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(1)Faculty of Physics and Applied Computer Science. Doctoral thesis. Marcin Guzik. Measurement of diffractive dijets photoproduction with the ZEUS detector at HERA. Supervisors: prof. dr hab. inż. Mariusz Przybycień dr inż. Leszek Adamczyk. Cracow, 2015.

(2) ii. Declaration of the author of this dissertation: Aware of legal responsibility for making untrue statements I hereby declare that I have written this dissertation myself and all the contents of the dissertation have been obtained by legal means. data, podpis autora. Declaration of the thesis Supervisor: This dissertation is ready to be reviewed. data, podpis promotora rozprawy.

(3) iii. Dedicated to my son Adam and his mother Agnieszka, for her kindness and endless support..

(4) iv. Streszczenie. W rozprawie tej przedstawiona jest analiza danych zebranych przez eksperyment ZEUS w latach 2003 - 2007 czyli w okresie po modernizacji aparatu pomiarowego znanym jako HERA II. Dane te pochodzą ze zderzeń elektronów (lub pozytonów) o energii 27.5 GeV z protonami o √ energii 920 GeV dając energię zderzenia w układzie środka masy s = 318 GeV. Ich całkowita świetlność wynosi około 372 pb−1 . Dane te zostały zbadane pod kątem pomiaru przekroju czynnego na dyfrakcyjną fotoprodukcję dżetów. Obszar kinematyczny analizy przedstawionej w tej rozprawie to: Q2 < 1. GeV2 , 0.2 < y < 0.85, ETjet1 > 7.5 GeV, ETjet2 > 6.5 GeV, |η jet1 | < 1.5, |η jet2 | < 1.5, xIP < 0.025, |t| < 1 GeV2 , gdzie dżety były zrekonstruoawne używając inkluzywnego algorytmu kT . Jest to obszar dyfrakcyjnej fotoprodukcji jetów, który jest szczególnie ciekawy gdyż pozwala na badanie łamania faktoryzacji w oddziaływaniach hadronowych. Faktoryzacja w dyfrakcji w przypadku zderzeń ep okazała się być w granicach niepewności spełniona gdy oddziaływanie może zostać potraktowane jako „twarde” to znaczy zachodzi w obecności dużej skali pozwalającej na zastosowanie rachunku perturbacyjnego. Jednakowoż inne eksperymenty np. Tevatron zaobserwowały łamanie dyfrakcyjenej faktoryzacji w zderzeniach p¯ p gdzie dyfrakcyjne PDFy otrzymane na podstawie danych z HERA przeszacowywały przekrój czynny niemal o rząd wielkości. Zjawisko to zostało później wytłumaczone jako spowodowane poprzez efekty absorbcyjne występujące podczas wielokrotnego rozpraszania i prowadzące do powstania nowych cząstek, które to wtórnie zapełniają przerwę w rapidity ukrywając eksperymentalny obraz dyfrakcji. W przypadku fotoprodukcji wymieniany foton może oddziaływać bezpośrednio tak jak w przypadku głęboko nieelastycznego rozpraszania ale ponieważ Q2 ≈ 0 może też dzięki zasadzie Heisenberg’a fluktuować do stanu q q¯ o relatywnie długim czasie życia. W tym drugim przypadku foton wchodzi w oddziaływanie w taki sposób jak hadrony to znaczy dostarczając tylko część swojego czteropędu do reakcji i służąc jako źródło partonów, z których tylko jeden bierze udział podczas rozpraszania. Zjawisko to, znane jako resolved PHP, pozwala na testowanie łamania faktoryzacji w reakcjach typu hadronowego używając zderzeń ep. Łamanie faktoryzacji powinno prowadzić do tłumienia przekroju czynnego w regionie resolved PHP w porównaniu do regionu oddziaływania bezpośredniego fotonu, znanego jako direct PHP, co może być zaobserwowane dzięki pomiarowi różniczkowego przekroju czynnego w funkcji zmiennej xγ , która rozdziela oba regiony umowną granicą xγ = 0.75, powyżej której oddziaływanie jest eksperymentalnie uznawane za bezpośrednie. Pomiary takie zostały dokonane wcześniej przez eksperymenty H1 i ZEUS. W obu przypadkach zmierzone przekroje czynne były mniejsze od przewidywań teoretycznych o pewien czynnik skalujący niezależny od xγ ale wartość tego czynnika różniła się pomiędzy eksperymentami od około 0.5 dla H1 do 0.7 otrzymanego przez ZEUS. Celem tej pracy jest otrzymanie przekrojów czynnych na dyfrakcyjną fotoprodukcję dżetów używając danych o większej świetlności oraz ich konfrontacja z wynikami otrzymanymi w poprzednich pomiarach. Pierwszy rozdział pracy zawiera krótkie wprowadzenie do badanego problemu. W kolejnym zawarty jest opis teoretycznych aspektów, które są istotne dla badanego procesu dyfrakcyjnej fotoprodukcji dżetów. Następnie w rozdziale drugim zawarty jest opis akceleratora HERA i eksperymentu ZEUS – układu pomiarowego, za pomocą którego zebrane zostały dane użyte w toku analizy. Kolejny rozdział zawiera opis użytych próbek Monte Carlo. Następne rozdziały dotyczą już bezpośrednio analizy. Rekonstrukcja wielkości kinematycznych jest opisana w rozdziale czwartym a selekcja przypadków w rozdziale piątym. W rozdziale szóstym omówiony jest opis danych przez Monte Carlo oraz jego dopasowanie w celu poprawy tego opisu. Kolejny.

(5) v rozdział zawiera opis algorytmu użytego do pomiaru przekroju czynnego oraz same przekroje otrzymane przy jego pomocy. Wnioski przedstawione są w rozdziale ostatnim. W toku analizy wygenerowane zostało Monte Carlo sygnałowe, niezbędne do odtworzenia rozkładów wybranych wielkości dżetowych, a także wielkości kinematycznych opisujących dyfrakcyjną fotoprodukcję na poziomie hadronowym. Przeprowadzone zostało jego dalsze dopasowanie i przeważenie na poziomie hadronowym w celu uzyskania lepszego opisu danych przez Monte Carlo. Wykonane zostało szczegółowe porównanie danych i symulacji Monte Carlo, zarówno sygnałowego jak i dla tła, w którym to uwzględniono niedyfrakcyjną fotoprodukcję oraz dyfrakcyjną elektroprodukcję i protonową dysocjację. Wynikiem analizy są różniczkowe przekroje czynne z uwzględnieniem efektów systematycznych w szacowaniu błędów pomiarowych. Ponadto autor dokonał porównania dostępnych danych z HERA I z tymi z HERA II w celu dokładniejszego zbadania i zrozumienia wpływu modernizacji detektora na uzyskane wyniki, a w szczególności na pomiar przyczynku do tła pochodzącego od protonowej dysocjacji. Podobnie jak w przypadku wcześniejszych wyników otrzymanych przy użyciu danych z HERA w analizie tej nie zaobserwowano zmniejszenia przekroju czynnego dla resolved PHP w stosunku do direct PHP. Niestety wynik tej analizy nie pozwala na rozstrzygnięcie kwestii stopnia supresji przekroju czynnego zmierzonego przez eksperyment ZEUS i H1 ponieważ normalizacja otrzymanych wyników obciążona jest dużą niepewnością wynikającą z ograniczonych możliwości aparaturowych pomiaru protonowej dysocjacji, takich jak brak komponentów detektora do pomiarów rozproszonego protonu lub jego zdysocjowanego stanu, prowadzącymi do dużej niepewności podczas jej estymacji. Jako możliwe rozwiązanie autor proponuje powtórne przeprowadzenie analizy danych z okresu HERA I włączając w to pełną analizę Monte Carlo. Podczas studiów doktoranckich autor przeprowadził również analizę dotyczącą dyfrakcyjnej produkcji ekskluzywnych par dżetów w oddziaływaniach głęboko nieelastycznych. Była to druga analiza zgodnie ze zwyczajami Współpracy ZEUS będąca warunkiem koniecznym publikacji wyników. Analiza ta została w rozszerzonej formie zaakceptowana i ukarze się jako publikacja Współpracy ZEUS Production of exclusive dijets in diffractive deep inelastic scattering at HERA, arXiv:1505.05783 [hep-ex]. Wyniki tej analizy nie mieszczą się w ramach tej pracy, jako że jej celem jest pomiar dyfrakcyjnej fotoprodukcji inkluzywnych jetów..

