%================================================================ %\RequirePackage{lineno} \documentclass[twocolumn,a4paper]{sn-jnl} \usepackage{epsfig} \usepackage{hhline} \usepackage{amsmath} \usepackage{amssymb} \usepackage{mathrsfs} \usepackage{placeins} %\usepackage[cal=boondox]{mathalfa} % different mathcal %\usepackage{calc} %\usepackage{accents} %\newcommand{\dbtilde}[1]{\accentset{\approx}{#1}} % double-tilde \DeclareMathOperator*{\argmin}{arg\,min} \usepackage{ bbold } %\usepackage{cancel} \usepackage{color} \usepackage{xspace} %\usepackage{xfrac} \usepackage{paralist} % compactitem \usepackage{caption} %\captionsetup[figure]{font=footnotesize,labelfont=footnotesize} \captionsetup{font=small,labelfont=small} \usepackage{bm} % 'bold' math symbols (better use \usepackage{newtxtext,newtxmath}, if available on latex distribution) \usepackage{acronym} \usepackage{adjustbox} %%%%%%%%%%%%% Comment the next two lines to remove the line numbering %\usepackage[]{lineno} %\linenumbers 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\setlength{\marginparwidth}{0.9\dinmargin} \marginparsep 8pt \marginparpush 5pt \topmargin -42pt \headheight 12pt \headsep 30pt \footskip 24pt \parskip 3mm plus 2mm minus 2mm \setlength{\parindent}{0.0cm} \newcommand{\picob}{\mbox{{\rm ~pb}}} \newcommand{\QQ} {\mbox{${Q^2}$}} %===============================title page============================= % Some useful tex commands % %\def\GeV{\hbox{$\;\hbox{\rm GeV}$}} %\def\MeV{\hbox{$\;\hbox{\rm MeV}$}} %\def\TeV{\hbox{$\;\hbox{\rm TeV}$}} \renewcommand{\d}{\text{d}} \newcommand{\pb}{\rm pb} \newcommand{\cm}{\rm cm} \newcommand{\hdick}{\noalign{\hrule height1.4pt}} \begin{document} %\pagestyle{empty} \newcommand{\GeVsq}{\ensuremath{\mathrm{GeV}^2} } \newcommand{\GeV}{\ensuremath{\mathrm{GeV}} } \newcommand{\pt}{\ensuremath{P_{\text{T}}}} \newcommand{\PP}{\ensuremath{\mathcal{P}}} %\newcommand{\ptAvg}{\ensuremath{\langle P_T^{jet}\rangle}} \newcommand{\Qsq}{\ensuremath{Q^{2}}\xspace} \newcommand{\xbj}{\ensuremath{x_{\text{Bj}}}\xspace} %% unfolding \newcommand{\chisq}{\ensuremath{\chi^{2}}} \newcommand{\hchisq}{\ensuremath{\hat\chi^{2}}} \newcommand{\chisqA}{\ensuremath{\chi_{\rm A}^{2}}} \newcommand{\chisqL}{\ensuremath{\chi_{\rm L}^{2}}} \newcommand{\ndf}{\ensuremath{n_{\rm dof}}} \newcommand{\V}{\ensuremath{\mathbb{V}}} \newcommand{\W}{\ensuremath{\mathbb{W}}} \newcommand{\M}{\ensuremath{\tilde{M}}} %\newcommand{\Y}{\ensuremath{\tilde{Y}}} \newcommand{\Y}{\ensuremath{Y\M}} \newcommand{\U}{\ensuremath{\Upsilon}} % vectors \newcommand{\dt}{\ensuremath{\bm{d}}} \newcommand{\mb}{\ensuremath{\bm{\bar{m}}}} \newcommand{\mt}{\ensuremath{\bm{\tilde{m}}}} \newcommand{\s}{\ensuremath{\bm{s}}} \newcommand{\ahut}{\ensuremath{\bm{\hat{a}}}} \newcommand{\ptjet}{\ensuremath{P_{\rm T}^{\rm jet}}} \newcommand{\Mjj}{\ensuremath{m_{12}}} \newcommand{\mz}{\ensuremath{m_{\rm Z}}\xspace} \newcommand{\as}{\ensuremath{\alpha_{\rm s}}\xspace} \newcommand{\asmz}{\ensuremath{\as(\mz)}\xspace} \newcommand{\asmzPDF}{\ensuremath{\as^{\rm PDF}(\mz)}\xspace} \newcommand{\asmzf}{\ensuremath{\as^{\Gamma}(\mz)}\xspace} \newcommand{\asmur}{\ensuremath{\alpha_{\rm s}(\mur)}\xspace} \newcommand{\chad}{\ensuremath{c_{\rm had}}\xspace} \newcommand{\ord}{\ensuremath{\mathcal{O}}\xspace} \newcommand{\PDFasResult}{0.1142} \newcommand{\HonePDF}{H1PDF2017\,{\protect\scalebox{0.8}{[NNLO]}}} \newcommand{\TODO}{\color{red} Todo.} \newcommand{\NEW}{\color{blue}} \newcommand{\tQ}{\ensuremath{\tau_{zQ}}\xspace} \newcommand{\tb}{\ensuremath{\tau_1^b}\xspace} \newcommand{\pzb}{\ensuremath{P_{z}^\text{Breit}}\xspace} \newcommand{\pzbi}{\ensuremath{P_{z,i}^\text{Breit}}\xspace} %============================================================ % pmj definitions \newcommand{\redtext}[1]{\textcolor{red}{#1}} \newcommand{\bluetext}[1]{\textcolor{blue}{#1}} %\newcommand{\greentext}[1]{\textcolor{green}{#1}} \definecolor{myGreen}{rgb}{0.0,0.6,0.1} \newcommand{\greentext}[1]{\textcolor{myGreen}{#1}} \newcommand{\commenting}[2]{\redtext{(#1: #2)}} \newcommand{\reply}[2]{\greentext{(#1: #2)}} \newcommand{\qsqr}{\ensuremath{Q^2}} \newcommand{\sqrts}{\ensuremath{\sqrt{s}}} \newcommand{\ep}{\ensuremath{ep}} \newcommand{\invpb}{\ensuremath{\mathrm{pb}^{-1}}} %============================================================ \hyphenation{mig-ra-tion} \hyphenation{ge-ne-ra-tors} \received{March 2024} \printhistory \motto{}{To be submitted to EPJC \hfill DESY-24-035} \title{Measurement of the 1-jettiness event shape observable in deep-inelastic electron-proton scattering at HERA} %\end{Large} \subtitle{H1 Collaboration} \author[50]{\fnm{V.}\sur{Andreev}} \author[34]{\fnm{M.}\sur{Arratia}} \author[46]{\fnm{A.}\sur{Baghdasaryan}} \author[9]{\fnm{A.}\sur{Baty}} \author[40]{\fnm{K.}\sur{Begzsuren}} \author[17]{\fnm{A.}\sur{Bolz}} \author[30]{\fnm{V.}\sur{Boudry}} \author[15]{\fnm{G.}\sur{Brandt}} \author[26]{\fnm{D.}\sur{Britzger}} \author[7]{\fnm{A.}\sur{Buniatyan}} \author[50]{\fnm{L.}\sur{Bystritskaya}} \author[17]{\fnm{A.J.}\sur{Campbell}} \author[47]{\fnm{K.B.}\sur{Cantun Avila}} \author[28]{\fnm{K.}\sur{Cerny}} \author[26]{\fnm{V.}\sur{Chekelian}} \author[36]{\fnm{Z.}\sur{Chen}} \author[47]{\fnm{J.G.}\sur{Contreras}} \author[32]{\fnm{J.}\sur{Cvach}} \author[23]{\fnm{J.B.}\sur{Dainton}} \author[45]{\fnm{K.}\sur{Daum}} \author[38,42]{\fnm{A.}\sur{Deshpande}} \author[25]{\fnm{C.}\sur{Diaconu}} \author[38]{\fnm{A.}\sur{Drees}} \author[17]{\fnm{G.}\sur{Eckerlin}} \author[43]{\fnm{S.}\sur{Egli}} \author[17]{\fnm{E.}\sur{Elsen}} \author[4]{\fnm{L.}\sur{Favart}} \author[50]{\fnm{A.}\sur{Fedotov}} \author[14]{\fnm{J.}\sur{Feltesse}} \author[17]{\fnm{M.}\sur{Fleischer}} \author[50]{\fnm{A.}\sur{Fomenko}} \author[38]{\fnm{C.}\sur{Gal}} \author[17]{\fnm{J.}\sur{Gayler}} \author[20]{\fnm{L.}\sur{Goerlich}} \author[17]{\fnm{N.}\sur{Gogitidze}} \author[50]{\fnm{M.}\sur{Gouzevitch}} \author[48]{\fnm{C.}\sur{Grab}} \author[23]{\fnm{T.}\sur{Greenshaw}} \author[26]{\fnm{G.}\sur{Grindhammer}} \author[17]{\fnm{D.}\sur{Haidt}} \author[21]{\fnm{R.C.W.}\sur{Henderson}} \author[26]{\fnm{J.}\sur{Hessler}} \author[32]{\fnm{J.}\sur{Hladký}} \author[25]{\fnm{D.}\sur{Hoffmann}} \author[43]{\fnm{R.}\sur{Horisberger}} \author[49]{\fnm{T.}\sur{Hreus}} \author[18]{\fnm{F.}\sur{Huber}} \author[5]{\fnm{P.M.}\sur{Jacobs}} \author[29]{\fnm{M.}\sur{Jacquet}} \author[4]{\fnm{T.}\sur{Janssen}} \author[44]{\fnm{A.W.}\sur{Jung}} \author[17]{\fnm{J.}\sur{Katzy}} \author[26]{\fnm{C.}\sur{Kiesling}} \author[23]{\fnm{M.}\sur{Klein}} \author[17]{\fnm{C.}\sur{Kleinwort}} \author[38,22]{\fnm{H.T.}\sur{Klest}} \author[26]{\fnm{S.}\sur{Kluth}} \author[17]{\fnm{R.}\sur{Kogler}} \author[23]{\fnm{P.}\sur{Kostka}} \author[23]{\fnm{J.}\sur{Kretzschmar}} \author[17]{\fnm{D.}\sur{Krücker}} \author[17]{\fnm{K.}\sur{Krüger}} \author[24]{\fnm{M.P.J.}\sur{Landon}} \author[17]{\fnm{W.}\sur{Lange}} \author[42]{\fnm{P.}\sur{Laycock}} \author[2,39]{\fnm{S.H.}\sur{Lee}} \author[17]{\fnm{S.}\sur{Levonian}} \author[19]{\fnm{W.}\sur{Li}} \author[19]{\fnm{J.}\sur{Lin}} \author[17]{\fnm{K.}\sur{Lipka}} \author[17]{\fnm{B.}\sur{List}} \author[17]{\fnm{J.}\sur{List}} \author[26]{\fnm{B.}\sur{Lobodzinski}} \author[34]{\fnm{O.R.}\sur{Long}} \author[50]{\fnm{E.}\sur{Malinovski}} \author[1]{\fnm{H.-U.}\sur{Martyn}} \author[23]{\fnm{S.J.}\sur{Maxfield}} \author[23]{\fnm{A.}\sur{Mehta}} \author[17]{\fnm{A.B.}\sur{Meyer}} \author[17]{\fnm{J.}\sur{Meyer}} \author[20]{\fnm{S.}\sur{Mikocki}} \author[5]{\fnm{V.M.}\sur{Mikuni}} \author[27]{\fnm{M.M.}\sur{Mondal}} \author[49]{\fnm{K.}\sur{M\"uller}} \author[5]{\fnm{B.}\sur{Nachman}} \author[17]{\fnm{Th.}\sur{Naumann}} \author[7]{\fnm{P.R.}\sur{Newman}} \author[17]{\fnm{C.}\sur{Niebuhr}} \author[20]{\fnm{G.}\sur{Nowak}} \author[17]{\fnm{J.E.}\sur{Olsson}} \author[50]{\fnm{D.}\sur{Ozerov}} \author[38]{\fnm{S.}\sur{Park}} \author[29]{\fnm{C.}\sur{Pascaud}} \author[23]{\fnm{G.D.}\sur{Patel}} \author[13]{\fnm{E.}\sur{Perez}} \author[37]{\fnm{A.}\sur{Petrukhin}} \author[31]{\fnm{I.}\sur{Picuric}} \author[17]{\fnm{D.}\sur{Pitzl}} \author[33]{\fnm{R.}\sur{Polifka}} \author[34]{\fnm{S.}\sur{Preins}} \author[18]{\fnm{V.}\sur{Radescu}} \author[31]{\fnm{N.}\sur{Raicevic}} \author[40]{\fnm{T.}\sur{Ravdandorj}} \author[12]{\fnm{D.}\sur{Reichelt}} \author[32]{\fnm{P.}\sur{Reimer}} \author[24]{\fnm{E.}\sur{Rizvi}} \author[49]{\fnm{P.}\sur{Robmann}} \author[4]{\fnm{R.}\sur{Roosen}} \author[50]{\fnm{A.}\sur{Rostovtsev}} \author[8]{\fnm{M.}\sur{Rotaru}} \author[10]{\fnm{D.P.C.}\sur{Sankey}} \author[18]{\fnm{M.}\sur{Sauter}} \author[25,3]{\fnm{E.}\sur{Sauvan}} \author*[17]{\fnm{S.}\sur{Schmitt}} \email{stefan.schmitt@desy.de} \author[38]{\fnm{B.A.}\sur{Schmookler}} \author[6]{\fnm{G.}\sur{Schnell}} \author[14]{\fnm{L.}\sur{Schoeffel}} \author[18]{\fnm{A.}\sur{Schöning}} \author[16]{\fnm{S.}\sur{Schumann}} \author[17]{\fnm{F.}\sur{Sefkow}} \author[26]{\fnm{S.}\sur{Shushkevich}} \author[17]{\fnm{Y.}\sur{Soloviev}} \author[20]{\fnm{P.}\sur{Sopicki}} \author[17]{\fnm{D.}\sur{South}} \author[30]{\fnm{A.}\sur{Specka}} \author[17]{\fnm{M.}\sur{Steder}} \author[35]{\fnm{B.}\sur{Stella}} \author[16]{\fnm{L.}\sur{St\"ocker}} \author[49]{\fnm{U.}\sur{Straumann}} \author[38]{\fnm{C.}\sur{Sun}} \author[33]{\fnm{T.}\sur{Sykora}} \author[7]{\fnm{P.D.}\sur{Thompson}} \author[5]{\fnm{F.}\sur{Torales Acosta}} \author[24]{\fnm{D.}\sur{Traynor}} \author[40,41]{\fnm{B.}\sur{Tseepeldorj}} \author[42]{\fnm{Z.}\sur{Tu}} \author[38]{\fnm{G.}\sur{Tustin}} \author[33]{\fnm{A.}\sur{Valkárová}} \author[25]{\fnm{C.}\sur{Vallée}} \author[4]{\fnm{P.}\sur{van Mechelen}} \author[11]{\fnm{D.}\sur{Wegener}} \author[17]{\fnm{E.}\sur{W\"unsch}} \author[33]{\fnm{J.}\sur{Žáček}} \author[36]{\fnm{J.}\sur{Zhang}} \author[29]{\fnm{Z.}\sur{Zhang}} \author[33]{\fnm{R.}\sur{Žlebčík}} \author[46]{\fnm{H.}\sur{Zohrabyan}} \author[29]{\fnm{F.}\sur{Zomer}} \affil[1]{\orgaddress{I. Physikalisches Institut der RWTH, Aachen, Germany}} \affil[2]{\orgaddress{University of Michigan, Ann Arbor, MI 48109, USA$^{f1}$}} \affil[3]{\orgaddress{LAPP, Université de Savoie, CNRS/IN2P3, Annecy-le-Vieux, France}} \affil[4]{\orgaddress{Inter-University Institute for High Energies ULB-VUB, Brussels and Universiteit Antwerpen, Antwerp, Belgium$^{f2}$}} \affil[5]{\orgaddress{Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA$^{f1}$}} \affil[6]{\orgaddress{Department of Physics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain}} \affil[7]{\orgaddress{School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom$^{f3}$}} \affil[8]{\orgaddress{Horia Hulubei National Institute for R\&D in Physics and Nuclear Engineering (IFIN-HH) , Bucharest, Romania$^{f4}$}} \affil[9]{\orgaddress{University of Illinois, Chicago, IL 60607, USA}} \affil[10]{\orgaddress{STFC, Rutherford Appleton Laboratory, Didcot, Oxfordshire, United Kingdom$^{f3}$}} \affil[11]{\orgaddress{Institut für Physik, TU Dortmund, Dortmund, Germany$^{f5}$}} \affil[12]{\orgaddress{Institute for Particle Physics Phenomenology, Durham University, Durham, United Kingdom}} \affil[13]{\orgaddress{CERN, Geneva, Switzerland}} \affil[14]{\orgaddress{IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France}} \affil[15]{\orgaddress{II. Physikalisches Institut, Universität Göttingen, Göttingen, Germany}} \affil[16]{\orgaddress{Institut für Theoretische Physik, Universität Göttingen, Göttingen, Germany}} \affil[17]{\orgaddress{Deutsches Elektronen-Synchrotron DESY, Hamburg and Zeuthen, Germany}} \affil[18]{\orgaddress{Physikalisches Institut, Universität Heidelberg, Heidelberg, Germany$^{f5}$}} \affil[19]{\orgaddress{Rice University, Houston, TX 77005-1827, USA}} \affil[20]{\orgaddress{Institute of Nuclear Physics Polish Academy of Sciences, Krakow, Poland$^{f6}$}} \affil[21]{\orgaddress{Department of Physics, University of Lancaster, Lancaster, United Kingdom$^{f3}$}} \affil[22]{\orgaddress{Argonne National Laboratory, Lemont, IL 60439, USA}} \affil[23]{\orgaddress{Department of Physics, University of Liverpool, Liverpool, United Kingdom$^{f3}$}} \affil[24]{\orgaddress{School of Physics and Astronomy, Queen Mary, University of London, London, United Kingdom$^{f3}$}} \affil[25]{\orgaddress{Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France}} \affil[26]{\orgaddress{Max-Planck-Institut für Physik, München, Germany}} \affil[27]{\orgaddress{National Institute of Science Education and Research, Jatni, Odisha, India}} \affil[28]{\orgaddress{Joint Laboratory of Optics, Palacký University, Olomouc, Czech Republic}} \affil[29]{\orgaddress{IJCLab, Université Paris-Saclay, CNRS/IN2P3, Orsay, France}} \affil[30]{\orgaddress{LLR, Ecole Polytechnique, CNRS/IN2P3, Palaiseau, France}} \affil[31]{\orgaddress{Faculty of Science, University of Montenegro, Podgorica, Montenegro$^{f7}$}} \affil[32]{\orgaddress{Institute of Physics, Academy of Sciences of the Czech Republic, Praha, Czech Republic$^{f8}$}} \affil[33]{\orgaddress{Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic$^{f8}$}} \affil[34]{\orgaddress{University