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Home , Cosmic ray, Deuterium

CERN-THESIS-2019-034 26/04/2019 etrnadatdueo rdcini galactic in production antideuteron and Deuteron NVRIA AINLAUT NACIONAL UNIVERSIDAD U AAOTRPRE RD DE: GRADO EL POR OPTAR PARA QUE IDDD M DE CIUDAD OGAOE INISF CIENCIAS EN POSGRADO r ruoAeadoMnhc Rocha Menchaca Alejandro Arturo Dr. IMRSDLCOMIT DEL MIEMBROS OTRE INIS(F CIENCIAS EN DOCTOR ig arcoG´omez Coral Mauricio Diego r nre advlEspinosa Andr´es Sandoval Dr. NTTT EF DE INSTITUTO UO PRINCIPAL: TUTOR cosmic-rays r alnGrabski Varlen Dr. nttt eF´ısica de Instituto F´ısica de Instituto nttt eF´ısica de Instituto PRESENTA: XC 9d bi e2019 de abril de 29 EXICO ´ Tesis ´ ISICA NM EM DE ONOMA TUTOR: E ´ ´ ´ ISICA) ´ ISICAS EXICO ´ Producci´onde deuterio y antideuterio en rayos c´osmicosgal´acticos

Diego Mauricio G´omezCoral

29 de abril de 2019 To my beloved family: Glorita, Hernando, Carito, and Mart´ın. Abstract

The increasing role of antideuterons in the indirect search for dark and new cosmic-ray propagation features from deuterons have generated that the study of the cosmic-ray deuteron and antideuteron receives an increasing interest in current investigations. To de- termine such fluxes precisely, it is necessary to consider the contribution from the nuclear in- teractions of primary cosmic rays with the interstellar matter. For that purpose, in this thesis, the production of cosmic-ray deuterons and antideuterons in the is calculated using advanced Monte Carlo (MC) generators and an updated database of measurements. Deuteron and antideuteron production in interactions is modeled through the coa- lescence model, which has demonstrated to reproduce data successfully. The coalescence mech- anism is simulated with an event-by-event afterburner in tandem with MC generators such as EPOS-LHC, QGSJET-II-04, and FTFP-GEANT4. These estimates depend on a single parame- ter (p0) obtained from a fit to the latest collider data, including ALICE-LHC current results. It was found that p0 for antideuterons is not a constant at all as previous works suggested, and as a result, the antideuteron production can be at least 20 times smaller in the low collision region than earlier estimations. CR deuterons and antideuterons propagation in the Galaxy is calculated using GALPROP, along with a set of parameters that fit AMS-02 latest observations in , and - to- ratio. The resulting antideuteron flux, employing EPOS-LHC, shows a larger mag- nitude compared to previous studies and a slightly different shape in the energy distribution as a consequence of the coalescence parameter (p0) energy dependence. Deuteron flux evaluation with QGSJET-II-04 shows an improved description of measurements at high energy compared to the result without considering coalescence. A simulation of the antideuteron production in AMS-02 as a consequence of CR collisions with the detector materials was conducted. It was concluded, that it is unlikely an antideuteron generated in the detector can be misidentified as an antideuteron from an external source in the lifetime of AMS-02 operations.

iii iv Resumen

El creciente inter´esen la b´usquedaindirecta de materia oscura a trav´esde la detecci´onde antideuterones, as´ıcomo la posibilidad de encontrar nuevas caracter´ısticasen la propagaci´onde rayos c´osmicosen la galaxia usando la raz´onentre deuterones y helio, motivan la realizaci´on de este trabajo. En este contexto, se vuelve fundamental entender y calcular con las her- ramientas m´asmodernas y con los conocimientos m´asrecientes la producci´onde deuterones y antideuterones a trav´esde colisiones nucleares entre los rayos c´osmicosprimarios y el medio inter- estelar. Por estos motivos en este trabajo se estima la producci´onde deuterones y antideuterones en la galaxia usando generadores Monte Carlo (MC) y una base de datos actualizada de medidas realizadas en experimentos sobre colisiones de part´ıculas. La producci´onde deuterones y antideuterones en la galaxia se describe con el modelo de coalescencia, el cual a demostrado ser acertado en la predicci´onde los datos experimentales. La implementaci´onde la simulaci´ondel modelo de coalescencia se realiza usando un gratinador o “afterburner”, en conjunto con los generadores Monte Carlo (EPOS-LHC, QGSJET-II-04 y FTFP-GEANT4). Las predicciones de la simulaci´ondependen de un solo par´ametro,el momento de coalescencia (p0), el cual se obtiene de los datos sobre colisiones incluyendo los resultados actuales de ALICE-LHC. En este trabajo se encontr´oque para antideuterones p0 no es constante a todas las energ´ıascomo se afirmaba en trabajos previos. Como consecuencia, la secci´oneficaz de producci´onde antideuterones puede llegar a ser 20 veces menor a lo encontrado anteriormente en la regi´onde baja energ´ıa. La propagaci´onde deuterones y antideuterones en la galaxia fue calculada con GALPROP, usando par´ametrosque fueron ajustados a datos recientes del experimento AMS-02. El flujo de antideuterones estimado con EPOS-LHC, muestra una magnitud mayor comparada con c´alculos anteriores, as´ı como tambi´enuna forma ligeramente diferente en la distribuci´onde energ´ıa. Dicha diferencia es consecuencia de p0 y su dependencia con la energ´ıa.Por otro lado, el flujo de deuterones en rayos c´osmicoscalculado con QGSJET-II-04 y el modelo de coalescencia describe apropiadamente los datos a altas energ´ıas. Por ´ultimo,se realiz´ouna simulaci´onde la producci´onde antideuterones en AMS-02 como consecuencia de las colisiones de rayos c´osmicoscon los materiales del detector. Se concluy´o que es muy poco probable que un antideuteron producido en el detector se confunda con un antideuteron proveniente de la Galaxia durante el tiempo de operaci´onde AMS-02.

v vi Acknowledgments

I want to express my sincere gratitude to my advisor Prof. Arturo Menchaca for his continuous support developing ideas for the construction of this work, as well as his comments, critics and corrections to this manuscript. Furthermore, I thank him for encouraging me to recognize my abilities and to pursue a research career. My sincere thanks also goes to the rest of my advisor committee: Prof. Andr´esSandoval and especially Prof. Varlen Grabski, who helped me many times to solve arising questions during the elaboration of this work. Besides, I want also to thank the thesis evaluation committee: Dra. Irais Bautista, Dra. Catalina Espinoza, Dr. Gustavo Medina Tanco, and Dr. Jos´eVald´es. I am also grateful to the cosmic-ray group at the University of Hawaii at Manoa led by Prof. Philip von Doetinchem, and its members Dr. Amaresh Datta, and Anirvan Shukla. A special mention to the computation department of the Institute of Physics, UNAM in particular to Carlos Ernesto L´opez Natar´en.I also thank T. Pierog, C. Baus, and R. Ulrich for providing the Monte Carlo package and for answering my questions about the program. The GALPROP team for allowing me to use and to edit the propagation code. Vladimir Uzhinskii and Dennis Wright who observed and commented on some of the results from this work and made suggestions about the afterburner implementation in GEANT4. I gratefully acknowledge the financial support by CONACyT through a Ph.D. fellowship and PNPC project, and by UNAM projects: PAPIIT-DGAPA IN109617 and PAEP. A special men- tion to the institutions involved and their people: Instituto de F´ısica(IFUNAM) and Posgrado en Ciencias F´ısicasUNAM (PCF). My gratitude goes to the ALICE and AMS-02 collabora- tions for allowing me to participate in such amazing experiments and to show me a of knowledge and exciting open questions in physics and astrophysics from which I am still learning. Last but by no means the least, I would like to thank my family: Glorita, Hernando, Carito y Mart´ınfor their unconditional love and support during my studies and in my . Special gratitude to my Colombian friends: Tatiana, Juano and Myriam. To all my friends and col- leagues who made my stay in Mexico a wonderful experience. Among them, los lovers: Guerito, Gusi, Cesar, Vladis, R2, Yorch, Peter, Jefe, Miguelover, Javier, Richi y Sult´an.Los del ciruelo: Francisco, Richard, Rafa, and Chava. My friends from the institute: Karina, David, Mario, Mariana, To˜no,Javi, Zu˜niga,Bora, and Sa´ul.