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(7) Contents. Introduction. ix. 1 Theoretical framework 1.1 Quark Parton Model and Deep Inelastic Scattering 1.1.1 The DGLAP Equation . . . . . . . . . . . . 1.1.2 The BFKL Equation . . . . . . . . . . . . . 1.2 Hadronisation . . . . . . . . . . . . . . . . . . . . . 1.3 Diffraction . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Soft Diffraction . . . . . . . . . . . . . . . . 1.4 Hard Diffraction . . . . . . . . . . . . . . . . . . . . 1.5 Diffractive DIS . . . . . . . . . . . . . . . . . . . . 1.6 Photoproduction of Dijets in Diffraction . . . . . . 1.7 Notation and Definitions of Kinematic Variables . .. . . . . . . . . . .. 1 . 1 . 4 . 4 . 5 . 5 . 7 . 7 . 8 . 9 . 12. . . . . . . .. . . . . . . .. 15 15 17 19 21 21 22 23. 3 Monte Carlo Samples 3.1 Diffractive Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Non-diffractive Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proton-dissociative Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25 25 26 26. 4 Event Reconstruction 4.1 Track Reconstruction and Vertexing . . 4.2 Calorimeter Reconstruction . . . . . . 4.3 EFO Reconstruction . . . . . . . . . . 4.4 Scattered Lepton Identification . . . . 4.5 Jet Reconstruction . . . . . . . . . . . 4.6 Reconstruction of Kinematic Variables. 27 27 28 30 31 32 33. 2 HERA and the ZEUS Detector 2.1 HERA . . . . . . . . . . . . . . . . . . . 2.2 The ZEUS Detector . . . . . . . . . . . . 2.2.1 The central tracking detector . . 2.2.2 The silicon micro vertex detector 2.2.3 The uranium calorimeter . . . . . 2.2.4 Luminosity Monitor . . . . . . . 2.2.5 Trigger . . . . . . . . . . . . . . .. . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . .. . . . . . .. 5 Data Sample and Event Selection 37 5.1 Data Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2 Trigger selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 vii.

(8) viii. CONTENTS 5.3 5.4 5.5 5.6 5.7 5.8 5.9. Quality Cuts . . . . . . . . . . PHP Cuts . . . . . . . . . . . . Dijet Cuts . . . . . . . . . . . . Diffractive Cuts . . . . . . . . . Cosmic Cuts . . . . . . . . . . . Summary of the Selection Cuts Background Estimation . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 38 38 39 41 41 41 42. 6 Description of Data by the Monte Carlo Simulation 6.1 Normalisation of MC . . . . . . . . . . . . . . . . . . 6.2 Reweighting of MC . . . . . . . . . . . . . . . . . . . 6.2.1 Reweighting of Signal MC in xγ and zIP . . . 6.2.2 The Distribution of ηmax of EFOs . . . . . . 6.3 Estimation of Proton-dissociative Background . . . . 6.4 Control Distributions . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 43 43 44 45 45 48 51. . . . . . . . . .. 55 55 56 59 63 68 68 69 73 74. 7 Measurement of Cross-Section 7.1 Control Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Acceptance, Purity and Efficiency . . . . . . . . . . . . . . . . . . . 7.3 Unfolding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Comparison of Acceptance and SVD Unfolding Methods . . 7.5.2 Comparison of Single and Double Differential Cross Sections 7.5.3 Comparison of HERA I and HERA II Result . . . . . . . . . 7.5.4 Comparison with H1 Result . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 8 Summary and Conclusions. 79. Appendices. 79. A Kinematic variables and their reconstruction. 81. B Double Differential Cross Sections. 83. C Single Differential Cross Sections. 93.

(9) Introduction Particle physics like all branches of natural science strives to understand the universe, its specific goal is to describe the structure of matter and interactions between its constituents on the deepest known level of elementary particles and fundamental forces. In the standard model, the strong force is described by the quantum chromodynamics, QCD, a non-abelian gauge theory based on the SU (3) group with running coupling constant, αS , which depends on the energy scale of the interaction exhibiting quick raise when the scale declines below ΛQCD . This raise of αS renders the usage of the perturbative approach to the QCD in certain kinematic regions impossible. One of them is the region of diffractive photoproduction with 0 ≈ Q2 < ΛQCD which is usually described by the Regge phenomenology yet allows the perturbative description when the hard scale is delivered by the hard final state like jets or heavy flavour production. In such case the diffractive hard factorisation can be used to obtain predictions thanks to the separation of the non-perturbative component into the parametrisation known as diffractive parton density function and allowing pQCD approach. The diffractive factorisation, just as the QCD factorisation in non diffractive case, proved to be useful in predicting and describing diffractive processes in presence of the hard scale in ep collisions at HERA. Yet in the p¯ p collisions at Tevatron the theoretical predictions made based on diffractive factorisation and HERA dPDFs overestimated the cross section by the factor of the order of magnitude. The theoretical explanation of this phenomena assumes that it results from absorptive effects caused by multiple rescattering producing secondary particles which repopulate the rapidity gap spoiling the experimental signature of the diffractive event. These makes the region of diffractive dijets in PHP most interesting as a test ground for breaking the QCD factorisation. The fact that in PHP Q2 ≈ 0 coupled with the Heisenberg’s principle may lead to the fluctuation of photon into a q q¯ pair. That is why in diffractive photoproduction photon interacts not only in the direct way transferring its whole momentum to the quarks but also like hadron in so called resolved way i.e. transferring only part of its 4-momentum and acting as a source of quarks from which only one takes part in the hard scatter. The study of these two regimes experimentally separated by the value of xγ = 0.75 with xγ < 0.75 being resolved, or at least resolved enriched, photoproduction and xγ > 0.75 being direct enriched photoproduction, can improve the understanding of the factorisation breaking in hadron-hadron collisions. The expected result is that the cross section for the resolved PHP will be more strongly suppressed than the cross section in direct PHP. Such measurements were performed by H1 Collaboration and by the ZEUS Collaboration on HERA I data in slightly different kinematic regions showing no xγ dependence of suppression factor but differing in the values of the global suppression. The aim of this analysis is to obtain differential cross sections on diffractive photoproduction of dijets from ZEUS HERA II data and compare them with previous results of H1 and ZEUS and, if possible, help to understand the difference in global suppression factor. In first chapter of this thesis the short overview of the theoretical ideas relevant to the measurement of diffractive photoproduction is given. The description of the HERA accelerator and the ZEUS detector – the measuring apparatus with which the data for the analysis were ix.

(10) x. INTRODUCTION. collected, is presented in the following chapters. After that the Energy Flow Objects (EFOs), used in the reconstruction of kinematic variables, and the reconstruction of kinematic variables itself are described. Last chapters are focused on the analysis and its results showing the comparison of data and Monte Carlo, the estimation of background, explaining the methods of reconstruction of cross sections and showing results of this reconstruction in comparison to the previous H1 and ZEUS results. The summary and conclusions are presented in the final chapter..

(11) Chapter 1 Theoretical framework 1.1. Quark Parton Model and Deep Inelastic Scattering. The results of SLAC experiment of electron-proton scattering led to the idea that proton has a structure [1, 2]. Model proposed by Feynman as explanation of the observed phenomena assumed that proton consists of non-interacting point-like constituents – partons, later identified as quarks. This idea known as Quark Parton Model (QPM) was pretty successful in explaining Deep Inelastic Scattering (DIS) that is interaction of lepton with proton at large lepton momentum transfer mediated by neutral boson. Here the case of electron (or positron) will be discussed in more detail, e + p → e + X. The kinematics of DIS is described by variables s, Q2 , W , x and y, which are defined in section 1.7. The diagrammatic presentation of DIS and its kinematics is shown in Fig. 1.1. In electroweak theory the role of the mediating neutral boson can be played by photon or by Z0 but since the propagator T in the amplitude takes the form T ∝. 1 2 Q2 + Mboson. and MZ20 ∝ 104 GeV2 the contribution from Z0 is significant only for high values of Q2 . Neglecting Z0 contribution the DIS processes takes the form of an incoherent elastic scattering of leptons from charged partons within the proton. Such interaction with a single point-like parton or quark can be described by Quantum Electrodynamic (QED). Its cross section is given by the Rosenbluth formula d2 σ (eqi → eqi ) αe2 Q2i Q2 Q2 4 θ 2 θ = cos + sin δ Ee − Ee0 − , dΩdEe0 2 2m2i 2 2mi 4Ee2 sin4 2θ where Qi stands for the charge of given parton and θ is the scattering angle of the electron. Assuming that the fi (x, Q2 ) is the classical probability density function of the photon to hit the quark of type i which has a fraction x of proton momentum the above lead to the final result for the DIS cross section: αe2 Q2i d2 σ (ep → eX) X 2mp 2 1 2 2 θ 4 θ = fi x, Q x cos + sin , 2 2 sin4 θ dΩdEe0 Q 2 m 2 4E p e 2 i where i runs over all quark flavours. This can be rewritten as 2θ 4θ d2 σ (ep → eX) αe2 Q2i 4πxmp 4π 2 2 = F2 x, Q cos + F1 x, Q sin , dΩdEe0 Q2 2 mp 2 4Ee2 sin4 2θ 1. (1.1).