of California, Riverside, CA 92521, USA}} \affil[35]{\orgaddress{Dipartimento di Fisica Università di Roma Tre and INFN Roma 3, Roma, Italy}} \affil[36]{\orgaddress{Shandong University, Shandong, P.R.China}} \affil[37]{\orgaddress{Fakultät IV - Department für Physik, Universität Siegen, Siegen, Germany}} \affil[38]{\orgaddress{Stony Brook University, Stony Brook, NY 11794, USA$^{f1}$}} \affil[39]{\orgaddress{Physics Department, University of Tennessee, Knoxville, TN 37996, USA}} \affil[40]{\orgaddress{Institute of Physics and Technology of the Mongolian Academy of Sciences, Ulaanbaatar, Mongolia}} \affil[41]{\orgaddress{Ulaanbaatar University, Ulaanbaatar, Mongolia}} \affil[42]{\orgaddress{Brookhaven National Laboratory, Upton, NY 11973, USA}} \affil[43]{\orgaddress{Paul Scherrer Institut, Villigen, Switzerland}} \affil[44]{\orgaddress{Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA}} \affil[45]{\orgaddress{Fachbereich C, Universität Wuppertal, Wuppertal, Germany}} \affil[46]{\orgaddress{Yerevan Physics Institute, Yerevan, Armenia}} \affil[47]{\orgaddress{Departamento de Fisica Aplicada, CINVESTAV, Mérida, Yucatán, México$^{f9}$}} \affil[48]{\orgaddress{Institut für Teilchenphysik, ETH, Zürich, Switzerland$^{f10}$}} \affil[49]{\orgaddress{Physik-Institut der Universität Zürich, Zürich, Switzerland$^{f10}$}} \affil[50]{\orgaddress{Affiliated with an institute covered by a current or former collaboration agreement with DESY}} %\maketitle %\begin{abstract} %\noindent %{\em \today --- To be submitted to EPJ C ---} \abstract{\unboldmath% The H1 Collaboration reports the first measurement of the 1-jettiness event shape observable \tb in neutral-current deep-inelastic electron-proton scattering (DIS). The observable \tb is equivalent to a thrust observable defined in the Breit frame. The data sample was collected at the HERA $ep$ collider in the years 2003--2007 with center-of-mass energy of $\sqrts=319\,\GeV$, corresponding to an integrated luminosity of 351.1\,\invpb. Triple differential cross sections are provided as a function of \tb, event virtuality \Qsq, and inelasticity $y$, in the kinematic region $\Qsq>150\,\GeVsq$. Single differential cross section are provided as a function of \tb\ in a limited kinematic range. Double differential cross sections are measured, in contrast, integrated over \tb\ and represent the inclusive neutral-current DIS cross section measured as a function of \Qsq\ and $y$. The data are compared to a variety of predictions and include classical and modern Monte Carlo event generators, predictions in fixed-order perturbative QCD where calculations up to $\mathcal{O}(\as^3)$ are available for \tb\ or inclusive DIS, and resummed predictions at next-to-leading logarithmic accuracy matched to fixed order predictions at $\mathcal{O}(\as^2)$. These comparisons reveal sensitivity of the 1-jettiness observable to QCD parton shower and resummation effects, as well as the modeling of hadronization and fragmentation. Within their range of validity, the fixed-order predictions provide a good description of the data. Monte Carlo event generators are predictive over the full measured range and hence their underlying models and parameters can be constrained by comparing to the presented data. %\end{abstract} } \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Introduction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} Measurements in high-energy lepton-proton deep-inelastic scattering (DIS) have played an important role in understanding the structure of Quantum Chromodynamics (QCD)~\cite{Abramowicz:1998ii,Klein:2008di,Newman:2013ada,Gross:2022hyw}. Inclusive neutral-current DIS cross section measurements probe the distribution of partonic constituents of the proton, and test %the scale dependence of perturbative QCD (pQCD). perturbative QCD (pQCD) over a wide range of energy scale. %, and further studies the $x$ dependence of the quark PDFs. Beyond those inclusive cross sections, dedicated measurements of the shape and substructure of the hadronic final state (HFS) provide rigorous tests of pQCD calculations. Observables related to the HFS are among others the properties of jets, heavy-quark production, or event shape quantities. They are sensitive to the strong coupling constant and the gluon content of the proton. In addition, they can be used to test the modeling of non-perturbative (NP) effects, particularly hadronization and fragmentation. However, comprehensive measurements of HFS observables over the full HFS phase space in neutral-current DIS were not performed in the past due to experimental and theoretical limitations. %Measurements of HFS properties are commonly carried out using jets, %heavy-quark cross sections, or more inclusively with event-shape %observables. % ----------------------------------------- % Event shapes have been studied extensively in $e^+e^-$ collisions~\cite{ALEPH:1996sio,ALEPH:2003obs,L3:1995eyy,L3:1997bxr,L3:1998ubl,L3:2002oql,L3:2004cdh,DELPHI:1999vbd,DELPHI:2003yqh,DELPHI:2004omy,SLD:1994idb,MovillaFernandez:1997fr,Biebel:1999zt,JADE:1999zar,MovillaFernandez:2001ed,Pahl:2009zwz,CELLO:1989okb}, and in hadron-hadron collisions~\cite{CDF:2011yfm,CMS:2011usu,ALICE:2012cor,ATLAS:2012tch,CMS:2013lua,CMS:2014tkl,ATLAS:2016hjr,CMS:2018svp,ATLAS:2020vup,Alvarez:2023fhi,ATLAS:2023tgo}. Several event shape observables were also measured in neutral-current (NC) DIS using data from the HERA-I data taking period (1992-2000)~\cite{H1:1997hbl,H1:1999wfh,ZEUS:2002tyf,H1:2005zsk,ZEUS:2006vwm}, which demonstrated sensitivity to the strong coupling constant \asmz, as well as to hadronization and resummation effects. Nevertheless, event shapes have not been studied to date as extensively in DIS as in $e^+e^-$ and hadronic collisions due to the more limited precision in predicting event shapes in DIS as compared to $e^+e^-$ collisions~\cite{Kluth:2006bw,Becher:2008cf,Abbate:2010xh,LL2022}. In this article, the H1 Collaboration reports the first measurement of the 1-jettiness event shape observable \tb\ in $ep$ collisions. This variable has theoretical advantages over previously studied event shape observables, since it is free of non-global logarithms~\cite{Dasgupta:2001sh,Dasgupta:2003iq} and thus can be calculated with high theoretical accuracy. Furthermore, it is closely related to event shapes in $e^+e^-$ collisions. A traditional event shape observable is thrust $T$~\cite{Brandt:1964sa,Farhi:1977sg}, which quantifies the momentum distribution of the HFS along a defined axis. There is freedom in choosing the projection axis, normalization and reference frames to analyzing the thrust observable in DIS. A common choice of these conditions is given by the Breit frame of reference~\cite{Streng:1979pv}, with the polar angle dividing the event into two hemispheres. These are referred to as the \emph{current} (or \emph{jet}) hemisphere, $\mathcal{H_C}$, and the \emph{target fragmentation} (or \emph{beam}) hemisphere, with polar angles smaller or larger than $\tfrac{\pi}{2}$, respectively\,\footnote{This sign convention for the $z$ axis in the Breit frame is opposite to that of the \emph{HERA laboratory coordinate system}, where the proton moves along the positive $z$ direction, while in the Breit frame as defined here it moves along the negative $z$ direction. The photon then moves in the positive $z$ direction~\cite{Streng:1979pv}. Some HERA papers define the $z$-axis in the Breit frame with opposite sign. }. % %In the Breit frame, the 3-momenta of the exchanged photon %$q$ and the incoming proton $P$ fulfill the condition $2x\vec{P} + \vec{q}=0$. In the Breit frame, the photon momentum\,\footnote{In NC DIS, the interaction is mediated by a photon, $\gamma Z$ interference, or $Z$ exchange, which is denoted \emph{photon exchange} in the following. The photon four-momentum $q$ is determined from the incoming and outgoing lepton four-vectors, $q=k-k^\prime$. The photon virtuality is $\Qsq=-q\cdot q$.} % is aligned with the positive $z$ axis, i.e.\ $q_{b}=(0,0,Q;0)$. % A natural choice for the axis of Thrust is the photon axis with normalization $\tfrac{Q}{2}$, and thus this variant of thrust is computed as %--- \begin{equation} T_{zQ} = 2 \sum_{i\in\mathcal{H_C}} \frac{P_{z,i}}{Q}\,, \label{eq:Thrust} \end{equation} %--- \noindent where the sum runs over all HFS particles in the Breit frame current hemisphere $\mathcal{H_C}$~\cite{Feynman:1973xc}. This variant of Thrust is a scaling variable~\cite{Streng:1979pv}. % % $P_{z,i}=\hat{z}_b\cdot p_i=\tfrac{q\cdot p_i}{Q}$. % A generalized set of inclusive observables defined with respect to the Breit frame axis, named \emph{current jet thrust} observables, is discussed in Ref.~\cite{Bouchiat:1980nz}. Power corrections, next-to-leading order QCD corrections, and resummed predictions for such observables have been calculated~\cite{Webber:1995ka,Graudenz:1997sp,Dasgupta:1997ex,Dasgupta:1998xt,Antonelli:1999kx,Dasgupta:2002dc,Dasgupta:2003iq,Gehrmann:2019hwf}. A change in notation has likewise been introduced, % %--- \begin{equation} \tQ = 1-T_{zQ}. \label{eq:tQ} \end{equation} % \noindent It was shown that \tQ\ is infrared and collinear safe, that it fulfills the criteria required for analytic or automatized resummation, and that it is free of non-global logarithms~\cite{Streng:1979pv,Banfi:2003je,Banfi:2004yd,caesar}. In the framework of soft-collinear effective theory (SCET), \tQ\ is one variant of a more general class of global event-shape observables called 1-jettiness~\cite{Dasgupta:1998xt,Stewart:2010tn,Kang:2013nha,Kang:2014qba}, \begin{equation} \tQ = \tb = \frac{2}{\Qsq}\sum_{i\in X}\min\left(\xbj P\cdot p_i , (q+\xbj P)\cdot p_i\right)\,, \label{eq:tau1b} \end{equation} \noindent where in this case $i$ runs over {\it all} final state particles, which renders the calculation of \tb\ free of non-global logarithms. Here, \xbj{} is the Bjorken-$x$ scaling variable, $q$ is the exchanged photon four-momentum, and $P$ is the incoming proton four-momentum. The observable \tb\ is a special case of a general class of $N$-jettiness observables~\cite{Stewart:2010tn,Jouttenus:2011wh,Kang:2012zr,Kang:2013lga,Cao:2024ota}. %which divides the Breit frame into two hemispheres.} Using momentum conservation, \tb\ can be rewritten % \begin{equation} \tb = 1- 2\cdot\sum_{i\in X}\max\left( 0, \frac{q\cdot p_i }{q\cdot q}\right) \\ = 1- 2\cdot\sum_{i\in \mathcal{H_C}} \frac{q\cdot p_i }{q\cdot q}\,, \label{eq:tau1bH1} \end{equation} \noindent where the sum runs over all HFS particles in the first expression, but only over particles in the Breit frame current hemisphere $\mathcal{H_C}$ in the second expression. % Following Eq.~\eqref{eq:tau1bH1} the 1-jettiness is proportional to the sum of particle 4-momenta radiated into $\mathcal{H}_C$, and projected onto the photon four-momentum. % %The observable \tb\ is equivalent to a Thrust observable, and it is a %scaling variable similar to Bjorken $x$, %$\xbj=-\tfrac{1}{2}\tfrac{q\cdot q}{q\cdot P}$. % It ranges from zero to unity, with $\tb\sim0$ indicating an event structure with a single collimated jet emitted into $\mathcal{H_C}$ along the photon direction. There are DIS event configurations at low $\xbj$ where the current hemisphere is empty, corresponding to $\tb=1$~\cite{Streng:1979pv,Dasgupta:1997ex,Kang:2014qba}. When the full range of \tb\ is considered, $\tb\in[0,1]$, each event in NC DIS has an associated value of \tb. %In this article, a first measurement of the 1-jettiness \tb\ is %presented, and also a first triple differential measurement as %unctions of the photon virtuality, the inelasticity and the 1-jettiness, %\begin{equation} % \Qsq = -q\cdot q\,,~~~~~~ % y = \frac{q\cdot P}{k\cdot P}\,~~~~~\text{and}~~~~ % \tb =1- 2\cdot\sum_{i\in X} \max\left( 0,\frac{q\cdot p_i }{q\cdot q}\right)\,, %\end{equation} %in a large NC DIS kinematic range is performed. % {\color{blue} {\TODO{rephrase:}} This article presents a first measurement of \tb, which constitutes the first triple-differential measurement of an hadronic event shape observable over the full phase space of selected NC DIS event kinematics. It is made possible by recent theoretical developments and improved experimental reconstruction techniques and is based on data recorded with the H1 detector at the HERA collider for \ep\ collisions at $\sqrts=319\,\GeV$ with an integrated luminosity of $351.1\,\invpb$. Differential cross sections as a function of {\tb}, as well as the triple-differential cross section as a function of photon virtuality $\Qsq$, event inelasticity $y$, and {\tb}, are reported over a large kinematic range. Inclusive DIS cross sections as a function of \Qsq\ and $y$, obtained by integrating the triple-differential cross section, are also measured. The data are compared to theoretical calculations based on Monte Carlo (MC) event generators and pQCD, probing their sensitivity to parton showering, resummation, \as, and non-perturbative effects. % } % %{\TODO{add triple-differential here?