Zeitschriftenartikel: „Next-to-leading order (NLO)“

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CONTOGOURIS, A. P., und Z. MEREBASHVILI. „APPROXIMATE NEXT-TO-LEADING ORDER AND NEXT-TO-NEXT-TO-LEADING ORDER CORRECTIONS“. International Journal of Modern Physics A 18, Nr. 06 (10.03.2003): 957–66. http://dx.doi.org/10.1142/s0217751x03013983.

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For processes involving structure functions and/or fragmentation functions, arguments that over a range of a proper kinematic variable, there is a part that dominates the next-to-leading order (NLO) corrections, are briefly reviewed. The arguments are tested against more recent NLO and in particular complete next-to-next-to-leading order (NNLO) calculations. A critical examination of when these arguments may not be useful is also presented.

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CHIRILLI, GIOVANNI ANTONIO. „SMALL-x EVOLUTION IN THE NEXT-TO-LEADING ORDER“. Modern Physics Letters A 24, Nr. 35n37 (07.12.2009): 3052–61. http://dx.doi.org/10.1142/s0217732309001261.

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After a brief introduction to Deep Inelastic Scattering in the Bjorken limit and in the Regge Limit we discuss the operator product expansion in terms of non local string operator and in terms of Wilson lines. We will show how the high-energy behavior of amplitudes in gauge theories can be reformulated in terms of the evolution of Wilson-line operators. In the leading order this evolution is governed by the non-linear Balitsky-Kovchegov (BK) equation. In order to see if this equation is relevant for existing or future deep inelastic scattering (DIS) accelerators (like Electron Ion Collider (EIC) or Large Hadron electron Collider (LHeC)) one needs to know the next-to-leading order (NLO) corrections. In addition, the NLO corrections define the scale of the running-coupling constant in the BK equation and therefore determine the magnitude of the leading-order cross sections. In Quantum Chromodynamics (QCD), the next-to-leading order BK equation has both conformal and non-conformal parts. The NLO kernel for the composite operators resolves in a sum of the conformal part and the running-coupling part. The QCD and [Formula: see text] kernel of the BK equation is presented.

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BALITSKY, IAN. „PHOTON IMPACT FACTOR AND kT FACTORIZATION IN THE NEXT-TO-LEADING ORDER“. International Journal of Modern Physics: Conference Series 20 (Januar 2012): 187–99. http://dx.doi.org/10.1142/s2010194512009233.

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The photon impact factor for the BFKL pomeron is calculated in the next-to-leading order (NLO) approximation using the operator expansion in Wilson lines. The result is represented as a NLO kT-factorization formula for the structure functions of small-x deep inelastic scattering.

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Triantafyllopoulos, D. N. „Forward particle production in proton-nucleus collisions at next-to-leading order“. EPJ Web of Conferences 192 (2018): 00014. http://dx.doi.org/10.1051/epjconf/201819200014.

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We consider the next-to-leading order (NLO) calculation of single inclusive particle production at forward rapidities in proton-nucleus collisions and in the framework of the Color Glass Condensate (CGC). We focus on the quark channel and the corrections associated with the impact factor. In the first step of the evolution the kinematics of the emitted gluon is kept exactly (and not in the eikonal approximation), but such a treatment which includes NLO corrections is not explicitly separated from the high energy evolution. Thus, in this newly established “factorization scheme”, there is no “rapidity subtraction”. The latter suffers from fine tuning issues and eventually leads to an unphysical (negative) cross section. On the contrary, our reorganization of the perturbation theory leads by definition to a well-defined cross section and the numerical evaluation of the NLO correction is shown to have the correct size.

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Carloni Calame, Carlo M., Mauro Chiesa, Guido Montagna, Oreste Nicrosini und Fulvio Piccinini. „Muon-electron scattering at next-to-leading order accuracy“. EPJ Web of Conferences 212 (2019): 05002. http://dx.doi.org/10.1051/epjconf/201921205002.

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The next-to-leading order electro-weak radiative corrections to the µ±e- → µ±e- process are reviewed and their relevance is discussed for the MUonE experiment, proposed at CERN. The aim of MUonE is the high precision measurement of the QED running coupling constant in the space-like region, from which the full hadronic contribution can be extracted and used to provide a new and independent determination of the leading-order hadronic correction to the muon g − 2. In this context, the required accuracy demands that radiative corrections are accounted for at the highest level of precision and implemented into a Monte Carlo event generator for data analysis. The first step towards the final goal of theoretical precision, which will require the full set of NNLO corrections and resummation of higher orders, is the inclusion of NLO electro-weak corrections.

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FIGY, TERRANCE. „NEXT-TO-LEADING ORDER QCD CORRECTIONS TO LIGHT HIGGS PAIR PRODUCTION VIA VECTOR BOSON FUSION“. Modern Physics Letters A 23, Nr. 24 (10.08.2008): 1961–73. http://dx.doi.org/10.1142/s0217732308028181.

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We present the NLO QCD corrections for light Higgs pair production via vector boson fusion at the LHC within the CP conserving type II two-Higgs doublet model in the form of a fully flexible parton-level Monte Carlo program. Scale dependences on integrated cross sections and distributions are reduced with QCD K-factors of order unity.

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BUTENSCHOEN, MATHIAS, und BERND A. KNIEHL. „NEXT-TO-LEADING ORDER TESTS OF NON-RELATIVISTIC-QCD FACTORIZATION WITH J/ψ YIELD AND POLARIZATION“. Modern Physics Letters A 28, Nr. 09 (21.03.2013): 1350027. http://dx.doi.org/10.1142/s0217732313500272.

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We report on recent progress in testing the factorization formalism of non-relativistic quantum chromodynamics (NRQCD) at next-to-leading order (NLO) for J/ψ yield and polarization. We demonstrate that it is possible to unambiguously determine the leading color-octet (CO) long-distance matrix elements (LDMEs) in compliance with the velocity scaling rules through a global fit to experimental data of unpolarized J/ψ production in pp, [Formula: see text], ep, γγ, and e+e-collisions. Three data sets not included in the fit, from hadro-production and from photo-production in the fixed-target and colliding-beam modes, are nicely reproduced. The polarization observables measured in different frames at DESY HERA and CERN LHC reasonably agree with NLO NRQCD predictions obtained using the LDMEs extracted from the global fit, while measurements at the FNAL Tevatron exhibit severe disagreement. We demonstrate that the alternative LDME sets recently obtained, with different philosophies, in two other NLO NRQCD analyses of J/ψ yield and polarization also fail to reconcile the Tevatron polarization data with the other available world data.