(12) 2. CHAPTER 1. THEORETICAL FRAMEWORK Q2. e0 (k 0 ). e (k). γ∗. s W2. p (P ) X. Figure 1.1: The schematic diagram of neutral current DIS via photon exchange and its kinematics. For the definition of kinematic variables see section 1.7. where F1 and F2 stand for structure functions of the proton and are given by the pair of equations X F2 x, Q2 = x Q2i fi x, Q2 i. and 1 F1 x, Q2 = F2 x, Q2 . 2x The last equation is known as the Callan-Gross relation and here has been derived from QPM model. Similar equation can be also obtained using QCD. Initially parton model assumed that fi (x, Q2 ) depends only on x which led to and explained the Bjorken scaling (or scale invariance) that is the fact that the DIS cross section is approximately independent of Q2 at fixed x. The scale invariance was observed at SLAC in late 60’s . The results from SLAC showing the Q2 dependence of F2 is presented in Fig. 1.2.. Figure 1.2: Results from SLAC confirming scale invariance of PDFs [3]. The vW2 for the proton as a function of Q2 , denoted on the plot with small letter. The vW2 is proportional to F2 ..

(13) 1.1. QUARK PARTON MODEL AND DEEP INELASTIC SCATTERING. 3. As stated in [4]: The physical justification of PDFs is that the momentum sloshes around 1 among proton constituents at time scales ΛQCD ∝ m1p . These time scales are much slower than the time scales Q1 which the photon probes. The separation of scales Q >> ΛQCD allows us to treat the parton wavefunctions within the proton as being decoherent, giving the probabilistic interpretation. The probabilistic interpretation of PDFs and momentum conservation XZ xfi x, Q2 dx = 1 i. backed up by the experimental results led to the following conclusion about valence quark momenta: Z x (fu (x) + fd (x)) dx ≈ 0.38. With today’s insight from QCD it is clear that apart from three valence quarks there is a sea of quark-antiquark pairs and gluons binding the quarks and restoring the observed colour confinement and carrying the rest of the proton momentum. Historically this missing momentum and the need for the explanation of breaking of the scale invariance that is the rise of F2 with Q2 for small x coupled with other experimental data opened the way for the new theory Quantum Chromodynamics (QCD). QCD is a non-abelian gauge SU (3) group invariant theory dealing with six quark-fermion fields carrying a so called colour quantum number and with a multiplet of eight gauge fields, identified with gluons. The QCD predicts the same form (1.1) of the cross section for the inclusive DIS as the one presented before and obtained from QPM model. The QPM can be treated as QCD’s 0-th order approximation, the approximation of non-interacting quarks. The fact that QCD predicts the same form of cross section (1.1) gives new meaning to the PDFs which are now not necessary independent of Q2 . The observed weak dependence of Q2 for DIS can be explained by the running of the coupling constant αS that is the fact that αS changes with Q2 . For the small values of Q2 αS becomes vary large leading e.g. to the confinement of quarks and for high Q2 it becomes small allowing the approximation of non-interacting quarks, these phenomena are known respectively as asymptotic slavery and asymptotic freedom. For the small coupling constant or high scale of the interaction, in case of DIS represented by Q2 , it is possible to use perturbative QCD that is expansion in orders of αS , to calculate the cross sections. With this approach depending on the order, n, of the perturbative expansion F2 (x, Q2 ) can be expanded up to terms of the form αSn logm Q2 and αSn logm (1/x), with m < n. Different terms of the expansion dominate in different kinematic regions. This all leads to the new QCD improved version of the QPM model that is to the phenomenological approach assuming factorization and universality of PDFs which are no longer representing probabilities but only act as parametrisations. Within this model the cross-section of any process can be expressed in the form σ = H ⊗ f, where H represents QCD calculable part and f stands for the universal (the same for any process) PDF. However factorisation can be derived from the QCD only in special cases its usage as a phenomenological approach enables understanding of the DIS data. Such PDFs are subject to the QCD evolution that is parton distributions at one point in the Q2 -x plane can be extrapolated or related to parton distributions at different point via evolution equations. Below two different schemes of evolution and different evolution equation are described..

(14) 4. CHAPTER 1. THEORETICAL FRAMEWORK. Figure 1.3: Evolution of parton densities. Evolution of PDFs is usually considered either in 1/x (BFKL equation) or in Q2 (DGLAP equation). The systems which undergo the evolution are in relatively dilute regime. They can evolve towards dense regime and undergo transition to saturation region which is characterized by saturation scale that depends on log (1/x).. 1.1.1. The DGLAP Equation. The DGLAP evolution equation named after authors (Dokshitzer, Gribov, Lipatov, Altarelli, Parisi) [6] resums large logarithms in structure functions. It is a set of integro-differential equations   X Z 1 x x 2 2 P P qi g ξ d αS dξ  qi qj ξ fj (ξ, Q2 ) fi (x, Q )  Q 2 , = fg (ξ, Q2 ) fg (x, Q2 ) dQ π x ξ Pgg x Pgqj x j ξ. ξ. . where P†? xξ are the splitting functions which can be derived from cross sections for the processes g → gg, g → q q¯, q → qg and can be interpreted as the probabilities of a quark (or gluon) with momentum fraction, x, to originate via Bremsstrahlung from an initial quark (or gluon) with momentum fraction ξ. The DGLAP evolution is applicable in perturbative regime and when all terms involving powers of log (1/x) are negligible (see Fig. 1.3). This type of evolution explains weak logarithmic Q2 dependence of the DIS cross-section observed at HERA.. 1.1.2. The BFKL Equation. The Balitski, Fadin, Kuraev and Lipatov equation (BFKL) sums up terms αSn logm x1 neglected in the DGLAP approach and describes the evolution of low x, to be precise in 2 in the region Q 1 region allowing usage of pQCD and where log Q2 << log x (see Fig. 1.3). The evolution 0. takes place in x at fixed Q2 . It assumes that splitting g → gg dominates in the evolution. The BFKL evolution equation takes form df = KL ⊗ f, dx where KL represents so called Lipatov kernel and the ⊗ stands for the integration over x and the transverse momentum of the gluons. An analytical solution at fixed Q2 leads to the prediction x.

(15) 1.2. HADRONISATION. 5. that in the low x region F2 (x, Q2 ) ∝ x−λ with λ ≈ 0.5.. 1.2. Hadronisation. The term hadronisation stands for a process of colour neutralization of partons, or equivalently the transition of the partons into observable hadrons. Such a process takes place at the longdistances in comparison with the distances of the interaction and do not involve the hard scale rendering the perturbative calculations not possible. Only approximative phenomenological approaches usually exploring Monte Carlo methods are used to describe the process of hadronisation. All models and experiments assume that short and long distance interactions are time separated and thanks to this the information about parton level and interaction is ‘frozen’ in hadrons and can be retrieved. First and the simplest model of hadronisation was the Independent Fragmentation Model proposed by Feynman [7]. The main idea standing behind this model is that a parton creates the so called ‘primary meson’ with one of the quarks from the quark-antiquark pairs generated from the vacuum. Such primary meson takes energy fraction of the parton, z, and the process continues with leftover quark with the rest of the energy that is with energy fraction 1–z. The process of the creation of primary mesons happens according to the probability density function f (z) which is independent of energy. It stops at lower energy cut-off. Next worth mentioning scheme of hadronisation is the Lund String Model. This is modern widely used model of hadronisation for the first time proposed in [8] and still being developed. The basics of this model include potential 4 αS r12 + κr V (r) ≈ − 3 r describing string formation between initial quark-antiquark pair at distance r, where κ stands for string tension. Such string breaks up if the potential energy is large enough creating new quark-antiquark pair or pairs. If energy of the pair is low enough a hadron is formed. Gluons in this model are present as ‘kinks’ in the string carrying energy and momentum, with forced ratio of gluons to quarks equal 2. The schematic representation of this model is shown in 1.4a on the following page. The last model of hadronisation mentioned in this thesis is the competing Cluster Fragmentation Model [9], which introduces forced g to qq branchings, then forms clusters from colour singlet combinations that decay isotropically in phase space to two hadrons. One of the characteristics of this model is the fact that colour flow during hadronisation is confined what leads to the more universal spectrum of hadrons. The schematic depiction of this model is presented in 1.4b on the next page.. 1.3. Diffraction. Interaction in which only the quantum numbers of vacuum are exchanged is called diffractive. This definition is almost impossible to apply in experimental physics hence there exist different one attributed to Bjorken. According to this experimental definition events with large rapidity gap LRG which is not exponentially suppressed are called diffractive. Rapidity, y, of a given particle with 4-momentum (E, px , py , pz ) is defined as 1 E + pz y := log . 2 E − pz.