}} %This article presents a first measurement of a hadronic event shape %observable covering the full phase space of selected NC DIS event %kinematics, and simultaneous as a function of the inclusive DIS observables. %This measurement is made possible by recent theoretical %developments and improved experimental reconstruction techniques. %This first measurement } of \tb\ in neutral-current DIS is based on %data recorded with the H1 detector at the HERA collider for %\ep\ collisions at $\sqrts=319\,\GeV$ with an integrated luminosity of %$351.1\,\invpb$. %We report the differential cross section as a function of {\tb}, as well as the %triple-differential cross section as a function of photon virtuality %$\Qsq = -q\cdot q$, event inelasticity $y$, and {\tb}, over a large NC %DIS kinematic range. %Inclusive DIS cross sections as a function of %\Qsq\ and $y$, obtained by integrating the triple-differential %cross section, are also reported. %We compare these data to theoretical %calculations based on Monte Carlo (MC) event generators and pQCD with %different accuracy, %and discuss their sensitivity to parton showering, resummation, \as, %and non-perturbative effects. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The measurement % Experimental setup % Data taking, etc.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Experimental setup} The data were taken with the H1 detector in the years 2003 to 2007 using electron or positron\,\footnote{The term `electron' is used in the following to refer to both electrons and positrons.} beams that were collided with a proton beam at a center-of-mass energy of $\sqrt{s}=319\,\GeV$. The data correspond to an integrated luminosity of $\mathcal{L}=351.1\,$pb$^{-1}$~\cite{H1:2012wor}. %% final lumi value from Dec22: 351.10771 The H1 experiment~\cite{H1:1993ids,H1CalorimeterGroup:1993boq,H1:1996prr,H1:1996jzy,H1SPACALGroup:1996ziw,Pitzl:2000wz} is a general purpose particle detector with full azimuthal coverage around the electron--proton interaction region. The H1 Collaboration uses a right handed coordinate system, where the proton beam direction defines the positive $z$ axis. The nominal interaction point is located at $z=0$. The H1 detector consists of several subsystems. The main subsystems used for this analysis are the tracking detectors, the liquid argon (LAr) calorimeter, and the backward calorimeter (SpaCal). All these systems are situated inside a superconducting solenoid that provides a magnetic field of 1.16\,T. The central tracking system consists of drift and proportional chambers, together with silicon-strip detectors close to the interaction region and covers the polar angular range $15^\circ<\theta<165^\circ$. %%Silicon vertex detector: 30◦ < θ < 150◦ The transverse momentum resolution of charged particles is $\sigma_{p_\text{T}} /p_\text{T} = 0.2\,\%\,p_\text{T}/\GeV \oplus 1.5\,\%$. % \commenting{PMJ May 8}{Need sentence on tracking efficiency.} % -> DB: The energy scale of the particle candidates is calibrated % using the pt-balance method (pT of the electron must balance the HFS) % The calibrated object then account for single-particle inefficiencies in % the reconstruction. Consequently, when using our particle-candidates % (as an inclusive sample) then no efficiencies need to be accounted % for. That is different, when single-tracks (e.g. in charged particle % multiplicity measurements) are considered. The LAr sampling calorimeter consists of an electromagnetic section made of lead absorbers and an hadronic section with steel absorbers, covering the polar angular range $4^\circ <\theta < 154^\circ$. The energy resolution is $\sigma_E /E = 11\%/\sqrt{E/\GeV} \oplus 1\%$ for electrons and $\sigma_E /E \simeq 55\%/ \sqrt{E/\GeV} \oplus 3\%$ for charged pions. The LAr calorimeter is used for triggering and particle reconstruction in this analysis. The SpaCal (`Spaghetti Calorimeter') is a lead-scintillating fiber calorimeter with electromagnetic and hadronic sections, covering the backward direction with polar angular range $153^\circ < \theta < 177^\circ$. The electromagnetic energy resolution $\sigma(E)/E$ is $7 \%/\sqrt{E/\GeV}\oplus 1\%$, and in the hadronic section it is $(56\pm13)\,\%$ for charged pions. Online triggering and event selection follow the procedure described in Ref.~\cite{H1:2014cbm}. Events are triggered by a high-energy cluster in the LAr calorimeter, with the scattered electron identified using isolation criteria. Events are accepted if the scattered electron has energy $E_{e}^{\prime}>11\,\GeV$ and is found in the high-efficient regions of the LAr calorimeter trigger system, which corresponds to about 90\% of its $\eta$--$\varphi$ coverage. The trigger efficiency for inclusive DIS events is greater than 99\,\%. In order to suppress non-collision backgrounds from cosmic muons, beam-gas interactions, and high energetic muons produced off the proton beam in the HERA tunnel, a triggered event must fulfill certain requirements~\cite{H1:2012qti,H1:2014cbm}. Events with an energy deposition in the range $\theta>175^\circ$ %\bluetext{$\theta>3.05$} \commenting{PMJ May 8}{Degrees? Is there an upper limit?} -> yes: 177°, see the SpaCal acceptance above are found to be sensitive to non-collision backgrounds and are removed. Events with a topology similar to QED Compton events are also removed~\cite{H1:2014cbm}. Tracks of charged particles are reconstructed from hits in the tracking detectors. The energy depositions of charged and neutral particles in the calorimeters are clustered and calibrated. Particle candidate four-vectors are then reconstructed offline using an energy-flow algorithm which combines information from tracks with that from clusters~\cite{energyflowthesis,energyflowthesis2,energyflowthesis3}. The energy of particle candidates is calibrated using a neural-network based shower-classification algorithm and a dedicated jet-calibration sample~\cite{Kogler:2011zz}. The scattered electron candidate is identified as the electromagnetic cluster in the LAr calorimeter which has the highest energy in the event, and which satisfies isolation criteria and is matched with a track~\cite{H1:2003xoe}. %\commenting{PMJ May 8}{What about granularity of LAr and SpaCal, especially their hadronic sections - is this important for the measurement of event shapes? Particle flow utilizes the spatial precision of tracks, but tracking efficiency is less than unity (what's the number?) and particle flow energy matching is not perfect. Does the residual effect of the granularity affect the resolution in \tb\ or are such effects negligible? A comment about this here would be helpful.} % DB: I have to think about that. Radiative photons may distort the kinematic reconstruction. Isolated high-energy depositions in the central or backward part of the electromagnetic calorimeters ($\theta>\frac{\pi}{3}$) are found to have a good association with photons radiated off the incoming or scattered electron. Such clusters are treated in the same manner as radiated photons at particle level. If their angular distance to the scattered electron candidate is smaller than the distance to the negative $z$-axis, they are recombined with the scattered electron to form a dressed scattered electron with four-vector $p_e$; otherwise, their energy deposition is removed from the event record. % This procedure suppresses photons from initial-state QED radiation, which effectively reduce the energy of the incoming lepton. In addition, it provides well-reconstructed observables in the presence of final-state QED radiation. %, which %otherwise reduces the energy of the outgoing electron. % % In addition, good consistency between detector-level and % particle-level quantities for events with photon radiation is % achieved. % All remaining particle candidates which are not classified as the scattered electron, comprise the hadronic final state (HFS), with their sum corresponding to the HFS four-vector, $p_h$. The incoming electron energy is reconstructed by the $\Sigma$ method~\cite{Bassler:1994uq}, which makes use of the energy and longitudinal momentum of $p_h+p_e$. The photon four-vector $q$ is calculated using the incoming electron energy and $p_e$. HFS particles satisfying $p_i\cdot q<0$ are associated to the current hemisphere $\mathcal{H}_C$ in the Breit frame, with their sum forming the current-hemisphere four-vector $p_c$. % q was introduced above % % $p_{e_0}=(\tfrac{\Sigma_e+\Sigma_h}{2};0,0,-\tfrac{\Sigma_e+\Sigma_h}{2})$, % neglecting the electron mass, and using $\Sigma_i:=E_i-P_{z,i}$. % % $k-k^\prime$, using the electron beam four-vector as used in the % $\Sigma$-method~\cite{Bassler:1994uq}.% % The I$\Sigma$ method has highest precision of a large kinematic range. % In the I$\Sigma$ method the electron beam is defined as % $p_{e_0}=(\tfrac{\Sigma_e+\Sigma_h}{2};0,0,-\tfrac{\Sigma_e+\Sigma_h}{2})$, % neglecting the electron mass, and using $\Sigma_i:=E_i-P_{z,i}$. The DIS kinematic observables are calculated from the four vectors $p_e$ with energy $E_e$ and transverse momentum $P_{\text{T},e}$, and $p_h$, using the I$\Sigma$ method~\cite{Bassler:1994uq,Bassler:1997tv}, % \begin{equation} % \Qsq_\Sigma = \frac{E^2_{e^\prime}\sin^2\theta_{e^\prime}}{1-y_\Sigma}\,,~~~ \begin{array}{cc} y_\Sigma = \frac{\Sigma_h}{\Sigma_h + \Sigma_e}\,, & % \label{eq:Q2is} \Qsq_\Sigma = \frac{P_{\text{T},e}^2}{1-y_\Sigma}, \\ % \label{eq:yis} % x = x_{\text{I}\Sigma} = \frac{E_{e^\prime}}{E_{p_0}}\frac{\cos^2\frac{\theta_{e^\prime}}{2}}{y_\Sigma}\,. \multicolumn{2}{c}{ x_{\text{I}\Sigma} = \frac{E_{e}}{E_{p_0}}\frac{\cos^2\frac{\theta_{e}}{2}}{y_\Sigma}\,,} % \label{eq:xis} \end{array} \label{eq:DIS} \end{equation} \noindent and for certain purposes the electron method is employed instead, \begin{equation} % \Qsq_e = 4E_0E_{e^\prime}\cos^2\frac{\theta_{e^\prime}}{2} = \frac{2E_0 P_{T,e}^2}{\Sigma_e} y_e=1-\frac{\Sigma_e}{2E_{e_0}} ~~~\text{using}~~~ \Qsq_e = \frac{P_{\text{T},e}^2}{1-y_e}\,, \label{eq:DISeMethod} \end{equation} %The quantity $\Sigma$ is defined as $\Sigma = E_h-P_{z,h} = \sum_{i\in X} (E_i - P_{z,i})$, %the polar angle of the scattered electron is denoted $\theta_{e^\prime}$, and $E_{p_0}$, where the variables $\Sigma_i$ denote $\Sigma_i = E_i-P_{z,i}$ for $i=e,h$. The electron and proton beam energies are $E_{e_0}=27.6\,\GeV$ and $E_{p_0}=920\,\GeV$, respectively. The I$\Sigma$ method achieves good resolution in $y$ over the entire kinematic range~\cite{Bassler:1994uq,Bassler:1997tv,Arratia:2021tsq}. Inserting $-Q^2_\Sigma$ for $q\cdot q$ in Eq.~\eqref{eq:tau1bH1} provides best resolution for the calculation of $\tb$~\cite{Arratia:2022wny}. However, the electron method provides higher resolution in $Q^2$ and is therefore used for $Q^2$ dependent measurements, $\propto\frac{\text{d}\sigma}{\text{d}\Qsq}$. %purities for most of the bins in the measured phase space, as %compared to other reconstruction methods. The electron-beam energy does not enter the equations of the I$\Sigma$ method, which thereby is largely insensitive to initial state QED radiative effects. The hadronic angle is defined in this analysis as $\gamma_h=2\arctan{\frac{\Sigma_h}{P_{\text{T},e}}}$. To achieve high resolution and to reduce initial-state QED radiation effects, events are selected as follows: the ratio of the transverse momentum of the HFS and scattered lepton satisfies $0.6\frac{\pi}{2}$, or with $\theta_c > 3.05$, or with $\theta_c>2.6$ where $\beta$ of the boost vector $b=2\xbj P+q$ ($\beta=\tfrac{\vec{b}^2 }{b_E^2}$) fulfills $\beta>0.9$, are found to be events with an initial state radiated photon or where a radiated photon was converted to hadrons; \item Events with $\theta_c<0.12$ cannot be reconstructed well due to limited acceptance in the forward direction; \item Events at low \tb ($\tb<0.3$) are found to have poor resolution when $\theta_c/\gamma_h>1.3$; \item Events at large \tb and low \Qsq ($\tb>0.65$ and $\Qsq<700\,\GeVsq$) have poor resolution when $\beta>0.9$ and $P_{\text{T},c}/P_{\text{T},e}<0.35$; %\item Events with $\Qsq<700\,\GeVsq$, $\beta>0.9$, and $P_{T,c}/P_{T,e}<0.35$ have low resolution at larger \tb, $\tb>0.65$; \item Events at very high value \tb ($\tb>0.95$) have poor resolution when $\beta>0.97$; \item Events at low \Qsq ($\Qsq<700\,\GeVsq$) have poor resolution for $\eta_c+\ln{(\tan{\frac{\gamma_h}{2}})}>0.3$. \item Events with $y_e>0.94$ are removed in order to suppress background contributions from photoproduction. % removed? \item Events with $\tb>0.65$ and $\theta_c<0.8$, have sizable contributions from low \tb\ migrations; %not mentioned: \item $\log_{10}(P_{T,\text{HFS}} - P_{T,e}) > 0.7$ %not mentioned: \item $E_c>0$ and $\theta_c< 0.7$ and Econe0p07 $> 0$ \end{compactitem} % % \begin{comment} % \color{red} % The