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\begin{document} \received{March 2024} \printhistory \motto{}{Published in EPJC Letters \hfill DESY-24-034} \title{Observation and differential cross section measurement of neutral current DIS events with an empty hemisphere in the Breit frame} \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}} \abstract{\unboldmath% The Breit frame provides a natural frame to analyze lepton--proton scattering events. In this reference frame, the parton model hard interactions between a quark and an exchanged boson defines the coordinate system such that the struck quark is back-scattered along the virtual photon momentum direction. In Quantum Chromodynamics (QCD), higher order perturbative or non-perturbative effects can change this picture drastically. As Bjorken-$x$ decreases below one half, a rather peculiar event signature is predicted with increasing probability, where no radiation is present in one of the two Breit-frame hemispheres and all emissions are to be found in the other hemisphere. At higher orders in $\alpha_s$ or in the presence of soft QCD effects, predictions of the rate of these events are far from trivial, and that motivates measurements with real data. We report on the first observation of the empty current hemisphere events in electron--proton collisions at the HERA collider using data recorded with the H1 detector at a center-of-mass energy of 319\,GeV. The fraction of inclusive neutral-current DIS events with an empty hemisphere is found to be $0.0112 \pm 3.9\,\%_\text{stat} \pm 4.5\,\%_\text{syst} \pm 1.6\,\%_\text{mod}$ in the selected kinematic region of $150<\Qsq<1500\,\GeVsq$ and inelasticity $0.14\xbj$ entering the hard interaction, then producing a massive dijet system. The larger $x$ than \xbj, where the latter defines the Breit frame, the more is the the dijet system boosted into the beam hemisphere of the Breit frame, possibly leaving the current hemisphere empty. % The probability of this event configuration increases as \xbj is decreasing below a value of $\frac{1}{2}$. % When considering even higher order corrections in pQCD with massless partons, the number of empty hemisphere events will successively decrease with the order in $\alpha_s$, since the probability of (infinitly soft) radiation into the current hemisphere rises. On the other hand, QCD confinement demands that the massless partons fragment into a finite number of massive particles, a process which is difficult to predict in great detail. Consequently, an accurate prediction of empty hemisphere events in higher order pQCD or at the particle level is considerably less trivial than in the two parton picture.\\ \indent In this letter we present the first measurement of the production rate and differential distributions of events with an empty current hemisphere in the Breit frame using data recorded in electron--proton and positron--proton collisions. % --------------------------------------------------------------------- % METHODOLOGY % --------------------------------------------------------------------- \section{Experimental method} %% -------------------------------------------- %% %% experimental setup %% -------------------------------------------- %% Data were recorded with the H1 detector~\cite{H1:1993ids,H1CalorimeterGroup:1993boq,H1:1996prr,H1:1996jzy,H1SPACALGroup:1996ziw,Pitzl:2000wz} at the HERA collider at DESY in the years 2003 to 2007, where electron or positron beams with energy of 27.6\,GeV collided with a proton beam of $E_p=920\,\GeV$, resulting in a center-of-mass energy of $\sqrt{s}=319\,\GeV$. The analyzed data sample corresponds to an integrated luminosity of 351.1\,pb$^{-1}$~\cite{H1:2012wor}, of which about one half was taken with an electron beam, and the other half with a positron beam. The H1 experiment consists of a series of subdetectors positioned around the $ep$ interaction point, including tracking detectors, calorimeters and the muon system. The central tracking system covers the polar angular range $15^\circ<\theta<165^\circ$ and consists of silicon vertex detectors, drift and proportional chambers. Charged particles are measured with a transverse momentum resolution of $\sigma_{p_\text{T}}/p_\text{T}=0.2\%p_\text{T}/\GeV\oplus1.5\%$. Two main calorimeters are used: a liquid argon calorimeter (LAr) and a lead-scintillating fiber calorimeter (Spacal). The LAr consists of electromagnetic sections made of lead absorbers and hadronic sections with steel absorbers. It covers the range $4^\circ<\theta<154^\circ$ and full azimuthal angle. Its 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 Spacal consists of an electromagnetic and hadronic