(16) 6. CHAPTER 1. THEORETICAL FRAMEWORK. (a) String model. (b) Cluster model. Figure 1.4: Schematic representations of hadronisation models. a. a0. a. Xa. a. Xa. b. b0. b. b0. b. Xb. (a). (b). (c). Figure 1.5: Diffractive interactions: (a) Elastic scattering; (b) Single dissociative diffractive scattering; (c) Double dissociative diffractive scattering. For highly relativistic particles y can be approximated by so called pseudorapidity η θ y ≈ η := log tan , 2 where θ stands for the polar angle of the momentum of the particle. In massless case these two variables are equal. The theoretical and experimental definitions of diffraction are not equivalent. Diffractive interactions can be divided into: - elastic scattering: a + b → a0 + b0 ; - single dissociative diffractive scattering: a + b → Xa + b0 ; where only a dissociates into the state Xa - double dissociative diffractive scattering: a + b → Xa + Xb ; where both a and b dissociate into the states Xa and Xb respectively. Schematic diagram of each type of diffractive interaction is presented in Fig. 1.5. Diffraction can be also divided into two regimes. The soft diffraction regime historically described by the Regge phenomenology [10, 11, 12] and the hard diffraction in which the pomeron is realised via pQCD..

(17) 1.4. HARD DIFFRACTION. 1.3.1. 7. Soft Diffraction. In non-perturbative regime the Regge phenomenology – the generalisation of the Yukawa type model based on the S-matrix approach gives the description of the elastic scattering a+b → a+b in terms of the Mandelstam variables s and t. This theory assumes that all resonances or exchanges, all possible particles with quantum numbers which are appropriate to act as a mediators, contribute to the scattering amplitude S. S can be expressed as S (s, t) = 1 + iT (s, t) , where T is a transition matrix. Such resonances can be treated as angular momentum excitations of some ‘ground state,’ as they can differ only in spin. It occurred that such particles lie on straight lines in the angular momentum, α (t), and t plane, where the angular momentum is treated as a complex and continuous variable which takes real values only for stable hadrons and its imaginary part is related to the decay width. These lines parametrised as α (t) = α (0) + α0 t are known as Regge trajectories. The transition matrix in such a situation takes the asymptotic form X βak (t) βak (t) sα(t) , T (s, t) = k. where the sum runs over all possible trajectories and β’s are called form factors and account for the non-point-like nature of the hadrons. It leads to the elastic scattering cross section dσ X 2 2 = βak (t) βak (t) s2α(t)−2 . dt k The elastic and total cross-sections are related by the Optical Theorem thanks to which 1 ab σtot ≈ Im (T (s, 0)) ∝ sα(0)−1 . s Apart from mesons and their trajectories called Reggeons, IR, the special trajectory called Pomeron, IP , was introduced to the Regge model to reproduce the rise of the total cross section in high centre of mass energy region. Processes in which this pomeron trajectory is exchanged are called diffractive. The trajectory of the pomeron was found experimentally [13] and takes form αIP = 1.08 + 0.25 GeV−2 t. The sub-leading contribution from the reggeons’ trajectories to the diffractive processes at HERA is small and observable mainly at high xIP values.. 1.4. Hard Diffraction. Notion of pomeron itself originates from non-perturbative theory of Regge where it is understood as diffractive exchange but unlike the other exchanges is not an observable particle. Hard diffraction is here understood as the concept of so called pQCD pomeron that is the pQCD realization of the pomeron by combinations of gluons and quarks. Such perturbative pomeron exchange can be realised by two gluon exchange (see Fig. 1.6b) or in higher order a gluon ladder carrying quantum numbers of vacuum (see Fig. 1.6c). Hard diffraction requires a presence of hard scale to manifest itself. This scale may be realised by t, Q2 , mass or pT . The above.

(18) 8. CHAPTER 1. THEORETICAL FRAMEWORK. mentioned idea of exchange of the gluon ladder with quantum numbers of vacuum is known as BFKL pomeron model. The Regge trajectory of BFKL pomeron is given by αIP ≈ 1.4 + 0 GeV−2 t. γ. X. γ. X. γ. X. p. p0. p. p0. p. p0. (a) Pomeron.. (b) Two gluon exchange.. (c) pQCD Pomeron.. Figure 1.6: Different representations of pomeron. Pomeron may have a non-perturbative nature like in case of Ingelman-Schlein model [14] inspired by Regge phenomenology and schematically depicted in (a) or purely perturbative character like in case of two gluon exchange model (b) or BFKL pomoeron model where it is realised by the gluon ladders (b) and/or (c). Different representations of the pomeron in γ − p interactions are presented in Fig. 1.6.. 1.5. Diffractive DIS. Diffractive DIS will be understood in this thesis as one of the two diffractive processes: • e+p → e+X+p only photon dissociates (single diffraction); X represents the dissociated photon state; • e+p → e+X +Y photon and proton dissociates (double diffraction); X and Y represent the states of dissociated photon and proton; with high momentum transfer Q2 . To describe the diffractive DIS additional kinematic variables t, β, xIP , MX and in case of double diffraction MY are needed. They are defined in section 1.7 and schematically depicted in the diffractive DIS diagram in Fig. 1.7. The Born level cross section for non-diffractive DIS takes form d2 σ (ep → eX) 4παe2 y2 = 1−y+ F2 x, Q2 , 2 2 2 dxdQ xQ 2 (1 + R (x, Q )) where the R (x, Q2 ) is defined as the ratio of photoabsorption cross-section for longitudinally polarised photons, σL , to the cross-section for transversely polarised ones, σT σL (x, Q2 ) R x, Q2 = σT (x, Q2 ) and is negligible in the kinematic region of HERA, therefore d2 σ (ep → eX) 4παe2 y2 2 = 1 − y + F x, Q . 2 dxdQ2 xQ2 2.

(19) 1.6. PHOTOPRODUCTION OF DIJETS IN DIFFRACTION Q2. 9. e0 (k 0 ). e (k) γ∗. s. X W. 2. p (P ) t. p0 (P 0 ) (or Y (P 0 )). Figure 1.7: The diagram of diffractive DIS process. Any type of colorless exchange is represented by the zigzag which in case of single diffraction leads to an intact leading proton in the final state or in case of double diffraction to the dissociative proton state Y . Presented kinematic variables are defined in section 1.7. By analogy to the previous equation it is possible to express the Diffractive DIS cross-section for e + p → e + X + p as 4παe2 y2 d4 σ (ep → eXp) D(4) 2 = F β, Q , x , t . 1 − y + IP 2 dβdQ2 dxIP dt βQ2 2 This equation can be treated as the defining one for the diffractive structure D(4) function, F2 (β, Q2 , xIP , t). The superscript (4) represents the dimensionality of the domain of diffractive structure function. Often it is difficult or even impossible to measure t and then the t dependence is integrated out. Diffractive structure function is in such situation defined D(3) as a function of β, Q2 and xIP , F2 (β, Q2 , xIP ) Z D(3) D(4) 2 F2 β, Q , xIP = dtF2 β, Q2 , xIP , t . Apart from QCD factorisation explained before usually the further one is assumed. It is factorisation of the diffractive vertex thanks to which, if it holds, the diffractive structure function can be expressed as the product of the pomeron flux, fIP/p (xIP , t), and the pomeron structure function, F2IP (β, Q2 ): D(4). F2. β, Q2 , xIP , t = fIP/p (xIP , t) F2IP β, Q2 .. Like in the elastic case if the diffractive factorisation holds all non-perturbative information should be contained in universal diffractive PDFs and dPDFs experimentally determined for one reaction could be used to obtain cross section for any other diffractive processes after the evolution with a hard scale.. 1.6. Photoproduction of Dijets in Diffraction. The main interest of this thesis lies in the photoproduction – process in which leptons are scattered at small angles and with small Q2 . The limit of Q2 defining PHP in ZEUS experiment.