analysis cuts the following events: % \begin{compactitem} % \item $\theta_c > 3.05$ % \item $E_c>0$ and ($\theta_c - \gamma_h) > \pi/2$ % \item $E_c>0$ and $\theta_c< 0.12$ % \item $E_c>0$ and $\tb < 0.3$ and $\theta_c/\gamma_h> 1.3$ % \item $\theta_c > 2.6$ and $\beta > 0.9$ % \item $E_c>0$ and $\Qsq < 700$ and $\tb > 0.65$ and $\beta > 0.90$ and $P_{T,c}/P_{T,e}< 0.35$ and $\theta_c< 0.8$ % \item $E_c>0$ and $\Qsq < 700$ and dEta $> 0.3$ % \item ($E_c=0$ or $\tb>=0.95$ ) and $\beta> 0.97$ % \item $\log_{10}(P_{T,\text{HFS}} - P_{T,e}) > 0.7$ % \item $E_c>0$ and $\theta_c< 0.7$ and Econe0p07 $> 0$ % \end{compactitem} % \end{comment} % After the application of all acceptance, background and cleaning cuts, about 35~to 80\,\% of the events generated within the phase space of the measurement are accepted at the detector-level. % (so-called \emph{acceptance}) % --- kinematic range of the measurements The final selected NC DIS kinematic range of the analysis is shown in Figure~\ref{fig:kineplane} as a function of $\xbj$ and \Qsq. % \begin{figure}[h] \centering \includegraphics[height=0.48\textwidth,trim={0 0 0 0 },clip]{d24-035f1} \caption{ \label{fig:kineplane} DIS kinematic plane as a function of $\xbj$ and \Qsq. The kinematic range of the measurement is displayed as colored area. The dark shaded area indicates the kinematic range of the reported single-differential cross section ($200\leq\Qsq<1700\,\GeVsq$ and $0.2\leq y<0.7$). } \end{figure} % The kinematic range is limited by acceptance, resolution, and trigger: The polar-angle acceptance of the LAr calorimeter, $\theta\lesssim154^\circ$, defines the \Qsq\ region of the measurement, $\Qsq>150\,\GeVsq$; The electron energy acceptance corresponds to $y\lesssim0.7$ for $Q^2\lesssim1000\,\GeVsq$; The inelasticity is required to be $y>0.05$, since acceptance and resolution deteriorate for lower $y$, where the hadronic angle $\gamma_h$ becomes small and the Lorentz-factor $\beta$ of the boost vector approaches unity. %\commenting{PMJ May 8}{Above discussion about event and kinematic %selection is somewhat confusing - there are many cuts for different %reasons. I wonder whether it can all be presented in a more uniform %and coherent way.} % DB: I tried to improve. Often, such a discussion is not included, % but I find it useful to explain the experimental constraints at the % phase space boundaries. % --- special binnings The binning used in this analysis is also presented in Table~\ref{tab:binnings}. %and Figure~\ref{fig:kineplane}. % % A graphical display of the \Qsq--$y$ bins in the DIS kinematic planea function of \Qsq\ and $x$ is . % %\begin{table}[tbhp] \begin{table*}[t] \footnotesize %\scriptsize \begin{center} \begin{tabular}{lccc} \toprule Observable & Binning name & Notation & Binning \\ \midrule \tb & \tb nominal & $b_{\tb}^{(11)}$ & $[0,0.05,0.1,0.15,0.22,0.3,0.4,0.54,0.72,0.86,0.98,1.0+\epsilon]$ \\ \tb & \tb coarse & $b_{\tb}^{(8)}$ & $[0,0.05,0.1,0.15,0.22,0.3,0.4,0.6,1.0+\epsilon]$ \\ \tb & \tb coarse low $y$ & $b_{\tb}^{(7)}$ & $[0,0.1,0.15,0.22,0.3,0.4,0.6,1.0+\epsilon]$ \\ $y$ & $y$ & $b_{y}$ & $[0.05, 0.1, 0.2, 0.4, 0.7, 0.94]$ \\ $\Qsq$& $\Qsq$ & $b_{\Qsq}$ & $[150, 200, 280, 440, 700, 1100, 1700, 3500, 8000, 20000[$ \GeVsq \\ \bottomrule \end{tabular} \end{center} \caption{ \label{tab:binnings} Binnings used in the analysis. } \end{table*} % Over most of the phase space the binning is dictated by the resolution of $\tb$. Coarser binning is chosen at the boundaries of the kinematic range due to the following factors: % \begin{compactitem} \item low $y$ or large \Qsq: large values of $\tb$ have low statistics or lower resolution; \item low $y$ and $\tb<0.05$: very low cross section which cannot be resolved. \end{compactitem} Altogether 308 cross section values are measured. The respective highest \tb\ bins ($0.98\leq\tb\leq1$ and $0.6<\tb\leq1$) include events with $\tb=1$, in which the current hemisphere is empty. Such events are related to low values of $x$~\cite{Streng:1979pv,Dasgupta:1997ex,Kang:2014qba}, % least smaller than 0.5 and are present because \tb\ is defined in the Breit frame. They are absent for event shape observables defined in the partonic center-of-mass frame, like thrust in $e^+e^-$ collisions~\cite{Dasgupta:2003iq,Kang:2014qba}. % At detector-level in data and particle-level\footnote{The % particle-level used for the unfolding includes higher-order QED % radiation at the lepton-vertex. In these \emph{radiative} MC % events, ISR photons are not included into the calculation of % $(E-P_z)_\text{tot}$ at particle-level. In contrast, in non-radiative MC % this cut is not applicable and one always finds $(E-P_z)_\text{tot}=2E_e$.}, % in radiative MC % the data are required to have % a longitudinal energy-momentum balance between $48<(E-P_z)_\text{tot}<65\,\GeV$, which is % defined from a sum over all particles as % $(E-P_z)_\text{tot}=\sum_{e^\prime,i\in X}(E_i-P_{z,i})$. % This requirement reduces effects from initial state QED radiation (ISR). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Monte Carlo simulations and models %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Monte Carlo simulations and model predictions} MC event generators are used to correct the data for detector acceptance and resolution effects, and for contributions from $ep$ collisions outside the phase-space of this analysis. % other hard processes. The generated events are processed with a detailed simulation of the H1 detector based on GEANT3~\cite{Brun:1987ma}, supplemented by fast shower simulations~\cite{Fesefeldt:1985yw,Grindhammer:1989zg,Gayler:1991cr,Kuhlen:1992ey,Grindhammer:1993kw,Glazov:2010zza}. The simulated data are reconstructed and processed with the same analysis algorithm as the real data~\cite{DPHEPStudyGroup:2009gfj,Steder:2011zz,South:2012vh,DPHEPStudyGroup:2012dsv,Britzger:2021xcx,DPHEP:2023blx,Brun:1997pa}. Detector effects are corrected using \emph{regularized unfolding}, based on the simulation of NC DIS events using the MC event generators Djangoh~1.4~\cite{Charchula:1994kf} and Rapgap~3.1~\cite{Jung:1993gf}. Djangoh uses Born level matrix elements for NC DIS and dijet production and applies the color dipole model from Ariadne~\cite{Lonnblad:1992tz} for higher-order emissions. % Rapgap implements Born level matrix elements for NC DIS and dijet production and uses the leading logarithmic approximation for parton shower emission. % Both generators are interfaced to Heracles~\cite{Kwiatkowski:1990es} for higher order QED effects at the lepton vertex. Both generators utilise the CTEQ6L parton distribution function (PDF) set~\cite{Pumplin:2002vw} and the Lund hadronization model~\cite{Andersson:1983ia,Sjostrand:1994kzr}. Hadronization model parameters were determined by the ALEPH Collaboration~\cite{ALEPH:1996oqp}. %% backgrounds Contributions to the event yield from processes other than NC DIS are simulated with a variety of MC event generators. Photoproduction events are simulated using Pythia~6.2~\cite{Sjostrand:1993yb,Sjostrand:2001yu}. Events with di-lepton production are generated using Grape~\cite{Abe:2000cv}. A sample of QED Compton events is simulated using the program Compton~\cite{Courau:1992ht}. Deeply virtual Compton scattering (DVCS) is simulated with Milou~\cite{Perez:2004ig}. % TinTin//Milou NC DIS events for lower values of \Qsq\ and for charged current DIS are simulated with Djangoh. % Overall, more than $3.4\times10^{8}$ signal events were simulated with Djangoh and Rapgap, and about $5.3\times10^{8}$ events were simulated in total. % all ev: 527281025 % The following predictions are compared to the measured cross sections. % \begin{itemize} % \item The DIS MC event generator Djangoh~1.4~\cite{Charchula:1994kf} % which implements higher-order QCD radiation through the % color-dipole model as implemented in % Ariadne~\cite{Lonnblad:1992tz}. The Lund string % fragmentation model~\cite{Andersson:1983ia,Sjostrand:1994kzr} with the ALEPH % tune~\cite{ALEPH:1996oqp} %cite{ALEPH:2004fxe} ? % and the CTEQ6L PDF~\cite{Pumplin:2002vw} are used. % Higher-order QED radiation is simulated with the Heracles % program~\cite{Kwiatkowski:1990es}, although this is used only for % radiative MC events used for unfolding, but not for the cross section predictions. % \item The DIS MC event generator Rapgap~3.1~\cite{Jung:1993gf}, which % implements parton showers in the leading-logarithmic % approximation. % Similarly as Djangoh, Heracles, the Lund string fragmentation model with the % ALEPH tune, and the CTEQ6L is implemented. % \end{itemize} % ------------------------------------------------ % Models without detector simulation % ------------------------------------------------ % Additionally, a number of other predictions % are made using the set of state-of-the-art generators developed mostly % for pp collisions. The fully-corrected \tb\ cross section measurements are compared to a set of theoretical predictions. The following classes of prediction are studied: two common MC event generators as used at HERA; three modern general-purpose event generators which are widely used in high-energy physics; one dedicated event generator incorporating transverse-momentum dependent effects; and two variants of fixed-order predictions. In order to explore the sensitivity of the data to various QCD effects, in each MC event generator one element is varied, such as the parton shower model, the fixed-order prediction and its matching or merging procedure with the parton shower, the hadronization model, or the PDF set. The fixed-order predictions include variations of the renormalization and factorization scales. In addition, dedicated predictions for the inclusive NC DIS cross sections are studied. %, which as indicated by their error bands. %The following models are considered: % The following MC event generators from the HERA era are considered: \begin{itemize} \item Djangoh~1.4 and Rapgap~3.1, similar to those described above, but with radiative effects switched off in Heracles. %, but purely photonic first order %corrections are not used in Heracles. \commenting{PMJ May 8}{Confusing, rephrase} % ---------------------------------------------- % Pythia % Predictions from Pythia 8.3 [98, 99] are used for comparison using the % default implementation and two additional parton shower % implementations: VINCIA [100, 101] and Dire [102]. % VINCIA uses a pT-ordered model for QCD + QED showers based on the % antenna formalism while Dire implements a pT ordered dipole shower similar % to Ariadne. The NNPDF3.1 PDF set [103] is used for both default and % VINCIA implementation and the MMHT14nlo68cl PDF set [104] is used for % the Dire implementation. \end{itemize} The following modern MC event generators are studied: \begin{itemize} \item Pythia~8.3~\cite{Sjostrand:2014zea,Bierlich:2022pfr}, where the impact of the parton-shower is studied by employing three different parton-shower models: i) `default' dipole-like $p_\perp$-ordered shower with a local dipole recoil strategy (Pythia~8.310)~\cite{Cabouat:2017rzi}, ii) $p_\perp$-ordered Vincia parton shower based on the antenna formalism at leading color (Pythia~8.307)~\cite{Giele:2007di,Giele:2011cb,Giele:2013ema,Fischer:2016vfv}, and iii) the Dire~\cite{Hoche:2015sya,Hoche:2017iem,Hoche:2017hno} parton shower which is an improved dipole-shower with additional treatment of collinear enhancements (Pythia 8.307). All models use the Pythia~8.3 default Lund string model for hadronization~\cite{Bierlich:2022pfr} and the PDF4LHC21 PDF set~\cite{PDF4LHCWorkingGroup:2022cjn} for the hard PDFs. %The value of the strong coupling constant is set consistently to 0.118. The Vincia and Dire parton shower use a value of 0.118 for the strong coupling at the mass of the $Z$ boson. % ---------------------------------------------- % Powheg \item Powheg Box plus Pythia (Powheg+Pythia)~\cite{Banfi:2023mhz} implements predictions in NLO QCD matched to parton showers using the Powheg method~\cite{Nason:2004rx,Frixione:2007vw}, where the radiation phase space is parameterised according to the Frixione-Kunszt-Signer (FKS) subtraction technique~\cite{Frixione:1995ms}. Dedicated momentum mappings preserve the DIS kinematic variables. The NLO predictions (up to $\mathcal{O}(\as)$) are then interfaced to parton showers and hadronziation from Pythia~8.308. % ---------------------------------------------- % Herwig 7 \item Herwig~7.2~\cite{Bellm:2015jjp}, where the impact of different modelling of the hard interaction and its merging or matching with the parton-shower model is studied in three variants: The default prediction implements leading-order matrix elements supplemented with an angular-ordered parton shower~\cite{Gieseke:2003rz} and cluster hadronization model~\cite{Webber:1983if,Marchesini:1991ch}. The second variant utilises MC@NLO~\cite{Frixione:2002ik} which implements NLO matrix element corrections. Matching