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Brandenburg, A., S. Dittmaier, P. Uwer und S. Weinzierl. „Top quark pair + jet production at next-to-leading order: NLO QCD corrections to“. Nuclear Physics B - Proceedings Supplements 135 (Oktober 2004): 71–75. http://dx.doi.org/10.1016/j.nuclphysbps.2004.09.038.

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Ogul, Hasan, Kamuran Dilsiz, Emrah Tiras, Ping Tan, Yasar Onel und Jane Nachtman. „High Order QCD Predictions for Inclusive Production ofWBosons inppCollisions ats=13 TeV“. Advances in High Energy Physics 2016 (2016): 1–9. http://dx.doi.org/10.1155/2016/7865689.

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Predictions of fiducial cross sections, differential cross sections, and lepton charge asymmetry are presented for the production ofW±bosons with leptonic decay up to next-to-next-to-leading order (NNLO) in perturbative QCD. Differential cross sections ofW±bosons andWboson lepton charge asymmetry are computed as a function of lepton pseudorapidity for a defined fiducial region inppcollisions ats=13 TeV. Numerical results of fiducialW±cross section predictions are presented with the latest modern PDF models at next-to-leading order (NLO) and NNLO. It is found that the CT14 and NNPDF 3.0 predictions with NNLO QCD corrections are about 4% higher than the NLO CT14 and NNPDF 3.0 predictions while MMHT 2014 predictions with NLO QCD corrections are smaller than its NNLO QCD predictions by approximately 6%. In addition, the NNLO QCD corrections reduce the scale variation uncertainty on the cross section by a factor of 3.5. The prediction of central values and considered uncertainties are obtained using FEWZ 3.1 program.

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Gnech, Alex, Jordy de Vries, Sachin Shain und Michele Viviani. „Electric dipole moment of light nuclei in chiral effective field theory“. EPJ Web of Conferences 258 (2022): 06007. http://dx.doi.org/10.1051/epjconf/202225806007.

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CP-violating interactions at quark level generate CP-violating nuclear interactions and currents, which could be revealed by looking at the presence of a permanent nuclear electric dipole moment. Within the framework of chiral effective field theory, we discuss the derivation of the CP-violating nuclear potential up to next-to-next-to leading order (N2LO) and the preliminary results for the charge operator up to next-to leading order (NLO). Moreover, we introduce some renormalization argument which indicates that we need to promote the short-distance operator to the leading order (LO) in order to reabsorb the divergences generated by the one pion exchange. Finally, we present some selected numerical results for the electric dipole moments of 2H, 3He and 3H discussing the systematic errors introduced by the truncation of the chiral expansion.

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Saveetha, H., D. Indumathi und Subhadip Mitra. „Vector meson fragmentation using a model with broken SU(3) at the next-to-leading order“. International Journal of Modern Physics A 29, Nr. 07 (13.03.2014): 1450049. http://dx.doi.org/10.1142/s0217751x14500493.

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A detailed study of fragmentation of vector mesons at the next-to-leading order (NLO) in QCD is given for e+e- scattering. A model with broken SU(3) symmetry using three input fragmentation functions α(x, Q2), β(x, Q2) and γ(x, Q2) and a strangeness suppression parameter λ describes all the light quark fragmentation functions for the entire vector meson octet. At a starting low energy scale of [Formula: see text] for three light quarks (u, d, s) along with initial parametrization, the fragmentation functions are evolved through DGLAP evolution equations at NLO and the cross-section is calculated. The heavy quarks contribution are added in appropriate thresholds during evolution. The results obtained are fitted at the momentum scale of [Formula: see text] for LEP and SLD data. Good-quality fits are obtained for ρ, K*, ω and ϕ mesons, implying the consistency and efficiency of this model. Strangeness suppression in this model is understood both in terms of ratios of quark fragmentation functions alone as well as in terms of observables; the latter yield a suppression through the K*/ρ multiplicity ratio of about 0.23 while the x dependence of this suppression is also parametrized through the cross-section ratios.

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BALITSKY, IAN. „HIGH-ENERGY AMPLITUDES IN ${\cal N}\, = \,4$ SYM IN THE NEXT-TO-LEADING ORDER“. International Journal of Modern Physics A 25, Nr. 02n03 (30.01.2010): 401–10. http://dx.doi.org/10.1142/s0217751x10048706.

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The high-energy behavior of the of [Formula: see text] SYM amplitudes in the Regge limit can be calculated order by order in perturbation theory using the high-energy operator expansion in Wilson lines. At large Nc, a typical four-point amplitude is determined by a single BFKL pomeron. The conformal structure of the four-point amplitude is fixed in terms of two functions: pomeron intercept and the coefficient function in front of the pomeron (the product of two residues). The pomeron intercept is universal while the coefficient function depends on the correlator in question. The intercept is known in first two orders in coupling constant : LO BFKL intercept and NLO BFKL calculated in Ref. [1]. As an example of using the Wilson-line OPE, we calculate the coefficient function in front of the pomeron for the correlator of four Z2 currents in the leading and next-to-leading order.

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Balitsky, I. „NLO Hierarchy of Wilson Lines Evolution“. International Journal of Modern Physics: Conference Series 37 (Januar 2015): 1560056. http://dx.doi.org/10.1142/s2010194515600563.

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The high-energy behavior of QCD amplitudes can be described in terms of the rapidity evolution of Wilson lines. I present the hierarchy of evolution equations for Wilson lines in the next-to-leading order.

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Eskola, K. J., V. J. Kolhinen, P. V. Ruuskanen und R. L. Thews. „Effects of Shadowing on Drell–Yan Dilepton Production in High Energy Nuclear Collisions“. International Journal of Modern Physics E 12, Nr. 02 (April 2003): 197–209. http://dx.doi.org/10.1142/s0218301303001260.

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We compute cross sections for the Drell–Yan process in nuclear collisions at next-to-leading order (NLO) in αs. The effects of shadowing on the normalization and on the mass and rapidity dependence of these cross sections are presented. An estimate of higher order corrections is obtained from next-to-next-to-leading order (NNLO) calculation of the rapidity-integrated mass distribution. Variations in these predictions resulting from choices of parton distributions sets are discussed. Numerical results for mass distributions at NLO are presented for RHIC and LHC energies, using appropriate rapidity intervals. The shadowing factors in the dilepton mass range 2 < M < 10 GeV are predicted to be substantial, typically 0.5 - 0.7 at LHC, 0.7 - 0.9 at RHIC, and approximately independent of the choice of parton distribution sets and the order of calculation.