Diego G´omez Mexico City, April 2019

vii viii Agradecimientos

Agradezco profundamente a mi asesor el Dr. Arturo Menchaca por su constante apoyo en el desarrollo de ideas para la construcci´onde este trabajo, as´ıcomo su ayuda en la revisi´ony correcci´ondel mismo en varias etapas de su elaboraci´on.Tambi´enagradezco la motivaci´onque me ha transmitido para continuar un proceso acad´emico y de investigaci´on.Adem´as,agradezco a los dem´asmiembros del comit´etutor Dr. Andr´esSandoval Espinoza y especialmente al Dr. Varlen Grabski con quien acud´ınumerosas veces para consultarle inquietudes que surg´ıandurante el desarrollo del trabajo. Adem´as,quiero agradecer a los miembros del jurado evaluador: Dra. Irais Bautista, Dra. Catalina Espinoza, Dr. Gustavo Medina Tanco, y Dr. Jos´eVald´es. Tambi´enquiero expresar mi gratitud al grupo de rayos c´osmicos de la universidad de Hawaii encabezado por el Dr. Philip von Doetinchem, y sus miembros Dr. Amaresh Datta y Anirvan Shukla. Al grupo de computaci´oncient´ıfica del Instituto de F´ısica, especialmente a Carlos Ernesto L´opez Natar´en. Igualmente agradezco a T. Pierog, C. Baus, and R. Ulrich por facil- itarme el programa CRMC y resolver algunas dudas acerca de su funcionamiento, tambi´enal equipo de GALPROP por suministrar el c´odigoque se utiliz´oen este estudio. A Vladimir Uzhin- skii y Dennis Wright quienes tuvieron la amabilidad de observar y criticar algunos resultados de este trabajo as´ıcomo hacer sugerencias sobre la implementaci´ondel afterburner en GEANT4. Agradezco igualmente el apoyo financiero brindado por CONACyT a trav´esde la beca para doctorado nacional, a la UNAM con el proyecto PAPIIT-DGAPA: IN109617 y los apoyos PAEP y PNPC. Mi reconocimiento es tambi´enpara las instituciones que hicieron posible la realizaci´on de este proyecto, el Instituto de F´ısica(IFUNAM) y el posgrado en ciencias f´ısicasde la UNAM (PCF) junto al personal que trabaja en ellas. De igual manera agradezco a las colaboraciones de los experimentos ALICE y AMS-02 por permitirme participar en experimentos tan espectacu- lares, y por mostrarme un universo de conocimiento y preguntas abiertas en f´ısicade part´ıculas y astrof´ısicadel cual a´uncontinuo aprendiendo. Por ´ultimo,pero no menos importante quiero agradecer a mi familia: Glorita, Hernando, Carito y Mart´ınpor su apoyo y amor incondicional. Un especial agradecimiento a mis amigos de Colombia: Tatiana, Juano y Myriam. A todos mis amigos y amigas, compa˜nerosy compa˜neras que han hecho de mi estad´ıaen M´exico una experiencia fabulosa. Entre ellos los lovers: Guerito, Gusi, Cesar, Vladis, R2, Yorch, Peter, Jefe, Miguelover, Javier, Richi y Sult´an,espero que alg´un d´ıalevantemos esa copa. A los del ciruelo: Francisco, Richard, Rafa y Chava. A los del instituto: Karina, David, Mario, Mariana, To˜no,Zu˜niga,Bora y Sa´ul.

Diego G´omez Ciudad de M´exico, Abril 2019

ix x Contents

Abstract III

Acknowledgments VII

List of Figures XIV

List of Tables XXI

Introduction 1

1. Background 3 1.1. Galactic cosmic-rays (CRs) ...... 3 1.1.1. CR acceleration ...... 6 1.1.2. CR propagation ...... 9 1.2. Search for (DM) ...... 14 1.2.1. DM candidates ...... 17 1.2.2. DM detection ...... 19 1.3. in CRs as a signature for DM ...... 24 1.3.1. Experimental search for CR antinuclei ...... 29 1.4. High-energy hadron collisions ...... 33 1.4.1. Definitions and notation ...... 33 1.4.2. General features of hadron collisions ...... 36 1.5. Monte Carlo (MC) generators ...... 40

2. Deuteron and antideuteron production model in high-energy collisions and space 45 2.1. Coalescence model ...... 45 2.1.1. Analytical approach ...... 47 2.1.2. MC approach ...... 47 2.2. CR antideuterons as a signature for DM ...... 48 2.3. Deuteron as a probe for CR propagation ...... 51

xi CONTENTS

3. Deuteron and antideuteron production simulation 53 3.1. Simulation ...... 53 3.1.1. MC generators ...... 53 3.1.2. Afterburner ...... 56 3.2. Results ...... 57 3.2.1. p and ¯pproduction simulation and MC selection ...... 58 3.2.2. d and d¯ Production Simulation ...... 62

4. Galactic secondary deuteron and antideuteron flux 71 4.1. GALPROP ...... 71 4.1.1. Galactic structure ...... 71 4.1.2. Galactic source distribution ...... 72 4.1.3. Galactic magnetic field ...... 72 4.1.4. Interstellar distribution ...... 72 4.1.5. Isotopic abundances ...... 73 4.1.6. Transport equation ...... 74 4.1.7. Solar modulation ...... 76 4.2. Deuteron and antideuteron source terms ...... 77 4.2.1. Deuterons ...... 77 4.2.2. Antideuterons ...... 78 4.3. Deuteron and antideuteron interaction with matter ...... 80 4.3.1. Antideuteron inelastic cross-section ...... 80 4.4. Results ...... 82 4.4.1. GALPROP validation ...... 82 4.4.2. Deuteron flux ...... 85 4.4.3. Antideuteron flux ...... 86

5. Secondary antideuterons produced in AMS-02 89 5.1. Afterburner implementation in GEANT4 ...... 89 5.2. Simulation ...... 91 5.2.1. AMS-02 geometry ...... 91 5.2.2. Event generation ...... 93 5.2.3. Event selection ...... 93 5.3. Results ...... 94 5.3.1. Antideuteron selection ...... 95 5.3.2. Antideuteron misidentification ...... 97