section and records the energy deposited in the backward direction in the range $153^\circ<\theta<177.5^\circ$. In this analysis, it is used to measure the hadronic energy flow in the region. The tracking detectors and calorimeters are surrounded by a superconducting solenoid providing a magnetic field of 1.16\,T. The return yoke is equipped with a muon system. %% -------------------------------------------- %% %% data analysis and MC models %% -------------------------------------------- %% The analysis strategy follows closely event selection and methodologies used previously~\cite{H1:2012qti,H1:2014cbm}. The trigger requires an energy deposit in the electromagnetic section of the LAr. This trigger requirement limits the measurements to the kinematic region of $\Qsq\gtrsim150\,\GeVsq$ and inelasticity $y<0.7$. The scattered lepton candidate is identified by requiring that the electromagnetic cluster with the highest momentum is matched to a track and its energy exceeds $E_{e^\prime}>11\,\GeV$. A set of fiducial cuts and quality requirements~\cite{H1:2012qti,H1:2014cbm} suppress non-collision backgrounds and backgrounds from processes other than neutral-current DIS such as photoproduction, charged-current DIS, or QED Compton. The electromagnetic energy scale is calibrated in situ using the double-angle reconstruction method to predict the electron energy. The hadronic energy scale is calibrated by exploiting the transverse momentum balance of the lepton and hadronic final state using a dedicated selection of NC DIS events~\cite{Kogler:2011zz}. An isolated electromagnetic energy deposit without a track pointing to it is classified as a radiated photon from the lepton vertex. It is combined with the lepton candidate if the angular distance is closer to the lepton than to the beam axis, or it is removed from the list of reconstructed particle candidates. After removing tracks and clusters that are associated with the scattered lepton candidate~\cite{H1:2003xoe}, a particle-flow algorithm~\cite{energyflowthesis,energyflowthesis2,energyflowthesis3} is applied to combine information from the calorimeter and tracking systems and define the objects of the hadronic final state (HFS). The total longitudinal energy-momentum balance of all recorded particle candidates ($\Sigma_\text{tot}=\sum_i(E_i-P_{z,i})$) is required to be in the range $45<\Sigma_\text{tot}<62\,\GeV$, which suppresses events with hard initial state QED radiation and contributions from photoproduction. This ensures an optimal resolution is achieved of the DIS kinematic observables for the selected events. The kinematic quantities of the DIS process, i.e.~the inelasticity $y$, the virtuality of the exchanged boson \Qsq, and the Bjorken variable \xbj %=\frac{\Qsq}{2q\cdot P}$ are reconstructed using the I$\Sigma$ reconstruction method~\cite{Bassler:1994uq} as \begin{equation} y = \frac{\Sigma_h}{\Sigma_\text{tot}}~,~~~~~ \Qsq = \frac{P_{\text{T},e^\prime}^2}{1-y} ~,~~~~~ \xbj E_p = \frac{Q^2}{2\Sigma_h} \,,\nonumber \end{equation} where $\Sigma_h$ is the longitudinal energy momentum balance of the HFS, expressed as $\sum_{i\in\text{HFS}}(E_i-P_{z,i})$. The event selection is restricted to the region $\Qsq>150\,\GeVsq$ and $0.14 10\,$mm, whereas particles with lower lifetime are included by their decay products. % The detector effects on the simulated particle-level events are processed with a detailed simulation of the H1 detector, which is based on Geant~3~\cite{Brun:1987ma} and fast shower simulation programs~\cite{Fesefeldt:1985yw,Grindhammer:1989zg,Gayler:1991cr,Kuhlen:1992ey,Grindhammer:1993kw,Glazov:2010zza}. The simulated data are processed with the same offline analysis chain as the real data~\cite{Britzger:2021xcx}. The reconstructed kinematic observables $\xbj$, $y$ and \Qsq\ are compared with MC predictions in Fig.~\ref{fig:controlplots} for all NC events in the selected phase space and for events with a reconstructed empty current hemisphere. Good agreement between the simulations and data are observed. % \begin{figure}[htb!] % \includegraphics[width=0.99\linewidth,trim={10 0 30 0 },clip]{d24-034f3a.pdf} \includegraphics[width=0.99\linewidth,trim={10 0 30 0 },clip]{d24-034f3b} \\ \includegraphics[width=0.99\linewidth,trim={10 0 30 0 },clip]{d24-034f3c}\\ % \caption{ Event distributions as functions of \xbj\,(top), $y$\,(middle), and \Qsq\,(bottom) at the detector level in comparison with the simulated MC event samples from Djangoh and Rapgap that further include background events from photoproduction and lower \Qsq. The left panels display events after selection of all NC DIS events, and the right panels display events with an empty current