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Liu, Wen, Wen-Gan Ma, Lei Guo, Liang-Wen Chen, Chang Chen und Ren-You Zhang. „T-odd quark pair production and decay at γγ collider in the littlest Higgs model with T-parity in next-to-leading order QCD“. Modern Physics Letters A 30, Nr. 25 (30.07.2015): 1550125. http://dx.doi.org/10.1142/s0217732315501254.

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We calculate the complete next-to-leading order (NLO) QCD corrections to the [Formula: see text]-odd mirror quark pair [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] production in the littlest Higgs model with [Formula: see text]-parity (LHT) at a high energy [Formula: see text] collider. We present the dependence of the leading order (LO) and NLO QCD corrected cross sections on the colliding energy [Formula: see text]. Our calculation includes the subsequent full weak decays of the final [Formula: see text]-odd mirror quarks by adopting the narrow width approximation and the exclusive 2-jet event selection criterion. We provide the LO and QCD NLO kinematic distributions of final particles. We find that the NLO QCD correction is phase space dependent and modifies the LO cross section evidently. The [Formula: see text]-factor increases noticeably when [Formula: see text] approaches the threshold of the on-shell [Formula: see text]-pair production. We conclude that it is possible to separate the signature of the [Formula: see text]-odd quark pair production from possible Standard Model (SM) background by taking proper kinematic cut.

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Ogul, Hasan, und Kamuran Dilsiz. „Cross Section Prediction for Inclusive Production of Z Boson in pp Collisions at s=14 TeV: A Study of Systematic Uncertainty due to Scale Dependence“. Advances in High Energy Physics 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/8262018.

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Prediction of Z→l+l- production cross section (where l±=e±,μ±) in proton-proton collisions at s=14 TeV is estimated up to next-to-next-to-leading order (NNLO) in perturbative QCD including next-to-leading order (NLO) electroweak (EW) corrections. The total inclusive Z boson production cross section times leptonic branching ratio, within the invariant mass window 66<mll<116 GeV, is predicted using NNLO HERAPDF2.0 at NNLO QCD and NLO EW as σZTot=2111.69-26.92+26.31 (PDF) ±11 (αs) ±17 (scale) -30.98+57.41 (parameterization and model). Theoretical prediction of the fiducial cross section is further computed with the latest modern PDF models (CT14, MMHT2014, NNPDF3.0, HERAPDF2.0, and ABM12) at NNLO for QCD and NLO for EW. The central values of the predictions are based on DYNNLO 1.5 program and the uncertainties are extracted using FEWZ 3.1 program. In addition, the cross section is also calculated as functions of μR and μF scales. The choice of μR and μF for scale variation uncertainty is further discussed in detail.

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Hermansson-Truedsson, Nils. „Chiral Perturbation Theory at NNNLO“. Symmetry 12, Nr. 8 (30.07.2020): 1262. http://dx.doi.org/10.3390/sym12081262.

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Chiral perturbation theory is a much successful effective field theory of quantum chromodynamics at low energies. The effective Lagrangian is constructed systematically order by order in powers of the momentum p2, and until now the leading order (LO), next-to leading order (NLO), next-to-next-to leading order (NNLO) and next-to-next-to-next-to leading order (NNNLO) have been studied. In the following review we consider the construction of the Lagrangian and in particular focus on the NNNLO case. We in addition review and discuss the pion mass and decay constant at the same order, which are fundamental quantities to study for chiral perturbation theory. Due to the large number of terms in the Lagrangian and hence low energy constants arising at NNNLO, some remarks are made about the predictivity of this effective field theory.

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Chirilli, Giovanni Antonio. „Rapidity evolution of Wilson lines at the next-to-leading order: Balitsky–JIMWLK equation at NLO“. Nuclear Physics A 931 (November 2014): 1130–35. http://dx.doi.org/10.1016/j.nuclphysa.2014.09.073.

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19

Nomura, Daisuke. „Hadronic Leading Order Contribution to the Muon g-2“. EPJ Web of Conferences 179 (2018): 01016. http://dx.doi.org/10.1051/epjconf/201817901016.

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We calculate the Standard Model (SM) prediction for the muon anomalous magnetic moment. By using the latest experimental data for e+e- → hadrons as input to dispersive integrals, we obtain the values of the leading order (LO) and the next-to-leading-order (NLO) hadronic vacuum polarisation contributions as ahad, LO VPμ = (693:27 ± 2:46) × 10-10 and ahad, NLO VP μ = (_9.82 ± 0:04) × 1010-10, respectively. When combined with other contributions to the SM prediction, we obtain aμ(SM) = (11659182:05 ± 3.56) × 10-10; which is deviated from the experimental value by Δaμ(exp) _ aμ(SM) = (27.05 ± 7.26) × 10-10. This means that there is a 3.7 σ discrepancy between the experimental value and the SM prediction. We also discuss another closely related quantity, the running QED coupling at the Z-pole, α(M2 Z). By using the same e+e- → hadrons data as input, our result for the 5-flavour quark contribution to the running QED coupling at the Z pole is Δ(5)had(M2 Z) = (276.11 ± 1.11) × 10-4, from which we obtain Δ(M2 Z) = 128.946 ± 0.015.

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WANG, JIAN, GUOMING CHEN und WEIMIN WU. „THE IMPACT OF LO, NLO AND NNLO FOR THE HIGGS SEARCHING AT $\sqrt{s} = 7$ TeV OF LHC“. Modern Physics Letters A 25, Nr. 36 (30.11.2010): 3027–31. http://dx.doi.org/10.1142/s0217732310034146.

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Most of current Monte Carlo studies on the Higgs searching are based on LO, or NLO calculation. However, in recent years, the next-to-next-to-leading order (NNLO) corrections have been computed for some physics process, and found that the cross section increases the kinematics changes. As the results, the analysis results could be impacted by these high order QCD corrections. We use standard Monte Carlo generator for LO, as well as MC@NLO for NLO and ResBos for NNLO at 7 TeV of LHC to evaluate this impact for physics channel of the Higgs, mass at 165 GeV, to WW, then W decay to lepton and neutrino as the final states. We found the signal rate could be effected by ratio of 1:2.6:3.4 for LO, NLO and NNLO using the same standard H→WW→lνlν searching analysis process.6

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DAWSON, S., C. B. JACKSON, L. REINA und D. WACKEROTH. „Higgs Boson Production with Bottom Quarks at Hadron Colliders“. International Journal of Modern Physics A 20, Nr. 15 (20.06.2005): 3353–55. http://dx.doi.org/10.1142/s0217751x05026558.

