We develop a new approach to estimate the uncertainty due to missing higher orders in perturbative predictions (the perturbative “theory uncertainty”), which overcomes many inherent limitations of the currently prevalent methods based on varying unphysical renormalization scales. In our approach, the true underlying sources of the theory uncertainty, namely the missing higher-order terms, are identified and parameterized in terms of mutually independent theory nuisance parameters (TNPs). The TNPs are true parameters of the calculation, i.e., they have a well-defined true value that is not or only imprecisely known. This approach affords the theory uncertainty all benefits of a truly parametric uncertainty: it provides correct correlations and allows for consistent error propagation and combination. Furthermore, the TNPs can be profiled in fits, allowing the data to reduce the theory uncertainties. On the theory side, it allows maximally exploiting all available higher-order information to reduce the theory uncertainty, such as partial higher-order results or any nontrivial knowledge of the higher-order or all-order structure.

Germany

Frank Tackmann

Beyond scale variations: perturbative theory uncertainties from nuisance parameters

Measurements of Higgs boson processes by the ATLAS and CMS experiments at the LHC use Simplified Template Cross Sections (STXS) as a common framework for the combination of measurements in different decay channels and their further interpretation, e.g. to measure Higgs couplings. The different Higgs production processes are measured in predefined kinematic regions — the STXS bins — requiring precise theory predictions for each individual bin. In gluon-fusion Higgs production a main division is into 0-jet, 1-jet, and ≥ 2-jet bins, which are further subdivided in bins of the Higgs transverse momentum $$ {p}_T^H $$ . Requiring a fixed number of jets induces logarithms ln $$ {p}_T^{\textrm{cut}}/Q $$ in the cross section where $$ {p}_T^{\textrm{cut}} $$ is the jet- threshold and ∼ $$ {p}_T^H $$ ∼ the hard-interaction scale. These jet-veto logarithms can be resummed to all orders in perturbation theory to achieve the highest possible perturbative precision. We provide state-of-the art predictions for the $$ {p}_T^H $$ spectrum in exclusive 1-jet production and the corresponding 1-jet STXS bins in the kinematic regime $$ {p}_T^{\textrm{cut}} $$ ≪ $$ {p}_T^H $$ ∼ . We carry out the resummation at NNLL accuracy, using theory nuisance parameters to account for the few unknown ingredients at this order, and match to full NNLO. We revisit the jet-veto factorization for this process and find that it requires refactorizing the total soft function into a global and soft-collinear contribution in order to fully account for logarithms of the signal jet radius. The leading nonglobal logarithms are also included, though they are numerically small for the region of phenomenological interest.

Pedro Cal

United Kingdom

Matthew Lim

Darren Scott

Netherlands

Wouter Waalewijn

Jet veto resummation for STXS

We provide state-of-the-art precision QCD predictions for the fiducial and boson transverse momentum spectra at the LHC at N LL and approximate N LL in resummed perturbation theory, matched to available $$ \mathcal{O}\left({\alpha}_s^3\right) $$ fixed-order results. Our predictions consistently combine all information from across the spectrum in a unified way, ranging from the nonperturbative region of small transverse momenta to the fixed-order tail, with an emphasis on estimating the magnitude of residual perturbative uncertainties, and in particular of those related to the matching. Parametric uncertainties related to the strong coupling, the collinear PDFs, and the nonperturbative transverse momentum-dependent (TMD) dynamics are studied in detail. To assess the latter, we explicitly demonstrate how the full complexity of flavor and Bjorken -dependent TMD dynamics can be captured by a single, effective nonperturbative function for the resonant production of any given vector boson at a given collider. We point out that the cumulative $$ {p}_T^Z $$ cross section at the level of precision enabled by our predictions provides strong constraining power for PDF determinations at full N LO.