6. Conclusions 99

Appendices

A. Afterburner 103 A.1. selection ...... 103 A.2. Coalescence conditions ...... 104 A.3. Analysis ...... 105

xii CONTENTS

B. MC simulations vs accelerator data 107 B.1. Comparison of simulations to accelerator data (p, ¯p,d and d)¯ ...... 107 B.1.1. p+p and p+Be at plab = 19.2 GeV/c ...... 107 B.1.2. p+p at plab = 24 GeV/c ...... 108 B.1.3. p+C at plab = 31 GeV/c ...... 108 B.1.4. p+p, p+Be and p+Al at plab = 70 GeV/c ...... 108 B.1.5. p+p, p+C at plab = 158 GeV/c ...... 110 B.1.6. p+Be, p+Al at plab = 200 GeV/c ...... 110 B.1.7. p+p, p+Be at p = 300 and 400 GeV/c ...... 110 √ lab B.1.8. p+p at s = 45 and 53 GeV ...... 111 √ B.1.9. p+He at s = 110 GeV ...... 113 √ NN B.1.10. p+p at s = 900 and 7000 GeV ...... 113 B.2. (Anti)proton mismatch factorization for EPOS-LHC and FTFP-BERT ...... 113

C. Galdef file 115

Acronyms 117

Notation 119

Bibliography 121

xiii CONTENTS

xiv List of Figures

1.1. Element abundance in cosmic-rays compared to element abundance in the solar- system. Figure taken from Reference [23] ...... 4 1.2. Energy cosmic-ray spectrum. Figures were taken from Reference [22] (left) and Reference [3] (right). The inset in the left figure shows the H/He ratio at constant rigidity...... 5 1.3. (Left) Gamma-ray emission measured by HESS (color), and X-ray emission mea- sured by ASCA (lines) of the SNR RX J1713.7-3946 [37]. (Right) Thermal X-ray emission from the Tycho SNR measured by Chandra [38]...... 7 1.4. basic geometry from its side (not to scale). Figure adapted from [3]. .9 1.5. (Left) Galactic magnetic field model with an axisymmetric configuration. (Right) Galactic magnetic field model with an bisymmetric configuration. Figure taken from [46]...... 10 1.6. Boron-to-Carbon ratio measured by AMS-02 [49]; compared with previous exper- iments...... 11 1.7. Galactic geometry considered in the propagation model. Figure taken from [50]. . 12 1.8. Rotation curve as a function of distance to the center of the spiral galaxy NGC 3198. The disk line is the expected rotation curve when only from visible is considered. The halo line is the extra dark matter needed to reproduce the observed rotation curve. Figure taken from [58]...... 14 1.9. (Color online) (Left) Image of the gravitational lensing effect in the galaxy cluster CL0024+17, captured by the Hubble Space . (Right) Same image with the expected dark matter distribution superimposed in diffused blue. Figures obtained from [59]...... 15 1.10. (Color online) Mass distribution after the collision of two clusters of . The image combines the contribution from two different observation techniques. (Pink) X-ray: NASA/CXC/CfA/M.Markevitch et al. [63] ; Optical: NASA/STScI; Magellan/U.Arizona/D.Clowe et al. [60]; (Blue) Lensing Map: NASA/STScI; ESO WFI; Magellan/U.Arizona/D.Clowe et al. [60] ...... 16 1.11. Temperature fluctuations in the CMB detected by Planck at different angular scales on the sky. Dots are the measurements, while the curve represents the best fit of the standard model of cosmology to the Planck data. Figure taken from [64] 17 1.12. Standard Model . Areas of the circles are proportional to the . Figure taken from [18]...... 18

xv LIST OF FIGURES

1.13. (Left) Diagram of the elastic between a WIMP and detector nuclei. (Right) Energy deposited from the WIMP interaction is measured as heat, charge or . The figure shows the numerous experiments currently active and what they measure...... 21 1.14. WIMP cross-section per nucleon for -independent coupling as a function of WIMP mass. Exclusion regions set by different experiments are presented. The enclosed areas are regions of possible signal events explored by DAMA/LIBRA [77] and CDMS-Si, for more details see Reference [22]...... 22 1.15. WIMP cross-section per nucleon for spin-dependent coupling as a function of mass, for more details see Reference [22]...... 23 1.16. Schematic diagram for the annihilation of neutralinos into particle- pairs. Figure taken from [53]...... 24 1.17. (Left) to proton ratio measurements from AMS-02 [15], compared to a secondary production model, and a dark-matter-origin model [94]. Figure taken from [95]. (Right) Antiproton flux reevaluated by [93] compared to AMS-02 data [15]...... 25 1.18. (Left) Best fit to the antiproton-to-proton ratio AMS-02 data [15] using a hy- pothetical dark matter component. (Right) Best fit regions for a dark matter component of the antiproton flux. The best-fit region for the dark matter inter- pretation of the Galactic center gamma-ray excess (black region) is also shown. Figures are taken from [96]...... 26 1.19. (Left) fraction measured by previous experiments to AMS-02 such as HEAT [105] and PAMELA [106]. (Right) Latest AMS-02 results on the positron fraction presented in [108, 109]...... 27 1.20. (Left) Latest AMS-02 results [108, 109] for the positron fraction compared to a dark matter model [19]. (Right) AMS-02 data [14] for the positron fraction compared to a model of pulsars as sources of high-energy [125]...... 28 1.21. (Left) Estimated positron flux at from Geminga (blue solid line), compared to AMS-02 data (green dots). The shaded blue region indicates a 3σ (99.5% confidence) statistical uncertainty from simulations [127]. (Right) Total flux and positron fraction predicted by the production model [123], compared to AMS-02 data [14]...... 29 1.22. AMS-02 detector. (Left) Structure diagram of the detector. (Right) Real event reconstructed...... 30 1.23. GAPS detector. (Left) Structure diagram of the detector. (Right) Antiproton, antideuteron interaction picture with the detector. Figures taken from [128, 129]. 32 1.24. (Left) Rapidity distribution in p+p collisions at 15.4 GeV/c (circles) 26.7 GeV/c (triangles) [132]. (Right) xF distribution of in p+C collisions at 158 GeV/c...... 34 1.25. Diagrams of the different types of inelastic interactions. P indicates Pomeron exchange. (a) and (b) are single-diffractive events, while (c) is a double-diffractive collision, and (d) represents a central collision or non-diffractive. Figure taken from [133]...... 35 1.26. (a) Lowest-order qq¯ interaction (b) and (c) Lowest-order corrections to the - gluon coupling. (d) qq¯ colour field with V (r) ∼ λr...... 37 1.27. (Left) Hadronic reaction A + B → C + X at large pT , in terms of the parton sub-process a + b → c + d [132]. (Right) Proton structure function given at two Q2 values (6.5 GeV2 and 90 GeV2). For more details see [22]...... 38 1.28. Event structure in a Monte Carlo generator of the p+p collision...... 41

xvi LIST OF FIGURES

1.29. (Left) Motion and breakup of a string system with two transverse degrees of freedom suppressed. (Right) Color structure of a parton shower in the framework of the cluster model...... 42

2.1. (Left) Expected antideuteron flux from three different dark matter candidates [148, 163, 164] compared to expected secondary production [151] and the corresponding detector sensitivity limits. (Right) Expected antideuteron flux from heavy dark matter candidates [165, 166]...... 50 2.2. (Left) Deuteron flux measurements by [170, 171, 172, 173, 174]. Figure taken from [174]. (Right) Measurements of the ratio 2H/He by [170, 174, 175, 176, 177]. Figure taken from [177]...... 51