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22

ZHANG, HANZHONG, und ENKE WANG. „A NLO ANALYSIS ON AZIMUTHAL ANISOTROPY OF HIGH pT HADRON IN HEAVY-ION COLLISIONS“. International Journal of Modern Physics E 16, Nr. 10 (November 2007): 3185–92. http://dx.doi.org/10.1142/s021830130700918x.

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The azimuthal anisotropy of high pT hadron in high energy A + A collisions is studied within the next-to-leading order (NLO) perturbative QCD parton model. The effect of jet quenching is incorporated via a model for modified jet fragmentation functions due to radiative parton energy loss in dense medium. Because NLO contributions behave with stronger quenching effect than LO contributions, the NLO elliptic flow parameter is found to be larger than the LO in the medium pT region.

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Chen, Chong, Lei Guo, Wen-Gan Ma, Ren-You Zhang, Xiao-Zhou Li und Yu Zhang. „Possible effects of the large extra dimensions on ZZW production at the LHC“. Modern Physics Letters A 29, Nr. 31 (10.10.2014): 1450153. http://dx.doi.org/10.1142/s0217732314501533.

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We investigate the possible large extra dimensions (LED) effects induced by the Kaluza–Klein (KK) gravitons up to the QCD next-to-leading order (NLO) on ZZW production at the large hadron collider (LHC). The integrated cross-sections and some kinematic distributions are presented in both the Standard Model (SM) and the LED model. The numerical results demonstrate that the NLO QCD corrections are sizeable and remarkably reduce the leading order (LO) LED effect depending strongly on the phase space. The NLO LED relative discrepancies of the total cross-section could become sizable for the ZZW production, if we apply proper event selection criteria. We find that the LO result overestimates the LED effect and is insufficient to provide a believable theoretical prediction.

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Mangano, Michelangelo, und Andrea Petrell. „NLO Quarkonium Production in Hadronic Collisions“. International Journal of Modern Physics A 12, Nr. 22 (10.09.1997): 3887–97. http://dx.doi.org/10.1142/s0217751x97002048.

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We present some preliminary results on the next-to-leading order calculation in QCD of quarkonium production cross sections in hadronic collisions. We will show that the NLO total cross sections for P-wave states produced at high energy are not reliable, due to the appearance of very large and negative contributions. We also discuss some issues related to the structure of final states in colour-octet production and to high-pt fragmentation.

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KHANPOUR, H., ALI N. KHORRAMIAN und S. ATASHBAR TEHRANI. „DETERMINATION OF THE STRONG COUPLING CONSTANT FROM NLO QCD ANALYSIS OF PROTON STRUCTURE FUNCTION“. International Journal of Modern Physics A 26, Nr. 03n04 (10.02.2011): 658–59. http://dx.doi.org/10.1142/s0217751x11052396.

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In this article we present a determination of the strong coupling constant and parton distribution functions (PDFs) based on a next-to-leading order (NLO) perturbative QCD analysis of proton structure function. More precisely, we extract [Formula: see text] and PDFs by fitting perturbative QCD predictions to the data from the measurements of the proton structure function [Formula: see text] in deep inelastic scattering, which are based on perturbative QCD calculations up to NLO. We obtain at NLO [Formula: see text] in the variable-flavor number scheme.

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Baral, R. C., S. K. Tripathy, M. Younus, Z. Naik und P. K. Sahu. „Production of D-mesons in p + p and p + Pb collisions at LHC energies“. International Journal of Modern Physics E 25, Nr. 11 (November 2016): 1650092. http://dx.doi.org/10.1142/s0218301316500920.

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We present theoretical model comparison with published ALICE results for [Formula: see text]-mesons ([Formula: see text], [Formula: see text] and [Formula: see text]) in [Formula: see text] collisions at [Formula: see text] = 7 TeV and [Formula: see text] collisions at [Formula: see text][Formula: see text]TeV. Event generator Heavy-Ion Jet Interaction Generator (HIJING), transport calculation of AMPT and calculations from Next-to-Leading Order (NLO)(MNR) and Fixed-Order Next-to-Leading-Logarithmic (FONLL) have been used for this study. We found that HIJING and AMPT model predictions are matching with published [Formula: see text]-meson cross-sections in [Formula: see text] collisions, while both under predict the same in [Formula: see text] collisions. Attempts were made to explain the [Formula: see text] data using NLO-pQCD(MNR), FONLL and other above mentioned models.

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CHIRILLI, GIOVANNI ANTONIO. „NLO EVOLUTION OF STRUCTURE FUNCTIONS AT SMALL x“. International Journal of Modern Physics: Conference Series 04 (Januar 2011): 46–55. http://dx.doi.org/10.1142/s2010194511001553.

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Using the high-energy Operator Product Expansion of the T-product of two electromagnetic currents, we calculate the Photon Impact Factor for Deep Inelastic Scattering at small value of the Bjorken variable x B at the next-to-leading order (NLO) accuracy in αs. We provide for the first time an analytic expression in coordinate space and in Mellin space of the NLO impact factor for the forward unpolarized structure functions.

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Ocalan, Kadir. „EW radiative corrections to theory predictions of charge asymmetry for W-boson hadroproduction“. Physica Scripta 97, Nr. 7 (23.06.2022): 075305. http://dx.doi.org/10.1088/1402-4896/ac789a.

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Abstract The higher-order predictions of the lepton charge asymmetry A η l (the W-boson charge asymmetry A y W ) for the leptonic final state of the W-boson hadroproduction in proton–proton (pp) collisions are presented. The predictions from the state-of-the-art computations are reported for achieving adequate description of the A η l by including next-to-leading order (NLO) electroweak (EW) radiative corrections in combination with next-to-NLO (NNLO) quantum chromodynamics (QCD) radiative corrections. The combined predictions NNLO QCD+NLO EW and NNLO QCD × NLO EW, based on standard additive and factorised combination prescriptions in turn, are provided in the fiducial phase space of the pseudorapidity of the decay lepton (of the rapidity of the W-boson), comprising both central and forward detector acceptance regions as η l ≤ 4.5 (y W ≤ 4.5). The inclusion of the NLO EW effects for the A η l ( A y W ) constitutes additional input for the relative u- and d-quark densities in the proton, which is also of high importance in the domain of the high-precision studies. The predicted A η l distributions are compared with the actual measurements by CERN Large Hadron Collider (LHC) experiments at 8 TeV pp collisions energies. The combined predictions for the A η l ( A y W ) distributions are also provided in comparisons with the NNLO QCD predictions at both 13 TeV and 14 TeV energies. The impact of the NLO EW corrections for the A η l ( A y W ) distributions is extensively assessed by means of relative correction factor analysis with respect to the NNLO QCD predictions, in addition to a detailed K-factor analysis with respect to the leading order (LO) accuracy. The predicted results show that the NLO EW effects have larger impact in the forward η l region of the A η l contrary to the central η l region, and is sizable in some of the y W ranges of the A y W . The paper suggests inclusion of the presented EW corrections at NLO to have explicit accounting for the EW effects for the A η l ( A y W ) in phenomenological studies.