hemisphere in the Breit frame. } \label{fig:controlplots} \end{figure} It is instructive to study the number of jets (the \emph{jet multiplicity}) in events with an empty hemisphere, since these events are expected to be predominantly present at $\mathcal{O}(\alpha_s)$. To this end, jets are defined in the Breit frame from all HFS objects using the $k_t$ jet algorithm~\cite{Catani:1993hr} with a distance measure of $R=1.0$ and using the $P_t$-recombination scheme. Jets are required to have a transverse momentum $\pt$ greater than $7\,\GeV$ in the Breit frame and to fall within the acceptance of the LAr calorimeter, i.e. the polar angular range $9.4^\circ<\theta_\text{lab}^\text{jet}<154^\circ$, when boosted to the laboratory rest frame. The contributions from proton remnants or initial state radiation are highly suppressed due to limited detector acceptance. % The jet multiplicity at the detector level is displayed in Fig.~\ref{fig:njet}. \begin{figure}[htb!] \includegraphics[width=0.62\linewidth]{d24-034f4%% } \caption{ The number of jets in events with an empty current hemisphere. The jets are defined in the Breit frame and must exceed a transverse momentum greater than 7\,\GeV in the Breit frame. Further details are given in the caption of Fig.~\ref{fig:controlplots}. } \label{fig:njet} \end{figure} It is observed that the jet multiplicity distribution sharply peaks at two jets and no significant number of events without jets are observed. A few events have only one jet, which is attributed to the requirement of $P_\text{T}>7\,\GeV$, as a second jet can also be with a lower transverse momentum. %% -------------------------------------------- %% %% unfolding %% -------------------------------------------- %% The data are corrected for acceptance and resolution effects using a regularized matrix inversion algorithm as implemented in the TUnfold package~\cite{Blobel:1984ku,Schmitt:2012kp}. For each of the three observables, \xbj, $y$, and \Qsq, a detector response matrix is constructed from the average of the Djangoh and Rapgap simulations. The matrix contains information on migration probabilities between detector- and particle-level quantities as well as acceptance and reconstruction efficiencies. In order to determine the fraction of empty hemisphere events, two distributions have to be measured: the unfolded event count of empty hemisphere events and the number of events of the inclusive DIS sample. % The response matrix therefore has a $2\times 2$ structure, differentiating events with or without an empty current hemisphere on detector or particle level. % The two measured distributions are measured and unfolded simultaneously and the number of DIS events is obtained at the particle level by adding up the two respective entries. % This procedure accounts for the migrations between events with an empty current hemisphere and the ones with activities in both hemispheres. % Additional bins at the detector level account for migrations from outside the DIS phase space in $\Qsq$ and $y$. % A larger number of bins than the nominal measurement bins are unfolded and the final cross sections are obtained by combining them back into nominal bins. % These extra bins effectively divide events into smaller groups where radiation is close to the boundary of the two hemispheres. Such topologies are very sensitive to details of the QCD models and thus have to be resolved in order to minimize model uncertainties. % A very small fraction of events identified as background is subtracted from data prior to unfolding. The contributions from photoproduction processes were estimated using Pythia~6 and the ones from the charged-current and low-\Qsq neutral current DIS processes using Djangoh~1.4. The regularization parameter was set to $\tau=10^{-2.8}$. It has a negligible impact on the unfolded results. The fraction of empty hemisphere events in the inclusive DIS cross section, $r$, is then defined as \begin{equation} r := \frac{n_{(E_\mathcal{C}=0)}} {n_{(E_\mathcal{C}>0)} + n_{(E_\mathcal{C}=0)} } \cdot c_\text{QED} = \frac{\sigma_{(E_\mathcal{C}=0)}} {\sigma_\text{(NC DIS)}} \,, \end{equation} where $n_i$ are the counts of events with an empty hemisphere after unfolding ($E_\mathcal{C}=0$) and the ones with activity in the current hemisphere ($E_\mathcal{C}>0$). The fraction $r$ then represents the ratio of the production cross sections of empty hemisphere events to the inclusive neutral-current DIS cross section $\sigma_\text{(NC DIS)}$. The factor $c_\text{QED}$ accounts for higher order QED virtual and real corrections at the lepton vertex and QED Compton processes~\cite{H1:1999qgy,H1:2018mkk}, and also corrects for initial