3.1. Invariant differential cross-sections as a function of rapidity (y) are calculated with different MC models for a), and antiprotons b) in p+p collisions at 158 GeV/c. Results for two bins of transverse momentum (pT ) are compared with data from experiments NA49 [82] and NA61 [85]...... 55 3.2. Diagram of the afterburner implementation for the production of deuterons and antideuterons...... 57 3.3. Distributions of difference between measurements and the MC generators divided by the uncertainty (see Eq. 3.1) for proton production in p+p and p+A collisions. 58 3.4. Distributions of difference between measurements and the MC generators divided by the uncertainty (see Eq. 3.1) for antiproton production in p+p and p+A collisions. 59 3.5. Distributions, in two different energy regions, of the difference between measure- ments and EPOS-LHC divided by the uncertainty (see Eq. 3.1) for proton (left) and antiproton (right) production in p+p and p+A collisions...... 60 3.6. EPOS-LHC total antiproton production cross-section comparted to parametriza- tions from Duperray et al. and Korsmeier et al. in p+p collisions (left) and p+He, He+p, and He+He (right)...... 61 3.7. Antiproton and antideuteron invariant differential cross-sections in p+p collisions at 70 GeV/c as function of transverse momentum (pT ) calculated with EPOS- LHC, FTFP-BERT and parametrizations [160, 161]. The results are compared to data [196, 197, 198] (see text for details)...... 62

3.8. Extracted coalescence momentum p0 (symbols) for deuterons (a) and antideuterons (b) as function of the collision kinetic energy (T). Fit functions [Eqs. (3.4) and (3.5)] for EPOS-LHC (long-dashed red line) and FTFP-BERT (dashed blue line) are shown. Additionally, the p0 values obtained from the analytic coalescence model and the parametrization of Korsmeier et al. are included (dashed cyan line and dots). Also, the constant value of p0 = 79 MeV/c estimated by Duperray et al. is plotted (solid magenta line)...... 63 3.9. Deuteron (a) and antideuteron (b) total production cross-section in p+p colli- sions. Deuteron (c) and antideuteron (d) total production cross-section in p+He collisions. The expected antideuteron cross-section from Duperray’s parametriza- tion has been added. In the lower panels Duperray to MC predictions in an- tideuteron are compared. Vertical broken lines represent the antideuteron pro- duction threshold...... 65

xvii LIST OF FIGURES

2 3.10. (Left) Antiproton differential cross-section as a function of pT in ¯p+ p collisions at 32 GeV. Results from simulations with Monte Carlo generators are compared to data from [214]. (Right) Total antideuteron production cross-section in ¯p+ p, ¯p+He, He+p and He+He collisions as a function of kinetic energy of the collision T...... 66 3.11. Parametrization and data for deuteron production in the reaction 4He + p → d (Left) and 3He + p → d. Stripping (dotted red line), break-up (dashed blue line) and total (solid black line) parametrizations fitted to data are shown togheter with simulations results from QGSJET-II-04 (dotted-dash and long-dash green lines) and EPOS-LHC (dotted-dash and long-dash red lines) (see text for details). Data and parametrizations taken from [216]...... 68

4.1. Adopted radial distribution of atomic (HI), molecular (H2), and ionized (HII) at z = 0. Figure taken from [43]...... 73 4.2. Secondary source term for antideuterons as function of kinetic energy per nucleon. On the left side the total contribution from the most important collisions is shown. On the right side, the result is compared to previous works [151, 157]...... 79 4.3. (Left) Total cross-section data of thepd ¯ reaction and its parametrization (black solid line) [22]. The black dashed line correspond to the Glauber approximation used in [157]. The two dotted curves are estimations of the dp¯ total cross-section by means ofpp ¯ andpn ¯ cross-sections. (Right) Elastic cross-section data for the pp¯ andpn ¯ reactions and its parametrization (dotted line) [22]. The black line ¯ ¯pp is a model of the pd elastic cross-section using 2σel . The dashed-dotted line corresponds to the Glauber cross-section used by [157]. Figures taken from [158]. 81 4.4. (Left) Total, inelastic, elastic and inelastic non-annihilation cross-sections for d¯ + H as a function of the antideuteron kinetic energy per nucleon. (Right) Differential redistribution cross-section d¯+ p → q¯ + X as a function of the an- tideuteron kinetic energy per nucleon. Figures taken from [158]...... 82 4.5. Results obtained with GALPROP and the parameters in Table 4.1 for proton, Boron-to-Carbon ratio and Helium compared to the most recent data from AMS- 02 experiment...... 83 4.6. Comparison between the antiproton flux obtained by Winkler et al. [93] and the secondary source term in Winkler et al. propagated with GALPROP...... 84 4.7. (Left) Antiproton secondary source term for EPOS-LHC. (Right) Antiproton flux obtained with EPOS-LHC and GALPROP compared to Winkler et al. [93] and AMS-02 data...... 85 4.8. (Left) Deuteron flux produced using QGSJET-II-04 with coalescence (solid cyan line) and without coalescence (dotdashed cyan line) propagated with GALPROP and compared to data from CAPRICE [174] and AMS [173]. (Right) Deuteron-to- Helium ratio compared to data from CAPRICE [174], IMAX [175] and SOKOL [177]. 86 4.9. Antideuteron flux at the top of the obtained with EPOS-LHC and GALPROP. (Left) The result is compared to previous works from Ibarra et al. [151], Donato et al. [158] and Duperray et al. [157]. (Right) The result is compared to an- tideuteron fluxes expected from dark matter annihilation or decay [148, 163, 164]. Sensitivity limits for AMS-02 and GAPS are also included...... 87

5.1. Diagram of the afterburner implementation at the user level, for antideuteron production in GEANT4...... 90 5.2. Level 3 implementation framework in the hadronic category of GEANT4 [239]. . 91

xviii LIST OF FIGURES

5.3. Detector geometry model used in the GEANT4 simulation (not to scale). . . . . 92 5.4. Final AMS-02 acceptance for antideuterons signal [130]...... 94 5.5. (Left) Energy distribution of the antideuterons likely to be misidentified by the detector. (Right) Dispersion of the number of tracks per event vs kinetic energy of the antideuterons in that event...... 95 5.6. (Left) Dispersion of the number of tracks per event vs kinetic energy of the an- tideuterons in that event. (Right) Angular distribution of the secondary particles produced in an event, respect to the generated antideuteron...... 96 5.7. Event simulation of an antideuteron (red line) and the additional secondary tracks produced by the collision of a proton with the AMS-02 detector...... 96 5.8. Upper limit flux for antideuterons produced in the materials of AMS-02 detector, that might be misidentified with Galactic antideuterons...... 97

A.1. TTree scheme of the event information saved in a ROOT file...... 103

B.1. Double differential cross sections from MC models compared to data of protons and deuterons produced in p+p collisions at 19 GeV/c [194]...... 107 B.2. Double differential cross sections from MC models and Duperray’s parametriza- tion (pink line) compared to data of antiprotons produced in p+Be collisions at 19.2 GeV/c [193]...... 107 B.3. Double differential cross sections from MC models compared to data of protons and deuterons produced in p+p collisions at 24 GeV/c [194]...... 108 B.4. Double differential momentum distribution from MC models compared to data of protons produced in p+C collisions at 31 GeV/c [195]...... 108 B.5. Invariant differential cross section for protons and deuterons produced in p+p collisions at 70 GeV/c. Data taken from [196, 197, 198]...... 109 B.6. Invariant differential cross section for protons and deuteron produced in p+Be collisions at 70 GeV/c. Data taken from [198, 198]...... 109 B.7. Invariant differential cross section for antiprotons and antideuterons produced in p+Be collisions at 70 GeV/c. Data taken from [198, 198]...... 109 B.8. Invariant differential cross section for protons produced in p+p collisions at 158 GeV/c. Data taken from [82]...... 110 B.9. Invariant differential cross section for antiprotons produced in p+C collisions at 158 GeV/c. Data taken from [83]...... 110 B.10.Invariant differential cross section for protons and deuteron produced in p+Be collisions at 200 GeV/c. Data taken from [200, 205]...... 111 B.11.Invariant differential cross section for antiprotons and antideuterons produced in p+Be collisions at 200 GeV/c. Data taken from [200, 205]...... 111 B.12.Invariant differential cross section for protons and deuterons produced in p+Be collisions at 300 GeV/c. Data taken from [202, 203]...... 111 B.13.Invariant differential cross section for antiprotons and antideuterons produced in p+Be collisions at 300 GeV/c. Data taken from [202, 203]...... 111 B.14.Invariant differential cross section for protons and deuteron produced in p+p √ collisions at s = 53 GeV. Data taken from [204, 208]...... 112 B.15.Invariant differential cross section for antiprotons and antideuterons produced in √ p+p collisions at s = 53 GeV. Data taken from [204, 206, 207]...... 112 √ B.16.Differential cross section for antiprotons produced in p+He collisions at sNN = 110 GeV. Data taken from [86]...... 112