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CHIRILLI, GIOVANNI, BO-WEN XIAO und FENG YUAN. „THE NLO INCLUSIVE FORWARD HADRON PRODUCTION IN pA COLLISIONS“. International Journal of Modern Physics: Conference Series 20 (Januar 2012): 208–13. http://dx.doi.org/10.1142/s2010194512009257.

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Recently, by performing the complete next-to-leading order calculation, we have demonstrated the one-loop factorization for inclusive hadron productions in pA collisions in the saturation formalism. The differential cross section is written into a factorization form in the coordinate space at the next-to-leading order, while the naive form of the convolution in the transverse momentum space does not hold. The rapidity divergence with small-x dipole gluon distribution of the nucleus is factorized into the energy evolution of the dipole gluon distribution function, which is known as the Balitsky-Kovchegov equation. Furthermore, the collinear divergences associated with the incoming parton distribution of the nucleon and the outgoing fragmentation function of the final state hadron are factorized into the splittings of the associated parton distribution and fragmentation functions, which allows us to reproduce the well-known DGLAP equation. The hard coefficient function, which is finite and free of divergence of any kind, is evaluated at one-loop order. This result is important, not only for the phenomenological applications to the inclusive hadron production in p-A collisions at RHIC and future LHC experiment, but also for theoretically promoting the rigorous developments towards a complete QCD factorization in small-x physics.

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Escobedo, Miguel Ángel, und Tuomas Lappi. „The dipole picture and the non-relativistic expansion“. EPJ Web of Conferences 258 (2022): 04006. http://dx.doi.org/10.1051/epjconf/202225804006.

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We study exclusive quarkonium production in the dipole picture at next-to-leading order (NLO) accuracy, using the non-relativistic expansion for the quarkonium wavefunction. The quarkonium light cone wave functions needed in the dipole picture have typically been available only at tree level, either in phenomenological models or in the nonrelativistic limit. Here, we discuss the compatibility of the dipole approach and the non-relativistic expansion and compute NLO relativistic corrections to the quarkonium light-cone wave function in light-cone gauge.

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Xing, Hongxi, und Shinsuke Yoshida. „Introduction to the Transverse-Momentum-Weighted Technique in the Twist-3 Collinear Factorization Approach“. Advances in High Energy Physics 2019 (02.06.2019): 1–15. http://dx.doi.org/10.1155/2019/4825790.

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The twist-3 collinear factorization framework has drawn much attention in recent decades as a successful approach in describing the data for single spin asymmetries (SSAs). Many SSAs data have been experimentally accumulated in a variety of energies since the first measurement was done in the late 1970s and it is expected that the future experiments like Electron-Ion-Collider will provide us with more data. In order to perform a consistent and precise description of the data taken in different kinematic regimes, the scale evolution of the collinear twist-3 functions and the perturbative higher-order hard part coefficients are mandatory. In this paper, we introduce the techniques for next-to-leading order (NLO) calculation of transverse-momentum-weighted SSAs, which can be served as a useful tool to derive the QCD evolution equation for twist-3 functions and to verify the QCD collinear factorization for twist-3 observables at NLO, as well as obtain the finite NLO hard part coefficients.

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Mulyawan, R. G., A. Salam und I. Fachruddin. „Bootstrapping Energy-Energy Correlation in Planar 𝓝 = 4 Supersymmetric Yang-Mills“. Journal of Physics: Conference Series 2377, Nr. 1 (01.11.2022): 012049. http://dx.doi.org/10.1088/1742-6596/2377/1/012049.

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The two-point energy flow correlation, alternatively dubbed the energy-energy correlation (EEC), is a class of conformal field theory observable in the maximally supersymmetric Yang-Mills theory (𝓝 = 4) related to the event shapes in scattering experiments. It has been calculated up to the next-to-leading order (NLO) recently, showcasing the simplicity of the correlation function. This paper calculates the EEC using an approach based on its polylogarithmic functions. Using the amplitude bootstrap method, two ansatzes are made for the energy flow operators, namely the polylogarithm ansatz crafted using the Symbols method, and the polynomial ansatz based on the results from the NLO correction. The computation is carried out in the leading order (LO) and NLO order. After the computation is made, physical constraints are discovered and accordingly applied to the ansatz, namely the symmetry and end-point kinematics constraints. The resulting computation retrieved the energy flow correlation calculated previously using the Mellin-Barnes representation. The non-trivial nature of the result implies a simpler way to calculate the energy flow correlation without the conformal field theory-based approach.

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ZIELIŃSKI, MAREK. „STUDY OF DIRECT-PHOTON AND PION PRODUCTION“. International Journal of Modern Physics A 16, supp01a (Oktober 2001): 232–34. http://dx.doi.org/10.1142/s0217751x01006577.

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We present comparisons of high-pT photon and pion production in hadronic interactions with expectations from next-to-leading order pertubative QCD (NLO pQCD). We also comment on phenomenological models of kT smearing (which approximate effects of additional soft-gluon emission) and on the status of resummation calculations.

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Heinrich, Gudrun, und Jannis Lang. „SMEFT truncation effects in Higgs boson pair production at NLO QCD“. Journal of Physics: Conference Series 2438, Nr. 1 (01.02.2023): 012153. http://dx.doi.org/10.1088/1742-6596/2438/1/012153.

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Abstract We present results for Higgs boson pair production in gluon fusion at next-to-leading order in QCD, including effects of anomalous couplings within Standard Model Effective Field Theory (SMEFT). In particular, we investigate truncation effects of the SMEFT series, comparing different ways to treat powers of dimension-six operators and double operator insertions.

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VIJAYA KUMAR, K. B., YONG-LIANG MA und YUE-LIANG WU. „SPIN POLARIZABILITY OF THE NUCLEON IN THE EFFECTIVE FIELD THEORY OF HEAVY BARYON“. International Journal of Modern Physics A 21, Nr. 19n20 (10.08.2006): 3947–66. http://dx.doi.org/10.1142/s0217751x06031521.