(20) 10. CHAPTER 1. THEORETICAL FRAMEWORK. (a). (b). e. e. e. e γ γ. γ-remnant (u). g. q. (u). q. jet1. jet1. jet2 g. (v). jet2. q¯. g. IP -remnant. (v). q¯ IP -remnant. IP. IP. p. p. p. p. Figure 1.8: Examples of leading order diagrams of diffractive dijet photoproduction, (a) for direct photon, (b) for resolved photon. is customary set to 1 GeV. This limit is also applied in this thesis and the events with Q2 < 1 GeV2 are from now on termed photoproductive. For the photon of such a small virtuality not only is it possible to interact in a direct way like in DIS but also due to the Heisenberg’s uncertainty principle the fluctuation of γ to qq pair can occur. Such a pair lives relatively long what can lead to the resolved, i.e. not point-like interacting photon transferring only part of its 4-momentum and acting as a source of quarks from which only one takes part in the hard scatter. Both situations – the direct one, called direct PHP in which photon couples directly to the quarks via QED vertex and the resolved one in which resolved manifestation of the photon originates from the point-like QED process, γ → q q¯, and then the further interaction is carried out by QCD, called resolved PHP, are depicted in Fig. 1.8 for the case of diffractive PHP. Kinematics of diffractive PHP requires additional variable xγ describing the longitudinal momentum fraction of γ taken by scattered parton with 4-momentum u and defined as xγ =. Pu , Pk. based on which the resolved and direct PHP can be discerned. At leading order QCD xγ = 1 for direct PHP and xγ < 1 for resolved PHP. Unlike in DIS the hard scale in photoproduction of dijet as well as in diffractive photoproduction of dijet is not set by Q2 but rather by ET (or pT ) of dijets, which in this context are understood as two final state partons. For more information about jets and jet definition see section 4.5 on page 32. Since the DIS and diffractive DIS cross sections are proportional to 1/Q4 in both regions elastic and diffractive one photoproduction dominates. The aforementioned fluctuation of photon and its hadron-like behaviour lead to the necessity of considering its structure. The relative transverse momentum squared, p2T , between the quarks in process γ → q q¯ provides the scale for the QCD. The spectrum of pT ’s is wide enough to include situations in which the quarks create a bounded state as well as the situation in which they are asymptotically free. In the first case of small p2T and strongly coupled quarks the behaviour of the photon is non-perturbative and can be described by the phenomenological Vector Dominance Model (VDM) first proposed by Sakurai [15] and then generalised [16] which assumes that the photon interacting with hadrons has a hadronic component, |hi, consisting of vector mesons end their.

(21) 1.6. PHOTOPRODUCTION OF DIJETS IN DIFFRACTION. 11. resonances and can be described as a composite state of bare photon |γb i and |hi |γi = A |γb i + B |hi , with appropriate coefficients A and B. In the second case when relative transverse momentum squared of the q q¯ pair is large enough they are asymptotically free allowing to some extent the further QCD development as a result of which not only quarks but also gluons occur. This leads by the analogy to the proton to the definition of photon structure function F2γ and its PDFs. The photon PDFs have been extracted from the scattering data in e+ e− collisions. The photon flux, fγ/e (y), can be obtained form Weizs¨acker-Williams Approximation formula [17] ! 2 2 2 Q 1−y Q αem 1 + (1 − y) log max −2 1 − 2min , fγ/e (y) = 2 2π y Qmin y Qmax where Q2min = m2e y 2 / (1 − y), me is the mass of the electron and Q2max stands for the upper limit cut on the photoproduction applied in the selection of the measurement. Using the photon flux and photon PDFs, the pomeron flux and diffractive PDFs and assuming the QCD factorisation the cross section for diffractive photoproduction of dijets can be calculated as 0. dσ (ep → e + 2jets + X + Y )) =. XZ. fiIP zIP , µ2 fIP /p (xIP , t) ×. i,j. ×fjγ. 2. xγ , µ. . fγ/e (y) dˆ σ (ij → 2jets) dxIP dt dzIP dxγ dy,. where the sum runs over all contributing partons, dˆ σ (ij → 2jets) is a parton level cross section of the dijet process with incoming parton i from the diffractive exchange and an incoming parton j from the photon given by QCD, the scale µ is assumed to be equal on the proton and photon vertex, the rest of the variables are defined in section 1.7. The leading order (LO) graphs contributing to the diffractive photoproduction are presented in Fig. 1.9 for direct PHP and in Fig. 1.10 on the next page for resolved PHP. e. e. e. e γ. γ q. g. q. q¯ q¯. IP. IP. p. p. p. (a) γ+g →q+q. p. (b) γ + g → g + q, γ+g →g+q. Figure 1.9: Leading order diagrams of direct photon diffractive dijet photoproduction..

(22) 12. CHAPTER 1. THEORETICAL FRAMEWORK e. e. e. e. e. γ g. e. γ. e. e γ. γ. q. g. q. g. g. q¯. g. q¯. g. g. IP. IP. p. IP. p. p. p. p. (a) g + g → q + q¯. p. (b) q + q¯ → g + g. e. e. (d) g+g →g+g. e. e. γ. γ. g. g. p. (c) q + q → q + q, q + q¯ → q + q¯, q¯ + q¯ → q¯ + q¯. e. e. IP. p. e. e γ. γ g. g. g. g. IP. IP. p. IP. p. p. p. (f) g + q → g + q, g + q¯ → g + q¯. g. IP. p. p. (e) g + q → g + q, g + q¯ → g + q¯. g. p. (g) g + q → g + q, g + q¯ → g + q¯. p. (h) g + q → g + q, g + q¯ → g + q¯. Figure 1.10: Leading order diagrams of resolved photon diffractive dijet photoproduction. The process shown in 1.9b is known as QCD-Compton scattering.. 1.7. Notation and Definitions of Kinematic Variables. In this section the used notations and the definitions of kinematic variables are presented. Throughout the whole thesis following notations will be used for 4-momenta of incoming and scattered leptons and protons and final state hadrons. P = (Ep , px,p , py,p , pz,p ). ≡. 4-momentum of incoming proton. k = (Ee , px,e , py,e , pz,e ). ≡. 4-momentum of incoming lepton. P 0 = (Ep0 , px,p0 , py,p0 , pz,p0 ). ≡. 4-momentum of scattered proton (if exists). k 0 = (Ee0 , px,e0 , py,e0 , pz,e0 ). ≡. 4-momentum of scattered lepton. ph = (Eh , px,h , py,h , pz,h ). ≡. 4-momentum of final state hadron h. Using this notations the following variables describing kinematics of DIS can be defined..

(23) 1.7. NOTATION AND DEFINITIONS OF KINEMATIC VARIABLES q := k − k 0. ≡. 4-momentum transfer at lepton vertex. s := (k + P )2 q W := (P + q)2. ≡. squared center-of-mass energy. ≡. boson-proton center-of-mass energy. Q2 := −q 2. ≡. squared 4-momentum transfer at lepton vertex. x :=. Q2 2P q. ≡. longitudinal momentum fraction of proton taken by scattered parton. y :=. Pq Pk. ≡. inelasticity. 13. The diagram of DIS with shown kinematic variables is presented in Fig. 1.1. In order to describe the kinematics of diffractive DIS additional notations are needed.. X. ≡. hadronic final state - final state without scattered electron and proton (or dissociated proton system). Y. ≡. dissociated proton system. What is more pY and pX will be used as 4-momenta of Y and X system. With slight abuse of the notation the P 0 can be set to be equal to pY if proton dissociates. Then the additional variables describing kinematics of diffractive DIS can be defined. ≡. squared invariant mass of hadronic final state X. ≡. squared 4-momentum transfer at proton vertex. (P − P 0 )q Pq. ≡. longitudinal momentum fraction of p taken by IP. Q2 2(P − P 0 )q. ≡. longitudinal momentum fraction of IP taken by scattered parton. MX2 := (q + P − P 0 )2 t := (P − P 0 ) xIP := β :=. 2. The diagram of diffractive DIS together with describing its kinematics variables is presented in Fig. 1.7. In case of diffractive PHP also the v. ≡. 4-momentum of the parton from the pomeron. ≡. 4-momentum of the parton from the photon. and u are needed to define.

(24) 14. CHAPTER 1. THEORETICAL FRAMEWORK. zIP :=. qv (P − P 0 )q. ≡. longitudinal momentum fraction of IP taken by the parton participating in the hard process. ≡. longitudinal momentum fraction of γ taken by scattered parton with 4-momentum u.. and Pu xγ := Pk. Once again it should be stressed that in definitions of MX , t, zIP , xIP and β the 4-momentum of scattered proton should be replaced by the 4-momentum of the dissociated proton system Y in case of the proton dissociation or equivalently the definition of P 0 should be in this case extended by setting P 0 := pY ..