with the default angular-ordered parton shower is performed~\cite{Platzer:2011bc}. The third variant also utilises NLO matrix elements, but with dipole merging and a dipole parton shower~\cite{Platzer:2011bc}. Herwig-generated events are further processed with Rivet~\cite{Bierlich:2019rhm}. % i) default Herwig 7.2, % ii) Herwig 7.2 with the internal implementation of MC@NLO (NLO$\oplus$PS), % iii) Herwig 7.2 with dipole merging (matchbox~\cite{Platzer:2011bc}) %The different NLO matching schemes are selected Subtractive or %MC@NLO matching to the angular-ordered show % Multiplicative matching, based on the POWHEG formalismis also % available and may be switched on by including in the input file: Matchbox/Powheg-DefaultShower.in %which implements the cluster hadronization model. % ---------------------------------------------- % Sherpa: % https://sherpa-team.gitlab.io/sherpa/v3.0.0alpha1/manual/introduction.html % AHADIC++ is Sherpa’s hadronization package % CSSHOWER: The default shower is based on Catani–Seymour dipole % factorisation~\cite{Catani:1996vz,Schumann:2007mg} % Comix is a multi-leg tree-level matrix element generator, based on the color dressed Berends-Giele recursive relations~\cite{Duhr:2006iq} \item Sherpa~2.2~\cite{Sherpa:2019gpd,Gleisberg:2008ta}, where the modelling of hadronization effects can be studied by using two different hadronization models. The default Sherpa~2.2 predictions are based on multi-leg tree-level matrix elements from Comix~\cite{Duhr:2006iq} that are combined in the CKKW merging formalism~\cite{Catani:2001cc} with dipole showers~\cite{Catani:1996vz,Schumann:2007mg} and supplemented with the cluster hadronization as implemented in AHADIC++~\cite{Winter:2003tt}. As an alternative prediction, the parton-level calculation is supplemented with the Lund string fragmentation model~\cite{Sjostrand:2006za}. % {\color{red}Todo: decide on sherpa 2 NLO predictions} \item %Sherpa 3 will come very very likely, together with NLO matrix %elements, a dipole shower, and new cluster hadronization model Predictions from the pre-release version of Sherpa~3.0~\cite{Sherpa3} are provided by the Sherpa authors featuring a new cluster hadronization model~\cite{Chahal:2022rid} and matrix element calculation at NLO QCD obtained from OpenLoops~\cite{Buccioni:2019sur} with the Sherpa dipole shower~\cite{Schumann:2007mg} based on the truncated shower method~\cite{Hoeche:2009rj,Hoeche:2012yf}. The predictions are associated with scale uncertainties from a 7-point scale variation by factors of two. % {\color{red}Todo: decide on sherpa 3 scale uncertainties} \end{itemize} The following dedicated MC event generators are studied: \begin{itemize} \item Cascade~3~\cite{CASCADE:2021bxe} implements off-shell processes via the automated matrix element calculators KaTie~\cite{vanHameren:2016kkz} and Pegasus~\cite{Lipatov:2019oxs}, parton shower and hadronization through Pythia~6. This utilises transverse momentum dependent (TMD) PDF sets in the parton branching (PB) methodology~\cite{Hautmann:2017xtx}. Two PB TMD PDF sets are studied, denoted as \emph{set 1} and \emph{set 2}~\cite{Jung:2021mox}. They differ primarily in the scale choice for QCD evolution. % Es laesst sich erkennen, das set2 vorallem bei kleinen tau Werten besser fittet als set1, fuer kleine tau Werte sogar besser, als rapgap. % \item N3LL predictions (are available, but do not fit the data) [todo] % \item NNLO+PS (unlikely, since the code is in a tarball at Fermilab)~\cite{Hoche:2018gti} \end{itemize} The following exact QCD predictions are studied: \begin{itemize} \item Next-to-next-to-leading order (NNLO) predictions in perturbative QCD up to third order in \as ($\mathcal{O}(\as^3)$) are obtained for the process $ep\to e+2\text{jets}+X$ with the program NNLOJET~\cite{Ridder:2016nkl,Currie:2017tpe,Currie:2016ytq,Gehrmann:2019hwf}. The factorization and renormalization scales are chosen to be $\mu=Q$. Scale uncertainties are determined by the largest difference in the 7-point scale-variation prescription, with scale factors 0.5 and 2. % NNPDF3.1~\cite{NNPDF:2017mvq} The PDF set PDF4LHC21~\cite{PDF4LHCWorkingGroup:2022cjn} is used. Non-perturbative correction factors are applied to these parton-level predictions as multiplicative correction factors for hadronization effects. These NNLO predictions are valid only in the region where the $ep\to e+2\text{jets}+X$ process dominates and hadronization corrections are small. This corresponds to the region $\tb\gtrsim0.1$ and $\tb\neq1$. %From the same program, also NLO predictions are obtained and %displayed for comparison. The NNLO predictions can formally be brought to next-to-next-to-next-to-leading (N3LO) order for NC DIS using the projection to Born method~\cite{Currie:2018fgr,Gehrmann:2019hwf}. %The NNLO predictions are associated scale uncertainties, which are %obtained by the so-called 7-point scale variation, where the renormalization and %factorization scales in the hard calculation are independently %varied by factors of 2 up and down. % \item Cascade (maybe...) % \item Sherpa, if we are lucky % \item N3LO if we are lucky ~\cite{Gehrmann:2019hwf} or NNLO, % NNLO+PC % N3LO with P2B:~\cite{Currie:2018fgr} \item Analytic predictions are provided in NNLO$\otimes$NLL$^{\prime}\otimes$Had accuracy~\cite{Knobbe:2023ehi}, using NLO pQCD matrix elements for dijet production projected onto the NNLO NC DIS cross section~\cite{Hoche:2018gti} and supplemented with automatised next-to-leading logarithmic accuracy using the Caesar framework~\cite{caesar,Banfi:2003je,Banfi:2004yd} as implemented in the Sherpa framework~\cite{Gerwick:2014gya}. For these predictions, hadronization corrections are applied as a multiplicative correction matrix obtained from Sherpa~3.0~\cite{Reichelt:2021svh}. \end{itemize} Further predictions for \tb\ have been reported previously but are not available here. These include NNLO predictions supplemented with parton shower~\cite{Hoche:2018gti}, next-to-next-to-leading logarithmic and next-to-NLL predictions~\cite{Kang:2015swk}, analytic NLL predictions~\cite{Kang:2014qba}, and N3LO predictions using the projection to Born method and a dispersive model for hadronization effects~\cite{Currie:2018fgr,Gehrmann:2019hwf}. % NNLL+NLO for tau1a cite{Kang:2013lga} % first NLO calculation cite{Graudenz:1997sp} The following predictions are compared to the inclusive NC DIS cross sections measured in bins of $\Qsq$ and $y$: \begin{itemize} \item Predictions in NNLO pQCD are obtained with the program Apfel++~\cite{Bertone:2013vaa,Bertone:2017gds}, using the three loop splitting and coefficient functions~\cite{Moch:2004pa,Vogt:2004mw,Moch:2004xu,Vermaseren:2005qc} together with the PDF set NNPDF31\_nnlo\_as\_0118. The predictions are associated with PDF uncertainties and with scale uncertainties, where the latter are defined as 7-point scale variations in the coefficient functions. An alternative prediction is obtained with the H1PDF2017{\scriptsize NNLO} PDF set~\cite{H1:2017bml}. \item Predictions in approximate next-to-next-to-next-to-leading order (aN3LO) are obtained for inclusive NC DIS cross sections % (note: not for \tb) using the program Apfel++~\cite{Bertone:2013vaa,Bertone:2017gds} and using the MSHT PDFs at a3NLO order~\cite{McGowan:2022nag} and four-loup splitting functions from Refs.~\cite{Moch:2017uml,Vogt:2018miu,Moch:2021qrk}. % \item HELL~\cite{Bonvini:2016wki,Bonvini:2017ogt,Bonvini:2018xvt,Bonvini:2018iwt} \end{itemize} % % % Predictions from % Herwig 7.2 [105, 106] are calculated using the default implementation % parameters and alternative MatchBox [107] matching and merging schemes % and the CT14 [108] PDF set. % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Unfolding % cross section definition %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Data correction: regularized unfolding} The data are corrected for detector effects, background processes, and higher-order QED effects using regularized unfolding~\cite{Blobel:1984ku,Schmitt:2012kp}. This section presents the unfolding procedure and the measurement of the fully-corrected cross section. In addition, correction factors for hadronization and electroweak effects are discussed. % ------------------------------------------------------------------------------ % Control plots y, Q2, tau % ------------------------------------------------------------------------------ The signal Monte Carlo generators Djangoh and Rapgap are used together with the detailed simulation of the H1 detector to generate synthetic detector-level events along with corresponding particle-level events (MC event simulation). Within the scope of the analysis, the detector response is found to be well modelled in all studied aspects, % and with high granularity, % \commenting{PMJ May 9}{ref to a previous paper where this is % documented?}, % DB: I don't know about a single reference that would be % suitable. There older ones from the shower parameterization, but no % recent ones that have all details that we studied % I changed the sentence, that those studies were done by us. which is essential for obtaining accurate unfolding results. %Both MC generators provide an accurate description of the data. % Figure~\ref{fig:ControlPlots} shows comparison of these simulations to data for the observables $\Qsq$, $y$, and $\tb$. % \begin{figure*}[t] \centering \includegraphics[width=0.32\textwidth,trim={0 0 0 0 },clip]{d24-035f2a} \includegraphics[width=0.32\textwidth,trim={0 0 0 0 },clip]{d24-035f2b} \includegraphics[width=0.32\textwidth,trim={0 0 0 0 },clip]{d24-035f2c} \caption{ \label{fig:ControlPlots} Detector level distributions of $\Qsq$ (left), $y$ (middle), and \tb~(right) of the selected data. The definition of these observables is given by Eq.~\eqref{eq:DIS} and~\eqref{eq:DISeMethod}. %~\eqref{eq:Q2is} and \eqref{eq:yis}. The data are compared to the MC event simulations Djangoh and Rapgap. Both simulations include estimated background contributions from processes other than high-\Qsq\ NC DIS, as indicated by the green line (visible on logarithmic scale). } \end{figure*} % %The data are described well by the simulations. Both MC event simulations, although based on different physics models, are found to describe the data well. %in many aspects. %Selected examples Comparisons to other observables, related closely to the detector performance, are discussed in the following. % ------------------------------------------------------------------------------ % single particle control plots % ------------------------------------------------------------------------------ % pmj May 9: these sentences don't say something new, not needed. The sentence on only the current hemishpere contribution to \tb\ moved to the next para. %Accurate, unbiased unfolding requires detailed determination of the acceptance and resolution \redtext{and efficiency} ofall contributing particles. Precise modelling of the detector response is required for this purpose. Figure~\ref{fig:CurrentPz} shows the distribution of \pzb, the particle-candidate longitudinal momentum in the Breit frame, compared with simulations for reconstructed clusters and tracks. Good overall agreement between simulations and data is observed, for both clusters and tracks. \begin{figure*}[t] \centering \includegraphics[width=0.38\textwidth,trim={0 0 0 0 },clip]{d24-035f3a} \includegraphics[width=0.38\textwidth,trim={0 0 0 0 },clip]{d24-035f3b} \caption{ \label{fig:CurrentPz} The longitudinal momentum \pzb in the Breit frame of reconstructed particle candidates in the selected events. Only particle candidates with $\eta^\text{Breit}>0$ are displayed, since only those contribute to the actual calculation of \tb. Particle candidates are defined by an energy-flow algorithm taking clusters and tracks into account. Left: the \pzb\ distribution for clusters not associated with a track, right: the \pzb distribution for objects associated with a track. The detector-level data are compared to the MC event simulations Djangoh and Rapgap, which include further MC samples for background processes. Further details are given in the Figure~\ref{fig:ControlPlots} caption. } \end{figure*} % % %As seen in Eq.