xix LIST OF FIGURES

B.17.Invariant differential cross section for protons and deuteron produced in p+p √ collisions at s = 900 GeV. Data taken from [84, 155, 209]...... 113 B.18.Invariant differential momentum distribution for antiprotons and antideuterons √ produced in p+p collisions at s = 900 GeV. Data taken from [84, 155, 209]. . . . 113 0 B.19.(Color online) Extracted coalescence momentum p0 (symbols) for deuterons (a) and antideuterons (b) as function of the collision kinetic energy (T). Fit functions [Eqs. (3.4) and (3.5)] for EPOS-LHC (long-dashed red line) and FTFP-BERT (dashed blue line) are shown...... 114

xx List of Tables

1.1. Dark matter particle candidates. Particle physics motivations have been abbrevi- ated as GHP (gauge hierarchy problem) and NPFP (new physics flavor problem). √√ For more details see [18]. The symbol is for a generic detection signal while √ is for a possible signal. Dark matter detection techniques are introduced in the next section. Table taken from [18]...... 20

3.1. Experimental data list on proton and antiproton production in p+p and p+A collisions considered in this thesis to be compared to simulations...... 56 3.2. Experimental data list on deuteron and antideuteron production in p+p and p+A collisions considered in this thesis to be compared to simulations...... 58 3.3. Values of the parameters for the fitting functions 3.4 and 3.5...... 64

4.1. Propagation parameter values and spectral parameters obtained from Reference [229]. 74 4.2. Spectral parameters obtained from Reference [229]...... 75

xxi LIST OF TABLES

xxii Introduction

The Earth is continuously hit by a special kind of called Cosmic Rays (CRs). This extraterrestrial radiation travels millions of light years through the galaxy, interacting with the (ISM) to before reaching our . CRs were discovered by Victor Franz Hess in 1912, from his studies about air in balloons flights [1] (for an extensive historical review see reference [2]). Since then, the and the sources of CRs have been intensive fields of investigation. It is now known that CRs are atomic nuclei, mostly protons (86%) and helium (11%), but also elemental particles such as (2%) and (less than 1%) [3]. The origin of CRs is an unsolved astrophysical problem, but it is known that a good part of these particles is produced in very energetic astrophysical events in our Galaxy like explosions, su- pernova remnants (SNRs), pulsars, black holes, and active galactic nuclei (AGNs). In the last decades, antimatter has also been detected in CRs, specifically positrons, and antipro- tons [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15](and possibly antihelium [16]). Antimatter in CRs constitutes less than 0.01%, which is why its detection is challenging. In a universe dominated by matter, the production of antimatter would only be possible through the nuclear interactions of CRs with the ISM, so-called secondary production. However, experimental measurements on positron fraction have shown disagreement with the expected signal from this secondary pro- duction [6, 7, 9, 10, 11, 12, 13, 14], particularly evidence of an excess in the positron fraction above 10 GeV. This excess represents a lack in the knowledge of the CRs sources and can be interpreted as the underestimation of standard sources or as the presence of new ones. The existence of dark matter is established on different length scales, from galaxies to galaxy clusters to cosmic background [17]; however, its nature remains unknown. Models beyond Standard Model (SM) predict the existence of new particles with special features mak- ing them good candidates for dark matter [18], being the WIMPs (Weak Interacting Massive Particles) the best-suited option. As a consequence of dark matter cooling, due to the universe expansion, the candidates could annihilate or decay into SM particles, such as positrons, antipro- tons, and antideuterons. Hence, the detection of these particles is ideal in the indirect search for dark matter. As mentioned before, positrons and antiprotons have already been detected, and the latest results from AMS-02 about the positron fraction [14] showed a structure that might be interpreted as evidence for some form of dark matter [19]. However, there are other interpre- tations not related to dark-matter that also could explain this structure [20]. Furthermore, in the case of antiproton-to-proton ratio, the results are inconclusive, indicating how difficult it is to identify a dark matter signal based solely on positrons and antiprotons as a result of the high

1 contribution from secondary sources. An antideuteron production signal from dark matter annihilation is expected to populate the low energy region of the spectrum (< 1 GeV/n), because of the near-to-rest momentum of the dark matter particles. In contrast, the secondary antideuteron flux coming from CRs interactions is only dominant at higher energies, due to the kinematic characteristics of the collisions. This energy separation represents a promising panorama in comparison to positrons and antiprotons since the secondary signal is the background in the search for a possible exotic source of antimatter. In this context, it is necessary to calculate precisely the antideuteron production from sources not related to dark matter, in order to avoid misidentification, what constitutes the main aim of this thesis. The principal uncertainties in a secondary production are the production cross sections of the particles and the model propagation parameters of CRs through the Galaxy. Thus, in the light of new data from ALICE-LHC experiment in hadron collisions and AMS- 02 in CRs, as well as the improvements in Monte Carlo generators and galactic propagation tools, this work is intended to reduce the uncertainties and to estimate a more precise secondary antideuteron flux. The galactic propagation of CRs is also a highly relevant topic of research, in order to understand the composition and the processes that are taking place. Abundance measurements of stable CRs nuclei are an essential indicator of the amount of matter traversed by CRs from their sources to Earth since some nuclei are essentially produced only by interactions of primary CRs with the ISM. From the secondary-to-primary nuclei ratio, valuable information related to grammage and diffusion in the galactic magnetic field is extracted and corroborated with models. The best-known example is the Boron-to-Carbon ratio (B/C). The abundance of light nuclei such as deuterons and the ratio deuteron-to-helium can also provide information about the propagation mechanism. Deuterons are mostly produced by helium in the ISM. Compared to heavier nuclei its interaction mean free path is considerably larger than the escape mean free path of cosmic rays from the Galaxy. The latter contribute to study the transport features in the whole confinement volume. In order to extract correct conclusions from the secondary-to-primary ratios, it is appropriate to know as precisely as possible the production cross section of the primary interactions pro- ducing the secondary component. Thus, using published the data from ALICE in deuteron pro- duction and other available measures explained in the framework of the coalescence model [21], a new and more complete estimation of the deuteron flux is achieved in this work. This thesis is organized as follows. In Chapter 1, a background about CR acceleration and propagation, antimatter CR experiments, and high energy collisions and Monte Carlo generators is presented. In Chapter 2, the model describing deuteron and antideuteron production in high- energy collisions and space is reviewed, and the principal motivation of this thesis is explained. In Chapter 3, the simulation process about the production of deuterons and antideuterons in CR interactions is detailed, and the results obtained are shown. In Chapter 4, the deuteron and antideuteron secondary fluxes are estimated using GALPROP; the results are compared to previous studies. In Chapter 5, an estimation of the antideuteron production in the materials of the AMS-02 detector is developed. Finally, in Chapter 6 the conclusions are summarized.