(25) Chapter 2 HERA and the ZEUS Detector 2.1. HERA. The Hadron-Elektron Ring Anlage (HERA), the world’s first and only particle accelerator that collided leptons (electrons or positrons) and protons, was located 10-30 m underground beneath the Deutsches Elektronen-Synchrotron (DESY) and Altonaer Volkspark in Hamburg, Germany. Its construction started in 1984 and lasted seven years. Finally HERA commenced operations in May 1992, and the final collision took place on 30th June 2007. The HERA tunnel 6.3 km in circumference was not circular but had four straight 360 m long sections and contained two separate storage rings for leptons and protons. On each of the linear segments there was a hall containing different experiment. In the north one and the south one there were the two multi-purpose detectors, H1 [19] and ZEUS [18] measuring zero crossing-angle collisions of leptons and protons. In two remaining ones fixed target collisions experiments were conducted. The east hall belonged to HERMES [20] purpose of which was to measure nucleon spin via collisions of the electron beam with polarised gas. In the last one, the west hall in year 1999 and 2003 interaction of the proton beam with fixed wire targets were measured by HERA-B experiment [21] in order to investigate CP violation. Initially leptons at HERA were accelerated to the energy of 26.7 GeV and protons to 820 GeV but then in the course of years these energies were increased to 27.5 GeV for the leptons, starting from the year√1995, and to 920 GeV for the protons, at the end of 1997, giving final centre-ofmass energy s = 318 GeV. At the end of HERA operation proton beam energy was lowered in order to allow the measurement of the longitudinal structure function. Protons of 460 GeV and 575 GeV were used in these runs called respectively Low Energy Runs (LER) and Medium Energy Runs (MER). The existence of two separate rings for leptons and protons was necessary due to the difference in masses between aforementioned types of particles one of which is more than 1800 times heavier than the other. In order for the proton to reach desired energy of 820 or later 920 GeV and to keep it within HERA ring 650 superconducting magnets, superconducting dipole and quadrupole magnets, along with non-superconducting radio frequency cavities for the proton acceleration were used what was unusual at the time when HERA was planned. Each of superconductive magnets produced 4.7 T of magnetic field and to do so had to be kept in temperature of 4.4 K. 15.

(26) 16. CHAPTER 2. HERA AND THE ZEUS DETECTOR. Figure 2.1: Aerial photograph of the HERA collider and pre-accelerator PETRA location site at DESY, Hamburg.. Figure 2.2: Diagram of the HERA accelerator and pre-accelerators. Electrons entering HERA and its pre-accelerators system, scheme of which is presented in Fig. 2.2, were obtained from a hot metal filament and accelerated to 220 MeV in the linear accelerator LINAC I. During positron runs these electrons hit tungsten sheet at which point electron-positron pairs were produced through the bremsstrahlung radiation yielding positrons for the further acceleration. Then electrons or positrons were passed to 70 m long LINAC II where they were accelerated up to 450 MeV. As a next step leptons were injected into DESY II synchrotron where, in bunches of approximately 3.5 × 1010 leptons per bunch with 96 ns bunch spacing, they were accelerated to.

(27) 2.2. THE ZEUS DETECTOR. 17. 7.5 GeV then moved to PETRA II where they reached up to 14 GeV and finally to the HERA lepton storage ring. Protons were produced through acceleration of hydrogen ions – protides (H − ) in LINAC to 40 MeV and passing them through thin metal foil at which point the remaining electrons were stripped off. Then protons were subsequently moved to DESY III synchrotron where they were accelerated to 7.5 GeV in 11 bunches with a temporal spacing of 96 ns and PETRA where they reached energy of 40 GeV in 70 bunches and to the HERA proton storage ring. Whilst HERA could store 210 proton and 210 electron bunches some were left empty so as to study interactions between beam particles and residual gas molecules in the ring. The whole above described HERA pre-acceleration system consisted mostly of previously existing synchrotrons and linear accelerators which had been once high energy particle accelerators in their own right.. -1. Integrated Luminosity (pb ). HERA delivered 300. 250. 200. 150. 100. 50. 0. 0. 200. 400. 600. 800. 1000. 1200. 1400. days of running. Figure 2.3: Integrated luminosity delivered by the HERA accelerator.. The data analysed in this thesis includes both electron and positron high energy runs collected by the ZEUS detector during so called HERA II that is the period after HERA upgrade in 2000-2001 when in order to increase the luminosity superconducting magnets were installed near the interaction point to make beam cross section smaller. Luminosity delivered by HERA I, i.e. before year 2000, and HERA II as a function of days of running is presented in Fig. 2.3.. 2.2. The ZEUS Detector. The ZEUS Detector [18, 22] was along with H1 one of the two multi-purpose detectors which were designed to study the lepton-proton collisions produced by HERA. Due to the considerable difference between the energies of lepton and proton beams the centre of mass system followed the proton beam trajectory hence most of the final state particles emerging from the interaction were boosted in its direction. Thus the ZEUS detector was asymmetric and contained more appliances in forward direction..

(28) 18. CHAPTER 2. HERA AND THE ZEUS DETECTOR. Figure 2.4: Coordinate system used by the ZEUS collaboration. The ZEUS collaboration used a Cartesian coordinate system with origin in nominal interaction point with Z axis parallel to the beam line and oriented in proton beam direction, Y axis pointing up, X axis pointing to the centre of the accelerator. Polar angle was measured from Z axis.. Figure 2.5: Schematic of cross section of the ZEUS detector along the beam direction (along XZ plane). Blueprints of the ZEUS are presented in Fig. 2.5 and Fig. 2.6 on the next page. ZEUS consisted of several independent subdetectors built by institutes from more than 11 countries. Starting from the interaction point next to it was The silicon Micro Vertex Detector (MVD) installed in 2001 to enable heavy quark tagging via identification of displaced secondary vertices. It was surrounded by the tracking detectors that is the Central Tracking Detector (CTD), the Forward Tracking Detector (FTD) and the Rear Tracking Detector (RTD), which were enclosed by the uranium CALorimeter (CAL) and the BAcking Calorimeter (BAC). In order to measure high energy muons crossing the calorimeters there were additional MUon Identification chambers between the CAL and the BAC: Forward (FMUI), Barrel (BMUI), Rear (RMUI), and an additional detector called MUON built of limited streamer tubes outside the BAC. The last important and yet unmentioned component of the ZEUS detector was the VETO wall –.

(29) 2.2. THE ZEUS DETECTOR. 19. an iron wall equipped with scintillator detectors on both sides designed to protect the central detector from beam halo particles. More detailed description of chosen components of the detector is presented in following sections.. Figure 2.6: Schematic of cross section of the ZEUS detector perpendicular to the beam direction (along XY plane).. 2.2.1. The central tracking detector. The Central Tracking Detector (CTD) was a cylindrical multi-wire chamber filled with mixture of argon, CO2 and ethane and lying within 1.43 T solenoid, thanks to which it could distinguish charge of incident particles. Inside its active chamber, inner and outer radii of which were equal 18.2 cm and 79.4 cm respectively, located around nominal interaction point and stretched in z from −100 cm to 105 cm, 4608 sense wires (anodes) and 19 584 field wires (cathodes) were located and grouped in 9 sections called superlayers..

(30) 20. CHAPTER 2. HERA AND THE ZEUS DETECTOR. Figure 2.7: (a) schematic of the CTD cross section along XY plane with visible ‘cylinders’ of superlayers , (b) layout of a typical CTD cell.. Odd-numbered, counting from the nominal interaction point, so called axial superlayers were parallel to Z axis, while rest of (stereo) superlayers were oriented at a small stereo angle (±5◦ ) allowing measurement of r – φ and z coordinates. In addition inner three superlayers of the CTD incorporated a z-by-timing system which provided a measurement of the z position of a hit via the measurement of the difference between times of arrivals of the signal on both ends of the same wire. The CTD covered polar angle range of 15◦ < θ < 165◦ , its resolution reached 200 mrad for both polar and azimuthal angles while its track transverse momentum (pt ) relative resolution took form σ (pt ) = pt where pt is expressed in GeV.. s 2. 2. (0.0058pt ) + (0.0065) +. . 0.0014 pt. 2 ,.

