas expansions for the RG-evolved jet functions. In many cases, the series can be summed up yielding closed expressions.

Alejandro Bris

Secondary massive quarks with the Mellin-Barnes expansion

We generalize and update our former top quark mass calibration framework for Monte Carlo (MC) event generators based on the e + e  −  hadron-level 2-jettiness τ 2 distribution in the resonance region for boosted   <math> <mi>t</mi> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ t\overline{t} $$  production, that was used to relate the Pythia 8.205 top mass parameter   <math> <msubsup> <mi>m</mi> <mi>t</mi> <mi>MC</mi> </msubsup> </math>  $$ {m}_t^{\textrm{MC}} $$  to the MSR mass   <math> <msubsup> <mi>m</mi> <mi>t</mi> <mi>MSR</mi> </msubsup> <mfenced> <mi>R</mi> </mfenced> </math>  $$ {m}_t^{\textrm{MSR}}(R) $$  and the pole mass   <math> <msubsup> <mi>m</mi> <mi>t</mi> <mtext>pole</mtext> </msubsup> </math>  $$ {m}_t^{\textrm{pole}} $$ . The current most precise direct top mass measurements specifically determine   <math> <msubsup> <mi>m</mi> <mi>t</mi> <mi>MC</mi> </msubsup> </math>  $$ {m}_t^{\textrm{MC}} $$ . The updated framework includes the addition of the shape variables sum of jet masses τ  s  and modified jet mass τ  m , and the treatment of two more gap subtraction schemes to remove the   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (ΛQCD) renormalon related to large-angle soft radiation. These generalizations entail implementing a more versatile shape-function fit procedure and accounting for a certain type of (m  t /Q)2 power corrections to achieve gap-scheme and observable independent results. The theoretical description employs boosted heavy-quark effective theory (bHQET) at next-to-next-to-logarithmic order (N2LL), matched to soft-collinear effective theory (SCET) at N2LL and full QCD at next-to-leading order (NLO), and includes the dominant top width effects. Furthermore, the software framework has been modernized to use standard file and event record formats. We update the top mass calibration results by applying the new framework to Pythia 8.305, Herwig 7.2 and Sherpa 2.2.11. Even though the hadron-level resonance positions produced by the three generators differ significantly for the same top mass parameter   <math> <msubsup> <mi>m</mi> <mi>t</mi> <mi>MC</mi> </msubsup> </math>  $$ {m}_t^{\textrm{MC}} $$  value, the calibration shows that these differences arise from the hadronization modeling. Indeed, we find that   <math> <msubsup> <mi>m</mi> <mi>t</mi> <mi>MC</mi> </msubsup> </math>  $$ {m}_t^{\textrm{MC}} $$  agrees with   <math> <msubsup> <mi>m</mi> <mi>t</mi> <mi>MSR</mi> </msubsup> </math>  $$ {m}_t^{\textrm{MSR}} $$ (1 GeV) within 200 MeV for the three generators and differs from the pole mass by 350 to 600 MeV.

Bahman Dehnadi

Oliver Jin

Top quark mass calibration for Monte Carlo event generators — an update

In a previous work by one of the authors, it was demonstrated that the discrepancy between the fixed-order and contour-improved (CIPT) perturbative expansions for <math><mrow><mi>τ</mi></mrow></math>-lepton decay hadronic spectral function moments, which had been affecting the precision of <math><msub><mi>α</mi><mi>s</mi></msub></math> determinations for many years, is related to the CIPT expansion being inconsistent with the standard formulation of the operator product expansion. Even though the problem can be alleviated phenomenologically for the most part by employing a renormalon-free scheme for the gluon-condensate matrix element, the principal inconsistency of CIPT remains. The CIPT expansion is special because it is not a power expansion, but represents an asymptotic expansion in a sequence of functions of the strong coupling. In this article we provide a closer look at the mathematical aspects of the asymptotic sequence of the functions the CIPT method is based on, and we expose the origin of the CIPT inconsistency as well as the reasons for its apparent good convergence at low orders. Our results are of general interest, and may in particular provide a useful tool to check for the consistency of expansion methods that are similar to CIPT.

