We present a new formulation of the MiNNLOPS method to match NNLO QCD calculations with parton showers by using jettiness as a resummation variable. The full derivation for colour-singlet processes is presented using 0-jettiness starting from the NNLL ′  resummation formula. We show phenomenological results for Drell-Yan and Higgs-boson production at the LHC and compare our predictions to ATLAS and CMS data. Differences to the original MiNNLOPS formulation using the transverse momentum of the colour singlet as resummation variable are discussed. We further present a comparison of MiNNLOPS predictions with Geneva. Finally, we extend the formulation of the MiNNLOPS method to 1-jettiness which is applicable to processes with a colour singlet plus one jet in the final state.

Germany

Markus Ebert

Switzerland

Luca Rottoli

Marius Wiesemann

Giulia Zanderighi

United Kingdom

Silvia Zanoli

Jettiness formulation of the MiNNLOPS method

A correction to this paper has been published: https://doi.org/10.1007/JHEP06(2022)007

United States

Anjie Gao

Iain Stewart

Erratum to: Factorization for azimuthal asymmetries in SIDIS at next-to-leading power

The extraction of nonperturbative TMD physics is made challenging by prescriptions that shield the Landau pole, which entangle long- and short-distance contributions in momentum space. The use of different prescriptions then makes the comparison of fit results for underlying nonperturbative contributions not meaningful on their own. We propose a model-independent method to restrict momentum-space observables to the perturbative domain. This method is based on a set of integral functionals that act linearly on terms in the conventional position-space operator product expansion (OPE). Artifacts from the truncation of the integral can be systematically pushed to higher powers in ΛQCD/k  T . We demonstrate that this method can be used to compute the cumulative integral of TMD PDFs over   <math> <msub> <mi>k</mi> <mi>T</mi> </msub> <mo>≤</mo> <msubsup> <mi>k</mi> <mi>T</mi> <mi>cut</mi> </msubsup> </math>  $$ {k}_T\le {k}_T^{\mathrm{cut}} $$  in terms of collinear PDFs, accounting for both radiative corrections and evolution effects. This yields a systematic way of correcting the naive picture where the TMD PDF integrates to a collinear PDF, and for unpolarized quark distributions we find that when renormalization scales are chosen near   <math> <msubsup> <mi>k</mi> <mi>T</mi> <mi>cut</mi> </msubsup> </math>  $$ {k}_T^{\mathrm{cut}} $$ , such corrections are a percent-level effect. We also show that, when supplemented with experimental data and improved perturbative inputs, our integral functionals will enable model-independent limits to be put on the non-perturbative OPE contributions to the Collins-Soper kernel and intrinsic TMD distributions.

Johannes Michel

Zhiquan Sun

Disentangling long and short distances in momentum-space TMDs

Differential measurements of the semi-inclusive deep inelastic scattering (SIDIS) process with polarized beams provide important information on the three-dimensional structure of hadrons. Among the various observables are azimuthal asymmetries that start at subleading power, and which give access to novel transverse momentum dependent distributions (TMDs). Theoretical predictions for these distributions are currently based on the parton model rather than a rigorous factorization based analysis. Working under the assumption that leading power Glauber interactions do not spoil factorization at this order, we use the Soft Collinear Effective Theory to derive a complete factorization formula for power suppressed hard scattering effects in SIDIS. This yields generalized definitions of the TMDs that depend on two longitudinal momentum fractions (one of them only relevant beyond tree level), and a complete proof that only the same leading power soft function appears and can be absorbed into the TMD distributions at this order. We also show that perturbative corrections can be accounted for with only one new hard coefficient. Factorization formulae are given for all spin dependent structure functions which start at next-to-leading power. Prospects for improved subleading power predictions that include resummation are discussed.