(31) 2.2. THE ZEUS DETECTOR. 2.2.2. 21. The silicon micro vertex detector. The silicon Micro Vertex Detector (MVD) [23] comprising barrel (BMVD) and forward (FMVD) sections was installed inside the CTD in 2001 to enable heavy quark tagging by identifying displaced secondary vertices and to improve tracking performance of the ZEUS detector. The BMVD had a structure of three concentric cylindrical layers placed around Z axis and comprised 600 single-sided silicon strip detectors giving the angular coverage of 22◦ < θ < 130◦ while the FMVD consisted of four planes perpendicular to the beam axis called wheels, constructed as two back to back layers of 14 silicon sensors of the same structure as those in the BMVD. Wheels being positioned at z = 32, 45 and 75 cm stretched the polar angle coverage in the forward region to 7◦ .. 2.2.3. The uranium calorimeter. The CAL, high resolution uranium-scintillator CALorimeter, consisted of three parts Rear (RCAL), Barrel (BCAL) and Forward (FCAL) each of which were further divided into modules and then towers consisting of ElectroMagnetic (EMC) and HAdronic (HAC) Cells, that together covered 99.8% of solid angle in the forward direction and 99.5% in the back. More detailed information about angular coverage and structure of different parts of the CAL are presented in Tab. 2.1.. FCAL. BCAL. RCAL. θ-range. 2.2◦ — 39.9◦. 36.7◦ — 129.1◦. 128.1◦ — 176.5◦. Number of Cells. 2172. 2592. 1668. Number of Towers per Module. 11-23. 14. 11-23. Number of Modules. 24. 32. 24. EMC cell size [cm × cm]. 5 × 20. 5 × 20. 10 × 20. HAC cell size [cm × cm]. 20 × 20. 20 × 20. 20 × 20. Depth in Radiation Lengths (X0 ). 181.0. 129.0. 103.0. Depth in Absorption Lengths (λ). 7. 5. 4. Table 2.1: Properties of the CAL sections.. Towers differed between the CAL components (see Fig. 2.8 on the next page) but each of them pointed roughly in the direction of nominal interaction point and consisted of cells that despite varying sizes (see Tab. 2.1) had the same structure of alternating layers of 3.3 mm thick absorber plates made out of depleted uranium thinly covered with steel and 2.6 mm thick plastic scintillators. Thicknesses of aforementioned layers were selected in a way ensuring equal calorimeter response to leptons and hadrons of the same energy, which is of great importance for jet measurement as they usually consists of both leptons and hadrons. The ratio of this response was found to be equal to 1.0 ± 0.05%..

(32) 22. CHAPTER 2. HERA AND THE ZEUS DETECTOR. Figure 2.8: Layouts of FCAL, BCAL and RCAL towers. The BCAL towers were wedge-shaped to ascertain better φ-resolution. Photons produced in the scintillator layers reached two Photon MulTiplayers (PMTs) outside the CAL through two wave shifters located on opposite sides of each cell giving additional information about the position within the cell. The usage of depleted uranium as an absorbent with its high atomic mass resulting in low absorption and radiation length was crucial to constructing a compact calorimeter and at the same time its natural low radiation served as a stable and well understood signal for the calibration. σ(E) √ , √ for hadrons = 35% = 18% The relative energy resolutions were equal respectively σ(E) E E E E and leptons where energy is expressed in GeV.. 2.2.4. Luminosity Monitor. The precise evaluation of the luminosity in high energy physics experiments is of crucial importance, since the luminosity uncertainty propagates to the uncertainty of the measured cross section. The ZEUS detector conducted this arduous task measuring the rate of high energy photons from electron bremsstrahlung e + p → e + p + γ. This is a high rate process cross section of which can be precisely calculated using pure QED [24] and thus the one that was suited perfectly for this purpose.. Figure 2.9: Schematic of the ZEUS luminometers at HERA II. The ZEUS detector and nominal interaction point was on the left side. Before the year 2001 bremsstrahlung photons, leaving proton beam pipe at a very small angle through 0.095 radiation wave length thick copper-beryllium window located at z = −92.5 m in.

(33) 2.2. THE ZEUS DETECTOR. 23. the ZEUS coordinate system, were measured directly by the Photon CALorimeter (PCAL), i.e. a lead-scintillator sampling calorimeter with a detector measuring shower position [25] located at z = −107 m, after they crossed 12.7 m long vacuum pipe and set of graphite filters shielding from synchrotron radiation(SR). Upgrade of the HERA accelerator enabling it to operate in higher luminosity caused additional difficulties in the measurement of the latter. The increase in luminosity was gained by higher beam current and new beam focusing scheme which lead among others to the higher synchrotron radiation and to the pile-up that is larger number of overlaid bremsstrahlung occurrences impossible to separate using calorimeter technique. To overcome this problems additional SPECtrometers (SPEC) system was applied [26]. Its design was to measure separately electron and positron produced by bremsstrahlung photon conversion, γ → e+ + e− , taking place in the beam pipe exit window. Electrons and positrons from the pairs were spatially split by dipole magnet and directed to two electromagnetic calorimeters. The magnetic field of dipole magnet placed at z = −95 m separated this pairs from initial photon beam removing at the same time low energy pairs originating from the conversion of synchrotron radiation conversion. The usage of these two complementary methods led to the estimation of systematic uncertainty of the luminosity measurement at HERA II at 1.7% [24].. 2.2.5. Trigger. In order to reconcile the fact that the nominal event rate of 10.4 MHz corresponding to the one bunch crossing per 96 ns in HERA accelerator did not meet technological constraints of a recording rate and rendered storage of such an amount of data impossible the ZEUS trigger and Data AcQuisition system (DAQ) was developed. Its aim was to remove from the data so called background events that is those not originating form the beam lepton and proton interactions, but from other sources such as cosmic radiation or most frequently from interaction of beam particles with residual gas in the beam pipe. The ZEUS triggering process was divided into three stages performed by triggers of three different levels. The First Level Triggers (FLTs) [27] were hardware-based and specific to each of the main detector components. FLTs’ analogue data based information was prepared in 2 µs and passed to the Global First Level Trigger (GFLT) which reached the decision in 4.4 µs. That time corresponded to 46 bunches full account of which had to be stored in local buffer during the whole decision making process. As a result each event was rejected and deleted or digitalised giving the final event rate of ∼ 1 kHz. There was additional element of the FLT system called Fast Clear (FC), which, in parallel to the digitalisation, processed simultaneously the RCAL, BCAL and FCAL data. Calculations performed by the FC were based on cluster algorithm and led within 50 µs to the decision of writing or aborting currently processed event before the completion of digitisation. Then accepted and digitalised data were reaching the software based Second Level Triggers (SLTs) [28]. As in case of the FLTs, the information from each detector component was processed separately and then send to the Global Second Level Trigger (GSLT) where Decision was reached in 3 ms reducing the event rate to ∼ 100 Hz. At this stage each accepted event was transformed by the ZEUS Event Builder [29] to the ADAMO format and in this form send to The Third Level Trigger (TLT) that performed offline computer-based selection giving the final event rate of ∼ 1 Hz. In the end the accepted event was passed to the ZEUS PHYsics Reconstruction program (ZEPHYR), which with use of tables of calibration constant specific for each detector component interpreted detector signals as energies, times, and positions. In such a form event was written to tape for storage and further analysis. The total decision time was approximately equal 0.3 s..

(34) 24. CHAPTER 2. HERA AND THE ZEUS DETECTOR.

(35) Chapter 3 Monte Carlo Samples The data measured by the ZEUS detector must be corrected for detector inefficiencies and smearing to enable comparison with theoretical predictions as well as the comparison with results of other experiments. These corrections are usually made using Monte Carlo simulations (MC) which provide artificial events that are directly comparable with real data. Such procedure was also applied in this thesis. The process of MC generation in particle physics is split into two phases producing so called generator and detector level MC. The output of generator level simulation comprises the list of simulated event each of which consists of the table of hadron level particles created in accordance with a set theoretical calculation or empirical model and as such is independent of the experiment itself, apart from the specifics like beam energy or types of collided particles. In the second level the interaction of these particles with matter in the detector and their reconstruction is modelled giving simulated events that are specific for given experiment and its detector configuration and have the same format as the measured data. Such simulated data are then treated in the same way as the measured ones and subjected to the same reconstruction and selection procedure. The simulation of the process of the detection was performed using the MOnte Carlo for Zeus Analysis, Reconstruction and Trigger (MOZART) package consisting of the model of ZEUS detector and based on developed at CERN package GEANT 3.13 [43] the purpose of which is to simulate the interaction of particles with matter. The simulation of trigger was done by separate package called ZGANA [44].. 3.1. Diffractive Monte Carlo. Diffractive events for the purposes of this analysis have been generated with RAPGAP [45] in which BASES [46] package is used to perform Monte Carlo integration and generation of pseudorandom numbers. The main signal MC sample of diffractive photoproduction has been generated with proton structure functions CTEQ 6.1 [47], SaS-G-1D-LO [49] for the photon and H1 set B [48] for the pomeron. The generation was divided into four sub-samples corresponding to the four sub-processes constituting (diffractive) PHP: - light quark boson-gluon fusion (BGF) (γ + g → q + q¯); - heavy quark boson-gluon fusion (BGF) (γ + g → c + c¯); - resolved PHP (g + g → q + q¯, g + g → g + g, g + q → g + q, q + q¯ → q + q¯, q + q¯ → g + g, q + q → q + q); 25.

