NLO) And Next-to-next-to-leading Order Research Articles

The path to N3LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {N}^3\\hbox {LO}$$\\end{document} parton distributions

We extend the existing leading (LO), next-to-leading (NLO), and next-to-next-to-leading order (NNLO) NNPDF4.0 sets of parton distribution functions (PDFs) to approximate next-to-next-to-next-to-leading order (aN3LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {aN}^3\\hbox {LO}$$\\end{document}). We construct an approximation to the N3LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {N}^3\\hbox {LO}$$\\end{document} splitting functions that includes all available partial information from both fixed-order computations and from small and large x resummation, and estimate the uncertainty on this approximation by varying the set of basis functions used to construct the approximation. We include known N3LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {N}^3\\hbox {LO}$$\\end{document} corrections to deep-inelastic scattering structure functions and extend the FONLL general-mass scheme to Oαs3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( \\alpha _s^3\\right) $$\\end{document} accuracy. We determine a set of aN3LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {aN}^3\\hbox {LO}$$\\end{document} PDFs by accounting both for the uncertainty on splitting functions due to the incomplete knowledge of N3LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {N}^3\\hbox {LO}$$\\end{document} terms, and to the uncertainty related to missing higher corrections (MHOU), estimated by scale variation, through a theory covariance matrix formalism. We assess the perturbative stability of the resulting PDFs, we study the impact of MHOUs on them, and we compare our results to the aN3LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {aN}^3\\hbox {LO}$$\\end{document} PDFs from the MSHT group. We examine the phenomenological impact of aN3LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {aN}^3\\hbox {LO}$$\\end{document} corrections on parton luminosities at the LHC, and give a first assessment of the impact of aN3LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {aN}^3\\hbox {LO}$$\\end{document} PDFs on the Higgs and Drell–Yan total production cross-sections. We find that the aN3LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {aN}^3\\hbox {LO}$$\\end{document} NNPDF4.0 PDFs are consistent within uncertainties with their NNLO counterparts, that they improve the description of the global dataset and the perturbative convergence of Higgs and Drell–Yan cross-sections, and that MHOUs on PDFs decrease substantially with the increase of perturbative order.

Determination of diffractive PDFs from a global QCD analysis of inclusive diffractive DIS and dijet cross-section measurements at HERA

We present an updated set of {\tt SKMHS} diffractive parton distribution functions (PDFs). In addition to the diffractive deep-inelastic scattering (diffractive DIS) data sets, the recent diffractive dijet cross sections measurements by the H1 experiment from the HERA collider are added to the data sample. The new set of diffractive PDFs, entitled {\tt SKMHS23} and {\tt SKMHS23-dijet}, are presented at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) accuracy in perturbative QCD. Since the gluons directly contribute to jet production through the boson-gluon fusion process, the data on diffractive dijet production in inclusive DIS help to constrain the gluon density, allowing for the determination of both the quark and gluon densities with better accuracy. The NLO and NNLO theory predictions calculated using both {\tt SKMHS23} and {\tt SKMHS23-dijet} are compared to the analyzed data showing excellent agreements. The effect arising from the inclusion of diffractive dijet data and higher order QCD corrections on the extracted diffractive PDFs and data/theory agreements are clearly examined and discussed.

Higgs pt distribution in Higgs + jet production at hadron colliders within the noncommutative effective standard model

We use the noncommutative Higgs effective standard model to make a phenomenological prediction for the transverse momentum distribution of the Higgs boson produced in association with a jet at hadron colliders. We calculate at leading order in the noncommutative parameter [Formula: see text] as well as leading order in the strong coupling [Formula: see text], the one-loop [Formula: see text] distribution of the Higgs boson. As in the standard model, the fixed-order distribution suffers from large logarithms at small [Formula: see text] which require an all-orders resummation. We find that the large-[Formula: see text] region of the distribution is strongly affected by the non-commutativity, while small-[Formula: see text] region is not. Following this observation, we propose a simple matching method that allows us to compute a result that is also valid at small [Formula: see text] obtained with standard-model parton showers such as Pythia 8. We also compare our results with the next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) distributions in the standard model, in order to assess the importance of higher-order effects in the search for non-commutativity at colliders.