2 CHAPTER 1 Background

The study of cosmic-rays (CRs) has contributed to the understanding of fundamental princi- ples in physics, as the discovery of the main constituents of the universe. From particle physics and astrophysics to geophysical, solar and planetary phenomena, the cosmic-ray field is a cross- disciplinary area of intensive research on theoretical and experimental grounds. At this date, CRs continue to be an important field of exploration, revealing new astrophysical phenomena and helping to disentangle the mystery of dark matter nature. Matter structure is a pillar of study in particle physics and general sciences. Modern physics has established an impressive picture of this structure within the framework of the Standard Model (SM). It is now well-known are composed of elemental particles called that bound together thanks to the action of the strong interaction mediated by massless called gluons. Hadrons are categorized into baryons when they are made of three quarks and when they have one quark and one antiquark. Protons and are examples of hadrons and are examples of mesons (for a complete list of hadrons see [22]). Hadron structure and its properties are investigated through high-energy collision experi- ments. In particular, hadron collisions provide information about the strong interaction. In the context of CRs, most of the galactic matter is composed of baryons. Therefore an important fraction of the particles produced in the Galaxy is generated in baryon interactions, including deuterons and antideuterons. In section 1.1 cosmic-rays phenomenology is reviewed, and in Section 1.2 dark matter is re- visited. Special attention to the indirect search of dark matter using CR antimatter is presented in Section 1.3. The latest experimental results (including those from AMS-02) and new interpre- tations about CRs antimatter production are also presented in this section. Since the analysis of hadron interactions is crucial to understand CR formation, the most important experimental and theoretical properties are described in Section 1.4. In Section 1.5, an overview about Monte Carlo (MC) generators in high-energy collisions is presented.

1.1. Galactic cosmic-rays (CRs)

Cosmic-rays are highly energetic electrically charged particles that reach the Earth from interstellar space. They are composed principally of protons and helium, but their nuclear abundance is not limited by this two particles since practically all the elements of the peri- odic table are produced during the element formation (see Fig. 1.1). Cosmic-rays produced in astrophysical sources that arrive unperturbed to the Earth are usually called primary CRs.

3 CHAPTER 1. BACKGROUND

Particles produced by inelastic collisions of primaries with the interstellar medium (ISM) or by the decay of unstable nuclei are named secondary CRs. In Fig. 1.1 the element abundance of cosmic-rays is compared with the elements in the . Both have similar magnitudes, except for the light elements: , , and Boron (Z = 3−5); and for some elements below the Iron group (Z < 26). Lithium, Beryllium, and Boron, are more abundant in CRs than in the solar system because of the fragmentation suffered by heavy nuclei like Carbon and when they interact with the interstellar matter in their travel to Earth. Similarly, CR iron fragmentation causes a larger abundance in some elements with Z < 26 [23].

Figure 1.1: Element abundance in cosmic-rays compared to element abundance in the solar-system. Figure taken from Reference [23]

The energy spectrum of CRs is steep, following a power-law function with some minor structures. The most abundant particles lie in the lower energy region of the spectrum with kinetic energy less than 1 GeV per nucleon (see Fig. 1.2). Low energy cosmic-rays are commonly measured by long-term spacecraft missions with high-resolution detectors. These measurements are strongly affected by the magnetic field carried by the , whose activity oscillates following the 11-year . Fig. 1.2 shows how rapidly the CR spectrum decreases with energy. For example, the number of particles above 100 GeV is higher than the number of particles above 1011 GeV by 16 orders of magnitude. The flux at three different energies is shown on the right side of Fig. 1.2, indicating the approximate number of particles above that energy that hit the atmosphere. A detector with an effective detection area of 1 m2 (as AMS-02) will have to collect data around two years to observe a single particle above 107 GeV [3]. For this reason, CRs with energies larger than the hundreds of GeVs are more effectively measured by indirect techniques as extensive air showers via air fluorescence and atmospheric air Cherenkov light. The energy range covered by the CR spectrum is highly extended; in fact, the upper limit is still a topic of discussion and investigation. If the cosmic-ray flux at the highest energies is

4 1.1. GALACTIC COSMIC-RAYS (CRS)

Figure 1.2: Energy cosmic-ray spectrum. Figures were taken from Reference [22] (left) and Reference [3] (right). The inset in the left figure shows the H/He ratio at constant rigidity.

a consequence of the inelastic interactions of Ultra High Energy (UHE) cosmic-rays with the cosmic microwave background, there should be a rapid steepening of the spectrum (called the GZK feature) around 5 × 1019 eV [24, 25]. There are two positions in the CR spectrum where the slope changes appreciably: the knee (between 106 and 107 GeV/n) where the spectrum becomes steeper and the ankle (between 109 and 1011 GeV/n) where the spectrum becomes flatter. It is believed CRs below the knee are accelerated by supernova remnants (SNRs), between the knee and the ankle by other galactic sources, and beyond the ankle by extragalactic sources [3]. The region of interest throughout this thesis is that below the knee, where experimental discrimination of CRs species can be achieved with increasing precision. CRs in this region are expected to be produced by supernova explosions (SNE) in our galaxy. Although the evidence is not conclusive, the main reasons supporting this idea are compelling. Supernova explosions are the only known galactic source with sufficient energy to power CRs [26]. Besides, the CRs composition is a mixture of interstellar material and a 20% contribution from elements ejected in the SNE, such large amounts of matter are only carried in supernova remnants. There is observational evidence based on non- about the formation of CRs in SNRs [27]. The shocks produced in these astrophysical phenomena may use the first-order Fermi shock acceleration mechanism (see Section 1.1.1) to eject CRs with high efficiency, generating the spectral shapes for and hadrons to be similar, just as observed [28]. Supernovae are transient astronomical events that occur during the last evolutionary stages of a . When a massive star has consumed its hydrogen and helium, the gravitational force exceeds the radiation pressure producing the collapse of the star. As a result, the temperature of the core increases to the point where successive fusion processes initiate, forming heavier elements of the iron group. During this process, stellar create electron-positron pairs,

5 CHAPTER 1. BACKGROUND and at even higher temperatures -antineutrino pairs. These neutrinos escape from the core unbalancing the hydrostatic equilibrium of the star. Fusion of heavier elements is no longer possible without additional energy, thus at this point, the core is only supported by ionized electrons, and it starts to contract. Electrons interact with the protons in the star, producing neutrons in a deleptonization process. As the electrons are consumed the pressure decreases and the star suffers a dramatic collapse reducing its radius several thousand kilometers, causing the central density of the star to exceed the nuclear density. In an attempt to balance the nuclear density a shock that propagates outwards is produced. The shock traverses the collapsing core and pushes the outer layers at a high velocity leaving in the center a new proton- star [3, 23]. At this point, the deleptonization process continues releasing an important number of elec- tron neutrinos. However, this is not the only source of neutrinos, since the super-heated neutron star core emits an enormous quantity of energy (∼ 99%) as thermal neutrinos and antineutri- nos of all flavors. Although neutrinos have a very small interaction cross section, inside this hyper-dense media they scatter thousands of times before to scape. As a result, the neutrino and antineutrino burst is ejected with an energy between 20 to 30 MeV in the first 10 seconds of the explosion. Galactic supernova explosions are expected to happen 1 to 3 times per century, which is a high enough rate in the astronomical context. The resulting structure of the supernova explosion is the . Initially, the luminosity of the remnant increases due to the ionized expanding shell, but when the temperature of the ejected material decreases, hydrogen recombine in neutral with a very low opacity, allowing photons to reach the base of the shell and to reduce the luminosity. The supernova shell expands with a very high velocity (c/10), while the inner layers expand more slowly, forming a shock front that sweeps-up the interstellar matter. When the mass of the collected material is comparable to the mass of the supernova shell the remnant decelerates, entering into an adiabatic stage known as the Taylor-Shedov phase. In this phase, the expansion is determined by the energy of the supernova explosion ESN , and the density of the supernova environment ρSN . The radius of the remnant is expressed as a function 1/5 2/5 of its age as RSNR ∝ ESN t . In general, SNRs reach the Taylor-Shedov phase when they have a radius bigger than 1 pc, and more than 1000 years old [3].