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We have constructed a heavy baryon effective field theory with photon as an external field in accordance with the symmetry requirements similar to the heavy quark effective field theory. By treating the heavy baryon and antibaryon equally on the same footing in the effective field theory, we have calculated the spin polarizabilities γi, i = 1,…,4 of the nucleon at third order and at fourth-order of the spin-dependent Compton scattering. At leading order (LO), our results agree with the corresponding results of the heavy baryon chiral perturbation theory, at the next-to-leading order (NLO) the results show a large correction to the ones in the heavy baryon chiral perturbation theory due to baryon–antibaryon coupling terms. The low-energy theorem is satisfied both at LO and at NLO. The contributions arising from the heavy baryon–antibaryon vertex were found to be significant and the results of the polarizabilities obtained from our theory is much closer to the experimental data.

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Haidenbauer, Johann, und Ulf-G. Meißner. „Status of the hyperon-nucleon interaction in chiral effective field theory“. EPJ Web of Conferences 271 (2022): 05001. http://dx.doi.org/10.1051/epjconf/202227105001.

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The Jülich-Bonn group aims at an extensive study of the baryonbaryon (BB) interaction involving strange baryons (Λ, Σ, Ξ) within SU(3) chiral effective field theory. An overview of achievements and new developments over the past few years is provided. The topics covered are: 1) Derivation of the leading charge-symmetry breaking (CSB) interaction for the ΛN system and its application in a study of CSB effects in A=4 Λ-hypernuclei. 2) Updated results for the ΞN interaction at NLO and predictions for Ξ−p correlation functions. 3) Extension of the ΛN-ΣN interaction to next-to-next-to-leading order.

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Shen, Dandan, Huimin Ren, Fan Wu und Ruilin Zhu. „Bc → J/ψ tensor form factors at large momentum recoil“. International Journal of Modern Physics A 36, Nr. 19 (03.07.2021): 2150135. http://dx.doi.org/10.1142/s0217751x21501359.

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We present a next-to-leading order (NLO) relativistic correction to [Formula: see text] tensor form factors within nonrelativistic QCD (NRQCD). We also consider complete Dirac bilinears [Formula: see text] with [Formula: see text] matrices [Formula: see text] in the [Formula: see text] transition. The relation among different current form factors is given and it shows that symmetries emerge in the heavy bottom quark limit. For a phenomenological extension, we propose to extract the long-distance matrix elements (LDMEs) for [Formula: see text] meson from the recent HPQCD lattice data and the NLO form factors at large momentum recoil.

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Xing, Hongxi, Zhong-Bo Kang, Enke Wang und Xin-Nian Wang. „QCD Evolution of Nuclear Quark-Gluon Correlation Function“. International Journal of Modern Physics: Conference Series 37 (Januar 2015): 1560061. http://dx.doi.org/10.1142/s2010194515600617.

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We summarize the results on the next-to-leading order (NLO) calculations of transverse momentum broadening in semi-inclusive deeply inelastic e + A scattering (SIDIS) and Drell-Yan dilepton production (DY) in p + A collisions. The corresponding transverse momentum weighted differential cross sections are shown to factorize at NLO. Our calculations identify the QCD evolution equation for the quark-gluon correlation function, and also confirm the universality of the associated quark-gluon correlation function in SIDIS and DY. The evolution equation can be further applied to determine the QCD factorization scale and the energy dependence of the jet transport parameter [Formula: see text].

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Meng, Rui-Qing, Sheng-Quan Wang, Ting Sun, Chao-Qin Luo, Jian-Ming Shen und Xing-Gang Wu. „QCD improved top-quark decay at next-to-next-to-leading order“. European Physical Journal C 83, Nr. 1 (23.01.2023). http://dx.doi.org/10.1140/epjc/s10052-023-11224-4.

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AbstractWe analyse the top-quark decay at the next-to-next-to-leading order (NNLO) in QCD by using the Principle of Maximum Conformality (PMC) which provides a systematic way to eliminate renormalization scheme and scale ambiguities in perturbative QCD predictions. The PMC renormalization scales of the coupling constant $$\alpha _s$$ α s are determined by absorbing the non-conformal $$\beta $$ β terms that govern the behavior of the running coupling by using the Renormalization Group Equation (RGE). We obtain the PMC scale $$Q_\star =15.5$$ Q ⋆ = 15.5 GeV for the top-quark decay, which is an order of magnitude smaller than the conventional choice $$\mu _r=m_t$$ μ r = m t , reflecting the small virtuality of the QCD dynamics of the top-quark decay process. Moreover, due to the non-conformal $$\beta $$ β terms disappear in the pQCD series, there is no renormalon divergence and the NLO QCD correction term is greatly increased while the NNLO QCD correction term is suppressed compared to the conventional results obtained at $$\mu _r=m_t$$ μ r = m t . By further including the next-to-leading (NLO) electroweak corrections, the finite W boson width and the finite bottom quark mass, we obtain the top-quark total decay width $$\Gamma ^{\textrm{tot}}_t=1.3112^{+0.0190}_{-0.0189}$$ Γ t tot = 1 . 3112 - 0.0189 + 0.0190 GeV, where the error is the squared averages of the top-quark mass $$\Delta m_t=\pm 0.7$$ Δ m t = ± 0.7 GeV, the coupling constant $$\Delta \alpha _s(M_Z)=\pm 0.0009$$ Δ α s ( M Z ) = ± 0.0009 and the estimation of unknown higher-order terms using the PAA method with [N/M]=[1/1]. The PMC improved predictions for the top-quark decay are complementary to the previous PMC calculations for top-quark pair production and helpful for detailed studies of properties of the top-quark.

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Di Luzio, Luca, und Gioacchino Piazza. „a → πππ decay at next-to-leading order in chiral perturbation theory“. Journal of High Energy Physics 2022, Nr. 12 (09.12.2022). http://dx.doi.org/10.1007/jhep12(2022)041.

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Abstract We discuss the construction of the two-flavour axion-pion effective Lagrangian at the next-to-leading order (NLO) in chiral perturbation theory and present, as a phenomenological application, the calculation of the decay rate of a GeV-scale axion-like particle via the channel a → πππ. Through the NLO calculation, we assess the range of validity of the effective field theory and show that the chiral expansion breaks down just above the kinematic threshold. Alternative non-perturbative approaches are called for in order to extend the chiral description of axion-pion interactions.