Parton distribution functions and QCD coupling constant from LHC and non-LHC data

We present a new QCD analysis of parton distribution functions (PDFs) at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO). In the present paper, the role of special types of experimental measurements for extracting PDFs is investigated, and simultaneously the values of the strong coupling constant are determined. Essential elements of this QCD analysis are the HERA combined data as a base, heavy-quark cross-section measurement together with Tevatron data on jet production and H1 and ZEUS jet cross sections as ``non-LHC'' data, and especially the ``LHC'' dataset for top-quark and jet cross sections. These different datasets have allowed detailed information at low $x$, providing worthwhile information on the nucleon's flavor, and have played an important role for some PDFs at large-$x$ and strong coupling constant. Since the large-$x$ gluon PDF benefits from an accurate determination of quark PDFs, we have enough motivation to focus on the top-quark production and jet cross-section measurements from LHC to find the impact of these data on the gluon PDF and strong coupling constant. These experimental data have an impact on the relative uncertainties of PDF. The gluon PDF at large $x$ and the values of ${\ensuremath{\alpha}}_{s}({M}_{Z}^{2})$ are affected significantly. Although the main motivation of this paper is to focus on the gluon PDF and its uncertainty at large-$x$, giving notice to heavy PDF behavior in this region is also very important. We study the intrinsic charm (IC) using the BHPS model together with our extracted extrinsic charm PDF at large $x$, which can be very worthwhile for future experiments at the LHC.

QCD analysis of pion fragmentation functions in the xFitter framework

We present the first open-source analysis of fragmentation functions (FFs) of charged pions (entitled IPM-xFitter) computed at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) accuracy in perturbative QCD using the xFitter framework. This study incorporates a comprehensive and up-to-date set of pion production data from single-inclusive annihilation (SIA) processes, as well as the most recent measurements of inclusive cross-sections of single pion by the BELLE collaboration. The determination of pion FFs along with their theoretical uncertainties is performed in the Zero-Mass Variable-Flavor Number Scheme (ZM-VFNS). We also present comparisons of our FFs set with recent fits from the literature. The resulting NLO and NNLO pion FFs provide valuable insights for applications in present and future high-energy analysis of pion final state processes.

$${W}^{{\pm}}{/Z}$$ Cross Section Ratios Through (N)NLO Predictions in QCD

This paper presents $$W^{\pm}/Z$$ cross section ratio predictions for the $$W$$ and $$Z$$ boson productions in their subsequent electron decay channels with next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) calculations in perturbative QCD. The predicted ratios from the fiducial cross sections are primarily compared with the ATLAS measurement based on 13 TeV LHC proton–proton collisions. The predicted fiducial ratios for 14 TeV proton–proton collisions are also presented at (N)NLO accuracies. The 13 TeV and 14 TeV differential distributions for the ratios are provided as a function of the boson transverse momentum at NNLO using different parton distribution functions (PDFs). The 13 TeV and 14 TeV differential distributions for the ratios are also provided for the boson production in association with at least one jet as functions of the first leading jet transverse momentum and absolute pseudorapidity at NLO using various PDF sets. The focus is to predict the ratios more precisely in terms of the reduced theoretical scale uncertainties for the fiducial calculations and to assess sensitivity of the ratios to different PDF sets for the differential calculations. The 13 TeV ratio is predicted as 10.55 $$\pm$$ 0.35 at NLO and 10.57 $$\pm$$ 0.07 at NNLO and found to be in good agreement with the ATLAS data. The 14 TeV ratio is predicted consistently as 10.50 $$\pm$$ 0.36 at NLO and 10.37 $$\pm$$ 0.07 at NNLO. Estimated scale uncertainties are reduced to less than percent level at NNLO in the fiducial ratios. The differential ratios distributions are consistently predicted by using different PDF sets at both 13 TeV and 14 TeV. The differential ratios exhibit more sensitivity to choice of PDF sets in higher regions of the distributions.