Italy

Georgios Billis

United States

Johannes Michel

Drell-Yan transverse-momentum spectra at N

We construct a novel event generator for the process   <math> <mi>pp</mi> <mo>→</mo> <msup> <mi>ℓ</mi> <mo>−</mo> </msup> <msub> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mi>ℓ</mi> </msub> <msup> <msup> <mi>ℓ</mi> <mo>′</mo> </msup> <mo>+</mo> </msup> <msub> <mi>ν</mi> <msup> <mi>ℓ</mi> <mo>′</mo> </msup> </msub> </math>  $$ pp\to {\ell}^{-}{\overline{\nu}}_{\ell }{\ell^{\prime}}^{+}{\nu}_{\ell^{\prime }} $$ , which matches fixed-order predictions at next-to-next-to-leading order in the strong coupling to a parton shower program. The matching is achieved using the Geneva method, in this case exploiting a resummed calculation for the hardest jet transverse momentum at next-to-next-to-leading logarithmic accuracy obtained via soft-collinear effective theory and implemented in the C++ library SCETlib. This choice of resolution variable ensures that the introduction of a jet veto, commonly used by experimental analyses to reject multi-jet background events, does not result in the appearance of unmitigated large logarithms for low veto scales before showering. After validating our partonic results against publicly available fixed order and resummed calculations, we compare our predictions to measurements taken at the ATLAS and CMS experiments, finding good agreement. This is the first NNLO+PS accurate event generator to use the hardest jet transverse momentum as a resolution variable.

Alessandro Gavardi

Simone Alioli

NNLO+PS W + W  −  production using jet veto resummation at NNLL′

We present predictions for the photon energy spectrum in inclusive B → X  s  γ decays mediated by the electromagnetic penguin operator O 7 to N3LL′. We use soft-collinear effective theory (SCET) to resum the singular contributions in the peak region at large photon energy. In the tail region the resummed predictions are matched to fixed order at N3LO, where we include the known fixed-order contributions for O 7 up to   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (  <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> </math>  $$ {\alpha}_s^2 $$ ). We develop a method to suitably parametrize the still unknown   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (  <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>3</mn> </msubsup> </math>  $$ {\alpha}_s^3 $$ ) nonsingular corrections in terms of theory nuisance parameters, whose variations provide an estimate of the associated theory uncertainty. In this context, we also study different ways to treat higher-order cross terms in the matching. Another important aspect of our analysis is the short-distance scheme used for the b-quark mass m  b . We find that in the present context, the 1S mass scheme, which was previously used up to 2-loop order, fails to work at 3-loop order, because the mass scheme enters at a soft scale much smaller than m  b  here, for which the 1S scheme was not devised. Using instead the MSR mass scheme with R ~ 1 GeV, we obtain stable results with good perturbative convergence up to N3LL′.

Bahman Dehnadi

Ivan Novikov

The photon energy spectrum in B → X  s  γ at N3LL′

Double parton distributions are the nonperturbative ingredients needed for computing double parton scattering processes in hadron–hadron collisions. They describe a variety of correlations between two partons in a hadron and depend on a large number of variables, including two independent renormalization scales. This makes it challenging to compute their scale evolution with satisfactory numerical accuracy while keeping computational costs at a manageable level. We show that this problem can be solved using interpolation on Chebyshev grids, extending the methods we previously developed for ordinary single-parton distributions. Using an implementation of these methods in the C++ library ChiliPDF, we study for the first time the evolution of double parton distributions beyond leading order in perturbation theory.

Markus Diehl

Riccardo Nagar

Peter Plößl

Evolution and interpolation of double parton distributions using Chebyshev grids

Parton distribution functions (PDFs) are an essential ingredient for theoretical predictions at colliders. Since their exact form is unknown, their handling and delivery for practical applications relies on approximate numerical methods. We discuss the implementation of PDFs based on a global interpolation in terms of Chebyshev polynomials. We demonstrate that this allows for significantly higher numerical accuracy at lower computational cost compared with local interpolation methods such as splines. Whilst the numerical inaccuracy of currently used local methods can become a nontrivial limitation in high-precision applications, in our approach it is negligible for practical purposes. This holds in particular for differentiation and for Mellin convolution with kernels that have end point singularities. We illustrate our approach for these and other important numerical operations, including DGLAP evolution, and find that they are performed accurately and fast. Our results are implemented in the C++ library ChiliPDF.