~\eqref{eq:tau1bH1}, The measurement of \tb\ includes only particle candidates of the current hemisphere. Figure~\ref{fig:ContribTh} shows relative contributions from different polar angular regions $\theta$ and particle energies $E$. The largest contributions are from objects in the central detector ($25<\theta<153^\circ$), and from objects with large energy ($E>1.0\,\GeV$). Both types of object are measured well by the relevant H1 sub-detector components, and are modelled well by the MC event simulations. \begin{figure*}[t] \centering \includegraphics[width=0.48\textwidth,trim={0 0 0 0 },clip]{d24-035f4a} \includegraphics[width=0.48\textwidth,trim={0 0 0 0 },clip]{d24-035f4b} \caption{ \label{fig:ContribTh} Left: The normalized contribution to \tb\ for differently reconstructed particle candidates (tracks or clusters as defined in Figure \ref{fig:CurrentPz}) in three distinct polar regions $\theta$. The regions are chosen according to $\theta$-ranges, where different components of the H1 detector are relevant for particle reconstruction~\cite{H1:1996prr,H1:1996jzy,H1SPACALGroup:1996ziw,Pitzl:2000wz}. Right: The normalised contribution to \tb\ for different ranges of particle energies $E$ and for differently reconstructed particle candidates. The data are compared to the MC event simulations Djangoh and Rapgap. %Good agreement between the simulation and the data are observed. The relative contributions are obtained by calculating a weight $w=\sum_\text{contrib}\pzb/\sum_\text{all}\pzb$ for each event and each contribution. The resulting distributions are normalized to the \tb\ distribution. } \end{figure*} % \commenting{PMJ May 9}{This repeats what is said in L124-132 - are % both needed?} \bluetext{ % DB: Here we discuss the particle-level definition, where the % electron comes from the generated vertex % also, any radiated photon is exactly avaiable. % at detector-level, the electron is the highest energetic LAr cluster % and only high-energetic photons, and/or photons with sufficiently % large angle are identified. At the particle level (also denoted as \emph{generator} or \emph{hadron level}), the scattered electron is identified with the electron emitted at the DIS vertex. In the case of a photon radiated off the incoming or scattered lepton, the photon is recombined with the scattered electron if the angular distance is smaller than the angular distance to the $-z$-axis; otherwise it is removed from the event record. The HFS is formed from all remaining particles with proper lifetime $c\tau>10\,\text{mm}$. The DIS kinematic observables at the particle level are then computed using equations~\eqref{eq:DIS} and~\eqref{eq:DISeMethod}. % The triple-differential cross section is reported as a function of % $\tb$ in bins of $(\Qsq,y)$, and can be compared to fully % triple-differential predictions, $\tfrac{d^{3}\sigma}{dQ^2 dy d\tb}$, % through integration over the respective bin sizes in \Qsq and $y$, % \begin{equation} % \tfrac{d\sigma}{d\tb} (Q^2,y) = \int_{\Delta Q^2 \Delta y} \frac{d^{3}\sigma}{dQ^2 dy d\tb}. % \label{eq:xsectionbin} % \end{equation} The triple-differential cross section is reported as a function of \tb in bins of $(\Qsq,y)$, and is obtained as % % \begin{equation} % \sigma = \frac{N_\text{data} - N_\text{Bkg}}{\mathcal{L} \cdot % \Delta_\tau}\cdot c_\text{unfold}\cdot c_\text{QED}\,, % \end{equation} % \begin{equation} \frac{d\sigma_i}{d\tb} (Q^2,y) = \frac{\left(A^+(\vec{n}_\text{data} - \vec{n}_\text{Bkg})\right)_i} {\mathcal{L} \cdot \Delta_{\tb,i}}\cdot c_{\text{QED},i}\,, \end{equation} \noindent where $\vec{n}_\text{data}$ denotes a large data vector of event counts; $\vec{n}_\text{Bkg}$ is a corresponding vector with the estimated number of events from processes other than high-\Qsq inclusive NC DIS; $\Delta_{\tb,i}$ is the bin width of the $i$th bin of the respective \tb\ distribution; and $A^+$ denotes the regularized inverse of the detector response matrix. The parameter $c_\text{QED}$ is the QED correction factor, and $\mathcal{L}$ specifies the integrated luminosity of the measurement. For this analysis, $\mathcal{L}=351.1\,\text{pb}^{-1}$~\cite{H1:2012wor}, where the $e^+$ beam data contribute an integrated luminosity of 191.0\,pb$^{-1}$, and the $e^-$ beam data have an integrated luminosity of 160.1\,pb$^{-1}$. % The detector response matrix $A$ is determined using both signal MC event simulators. The TUnfold package~\cite{Schmitt:2012kp} is used to calculate the regularised inverse $A^+$ and to propagate its uncertainties. Finer binning is used at detector than at particle level; typically there are 12 bins in the detector-level histogram. The vector $\vec{n}$ has 4331 entries in the extended detector-level phase space, $0.0160\,\GeVsq$, % it's actually 60, but this would be very confusing here. %which are connected through the detector response matrix $A$ with and the vector $A^{+}\vec{n}$ has 572 entries at the particle level. The extended detector-level phase space in $y$ and \Qsq\ accounts for migrations into and out of the fiducial measurement phase space. In order to obtain the final 324 cross section points from the 572 values, two adjacent values are combined each, with exceptions made at lowest and largest \tb{}. This procedure stabilizes the unfolding and reduces dependencies on the MC models. % number of bins: % non empty in A: 324 + overflow % number of cross section points in cs.* 308 % missing points: 16 (from highest Q2) % plot: 7 * 5 % 8 * 8 % 11 * 19 % -> 308 % n panels: 5+15+17 = 32 The regularization procedure employs the so-called \emph{curvature} mode of TUnfold, which approximates second derivatives, and it acts on the differences of the unfolded results and the MC predictions as a bias distribution. The regularization is applied separately for each $\tb$ distribution in the individual ($Q^2,y$) intervals. The regularization parameter is set to a small value $\tau=10^{-3.2}$ as suggested by the analysis of Stein's Unbiased Risk Estimator, SURE~\cite{Stein:1981}. The regularization has a moderate impact on the result, and reduces large fluctuations and sizable negative correlations between adjacent data points in some regions of the analysis. The background vector $\vec{n}_\text{Bkg}$ consists of events from photoproduction, charged current DIS, di-lepton production, and DVCS processes. A small number of simulated events which are outside the enlarged phase space at particle level (e.g.\ $y<0.01$, or $Q^2<120\,\GeVsq$) but migrate into the enlarged detector-level phase space are treated as background, and are also included in $\vec{n}_\text{Bkg}$ (``\emph{acceptance correction}''). %% QED corrections The multiplicative factor $c_\text{QED}$ corrects the data for higher-order QED effects and for the positron charge. It is defined as the ratio of the cross section predicted without ($\sigma_\text{norad}^{e^-p}$) to the one with QED radiative effects ($\sigma_\text{rad}$). The factors are determined using the Djangoh event generator, where higher-order QED effects are implemented from Heracles~\cite{Kwiatkowski:1990es}. The factors are validated with Rapgap. The denominator $\sigma_\text{rad}$ includes radiative cross sections for electron--proton and positron--proton scattering with relative integrated luminosity weights as for real data. The numerator $\sigma_\text{norad}^{e^-p}$ is defined for non-radiative electron--proton scattering ($e^-p$). The resulting cross section measurements are therefore reported for $e^-p$ scattering.\,\footnote{At earlier HERA publications $e^+p$ cross sections were commonly reported.} % The effects of longitudinal polarization of the lepton beam are negligible, due to a nearly balanced mixture of running periods with opposite lepton beam helicities. % The QED correction $c_\text{QED}$ corrects for first-order real emission of photons off the lepton line, including QED Compton enhancement, photonic vertex correction, and purely photonic self-energy contributions at the external lepton lines. This defines the \emph{non-radiative cross section level} ($\sigma_\text{norad}^{e^-p}$), which is reported as the main result of the paper. %The triple-differential cross sections are reported for $e^-p$ %collisions, although the data include data taking periods with positrons %($e^+p$). %In this case, effects from first-order electroweak corrections are %also considered when determining the QED correction factors, since %those differ between $e^-p$ and $e^+p$ collisions. The data still include higher-order purely weak effects, self-energy corrections of the exchanged bosons, second order electroweak corrections, and photon-PDF induced contributions. %Since the leading order electroweak corrections are corrected for, %and when further assuming that emissions beyond the first order %correction are negligible, the DIS kinematic variables and \tb\ are %defined unambiguously. The QED correction factors $c_\text{QED}$ are largely independent of the kinematic variables $\tb$, \Qsq, and $y$ owing to the use of the I$\Sigma$ method and the treatment of radiated photons at the radiative particle level. Effectively, $c_\text{QED}$ corrects mainly for the cut on the longitudinal energy-momentum balance ($(E-P_z)_{\text{gen}}>45\,\GeV$), which has no effect at the non-radiative cross section level, but is active at the radiative particle-level. When the inverse of $c_\text{QED}$ is applied, the cross section at the radiative level discussed above is obtained (see also Ref.~\cite{Arratia:2022wny}), where the treatment of radiated photons is relevant. %% further QED correction factors To enable comparison with a range of theoretical calculations, additional correction factor are determined which optionally can be applied: % \begin{itemize} \item The correction factor $c_\text{NoZ}$ corrects further for all leading and subleading electroweak corrections, which are $\gamma Z$ and pure $Z$-exchange, and first order purely weak vertex, self-energy and box corrections. Hence, only leptonic and hadronic corrections to the photon self-energy remain ($\sigma_{\gamma-\text{only}}$). \item The correction factor $c_\text{Born}$ corrects further to the so-called \emph{DIS Born level} ($\sigma_\text{Born}^{e^-p}$), where first order purely weak vertex, self-energy and box corrections are corrected. In addition, corrections for leptonic and hadronic contributions to the photon and $Z$ self-energy are applied. \item The correction factor $c_{e^+p}$ corrects the reported $e^-p$ non-radiative cross sections to an $e^+p$ initial state ($\sigma_\text{norad}^{e^+p}$). \end{itemize} The QED correction factor and the optional correction factors can be summarized as: \begin{equation} \begin{array}{cc} c_\text{QED}=\frac{\sigma_\text{norad}^{e^-p}}{\sigma_\text{rad}} \,, & c_\text{NoZ}=\frac{\sigma_{\gamma-\text{only}}}{\sigma_\text{norad}^{e^-p}} \,, \\ c_\text{Born}=\frac{\sigma_\text{Born}^{e^-p}}{\sigma_\text{norad}^{e^-p}} \,, & c_{e^+p} =\frac{\sigma_\text{norad}^{e^+p}}{\sigma_\text{norad}^{e^-p}} \,. \end{array} \end{equation} %% 1D cross sections The single-differential cross section as a function of \tb\ in the NC DIS kinematic range, $200<\Qsq<1700\,\GeVsq$ and $0.23500\,\GeVsq$ ~\cite{Shushkevich:2012luo,H1:2012qti}. \item The energies of all clusters and tracks receive scale factors from a dedicated jet energy calibration~\cite{Kogler:2011zz,H1:2014cbm}. This calibration procedure results in two independent uncertainty contributions, whether clusters are contained in %high-$p_T$ jets or not. The two contributions are denoted as `jet energy scale uncertainty' (JES) and `remaining cluster energy scale uncertainty' (RCES), and both uncertainties are determined by varying the energy of the respective HFS objects by $\pm1\,\%$. %The JES affects high-energetic objects, %while the RCES affects objects with lower transverse momentum. As compared to the JES, the RCES typically affects objects with lower transverse momenta. % \item The polar-angle position of the LAr calorimeter with respect to the Central Tracking Detector (CTD) is aligned with a precision of 1\,mrad~\cite{H1:2012qti}. This uncertainty component is considered separately for the scattered electron and for the HFS objects. % %The photoproduction background is the most relevant background %contribution and is taken from simulation. An uncertainty of 50\,\% %is assigned on its normalisation. %The uncertainties of other background sources are negligible. % % \item % The vertex identification and electron track reconstruction are % closely related, and their efficiency % %the efficiency for the reconsruction of a track associated to the % %scattered electron, and for reconstructing the interaction vertex, % is studied with a control selection and using three different topological algorithms % to identify the nominal interaction vertex. % The difference between data and simulation is taken as an uncertainty of % 0.2\,\%~\cite{Shushkevich:2012luo}, which is considered to be % uncorrelated between data points. % % \item The unfolding procedure is associated with several uncertainties. Differences in the migration matrix $A$ when determined from Djangoh or Rapgap are denoted as `model' uncertainty. Half of the difference in each element $A_{ij}$ is propagated to the unfolded cross section. This applies also to events that migrate from outside the particle-level phase space into detector-level phase space, and to events that are generated in the particle-level phase space but are not reconstructed at detector-level. % %Events that migrate from outside the particle-level phase space into %detector-level