Néstor G. Gracia

André H. Hoang

Mathematical aspects of the asymptotic expansion in contour improved perturbation theory for hadronic tau decays

In this article we compute the cross section for the process   <math> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>−</mo> </msup> <mo>→</mo> <mi>Q</mi> <mover> <mi>Q</mi> <mo>¯</mo> </mover> <mo>+</mo> <mi>X</mi> </math>  $$ {e}^{+}{e}^{-}\to Q\overline{Q}+X $$ , with Q a heavy quark, differential in a given event shape e and the angle θ  T  between the thrust axis and the beam direction. These observables are usually referred to as oriented event shapes, and it has been shown that the θ  T  dependence can be split in two structures, dubbed the unoriented and angular terms. Since the unoriented part is already known, we compute the differential and cumulative distributions in fixed-order for the angular part up to   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (α  s ). Our results show that, for the vector current, there is a non-zero   <math> <mi>O</mi> <mfenced> <msubsup> <mi>α</mi> <mi>s</mi> <mn>0</mn> </msubsup> </mfenced> </math>  $$ \mathcal{O}\left({\alpha}_s^0\right) $$  contribution, in contrast to the axial-vector current or for massless quarks. This entails that for the vector current one should expect singular terms at   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (α  s ) as well as infrared divergences in real- and virtual-radiation diagrams that should cancel when added up. On the phenomenological side, and taking into account that electroweak factors enhance the vector current, it implies that finite bottom-mass effects are an important correction since they are not damped by a power of the strong coupling and therefore cannot be neglected in precision studies. Finally, we show that the total angular distribution for the vector current has a Sommerfeld enhancement at threshold.

Nestor Gracia

NLO oriented event-shape distributions for massive quarks

We calculate the small-momentum expansion of vector, axial-vector, scalar, and pseudo-scalar heavy-quark current correlators in the large-β 0 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, α  s , 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.

Brazil

Diogo Boito

Marcus Rodrigues

Small-momentum expansion of heavy-quark correlators in the large-β 0 limit and α  s  extractions

We present results for the 2-jettiness differential distribution for boosted top quark pairs produced in <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> collisions in the peak region accounting for QCD large-logarithm resummation at next-to-next-to-next-to-leading logarithmic order and fixed-order corrections to matrix elements at next-to-next-to-leading order calculated in the framework of soft-collinear effective theory and boosted heavy quark effective theory. Electroweak and finite-width effects are included at leading order. We study the perturbative convergence of the cross section in the pole and MSR mass schemes, with and without soft gap subtractions. We find that there is a partial cancellation between the pole mass and soft function renormalons. When renormalon subtractions concerning the top mass and the soft function are implemented, the perturbative uncertainties are, however, systematically smaller and an improvement in the stability of the peak position is observed. We find that the top MSR mass may be determined with perturbative uncertainties well below 100 MeV from the peak position of the 2-jettiness distribution. This result has important applications for Monte Carlo top quark mass calibrations.

Brad Bachu

United Kingdom

Aditya Pathak

Iain W. Stewart

Boosted top quarks in the peak region with <math><mrow><mi>N</mi><mrow><mmultiscripts><mrow><mi>L</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mi>L</mi></mrow></math> resummation

The strong coupling, <math><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub></math>, governs perturbative Quantum Chromodynamics (QCD) and is one of the free parameters of the Standard Model. We introduce a new method that allows a precise extraction of <math><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>Z</mi></mrow></msub><mo>)</mo></math> from dimensionless ratios of roots of moments of the charm-quark vector correlator. The ratios we use in our analysis have a rather weak logarithmic quark-mass dependence, starting at <math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></math>, and can be obtained from experimental data with good precision, since they benefit from positive correlations among the individual experimentally determined moments. We perform a careful and conservative error analysis with special emphasis on uncertainties related to the truncation of perturbation theory, treating the renormalization scales such as to ensure order-by-order convergence. Our final result, with expressions at <math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>)</mo></math>, is <math><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>Z</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>0.1168</mn><mo>±</mo><mn>0.0019</mn></math>.