1.1.1. CR acceleration As CRs are expected to be formed in SNE, there is rather convincing evidence that CRs are accelerated in SNRs situated in the Galactic disk, during the Taylor-Shedov phase [29, 30]. The strongest argument for SNRs as potential CRs accelerator is the power efficiency. Supernovae are classified according to their lines curves and absorption lines of different chemical elements. For example, supernovae type I have no hydrogen, otherwise they are classified as type II. 51 The kinetic energy liberated by a supernova type Ia is around ESN = 10 erg [3]. If a rate of three SNE per century is assumed, with an expanding velocity of 5 × 108 cm/s per century, then the power produced is around 1 × 1042 erg/s. The power required to produce the bulk of CRs can be roughly estimated assuming they are uniformly distributed around the Galactic 3 7 disk (VGD), with an of about 1 eV/cm , and characteristic time of τGD = 10 years. The value obtained is around 1 × 1040 erg/s which represents the 1 % of the SNR power. Refined calculations estimate that the total acceleration efficiency is between 5 and 10% [31]. Furthermore, the observation of gamma rays from SNRs close to molecular clouds is associated with a hadronic origin, namely a neutral production within the SN [32, 33]. Another observation is the bright X-ray rims in young SNRs as prove of the amplification of the local magnetic field probably because accelerated particles in streaming instabilities [34, 35, 36].

6 1.1. GALACTIC COSMIC-RAYS (CRS)

These estimations depend on the condition of the SNE. For example, in the case of a fast wind SN with low density, the adiabatic phase might start at later times, while for a core-collapse SN explosion, the Taylor-Sedov phase could start earlier. In Fig. 1.3 two different types of SNRs are presented. On the left side is the figure of RX J1713.7-3946, a SNR from a core-collapse SN explosion [37]. The color shows the gamma-ray emission and the lines the X-ray emission both with irregular morphology. The right figure is the Tycho SNR from a type Ia SN explosion in 1572 [38]. In this last figure, the green central region represents the X-ray emission, which is most of the contribution, while the emission in the rim is from due to very high-energy electrons.

Figure 1.3: (Left) Gamma-ray emission measured by HESS (color), and X-ray emission measured by ASCA (lines) of the SNR RX J1713.7-3946 [37]. (Right) Thermal X-ray emission from the Tycho SNR measured by Chandra [38].

From all the astrophysical observations it has been concluded that CRs are very likely to be accelerated in SNRs, at least for those particles below the knee. However, the intrinsic mechanisms about how CRs are accelerated in SNRs is still a developing field. The standard framework to explain the CR acceleration is associated with the impulse given by strong shock waves to charged particles, this process is known as the diffusive shock acceleration. The shock wave of the SNR is formed when the velocity expansion of the ejected material (Vej) is much higher than the sound velocity of the interstellar medium (cs). Typical velocities for shock waves are around 104 km/s while the sound velocity of the ISM is around 10 km/s, much less than the velocities of particles which are considered closer to the [31]. A shock wave is a discontinuity between two regions of gas flow, the ISM undisturbed region (named upstream) and the zone behind the shock wave with velocity Vej (named downstream). In the reference frame of the shock wave, the ISM undisturbed gas now moves to the shock front with a velocity u1 = Vej, and leaves the front with a smaller velocity u2 (upstream to downstream). In the reference frame of the upstream gas, the matter behind the shock moves to the front with a velocity u1 − u2 = 3/4Vej. The same difference in velocity is obtained when upstream gas moves toward the shock front as observed from the downstream gas reference system. Thus, a charged particle moving towards the shock front can be reflected by the moving magnetic inhomogeneities in the shock wave with an increased velocity u1 − u2. This process occurs every time the particles cross the shock back and forth, leading to an average energy gain

7 CHAPTER 1. BACKGROUND in a round trip of ∆E  4(u − u ) = 1 2 , (1.1) E 3c as calculated in [39] with a relativistic treatment and including the average over the scatter- ing angles. As can be observed, the fractional energy increment is proportional to first order in velocity (u1 − u2), this is why the process is known as first-order Fermi acceleration mechanism. From the properties of the shock acceleration, the spectral index of the accelerated particles can be calculated (see Reference [39]). The energy spectrum reads:

N(E)dE ∝ E−2dE. (1.2) Eq. (1.2) presents a power law energy spectrum with index 2, close to the observed in CRs measurements. Note that the shape of the accelerated particles depends only on the velocities of the media in the vicinity of the shock wave, and it does not depend on the diffusion coefficient. This acceleration mechanism is faster and more efficient than others, the reason why diffusive shock acceleration is presented as the most attractive mechanism to explain the bulk of CRs acceleration. The maximum energy achieved by charged particles with the shock acceleration mechanism, assuming a magnetic field of 10−10 T has been estimated in about 105 GeV per nucleon [3]. The second important process of particle acceleration in SNRs is the Fermi mechanism (or second-order Fermi acceleration mechanism). The main idea is that charged particles are accelerated in the interactions with magnetic clouds, gaining energy stochastically. Unlike the first order Fermi mechanism, in this process, the particle may lose energy depending on the velocity direction respect to a magnetic cloud. To find the average energy gain, magnetic clouds are considered as moving magnetic mirrors with velocity Vcl. When the particle enters the cloud, it will interact elastically with the magnetic fields inside, leaving the cloud in a different direction than the one it entered. The charged particle moves toward the cloud with a velocity v ≈ c making an angle θ with the normal of the mirror and with an energy E0. Averaging over a random distribution of angles (see Reference. [39]), the mean energy gain per collision is

∆E  8V 2 = cl . (1.3) E 3c2 Thus, the energy gain of the charged particle is proportional to the square (second order) of the velocity of the magnetic cloud. It is shown (see References [3, 39]) this stochastic acceleration produces a power law energy spectrum with an index that depends on the velocity of the cloud. However, the energy gained by the particles with this mechanism is very low, due to the small velocities of interstellar clouds compared to the velocity of light. Besides, charged particles lose energy by ionization of the interstellar gas, which means they will only be effectively accelerated above a minimum energy value. The latter suggest particles accelerated in shock waves could be injected in magnetic clouds for a second acceleration. Nonetheless, the model of diffusive shock acceleration has important limitations. For ex- ample, the high acceleration efficiency required to produce CRs as observed from Earth causes elevated pressures that affect the shock dynamics and the acceleration process directly. Ampli- fied magnetic fields are present in the acceleration process in a SNR, produced by the charged particles in the remnant. These amplified magnetic fields alter the diffusion coefficient of CRs motion, and they could also affect the compression of the shock, exerting a magnetic pressure on the incoming gas transforming the energy spectrum. An extended diffusive shock theory including non-linear effects has been proposed to solve the problems mentioned above [31, 40].