phase space are estimated with either Djangoh or %Rapgap predictions. %Half of the difference is considered as an uncertainty. %Similarly, events that are generated in the particle-level phase %space, but are not %reconstructed at detector-level are corrected for. %Half of the difference between the predicted events with Djangoh and %Rapgap are considered as an uncertainty. % Statistical uncertainties in $A$ from the limited event sample of the simulations are also considered. The size of the regularization parameter has an uncertainty of 50\,\% and that variation is propagated to the resulting cross sections~\cite{Schmitt:2012kp}. The considered variation covers alternative procedures to determine the regularization strength, including the minimum global correlation coefficient or the $L$-curve scan~\cite{Schmitt:2012kp}. % \item Other uncertainty components are found to be negligible on their own, and a conservative bin-to-bin uncorrelated uncertainty of 0.5\,\% is introduced to cover the sum of these. An example of the uncertainties included here is the vertex and electron track reconstruction efficiency, which has an uncorrelated uncertainty of 0.2\,\%~\cite{Shushkevich:2012luo}, The distance between the calorimeter cluster of the scattered electron and a vertex-associated track is not described perfectly by the simulation, and the corresponding efficiency correction introduces an uncorrelated uncertainty of about 0.1 to 0.2\,\%. % vertex-track-cluster link efficiency (vtc) The uncertainty associated with the choice of the regularization strength in the unfolding procedure is found to be below 0.1\,\%. The contributions from other processes than high-\Qsq\ NC DIS (\emph{backgrounds}) are estimated from MC simulations and a normalization uncertainty of 100\,\% is considered, which results in an uncertainty smaller than 0.1\,\%. % \item The normalization uncertainty is found to be 2.7\,\%, and is dominated by the uncertainty on the integrated luminosity~\cite{H1:2012wor}. Other normalization uncertainties are negligible in comparison. For example, the efficiency of the trigger is found to be higher than 99.5\,\%~\cite{H1:2014cbm,H1:2012qti} and the related uncertainty is smaller than $0.5\%$. Further normalization uncertainties are related to the electron identification, the noise suppression algorithm in the LAr, and to the track and vertex identification. % \item The QED correction factors were determined separately with Djangoh and Rapgap and very good agreement was found. While both MC generators implement QED corrections from Heracles, previous studies showed that these are consistent with the program Hector~\cite{Arbuzov:1995id} and EPRC~\cite{Spiesberger:1995pr}, and that the contribution from the two-photon exchange, which is not implemented in Heracles, is negligible~\cite{H1:2012qti,H1:2018mkk}. Consequently, the uncertainty of the QED correction factors represent only their statistical component, while other uncertainties would be of negligible size, when compared to other uncertainty sources. % \item Uncertainties of the hadronization correction factors, relevant for a subset of the comparisons only, are determined as half of the difference between the correction factor obtained from Djangoh and Rapgap. \end{compactitem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Results %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Results} Results for the single-differential and triple-differential 1-jettiness cross sections as well as for the double-differential inclusive NC DIS cross section are presented. % --- 1D cross sections \paragraph{\boldmath Single-differential cross sections as a function of $\tau_1^b$} The single-differential 1-jettiness cross sections $\tfrac{d\sigma}{d\tb}$ in NC DIS $ep$ scattering in the kinematic range $200\leq\Qsq<1700$\,\GeVsq and $0.2\leq y<0.7$ are measured for $e^+p$ and $e^-p$ collisions in Tables~\ref{tab:xs1D.em} and~\ref{tab:xs1D.ep} and displayed in Figure~\ref{fig:1Da}. As no significant differences are observed between the two cross sections, the measurement is repeated using the sum of both datasets, corrected to $e^-p$ scattering cross sections, in Table~\ref{tab:xs1D.em351} and Figure~\ref{fig:1Db}. % The statistical uncertainty in the data is typically around 2 to 4\,\% and the systematic uncertainty is of order 4\,\%. Larger uncertainties are seen for the lowest \tb\ bin. % The differential \tb\ cross section exhibits a distinct peak at $\tb\sim0.13$ and a tail towards high values of \tb. % The distinct \emph{DIS peak} is populated by DIS Born-level kinematics with a single hard parton, the position and shape of which are dominated by hadronization and resummation effects. The tail region is populated by events with hard radiation, including two-jet topologies in the far tail. % The cross section at $\tb\simeq1$ has a sizeable value, as it includes events with empty current hemisphere in the Breit frame. % and is related to events with low $x$. This event configuration can occur at low \xbj\ and is studied in a dedicated publication~\cite{EHEletter}. % The data are compared to predictions from NNLOJET up to $\mathcal{O}(\as^3)$ in the strong coupling; resummed predictions at NLL accuracy matched to fixed-order predictions at $\mathcal{O}(\as^2)$ and corrected for hadronization; various predictions from the modern MC event generators Pythia~8.3, Sherpa~2.2, Sherpa~3.0, Herwig~7.2, and Powheg Box plus Pythia; as well as to the dedicated MC event generators for DIS, Djangoh, Rapgap, and KaTie+Cascade. % The agreement between the predictions and data are very similar for $e^+p$ and $e^-p$ cross sections. % Further details are discussed below. %% --- 3D cross sections \paragraph{\boldmath Triple-differential cross sections as a function of $\tau_1^b$, \Qsq, and $y$} Triple-differential cross sections are presented in Tables~\ref{tab:xstaub1b.em351.150_0.05}--~\ref{tab:xstaub1b.em351.8000_0.20} and are displayed in Figure~\ref{fig:xs3Dall}. The ratio of the data and predictions to the predictions from Sherpa~3.0 are displayed in Figures~\ref{fig:xs3DRatioNNLO}--\ref{fig:xs3DRatioCascade}. % ane \ref{fig:xs3DRatio2} and \ref{fig:xs3DRatio3}. The triple differential cross sections are presented in the kinematic range $150<\Qsq<20\,000\,\GeVsq$ and $0.050.1$. % However, the hadronization corrections are be sizeable with values % between 1.25 and 1.40, which prohibits quantitative % conclusions from the fixed order predictions alone. % % Improve the following text. \paragraph{Comparison to modern MC event generators} The recent MC event generators Pythia~8.3, Powheg+Pythia, Herwig~7.2, Sherpa~2.2 and Sherpa~3.0 employ LO, multi-leg or NLO matrix elements matched to parton shower and hadronization models. Those generators generally provide a good description of the data. %\begin{compactitem} % \item Pythia: % -- pythia The MC predictions from Pythia provide an overall reasonable description of the data, with small but visible differences between the three parton shower models studied. % The default Pythia and Pythia+Vincia predictions overestimate the data at low $\tb<0.1$, whereas Pythia+Dire provides a good description. All three Pythia predictions are similar in the parton-shower region ($0.1<\tb<0.4$), but tend to underestimate the data in the tail region ($\tb>0.5$). The latter could be related to the fact that matrix elements for $eg\to eq\bar{q}$ enter at $\mathcal{O}(\as)$ only, and are not recovered properly in a parton shower emerging from $eq\to eq$ alone. %The latter could point towards missing dijet matrix %elements in the generator. % {\TODO{The default parton shower requires a large value of $\asmz$ for the shower, which is default in Pythia.}} The default Pythia predictions are a leading-order MC model and they benefit from a large value of $\asmz$ for the parton shower, which is default in Pythia, and thus raises the prediction at larger \tb and brings them closer to the data. % %Altogether, the Dire parton shower provides the best description of the data. Looking only at the single-differential cross sections, Pythia+Dire tends to overshoot the data in the parton-shower region, but from the triple-differential data it is evident that the discrepancy arises from events at lower values of \Qsq{}. At very large values of $y$ ($y>0.7$), the Pythia+Dire predictions fail to describe the data at large values of \tb. % %Pythia with the Dire parton shower is similar to Rapgap at low $y$ but %overshoots the data at low \tb\ while underestimating the high \tb\ region. %has additionally a too high peak at low \tb, and underestimates the data at high \tb. % %The other parton-shower models interfaced to Pythia are similar to %Dire, and the main difference is at low \tb, where all models %overestimate the data at lower \Qsq. % The Pythia+Vincia prediction provides a good description of the data in the range $0.1\lesssim\tb\lesssim0.5$. It has difficulties at larger values of \tb, presumably related to the fact that Vincia was developed for a symmetric collider setup and is not yet fully validated for the asymmetric beams of $ep$ collisions. The predictions from Powheg Box are lower than the default Pythia predictions at medium to large \tb, and overshoot the data at low \tb. Larger values of \asmz\ in the simulation, or corrections beyond NLO DIS, such as NLO dijet matrix elements, may be required to improve the description of the data. % % -- Herwig % \item Herwig: The Herwig variants are able to provide a good description of the data. Although the default predictions from Herwig~7.2 provide the worst description among all modern MC generators, a significant improvement is obtained with NLO matrix elements. The merging technique provides the best description among the Herwig predictions, with benefits particularly at larger \tb. All Herwig predictions fail to describe the lowest \tb\ cross section ($\tb<0.05$). %The Herwig NLO predictions with parton shower matching provides the best %description at lowest \tb\ ($\tb<0.1$) among all models. % The default Herwig predictions exhibit a prominent structure at medium \tb. % % -- Sherpa % \item Sherpa: The Sherpa~2.2 predictions provide a good description of the data, in particular at larger \tb. The two different hadronization models (String vs.\ Cluster) provide very similar predictions, which do not differ by more than the data uncertainties, albeit the string model is a bit closer to the data at medium \tb. Both Sherpa~2.2 predictions overshoot the data at lowest \tb ($\tb<0.1$), and undershoot the data at $\tb\simeq1$. The underprediction at high \tb{} and overprediction at low \tb{} seems to be a common feature of NLO+PS models, c.f.\ Powheg Box and Herwig. The Sherpa~3.0 predictions with NLO matrix elements, improved parton shower and cluster hadronization model provide a better description of the data than those from Sherpa~2.2, and the predictions provide an accurate description of the data within the uncertainties. Only at lower \Qsq\ and values of $\tb\sim0.3$ the Sherpa~3.0 predictions overestimate the data. A noteworthy improvement over other MC predictions is observed in the lowest \tb\ region, where also good agreement with data is observed. This could be related to the modelling of an intrisic $k_t$ in Sherpa~3.0. % %\end{compactitem} \paragraph{Comparison to MC event generators from the HERA era and dedicated DIS models} The DIS MC event generators Djangoh and Rapgap provide an overall satisfactory description of the data, although both have difficulties to describe the shape in the region $\tb<0.3$ accurately and both models underestimate the data in the range $0.1<\tb<0.3$. The high \tb region is well described. Djangoh is somewhat higher than Rapgap in that region, which is consistent with the observation of a harder $P_{\text{T}}$-spectrum of jets~\cite{H1:2016goa}. The triple-differential cross sections are well described by Djangoh and Rapgap. At low $y$, Rapgap underestimates the high \tb region more than Djangoh. Both models fail to describe the region $\tb<0.05$. %\paragraph{Dedicated MC event generators} % -- Cascade The KaTie+Cascade~3 predictions, employing TMD-sensitive PDF sets with two different choices for the QCD evolution scale, provide a reasonable description of the data. TMD PDF Set~2 provides one of the best descriptions at the low \tb region, whereas the predictions with TMD PDF Set~1 are a bit higher. However, at lowest \Qsq\ and large $y$, also predictions from Set~2 overshoot the data significantly. In the region of larger \tb, the KaTie+Cascade~3 predictions fail to describe the data, %understandably, since %gluon-induced processes are not included in those predictions. likely related to the absence of $eg\to eq\bar(q)$ processes in the hard matrix elements. % %The Cascade predictions show an excellent agreement with the data in %the central phase space