(36) 26. CHAPTER 3. MONTE CARLO SAMPLES - QCD Compton (γ + g → g + q),. and performed in kinematic range of Q2 < 1 GeV2 , y > 0.02 with dijet cuts on Et > 4 GeV and −2.5 < η < 2.5 applied to the two leading jets i.e. jets with highest transverse energy. Then the MC sample has been weighted to obtain better description of the data. The background from diffractive deep inelastic scattering was generated with saturation model (SATRAP) [50, 51] with the proton structure function CTEQ 5D [52]. In both cases hadronisation was modelled by JETSET [53] program which is based on Lund string fragmentation model [54].. 3.2. Non-diffractive Monte Carlo. Non-diffractive PHP events have been generated separately for light flavours direct and resolved sub-processes using PYTHIA [53] with the proton structure function CTEQ 4 [55], GRV G for the photon structure function and in kinematic range of Q2 < 2 GeV.. 3.3. Proton-dissociative Monte Carlo. For the simulation of a background from the events in which proton dissociates two different MC samples generated with RAPGAP and EPSOFT [57] were used. The first, bigger one, was produced using EPSOFT for the simulation of dissociative proton system. Since the production of dijets is not implemented in this generator, diffractive dijets without proton dissociation were simulated with RAPGAP and then the intact proton was replaced with a dissociated one. Such a solution can be justified by the factorisation hypothesis that is the assumption that the interactions at both the lepton and at the proton vertex factorises. The factorisation hypothesis has been verified for diffractive photoproduction in ep collisions at HERA [56]. Another smaller proton-dissociative MC was generated using only RAPGAP with implemented proton dissociation. This MC was intended as a check of the described above procedure of mixing RAPGAP and EPSOFT particles within one event. The RAPGAP proton dissociative MC was generated with the same settings as the signal RAPGAP MC..

(37) Chapter 4 Event Reconstruction This chapter describes how the information stored in the tape after triggering and data acquisition process is used to reconstruct a physics event.. 4.1. Track Reconstruction and Vertexing. The offline reconstruction of trajectories of charged particles (tracks) at ZEUS is performed by VCTRACK routine [30] and depending on the reconstruction mode is divided into two or three stages for REG and ZTT mode respectively. During the first Pattern Recognition Phase consisting of two χ2 fits tracks are parametrised as helices, axes of which are parallel to the beam pipe. Such a description is valid only under the assumptions of homogeneity of the magnetic field and that the lines of this field are parallel to the beam axis. First fit provides the parameters of the circle constituting an orthogonal projection of idealized helical trajectory onto XY plane while the second one the information about the Z position and pitch of the helix. In the second step known as Trajectory Fit Phase these rough tracks are improved taking into account the actual inhomogeneity in the magnetic field and kinks where the track passes between different detector components. It is done by modelling the tracks as series of linked local helices from which only the one, usually the inner most, is independent. The parameters of the others are treated as functions of the parameters of this independent one [33]. Each of the local helical trajectories accounts for local parameters of magnetic field and is expressed in local coordinate system with Z axes parallel to the magnetic field. The parameters of all helices are obtained simultaneously in one χ2 fit giving final tracks in REG tracking mode. Such REG tracks can be used in physics analysis and to determine the primary vertex position that is the point in space where the electron-proton interaction occurred. The latter is done by performing χ2 fit on tracks which have origins compatible with the position of the proton beam axis. After this, if in ZTT mode, tracks can be further corrected using Kalman filters technique [32]. In this approach hits’ positions on the subsequent measurements’ planes, e.g. next MVD silicon strip detector or next wire of CTD, are treated as states of discrete dynamic system and their errors as an additive white noise what allows to use Kalman filters to obtain smoother and more robust estimation of track parameters in a way described in [31]. Thanks to this step-by-step procedure, in which each hit is re-evaluated individually taking into account its uncertainty and predictions based on previous hits, it is possible to avoid using large fit sum matrices otherwise resulting in an increase in the computation time. Within this technique, cases of multiple Coulomb scattering or ionisation energy loss can be treated locally and in a way from which no long-range correlations of the observations arise, also material effects can be evaluated more precisely. What is more, it enables optimal estimation of parameters of the tracks at any point along the track, what makes the detection of outliers and, in some cases, 27.

(38) 28. CHAPTER 4. EVENT RECONSTRUCTION. the resolution of ambiguities [34] possible. After the Kalman filter based procedure of track reconstruction, vertexing is redone using a Deterministic Annealing Filter (DAF) described in [34].. 4.2. Calorimeter Reconstruction. At the first step of calorimeter reconstruction, cells of the calorimeter are clustered into Cell Islands. Clustering algorithm runs separately on FEMC, FHAC1, FHAC2, BEMC, BHAC1, BHAC2, REMC, RHAC cells. Part of the signal form the calorimeter may consist of so called noise that is may not originate from electron-proton interaction but from other sources such as nuclear decay of depleted uranium in the calorimeter or be caused by malfunction of PMTs and readout electronics. Such signal should not be used during the reconstruction hence is removed before clustering by a set of special cuts. The noise resulting from depleted uranium decay is suppressed by the threshold cut on the energy deposited in a cell. The value of this cut is equal 60 MeV for EMC cell and 100 MeV for so called isolated that is not surrounded by other cells with energy deposit, EMC cell, and 110 MeV and 150 MeV for HAC and isolated HAC cell respectively. The cut on energy imbalance applied to cells with energy deposit greater than 1 GeV removes cells in which one of PMTs gives signal as a result of high-voltage discharge while the other one is still functioning properly. This cut takes form |Eright − Elef t | > 0.7 (Eright + Elef t ) + 0.018 [GeV], where Eright and Elef t are signals form left and right PMT. If one of PMTs is dead then the value of its energy is set to be equal to the working one, which makes the energy imbalance method inapplicable. In such cases the faulty cell can be identified during quality checks and placed on a noisy cell list. Such a decision is reached after scrutinizing the frequency of fires of given cell as the faulty ones used to fire more frequently.. calorimeter cell reference cell nearest neighbour Figure 4.1: Schematic definition of nearest neighbour of a cell. Cell Islands are created by clustering a cell to its nearest neighbour with the highest energy deposit provided that the energy of this neighbour is higher than that of a given cell. A cell is considered to be a nearest neighbour of given cell if it is below, above on the left or on the right side of a given cell, the corner cells are not included (see Fig. 4.1). Cells on the opposite sides of the beam pipe are treated as nearest neighbours which can lead to the creation of the Cone Island the center of energy of which is inside the beam pipe..

(39) 4.2. CALORIMETER RECONSTRUCTION. 29. Figure 4.2: Schematic example of cell island clustering. In the last step Cell Islands are clustered to form Cone Islands starting from the ones consisting of the cells of the most external calorimeter components and proceeding toward the center of the detector. Clustering algorithm uses the angular separation between cells at the nominal interaction point denoted by d as a measure of distance and uses this distance to calculate parameter a given by the formula    exp (−643.6d4 + 483.2d3 − 103.7d2 − 6.527d) , if both cell island are hadronic a (d) =   exp (−650.5d4 + 509.2d3 − 100.1d2 − 11.75d) , otherwise. Then value of this parameter is used to decide whether to cluster given cell islands or not. The formula above for the parameter a was obtained from single pion MC simulations. The algorithm commences its operation finding for each HAC2 cell island HAC1 and EMC island which are the closest to the given one in the sense of above mentioned angular separation. Then distances between HAC2 and HAC1 islands, dh . and HAC2 and EMC island, de . are used to evaluate a (dh ) and a (de ). Out of two possible combination HAC2-HAC1 and HAC2-EMC only the pair of Cell Islands with the highest a is clustered together and it happens only if that a is greater than 0.05. In the next step HAC1 cell islands are connected to those from EMC. Once again for each of the HAC1 islands closest one in EMC is found and connected or not depending on whether the value of parameter a is greater than 0.05 or not. Then EMC islands are connected on similar basis but this time a must be greater than 0.2. Position of cone island is reconstructed using logarithmic weights compensating the exponential falloff of shower energy. They take as an input the energy deposit as well as its position in given cell – the position of the geometrical center of a cell shifted according to the magnitude of signal from both PMTs. Weight is given by the formula    max {0, 2 + log (Ei /Etot )} , for HAC cell wi =   max {0, 4 + log (Ei /Etot )} , for EMC cell.