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Brambilla, Nora, Miguel Ángel Escobedo, Ajaharul Islam, Michael Strickland, Anurag Tiwari, Antonio Vairo und Peter Vander Griend. „Heavy quarkonium dynamics at next-to-leading order in the binding energy over temperature“. Journal of High Energy Physics 2022, Nr. 8 (30.08.2022). http://dx.doi.org/10.1007/jhep08(2022)303.

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Abstract Using the potential non-relativistic quantum chromodynamics (pNRQCD) effective field theory, we derive a Lindblad equation for the evolution of the heavy-quarkonium reduced density matrix that is accurate to next-to-leading order (NLO) in the ratio of the binding energy of the state to the temperature of the medium. The resulting NLO Lindblad equation can be used to more reliably describe heavy-quarkonium evolution in the quark-gluon plasma at low temperatures compared to the leading-order truncation. For phenomenological application, we numerically solve the resulting NLO Lindblad equation using the quantum trajectories algorithm. To achieve this, we map the solution of the three-dimensional Lindblad equation to the solution of an ensemble of one-dimensional Schrödinger evolutions with Monte-Carlo sampled quantum jumps. Averaging over the Monte-Carlo sampled quantum jumps, we obtain the solution to the NLO Lindblad equation without truncation in the angular momentum quantum number of the states considered. We also consider the evolution of the system using only the complex effective Hamiltonian without stochastic jumps and find that this provides a reliable approximation for the ground state survival probability at LO and NLO. Finally, we make comparisons with our prior leading-order pNRQCD results and experimental data available from the ATLAS, ALICE, and CMS collaborations.

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Rein, Felix, Simone Rodini, Andreas Schäfer und Alexey Vladimirov. „Sivers, Boer-Mulders and worm-gear distributions at next-to-leading order“. Journal of High Energy Physics 2023, Nr. 1 (20.01.2023). http://dx.doi.org/10.1007/jhep01(2023)116.

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Abstract We compute the Sivers, Boer-Mulders, worm-gear (T and L) transverse momentum dependent distributions in terms of twist-two and twist-three collinear distributions in the small-b limit up to next-to-leading order (NLO) in perturbation theory.

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Feng, Feng, Yu Jia und Wen-Long Sang. „Next-to-leading-order QCD corrections to heavy quark fragmentation into $${}^1S^{(1,8)}_0$$ quarkonia“. European Physical Journal C 81, Nr. 7 (Juli 2021). http://dx.doi.org/10.1140/epjc/s10052-021-09390-4.

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AbstractWithin NRQCD factorization framework, in this work we compute, at the lowest order in velocity expansion, the next-to-leading-order (NLO) perturbative corrections to the short-distance coefficients associated with heavy quark fragmentation into the $${}^1S_0^{(1,8)}$$ 1 S 0 ( 1 , 8 ) components of a heavy quarkonium. Starting from the Collins and Soper’s operator definition of the quark fragmentation function, we apply the sector decomposition method to facilitate the numerical manipulation. It is found that the NLO QCD corrections have a significant impact.

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Contreras, Carlos, Eugene Levin und Rodrigo Meneses. „BFKL equation in the next-to-leading order: solution at large impact parameters“. European Physical Journal C 79, Nr. 10 (Oktober 2019). http://dx.doi.org/10.1140/epjc/s10052-019-7363-6.

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Abstract In this paper, we show (1) that the NLO corrections do not change the power-like decrease of the scattering amplitude at large impact parameter ($$b^2 \,>\,r^2 \exp ( 2{\bar{\alpha }}_S\eta (1 + 4 {\bar{\alpha }}_S) )$$b2>r2exp(2α¯Sη(1+4α¯S)), where r denotes the size of scattering dipole and $$\eta = \ln (1/x_{Bj} )$$η=ln(1/xBj) for DIS), and, therefore, they do not resolve the inconsistency with unitarity; and (2) they lead to an oscillating behaviour of the scattering amplitude at large b, in direct contradiction with the unitarity constraints. However, from the more practical point of view, the NLO estimates give a faster decrease of the scattering amplitude as a function of b, and could be very useful for description of the experimental data. It turns out, that in a limited range of b, the NLO corrections generates the fast decrease of the scattering amplitude with b, which can be parameterized as $$N\, \propto \,\exp ( -\,\mu \,b )$$N∝exp(-μb) with $$\mu \, \propto \,1/r$$μ∝1/r in accord with the numerical estimates in Cepila et al. (Phys Rev D 99(5):051502, 10.1103/PhysRevD.99.051502, arXiv:1812.02548 [hep-ph], 2019).

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Bergabo, Filip, und Jamal Jalilian-Marian. „Single inclusive hadron production in DIS at small x: next to leading order corrections“. Journal of High Energy Physics 2023, Nr. 1 (18.01.2023). http://dx.doi.org/10.1007/jhep01(2023)095.

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Abstract We calculate the one-loop corrections to single inclusive hadron production in Deep Inelastic Scattering (DIS) at small x in the forward rapidity region using the Color Glass Condensate formalism. We show that the divergent parts of the next to leading order (NLO) corrections either cancel among each other or lead to x (rapidity) evolution of the leading order (LO) dipole cross section according to the JIMWLK evolution equation and DGLAP evolution of the parton-hadron fragmentation function. The remaining finite parts constitute the NLO (αs) corrections to the LO single inclusive hadron production cross section in DIS at small x.

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Campbell, John M., Giuseppe De Laurentis, R. Keith Ellis und Satyajit Seth. „The pp → W(→ lν) + γ process at next-to-next-to-leading order“. Journal of High Energy Physics 2021, Nr. 7 (Juli 2021). http://dx.doi.org/10.1007/jhep07(2021)079.

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Abstract We present details of the calculation of the pp → W(→ lν)γ process at next-to-next-to-leading order in QCD, calculated using the jettiness slicing method. The calculation is based entirely on analytic amplitudes. Because of the radiation zero, the NLO QCD contribution from the gq channel is as important as the contribution from the Born $$ q\overline{q} $$ q q ¯ process, disrupting the normal counting of leading and sub-leading contributions. We also assess the importance of electroweak (EW) corrections, including the EW corrections to both the six-parton channel 0 →$$ \overline{u} d\nu {e}^{+}\gamma g $$ u ¯ dν e + γg and the five-parton channel 0 →$$ \overline{u} d\nu {e}^{+}\gamma $$ u ¯ dν e + γ . Previous experimental results have been shown to agree with theoretical predictions, taking into account the large experimental errors. With the advent of run II data from the LHC, the statistical errors on the data will decrease, and will be competitive with the error on theoretical predictions for the first time. We present numerical results for $$ \sqrt{s} $$ s = 7 and 13 TeV. Analytic results for the one-loop six-parton QCD amplitude and the tree-level seven-parton QCD amplitude are presented in appendices.