region, $\Qsq\sim 800\,\GeVsq$ and $y\sim0.3$. %While the data are well described in the peak region, the tail region is underestimated. % %% --- inlusive DIS \paragraph{\boldmath Double-differential cross sections for inclusive neutral-current DIS as a function of \Qsq\ and $y$} The inclusive NC DIS cross sections $\tfrac{d^{2}\sigma_\text{NC DIS}}{d\Qsq dy}$ are presented in Tables~\ref{tab:NCDIS2D.em} and~\ref{tab:NCDIS2D.ep} and are displayed in Figure~\ref{fig:NCDIS2D}, where NNLO and aN3LO predictions for inclusive DIS are compared with the data. For comparisons with the NC DIS predictions, the data are corrected to the DIS Born level by applying the factor $c_{\text Born}$. These cross sections are measured for $e^+p$ and $e^-p$ scattering by analyzing the data from the two lepton charges separately, including dedicated unfolding matrices and QED correction factors. The inclusive NC DIS cross sections $\tfrac{d^{2}\sigma_\text{NC DIS}}{d\Qsq dy}$ are obtained by integrating over \tb\ in every $(\Qsq,y)$ range. Hence, kinematic migrations due to limited resolution of the detector are corrected with high accuracy, since different configurations of the hadronic final state, represented by \tb, are considered in the unfolding. The inclusive NC DIS cross sections can be compared with predictions from structure function calculations, and serve as an important validation of the absolute size of the \tb\ cross sections. These data constitute the first bin integrated inclusive NC DIS cross section measurements at HERA in \Qsq\ and $y$, employing proper regularized unfolding of detector effects. The results can be compared with predictions based on structure functions and serve as a crosscheck in the determination of the absolute normalization of the triple-differential cross section measurements. The NC DIS measurements are also used together with the triple-differential cross sections to obtain normalized 1-jettiness event shape distributions~\cite{h1pubs}. % %Such a normalisation was performed for event %shape measurements in the past, and although several uncertainties %cancel (partially), such a normalisation will not add additional %information. % % {\color{red} comment on NC DIS cross sections: i) sensitive to PDFs, % ii) relation to previous publications, and iii) important cross % check and iv) the respective numbers for normalised \tb cross % sections.} \paragraph{Predictions for inclusive NC DIS cross sections} %% --- inlusive DIS The double-differential inclusive DIS $\tfrac{d^{2}\sigma_\text{NC DIS}}{d\Qsq dy}$ are compared to predictions in NNLO and a3NLO accuracy. It is observed that both the NNLO and the a3NLO predictions provide a very good description of the $e^+p$ and $e^-p$ data in the entire kinematic range, and small discrepancies are observed only at lowest values of $y$ or largest values of \Qsq. One can note that the H1 data~\cite{H1:2012qti,H1:2015ubc} are already included in the PDF determination, which is the basis of the predictions. Differences between the H1PDF2017{\scriptsize NNLO} PDF set and the NNPDF3.1 PDF set are small. % {\color{blue}Add discussion on PDF and scale uncertainties } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Summary and Conclusion %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Summary} The first measurement of the 1-jettiness event shape observable in neutral-current deep-inelastic electron--proton scattering is presented. The measured 1-jettiness observable \tb\ is equivalent to the classical event shape observable Thrust normalized with $\frac{Q}{2}$, \tQ. It quantities the degree to which the hadronic final state in the current hemisphere is collimated along the exchanged bosons four-vector. The data were recorded with the H1 experiment at the HERA collider operating with a center-of-mass energy of $\sqrt{s}=319\,\GeV$. A single-differential cross section is measured as a function of \tb\ in the kinematic region $200\leq\Qsq<1700\,\GeVsq$ and $0.2\leq y<0.7$. A triple-differential cross section as a function of \tb, \Qsq, and $y$ is reported in the kinematic range $0\leq\tb\leq1$, $150\leq\Qsq<20\,000\,\GeVsq$, and $0.05\leq y<0.94$. The data are unfolded to the particle level and corrected for higher-order QED radiative effects. % {\TODO{ Add the correct numbers for the uncertainties below}} Given typical accuracies of $4$--$20\,\%$ (single differential) and $6$--$30\,\%$ (triple-differentials), the data exhibit high sensitivity to the modelling of the hard interaction, to the parton shower and hadronization models in MC event generators. The unfolded data are compared to a variety of predictions. Most of the models studied provide a satisfactory description of the data. The classical MC event generators Djangoh and Rapgap, which were optimised for HERA physics, provide an overall good description. The more modern general purpose MC event generators Herwig~7.2, Pythia~8.3, Sherpa~2.2 and Sherpa~3.0 provide a good description of the data, although some problems remain to be resolved in regions of the phase-space specific for each model. When studying different parton shower or hadronization models, the potential of these data for the optimization of event generators is demonstrated. Fixed order pQCD predictions provide a good description of the data, but sizeable non-perturbative corrections for hadronization effects have to be applied. When incorporating NLL resummation, such predictions are seen to be in good agreement with the data over the full range of \tb{}. Future pQCD analyses may be able to constrain parton distribution functions of the proton (PDFs) or the strong coupling constant from the data. The \tb-integrated triple-differential cross sections are found to be in agreement with the previously measured double-differential inclusive NC DIS cross section. This provides an important consistency test of the data, but also a consistency check with the HERA legacy measurements of inclusive NC DIS. %From those double-differential DIS cross sections, normalised %\tb\ cross sections could be obtained. The inclusive NC DIS cross section is described by structure function calculations which are available up to aN3LO and which are free of hadronization effects. The 1-jettiness measurement as a function of \tb, \Qsq, and $y$ can be viewed as a generalised NC DIS cross section, and is the first triple-differential measurement of this kind. %NC DIS cross sections. In summary, it is believed that these data will be highly valuable to improve MC event generators, which are of importance to achieve the physics goals of the HL-LHC physics program~\cite{Dainese:2019rgk,ZurbanoFernandez:2020cco}. Further improvements can be achieved when complemented with recent jet substructure measurements~\cite{H1:2023fzk}. With an improved understanding of soft and non-perturbative effects, these data will become useful for the determination of PDFs or the value of the strong coupling constant $\asmz$. The presented measurement at HERA can be complemented in the future with measurements in electron--proton collisions at lower center-of-mass energies at the electron--ion collider in Brookhaven (EIC)~\cite{AbdulKhalek:2021gbh} or at higher energies at the LHeC or FCC-eh at CERN~\cite{FCC:2018byv,LHeC:2020van,AbdulKhalek:2021gbh}, or with measurements in $pp$ at the LHC. \section*{Acknowledgements} We are grateful to the HERA machine group whose outstanding efforts have made this experiment possible. We thank the engineers and technicians for their work in constructing and maintaining the H1 detector, our funding agencies for financial support, the DESY technical staff for continual assistance and the DESY directorate for support and for the hospitality which they extend to the non-DESY members of the collaboration. We express our thanks to all those involved in securing not only the H1 data but also the software and working environment for long term use, allowing the unique H1 data set to continue to be explored. The transfer from experiment specific to central resources with long term support, including both storage and batch systems, has also been crucial to this enterprise. We therefore also acknowledge the role played by DESY-IT and all people involved during this transition and their future role in the years to come. We would like to thank Valerio Bertone, Silvia Ferrario Ravasio, %Thomas Gehrmann, Ilkka Helenius, Am{\'e}lie Henke, Alexander Huss, Hannes Jung, Alexander Karlberg, Christopher Lee, Simon Plätzer, Christian Preuss, and Hubert Spiesberger for many valuable comments and discussions, for providing us with theoretical predictions, or for help with the predictions. We thank Iain Stewart for motivating this analysis in the `Future Physics with HERA data' workshop 2014~\cite{Stewart:2014}. \par $^{f1}$ supported by the U.S. DOE Office of Science \par $^{f2}$ supported by FNRS-FWO-Vlaanderen, IISN-IIKW and IWT and by Interuniversity Attraction Poles Programme, Belgian Science Policy \par $^{f3}$ supported by the UK Science and Technology Facilities Council, and formerly by the UK Particle Physics and Astronomy Research Council \par $^{f4}$ supported by the Romanian National Authority for Scientific Research under the contract PN 09370101 \par $^{f5}$ supported by the Bundesministerium für Bildung und Forschung, FRG, under contract numbers 05H09GUF, 05H09VHC, 05H09VHF, 05H16PEA \par $^{f6}$ partially supported by Polish Ministry of Science and Higher Education, grant DPN/N168/DESY/2009 \par $^{f7}$ partially supported by Ministry of Science of Montenegro, no. 05-1/3-3352 \par $^{f8}$ supported by the Ministry of Education of the Czech Republic under the project INGO-LG14033 \par $^{f9}$ supported by CONACYT, México, grant 48778-F \par $^{f10}$ supported by the Swiss National Science Foundation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % bib %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibliography{desy24-035} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\section*{Figures} % ------------------------------------------------------------- % cross section plots -- 1D % ------------------------------------------------------------- \begin{figure*}[t] \centering \includegraphics[width=0.495\textwidth,trim={20 0 30 0 },clip]{d24-035f5a} \includegraphics[width=0.495\textwidth,trim={20 0 30 0 },clip]{d24-035f5b} \caption{ The differential cross section $d\sigma/d\tb$ in the kinematic region $200\leq\Qsq/\,\GeVsq<1700$ and $0.2\leq y<0.7$ for $e^+p$ (left) and $e^-p$ scattering at $\sqrt{s}=319\,\GeV$. The data are corrected for detector effects (acceptance, resolution) and QED radiative effects. The statistical uncertainties are displayed as vertical error bars, systematic uncertainties as shaded areas. The data are compared to the MC predictions from NNLOJET, NNLO$\otimes$NLL$^{\prime}\otimes$Had, Pythia~8.3, Powheg Box, Herwig~7.2, Sherpa~2.2, Sherpa~3.0, Djangoh, Rapgap, and KaTie+Cascade~3. Selected parameters are varied in the predictions as indicated in brackets. Colored areas indicate theoretical uncertainties associated with some of the predictions (see text). The lower panel displays the ratio of predictions to data. } \label{fig:1Da} \end{figure*} \begin{figure*}[t] \centering \includegraphics[width=0.65\textwidth,trim={0 0 0 0 },clip]{d24-035f6} \caption{ The differential cross section $d\sigma/d\tb$ in the kinematic region $200\leq\Qsq/\,\GeVsq<1700$ and $0.2\leq y<0.7$ for $e^-p$ scattering at $\sqrt{s}=319\,\GeV$. The data are recorded with electron and positron beams, and corrected to $e^-p$ cross sections using Heracles. Further details are given in the caption of Figure~\ref{fig:1Da}. } \label{fig:1Db} \end{figure*} % ------------------------------------------------------------- % cross section plots -- 3D % ------------------------------------------------------------- \begin{figure*}[t] \centering \includegraphics[width=0.92\textwidth,trim={0 0 0 0 },clip]{d24-035f7} \caption{ The triple-differential cross section $\tfrac{d^3\sigma}{d\tb d\Qsq dy}$, presented as $\tfrac{d\sigma}{d\tb}$ in regions of \Qsq\ and $y$. The \Qsq\ and $y$ ranges are displayed on the left and top, respectively. Each panel displays the differential cross section $d\sigma/d\tb$ in that phase space. The data are displayed as full circles, the vertical error bars indicate statistical uncertainties, while systematic uncertainties are displayed as shaded area. The data are compared to predictions from NNLOJET, NNLO$\otimes$NLL$^{\prime}\otimes$Had, Pythia~8.3, Powheg Box, Herwig~7.2, Sherpa~2.2, Sherpa~3.0, Djangoh, Rapgap, and KaTie+Cascade~3. Selected parameters are varied in the predictions as indicated. The ratios of the data and predictions to the Sherpa~3.0 predictions are displayed in Figures~\ref{fig:xs3DRatioNNLO}--\ref{fig:xs3DRatioCascade}. At highest \Qsq ($8000<\Qsq/\GeVsq<20\,000\,$) only a single $y$ interval is measured, $0.2