lepton charge effects to an $e^-p$ initial state. The factor $c_\text{QED}$ is determined with Heracles as implemented in Djangoh 1.4. It is very close to unity, since the effects of QED corrections as well as electroweak effects cancel in the ratio to a large extent. To estimate systematic uncertainties, each source of uncertainties is varied independently in simulation and the difference between the nominal results and the ones obtained from the new unfolding matrix is taken as uncertainties attributed to the source, propagated linearly to the results \cite{Schmitt:2012kp}. Two sources of uncertainties on the energy scale of HFS objects, associated with the components contained within high \pt{} jets and the ones that are not, are treated separately by independently varying the energy of corresponding HFS objects by $\pm1\,\%$~\cite{H1:2014cbm}. The angular resolutions of the azimuthal angle of the HFS objects and scattered leptons, $\pm20\,$mrad and $\pm1\,$mrad~\cite{H1:2012qti}, respectively, are taken as sources of systematic uncertainties. The energy resolution of scattered leptons which varies from $\pm0.5\,\%$ in the backward and central regions to $\pm1\,\%$ in the forward regions is also taken into account. The uncertainty attributed to the modeling of the HFS in the event generator used for unfolding is evaluated by comparing the results when either Djangoh or Rapgap is used to determine the migration matrix. This \emph{model} uncertainty accounts for acceptance, efficiency, and migration effects due to differences in the modelling of the HFS in the MC event generators used. As the final cross sections are reported normalized to the inclusive NC DIS cross section, normalization uncertainties such as luminosity scale or trigger efficiency cancel, and several other uncertainties including those related to the reconstruction of the scattered lepton cancel to a large extent. For the determination of the uncertainties on $r$ after unfolding, systematic uncertainties are considered to be fully correlated across different bins, and statistical uncertainties have a non-zero correlation coefficient after unfolding. %% -------------------------------------------- %% %% predictions and hadronization uncertainties %% -------------------------------------------- %% The resulting cross section ratios are compared with predictions from Pythia~8.307~\cite{Sjostrand:2014zea,Bierlich:2022pfr,Pythia83}, which implements DIS matrix elements at leading order and various parton-shower models for higher-order emissions. Two different models are studied: The default `simple' dipole-like $p_\perp$-ordered shower, and the Dire parton shower~\cite{Hoche:2015sya,Hoche:2017iem,Hoche:2017hno}, which is a variant of a dipole shower model. % with additional treatment of collinear enhancements. Both models use the Pythia~8.3 default Lund string model for hadronization~\cite{Pythia83}. Predictions from Powheg Box in NLO pQCD are matched to parton shower and hadronization from Pythia 8.308~\cite{Banfi:2023mhz}. These predictions are referred to as Powheg+Pythia. Furthermore, predictions from the Sherpa~2.2~\cite{Sherpa:2019gpd,Gleisberg:2008ta} and Sherpa~3.0~\cite{Sherpa3} MC event generators are studied. The Sherpa~2.2 predictions employ multi-leg tree-level matrix elements~\cite{Duhr:2006iq} with up to three jets combined with dipole showers~\cite{Catani:2001cc,Schumann:2007mg} and use a cluster hadronization model~\cite{Winter:2003tt}. The Sherpa~3.0 predictions employ NLO QCD matrix elements from Openloops~\cite{Buccioni:2019sur}, combined with a dipole shower based on the truncated shower method~\cite{Schumann:2007mg,Hoeche:2009rj,Hoeche:2012yf}. Two hadronization models are implemented and studied: an improved cluster hadronization model (Cluster)~\cite{Chahal:2022rid} and the string fragmentation model of Pythia~8.3 (Lund)~\cite{Bierlich:2022pfr}.%~\cite{Sjostrand:2006za} % The Vincia and Dire parton shower use a value of 0.118 for the % strong coupling at the mass of the $Z$ boson. % -------------------------------------------------------- % % results plot % -------------------------------------------------------- % \section{Results} The fraction of NC DIS events with no particle candidates in the current hemisphere in the Breit frame is determined at an $ep$ center-of-mass energy of $\sqrt{s}=319\,\GeV$ in the kinematic region of $150<\Qsq<1500\,\GeVsq$ and $0.140$, while the target/fragmentation hemisphere has $p^\text{b}_z<0$. Interestingly, the four-vectors in the Breit frame can be expressed without evaluating the Bjorken scaling variable \xbj, c.f.~Eq.~\eqref{eq:b}. % $\xbj=\frac{-q\cdot q}{2q\cdot P}$ \end{document}