Open-source QCD analysis of nuclear parton distribution functions at NLO and NNLO

We present new sets of nuclear parton distribution functions (nPDFs) at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO). Our analyses are based on deeply inelastic scattering data with charged-lepton and neutrino beams on nuclear targets. In addition, a set of proton baseline PDFs is fitted within the same framework with the same theoretical assumptions. The results of this global QCD analysis are compared to existing nPDF sets and to the fitted cross sections. Also, the uncertainties resulting from the limited constraining power of the included experimental data are presented. The published work is based on an open-source tool, xFitter, which has been modified to be applicable also for a nuclear PDF analysis. The required extensions of the code are discussed as well.

Consistent treatment of charm production in higher-orders at tree-level within kT -factorization approach

We discuss production of $c \bar c$-pairs within $k_T$-factorization approach (off-shell initial partons) with unintegrated parton distribution functions (uPDFs). We present a consistent prescription which merges the standard leading-order (LO) $k_T$-factorization calculations for this process with tree-level next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) matrix elements. For the first time we include in this framework 2 $\to$ 3 and 2 $\to$ 4 processes with extra partonic emissions for single particle distributions as well as for correlation observables. The use of the KMR uPDF leads to a good description of the existing charm ($D$-meson) data already at the leading-order. On the other hand, a new Parton-Branching (PB) uPDF strongly underestimates the same experimental data. A direct inclusion of the higher-orders at tree-level leads to an overestimation of the data, especially for the KMR uPDF. This suggests a significant double-counting. We propose a simple method how to avoid the double-counting. Our procedure leads to a much better description of the experimental data when including the higher-order contributions. Then with the KMR uPDF we get similar results (both for single particle and correlation observables) as for the standard calculations of the 2 $\to$ 2 processes. For the PB uPDF inclusion of the higher-orders considerably improves description of the experimental data. We conclude that the LO calculation with the KMR uPDF effectively includes the higher-orders which is not the case for the PB uPDF.

On phenomenological study of the solution of nonlinear GLR-MQ evolution equation beyond leading order

We present a phenomenological study of the small-x behavior of gluon distribution function G(x,Q2) at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) in light of the nonlinear Gribov–Ryskin–Levin–Mueller–Qiu (GLR-MQ) evolution equation by keeping the transverse size of the gluons (∼1/Q) fixed. We consider the NLO and NNLO corrections, of the gluon–gluon splitting function Pgg(z) and strong coupling constant αs(Q2). We have suggested semi-analytical solutions based on Regge like ansatz of gluon density G(x,Q2), which are supposed to be valid in the moderate range of photon virtuality (Q2) and at small Bjorken variable (x). The study of the effects of nonlinearities that arise due to gluon recombination effects at small-x is very interesting, which eventually tames down the unusual growth of gluon densities towards small-x as predicted by the linear DGLAP evolution equation.

Detailed k-Factor Studies Using Inclusive Production of W Bosons in pp Collisions

The theoretical k-Factor describes the difference between leading and higher order cross sections and its proper usage has a crucial importance for theoretical predictions. In this paper, three different ways are considered to define the k-Factor of W bosons at $$\sqrt s $$ = 14 TeV in proton-proton collisions. First, we calculate k-Factors using inclusive W± boson leading order (LO), next-toleading order (NLO) and next-to-next-to-leading order (NNLO) quantum chromodynamics (QCD) cross sections with LO, NLO, NNLO parton distribution function (PDF) models, respectively. The second approach used is the calculation of k-Factors based on LO, NLO and NNLO QCD cross sections with the NNLO PDF model. The last method used here is the calculation of k-Factors using the LO QCD cross section with LO, NLO and NNLO PDF models. We further investigated the dependencies of the k-Factor on the renormalization (μR) and the factorization (μF) scales, the strong coupling constant (α s ), the charges of W boson and the collision energy of protons in a range of 7 to 100 TeV.