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Iancu, Edmond, und Yair Mulian. „Forward dijets in proton-nucleus collisions at next-to-leading order: the real corrections“. Journal of High Energy Physics 2021, Nr. 3 (März 2021). http://dx.doi.org/10.1007/jhep03(2021)005.

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Abstract Using the CGC effective theory together with the hybrid factorisation, we study forward dijet production in proton-nucleus collisions beyond leading order. In this paper, we compute the “real” next-to-leading order (NLO) corrections, i.e. the radiative corrections associated with a three-parton final state, out of which only two are being measured. To that aim, we start by revisiting our previous results for the three-parton cross-section presented in [1]. After some reshuffling of terms, we deduce new expressions for these results, which not only look considerably simpler, but are also physically more transparent. We also correct several errors in this process. The real NLO corrections to inclusive dijet production are then obtained by integrating out the kinematics of any of the three final partons. We explicitly work out the interesting limits where the unmeasured parton is either a soft gluon, or the product of a collinear splitting. We find the expected results in both limits: the B-JIMWLK evolution of the leading-order dijet cross-section in the first case (soft gluon) and, respectively, the DGLAP evolution of the initial and final states in the second case (collinear splitting). The “virtual” NLO corrections to dijet production will be presented in a subsequent publication.

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Zhang, Yu-Dong, Feng Feng, Wen-Long Sang und Hong-Fei Zhang. „Next-to-leading-order QCD corrections to a vector bottomonium radiative decay into a charmonium“. Journal of High Energy Physics 2021, Nr. 12 (Dezember 2021). http://dx.doi.org/10.1007/jhep12(2021)189.

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Abstract Within the framework of nonrelativistic QCD (NRQCD) factorization, we calculate the next-to-leading-order (NLO) perturbative corrections to the radiative decay Υ → ηc(χcJ) + γ. Both the helicity amplitudes and the helicity decay widths are obtained. It is the first computation for the processes involving both bottomonium and charmonium at two-loop accuracy. By employing the Cheng-Wu theorem, we are able to convert most of complex-valued master integrals (MIs) into real-valued MIs, which makes the numerical integration much efficient. Our results indicate the $$ \mathcal{O}\left({\alpha}_s\right) $$ O α s corrections are moderate for ηc and χc2 production, and are quite marginal for χc0 and χc1 production. It is impressive to note the NLO corrections considerably reduce the renormalization scale dependence in both the decay widths and the branching fractions for χcJ, and slightly improve that for ηc. With the NRQCD matrix elements evaluated via the Buchmüller-Tye potential model, we find the decay width for ηc production is one-order-of-magnitude larger than χcJ production, which may provide a good opportunity to search for Υ → ηc + γ in experiment. In addition, the decay width for χc1 production is several times larger than those for χc0,2. Finally, we find the NLO NRQCD prediction for the branching fraction of Υ → χc1 + γ is only half of the lower bound of the experimental data measured recently by Belle. Moreover, there exists serious contradiction between theory and experiment for Υ → ηc + γ. The discrepancies between theory and experiment deserve further research efforts.

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Lansberg, Jean-Philippe, Maxim Nefedov und Melih A. Ozcelik. „Matching next-to-leading-order and high-energy-resummed calculations of heavy-quarkonium-hadroproduction cross sections“. Journal of High Energy Physics 2022, Nr. 5 (Mai 2022). http://dx.doi.org/10.1007/jhep05(2022)083.

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Abstract The energy dependence of the total hadroproduction cross section of pseudoscalar quarkonia is computed via matching Next-to-Leading Order (NLO) Collinear-Factorisation (CF) results with resummed higher-order corrections, proportional to $$ {\alpha}_s^n{\ln}^{n-1} $$ α s n ln n − 1 (1/z), to the CF hard-scattering coefficient, where z = M2/$$ \hat{s} $$ s ̂ with M and $$ \hat{s} $$ s ̂ being the quarkonium mass and the partonic center-of-mass energy squared. The resummation is performed using High-Energy Factorisation (HEF) in the Doubly-Logarithmic (DL) approximation, which is a subset of the leading logarithmic ln(1/z) approximation. Doing so, one remains strictly consistent with the NLO and NNLO DGLAP evolution of the PDFs. By improving the treatment of the small-z asymptotics of the CF coefficient function, the resummation cures the unphysical results of the NLO CF calculation. The matching is directly performed in the z-space and, for the first time, by using the Inverse-Error Weighting (InEW) matching procedure. As a by-product of the calculation, the NNLO term of the CF hard-scattering coefficient proportional to $$ {\alpha}_s^2 $$ α s 2 ln(1/z) is predicted from HEF.

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Contreras, Carlos, Eugene Levin, Rodrigo Meneses und Michael Sanhueza. „Non-linear equation in the re-summed next-to-leading order of perturbative QCD: the leading twist approximation“. European Physical Journal C 80, Nr. 11 (November 2020). http://dx.doi.org/10.1140/epjc/s10052-020-08580-w.

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AbstractIn this paper, we use the re-summation procedure, suggested in Ducloué et al. (JHEP 1904:081, 2019), Salam (JHEP 9807:019 1998), Ciafaloni et al. (Phys Rev D 60:1140361999) and Ciafaloni et al. (Phys Rev D 68:114003, 2003), to fix the BFKL kernel in the NLO. However, we suggest a different way to introduce the non-linear corrections in the saturation region, which is based on the leading twist non-linear equation. In the kinematic region: $$\tau \,\equiv \,r^2 Q^2_s(Y)\,\le \,1$$ τ ≡ r 2 Q s 2 ( Y ) ≤ 1 , where r denotes the size of the dipole, Y its rapidity and $$Q_s$$ Q s the saturation scale, we found that the re-summation contributes mostly to the leading twist of the BFKL equation. Assuming that the scattering amplitude is small, we suggest using the linear evolution equation in this region. For $$\tau \,>\,1$$ τ > 1 we are dealing with the re-summation of $$(\bar{\alpha }_S\,\ln \tau )^n$$ ( α ¯ S ln τ ) n and other corrections in NLO approximation for the leading twist. We find the BFKL kernel in this kinematic region and write the non-linear equation, which we solve analytically. We believe the new equation could be a basis for a consistent phenomenology based on the CGC approach.

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