Factorization for azimuthal asymmetries in SIDIS at next-to-leading power

Transverse-momentum-dependent parton distribution functions (TMDs) can be studied from first principles by a perturbative matching onto lattice-calculable quantities: so-called lattice TMDs, which are a class of equal-time correlators that includes quasi-TMDs and TMDs in the Lorentz-invariant approach. We introduce a general correlator that includes as special cases these two Lattice TMDs and continuum TMDs, like the Collins scheme. Then, to facilitate the derivation of a factorization relation between lattice and continuum TMDs, we construct a new scheme, the Large Rapidity (LR) scheme, intermediate between the Collins and quasi-TMDs. The LR and Collins schemes differ only by an order of limits, and can be matched onto one another by a multiplicative kernel. We show that this same matching also holds between quasi and Collins TMDs, which enables us to prove a factorization relation between these quantities to all orders in α  s . Our results imply that there is no mixing between various quark flavors or gluons when matching Collins and quasi TMDs, making the lattice calculation of individual flavors and gluon TMDs easier than anticipated. We cross-check these results explicitly at one loop and discuss implications for other physical-to-lattice scheme factorizations.

Stella Schindler

Yong Zhao

Factorization connecting continuum &amp; lattice TMDs

Sudakov form factors appear ubiquitously in factorized cross sections where they allow one to resum large logarithms to all orders in perturbation theory. Their exact evaluation requires numerical integrals over anomalous dimensions, which in practice can hamper efficiency. Alternatively, one can use approximate analytic solutions, which provide fast evaluation at the cost of numerical precision and loss of properties such as renormalization group invariance. We provide an exact analytic expression of the QCD Sudakov form factor which allows one to obtain fast and numerically exact results.

Analytic results for Sudakov form factors in QCD

We present the analytic formula for the Energy-Energy Correlation (EEC) in electron-positron annihilation computed in perturbative QCD to next-to-next-to-next-to-leading order (N3LO) in the back-to-back limit. In particular, we consider the EEC arising from the annihilation of an electron-positron pair into a virtual photon as well as a Higgs boson and their subsequent inclusive decay into hadrons. Our computation is based on a factorization theorem of the EEC formulated within Soft-Collinear Effective Theory (SCET) for the back-to-back limit. We obtain the last missing ingredient for our computation — the jet function — from a recent calculation of the transverse-momentum dependent fragmentation function (TMDFF) at N3LO. We combine the newly obtained N3LO jet function with the well known hard and soft function to predict the EEC in the back-to-back limit. The leading transcendental contribution of our analytic formula agrees with previously obtained results in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 supersymmetric Yang-Mills theory. We obtain the N = 2 Mellin moment of the bulk region of the EEC using momentum sum rules. Finally, we obtain the first resummation of the EEC in the back-to-back limit at N3LL′ accuracy, resulting in a factor of ∼ 4 reduction of uncertainties in the peak region compared to N3LL predictions.

Bernhard Mistlberger

Gherardo Vita

The Energy-Energy Correlation in the back-to-back limit at N3LO and N3LL′

We compute the unpolarized quark and gluon transverse-momentum dependent fragmentation functions (TMDFFs) at next-to-next-to-next-to-leading order (N3LO) in perturbative QCD. The calculation is based on a relation between the TMDFF and the limit of the semi-inclusive deep inelastic scattering cross section where all final-state radiation becomes collinear to the detected hadron. The required cross section is obtained by analytically continuing our recent computation of the Drell-Yan and Higgs boson production cross section at N3LO expanded around the limit of all final-state radiation becoming collinear to one of the initial states. Our results agree with a recent independent calculation by Luo et al.

TMD fragmentation functions at N3LO

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.

Frank Tackmann

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

We demonstrate how to efficiently expand cross sections for color-singlet production at hadron colliders around the kinematic limit of all final state radiation being collinear to one of the incoming hadrons. This expansion is systematically improvable and applicable to a large class of physical observables. We demonstrate the viability of this technique by obtaining the first two terms in the collinear expansion of the rapidity distribution of the gluon fusion Higgs boson production cross section at next-to-next-to leading order (NNLO) in QCD perturbation theory. Furthermore, we illustrate how this technique is used to extract universal building blocks of scattering cross section like the N-jettiness and transverse momentum beam function at NNLO.