Study of the [formula omitted] asymmetry in the proton

The study of s−s¯ asymmetry is essential to better understand of the structure of nucleon and also the perturbative and nonperturbative mechanisms for sea quark generation. Actually, the nature and dynamical origins of this asymmetry have always been an interesting subject to research both experimentally and theoretically. One of the most powerful models can lead to s−s¯ asymmetry is the meson–baryon model (MBM). In this work, using a simplified configuration of this model suggested by Pumplin, we calculate the s−s¯ asymmetry for different values of cutoff parameter Λ, to study the dependence of model to this parameter and also to estimate the theoretical uncertainty imposed on the results due to its uncertainty. Then, we study the evolution of distributions obtained both at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) using different evolution schemes. It is shown that the evolution of the intrinsic quark distributions from a low initial scale, as suggested by Chang and Pang, is not a good choice at NNLO using variable flavor number scheme (VFNS).

Evidence of quasi-partonic higher-twist effects in deep inelastic scattering at HERA at moderate $$Q^2$$ Q 2

The combined HERA data for the inclusive deep inelastic scattering (DIS) cross sections for the momentum transfer Q^2 > 1,mathrm {GeV}^2 are fitted within the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) framework at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) accuracy, complemented by a QCD-inspired parameterisation of twist 4 corrections. A modified form of the input parton density functions is also included, motivated by parton saturation mechanism at small Bjorken x and at a low scale. These modifications lead to a significant improvement of the data description in the region of low Q^2. For the whole data sample, the new benchmark NNLO DGLAP fit yields chi ^2/mathrm{d.o.f. } simeq 1.19 to be compared to 1.46 resulting from the standard NNLO DGLAP fit. We discuss the results in the context of the parton saturation picture and describe the impact of the higher-twist corrections on the derived parton density functions. The resulting description of the longitudinal proton structure function F_{mathrm {L}} is consistent with the HERA data. Our estimates of higher-twist contributions to the proton structure functions are comparable to the leading-twist contributions at low Q^2 simeq 2 GeV^2 and x simeq 10^{-5}. The x-dependence of the twist 4 corrections obtained from the best fit is consistent with the leading twist 4 quasi-partonic operators, corresponding to an exchange of four interacting gluons in the t-channel.

NLO+NLL limits on W′ and Z′ gauge boson masses in general extensions of the Standard Model

QCD resummation predictions for the production of charged (W') and neutral (Z') heavy gauge bosons decaying leptonically are presented. The results of our resummation code at next-to-leading order and next-to-leading logarithmic (NLO+NLL) accuracy are compared to Monte Carlo predictions obtained with PYTHIA at leading order (LO) supplemented with parton showers (PS) and FEWZ at NLO and next-to-next-to-leading order (NNLO) for the $p_T$-differential and total cross sections in the Sequential Standard Model (SSM) and general SU(2)xSU(2)xU(1) models. The LO+PS Monte Carlo and NNLO fixed-order predictions are shown to agree approximately with those at NLO+NLL at small and intermediate $p_T$, respectively, and the importance of resummation for total cross sections is shown to increase with the gauge boson mass. The theoretical uncertainties are estimated by variations of the renormalisation/factorisation scales and of the parton densities, the former being significantly reduced by the resummation procedure. New limits at NLO+NLL on W' and Z' boson masses are obtained by reinterpreting the latest ATLAS and CMS results in general extensions of the Standard Model.

Parton distribution functions at LO, NLO and NNLO with correlated uncertainties between orders

Sets of parton distribution functions (PDFs) of the proton are reported for the leading (LO), next-to-leading (NLO) and next-to-next-to-leading-order (NNLO) QCD calculations. The parton distribution functions are determined with the HERAFitter program using the data from the HERA experiments and preserving correlations between uncertainties for the LO, NLO and NNLO PDF sets. The sets are used to study cross-section ratios and their uncertainties when calculated at different orders in QCD. A reduction of the overall theoretical uncertainty is observed if correlations between the PDF sets are taken into account for the ratio of \(WW\) di-boson to \(Z\) boson production cross sections at the LHC.

Nonperturbative renormalization of the chiral nucleon-nucleon interaction up to next-to-next-to-leading order

We study the nonperturbative renormalization of the nucleon-nucleon (NN) interaction at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) of chiral effective field theory. A systematic variation of the cutoff parameter is performed for values below the chiral symmetry breaking scale of about 1 GeV. The accuracy of the predictions is determined by calculating the $\chi^2$ for the reproduction of the NN data for energy intervals below pion-production threshold. At NLO, the NN data are described well up to about 100 MeV laboratory energy and, at NNLO, up to about 200 MeV---with, essentially, cutoff independence for cutoffs between about 450 and 850 MeV.