ChiliPDF: Chebyshev interpolation for parton distributions

We consider Drell-Yan production pp → V * X → LX at small q  T  ≪ Q, where q  T  and Q are the total transverse momentum and invariant mass of the leptonic final state L. Experimental measurements require fiducial cuts on L, which in general introduce enhanced, linear power corrections in q  T  /Q. We show that they can be unambiguously predicted from factorization, and resummed to the same order as the leading-power contribution. For the fiducial q  T  spectrum, they constitute the complete linear power corrections. We thus obtain predictions for the fiducial q  T  spectrum to N3LL and next-to-leading-power in q  T  /Q. Matching to full NNLO (  <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> </math>  $$ {\alpha}_s^2 $$ ), we find that the linear power corrections are indeed the dominant ones, and once included by factorization, the remaining fixed-order corrections become almost negligible below q  T  ≲ 40 GeV. We also discuss the implications for more complicated observables, and provide predictions for the fiducial ϕ * spectrum at N3LL+NNLO. We find excellent agreement with ATLAS and CMS measurements of q  T  and ϕ *. We also consider the   <math> <msubsup> <mi>p</mi> <mi>T</mi> <mi>ℓ</mi> </msubsup> </math>  $$ {p}_T^{\mathrm{\ell}} $$  spectrum. We show that it develops leptonic power corrections in q  T  /(Q − 2  <math> <msubsup> <mi>p</mi> <mi>T</mi> <mi>ℓ</mi> </msubsup> </math>  $$ {p}_T^{\mathrm{\ell}} $$ ), which diverge near the Jacobian peak   <math> <msubsup> <mi>p</mi> <mi>T</mi> <mi>ℓ</mi> </msubsup> </math>  $$ {p}_T^{\mathrm{\ell}} $$  ∼ Q/2 and must be kept to all powers to obtain a meaningful result there. Doing so, we obtain for the first time an analytically resummed result for the   <math> <msubsup> <mi>p</mi> <mi>T</mi> <mi>ℓ</mi> </msubsup> </math>  $$ {p}_T^{\mathrm{\ell}} $$  spectrum around the Jacobian peak at N3LL+NNLO. Our method is based on performing a complete tensor decomposition for hadronic and leptonic tensors. We show that in practice this is equivalent to often-used recoil prescriptions, for which our results now provide rigorous, formal justification. Our tensor decomposition yields nine Lorentz-scalar hadronic structure functions, which for Z/γ * → ℓℓ or W → ℓν directly map onto the commonly used angular coefficients, but also holds for arbitrary leptonic final states. In particular, for suitably defined Born-projected leptons it still yields a LO-like angular decomposition even when including QED final-state radiation. Finally, we also discuss the application to q  T  subtractions. Including the unambiguously predicted fiducial power corrections significantly improves their performance, and in particular makes them applicable near kinematic edges where they otherwise break down due to large leptonic power corrections.

Markus Ebert

Iain Stewart

Drell-Yan q  T  resummation of fiducial power corrections at N3LL

The rapidity-dependent jet veto observables $$ {\mathcal{T}}_{Bj} $$ and $$ {\mathcal{T}}_{Cj} $$ provide a tight jet veto at central rapidity, gradually transitioning to a loose veto at forward rapidities. They divide the phase space into exclusive jet bins in a different way to the traditional jet veto observable , and are advantageous to use under harsh pile-up conditions. We obtain predictions for the 0-jet gluon-fusion (ggF) Higgs cross section using both of these veto observables at NNLL +NNLO, and compare these predictions to the prior state-of-the-art of NLL +NLO. A significant reduction in perturbative uncertainty is observed going from NLL +NLO to NNLL +NNLO, with the NNLL +NNLO predictions lying inside the uncertainty band of the NLL +NLO predictions. We also investigate the relative sensitivities of ggF Higgs cross sections with $$ {\mathcal{T}}_{Bj} $$ , $$ {\mathcal{T}}_{Cj} $$ and jet vetoes to underlying event and hadronisation effects using an NLO+parton shower calculation. We find that the cross sections with $$ {\mathcal{T}}_{Bj} $$ and $$ {\mathcal{T}}_{Cj} $$ vetoes have a reduced sensitivity to underlying event and hadronisation effects compared to that with a veto.

India

Shireen Gangal

Switzerland

Jonathan Gaunt

Eleni Vryonidou

Higgs production at NNLL′+NNLO using rapidity dependent jet vetoes

We consider the higher-order resummation of Sudakov double logarithms in the presence of multiple coupled gauge in. The associated evolution equations depend on the coupled β functions of two (or more) coupling constants α  a  and α  b , as well as anomalous dimensions that have joint perturbative series in α  a  and α  b . We discuss possible strategies for solving the system of evolution equations that arises. As an example, we obtain the complete three-loop (NNLL) QCD⊗QED Sudakov evolution factor. Our results also readily apply to the joint higher-order resummation of electroweak and QCD Sudakov logarithms.