Parton distribution functions and benchmark cross sections at next-to-next-to-leading order

We present a determination of parton distribution functions (ABM11) and the strong coupling constant alpha_s at next-to-leading order and next-to-next-to-leading order (NNLO) in QCD based on world data for deep-inelastic scattering and fixed-target data for the Drell-Yan process. The analysis is performed in the fixed-flavor number scheme for n_f=3,4,5 and uses the MSbar-scheme for alpha_s and the heavy-quark masses. At NNLO we obtain the value alpha_s(M_Z) = 0.1134 +- 0.0011. The fit results are used to compute benchmark cross sections at hadron colliders to NNLO accuracy and to compare to data from the LHC.

Evolution of singlet structure functions from DGLAP equation at next-to-next-to-leading order at small-x

A semi-numerical solution to Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations at leading order (LO), next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) in the small-x limit is presented. Here we have used Taylor series expansion method to solve the evolution equations and, t- and x-evolutions of the singlet structure functions have been obtained with such solution. We have also calculated t- and x-evolution of deuteron structure functions $F_{2}^{d}$ , and the results are compared with the E665 data and NMC data. The results are also compared to those obtained by the fit to $F_{2}^{d}$ produced by the NNPDF collaboration based on the NMC and BCDMS data.

Neutrino mixing angles in sequential dominance to NLO and NNLO

Neutrinos with hierarchical masses and two large mixing angles may naturally originate from sequential dominance (SD). Within this framework we present analytic expressions for the neutrino mixing angles including the next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) corrections arising from the second lightest and lightest neutrino masses. The analytic results for neutrino mixing angles in SD presented here, including the NLO and NNLO corrections, are applicable to a wide class of models and may provide useful insights when confronting the models with data from high precision neutrino experiments. We also point out that for special cases of SD corresponding to form dominance (FD) the NLO and NNLO corrections both vanish. For example we study tri-bimaximal (TB) mixing via constrained sequential dominance (CSD) which involves only a NNLO correction and tri-bimaximal-reactor (TBR) mixing via partially constrained sequential dominance (PCSD) which involves a NLO correction suppressed by the small reactor angle and show that the analytic results have good agreement with the numerical results for these cases.

Measurement of [formula omitted] of Drell–Yan [formula omitted] pairs in the Z mass region from [formula omitted] collisions at [formula omitted

We report on a CDF measurement of the total cross section and rapidity distribution, dσ/dy, for γ∗/Z→e+e− events in the Z boson mass region (66<Mee<116 GeV/c2) produced in pp¯ collisions at s=1.96 TeV with 2.1 fb−1 of integrated luminosity. The measured cross section of 257±16 pb and dσ/dy distribution are compared with Next-to-Leading-Order (NLO) and Next-to-Next-to-Leading-Order (NNLO) QCD theory predictions with CTEQ and MRST/MSTW parton distribution functions (PDFs). There is good agreement between the experimental total cross section and dσ/dy measurements with theoretical calculations with the most recent NNLO PDFs.

Dynamical next-to-next-to-leading order parton distributions and the perturbative stability ofFL(x,Q2)

Recent measurements for ${F}_{2}(x,{Q}^{2})$ have been analyzed in terms of the ``dynamical'' and ``standard'' parton model approach at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) of perturbative QCD. Having fixed the relevant NLO and NNLO parton distributions, we present the implications and predictions for the longitudinal structure function ${F}_{L}(x,{Q}^{2})$. It is shown that the previously noted extreme perturbative NNLO/NLO instability of ${F}_{L}(x,{Q}^{2})$ is an artifact of the commonly utilized ``standard'' gluon distributions. In particular it is demonstrated that using the appropriate---dynamically generated---parton distributions at NLO and NNLO, ${F}_{L}(x,{Q}^{2})$ turns out to be perturbatively rather stable already for ${Q}^{2}\ensuremath{\ge}\mathcal{O}(2--3\text{ }\text{ }{\mathrm{GeV}}^{2})$.