8 1.1. GALACTIC COSMIC-RAYS (CRS)

However, there are still many issues about CRs acceleration to solve, and the new experimental developments are essential to understanding this process.

1.1.2. CR propagation

Interstellar medium Once CRs have been produced and accelerated in supernova explosions and supernova rem- nants, these charged particles diffuse in the interstellar medium of the Galaxy, and a fraction of them reaches the Earth. Before describing this diffusion process, there are some important properties of the Galaxy itself that it is necessary to mention. The Milky Way is a spiral galaxy where most of its luminous matter is concentrated within a thin disk of radius r ≈ 25-30 kpc and thickness 2h ≈ 200-300 pc. The Galaxy also has a bulge with radius ≈ 2-3 kpc located at the center of the disk. The solar system is located about 8.5 kpc away from the galactic center and about 15 pc above the mid-plane (see Fig. 1.4) [3]. Surrounding the disk there is a spheroidal galactic halo composed principally of field stars, globular clusters, and gas, with much lower density than the disk. A dark matter halo is proposed to extend throughout the Galaxy and far beyond the visible region to explain the dynamics of its rotation.

2h = 200-300 pc

c p k -3 2

8.5 kpc

25-30 kpc

Figure 1.4: Milky Way basic geometry from its side (not to scale). Figure adapted from [3].

The interstellar medium is mostly composed of hydrogen in the form of molecular gas (H2), atomic gas (HI) and ionized gas (HII). The molecular hydrogen is confined in cold and dense molecular clouds mostly in the Galaxy arms and the galactic center. Molecular hydrogen in the Galaxy is measured through the spatial distribution of the CO [41], whose density is proportional to that of H2. The atomic hydrogen distribution is observed through the 21 cm emission line measured using radio . The results show that at large-scale HI is extended to 30 kpc from the galactic center and the thickness layer varies from about 100 pc for r < 3 kpc to 230 pc, for 3 < r < 8.5 kpc to 3 kpc in the galactic boundaries. Ionized hydrogen is observed in the optical part of the spectrum in compact regions around massive stars. The exact structure of the matter distribution in the Galaxy is highly complex and has been reviewed with detail in Reference. [42]. Interstellar matter is exceedingly tenuous, for example near the the density varies from 2 × 10−26 g cm−3 in the hot medium to 2 × 10−18 g cm−3 in the highly dense molecular regions. However, an approximation of the interstellar matter distribution is a constant value of around 1 nucleon/cm3, which corresponds to an average density of about 2 × 10−24 g cm−3. In cosmic-ray propagation codes like GALPROP [43, 44, 45], the distribution of the interstellar hydrogen is defined using tables from dedicated measurements, interpolations,

9 CHAPTER 1. BACKGROUND and models.

Figure 1.5: (Left) Galactic magnetic field model with an axisymmetric configuration. (Right) Galactic magnetic field model with an bisymmetric configuration. Figure taken from [46].

An essential component of the interstellar medium that affects the CR transport is the magnetic field structure of the Galaxy. Besides deflecting CRs randomly, the galactic magnetic field takes part in stellar formation, galactic gravitational balance, and the creation of galaxies. The magnetic field structure of the Galaxy is challenging to investigate, and an important part of what is known comes from observations in other galaxies. Despite this complication, it is generally accepted the Milky Way has a magnetic field with two components. An organized large-scale disk field part that follows a spiral pattern just like the matter distribution [46], and a turbulent part that causes the observed CRs random motion. The regular component is modeled with a 2π symmetry (axisymmetric spiral (ASS)) or with a π symmetry (bisymmetric spiral (BSS)) (see Fig. 1.5), while the random component is modeled through turbulent fluid dynamics. The magnetic field magnitude is inversely proportional to the galactocentric distance (r), and its average strength is estimated as 5-6 µG. The magnetic field is considered to decrease exponentially with the thickness (h) on both sides of the galactic disk. Models of the Milky Way magnetic field are in continuous development alongside with experimental observations (see review [47]). Recent measurements suggest an additional magnetic component is required, and a halo magnetic field has been proposed [46, 48].

CR diffusion The strongest evidence for the diffusive motion of CRs in the Galaxy is the magnitude of the escape time (τesc), i.e., the time required for a charged particle to leave the Galaxy. The escape time depends on the energy of the particle, and it is obtained from the measured abundance of light elements such as Boron, Lithium, and Beryllium produced in nuclear fragmentation of primary CRs with the ISM (see Fig. 1.1). Additionally, a feature that also supports a diffusive motion is the isotropic distribution of CRs in the Galaxy. The most extended measurement of secondary-to-primary CRs is the boron-to-carbon ratio. Boron is produced almost entirely in nuclear interactions of carbon with the ISM, and carbon is a highly abundant primary nucleus after helium and hydrogen. Thus, the ratio of boron-to- carbon fluxes is directly related to the grammage traversed by CRs. This grammage can be estimated from the escape time and the ISM density. From [31]:

X(E) = nµvτesc(E), (1.4)

10 1.1. GALACTIC COSMIC-RAYS (CRS) where n is the mean gas density of the Galaxy, µ is the mean mass of the gas, and v is the speed of the particle. Fig. 1.6 shows a compilation of data of the boron-to-carbon ratio including the recently published AMS-02 data. From the comparison of the models with data, a grammage of X ≈ 10 g cm−2 is obtained for particles with an energy of 10 GeV per nucleon. Considering a Galaxy with half thickness h = 150 pc and a halo extension H, the mean density is approximated −2 −3 −1 −3 to n = ndiskh/H = 5 × 10 (ndisk/1 cm )(H/3 kpc) cm . With µ ≈ 1.4 mp and E=10 GeV, the escape time is

Figure 1.6: Boron-to-Carbon ratio measured by AMS-02 [49]; compared with previous experiments.

X(E)  H  τ (E) = = 90 106yr. (1.5) esc nµc 3 kpc This result shows the escape time is longer than the time required by a particle to travel straight through the Galaxy by at least three orders of magnitude. Then, a diffusion dynamics 2 interpretation is justified, and a diffusion coefficient can be introduced as τesc(E) = H /D(E). At E=10 GeV, D(E) ≈ 3 × 1028(H/3 kpc) cm2 s−1 [31]. The diffusion process depends on the particles energy (or rigidity R) as a power Rδ with δ = 0.3 - 0.6. As expected, at higher energies CRs have shorter escape times or higher escape probabilities. The microscopic interpretation of the CRs diffusion propagation is related to the ionized gas and the magnetic field produced by the particles in this plasma, which forms a magnetohydrody- namic fluid (MHD). CRs are scattered in the interaction with these MHD waves, resulting in an effective diffusion movement. Such interaction only occurs when the wavelength is comparable to the Larmor radius of the CR. MHD waves are classified in Alfv´en waves and magnetosonic waves. Alfv´enwaves are transverse oscillations propagating parallel to magnetic field lines and characterized by the Alv´envelocity Va, while magnetosonic waves are longitudinal oscillations propagating perpendicular to the magnetic field whose restoring force is the magnetic pressure. The mathematical formulation of the interaction between CRs and MHD waves results in the diffusion-reacceleration equation, an accepted model to describe CRs transport in the Galaxy. The general propagation equation for a particular CR species is [45, 50]

∂f(p, ~r, t) ∂ ∂ 1 = ∇·~ D (p, ~r)∇~ f − V~ f + p2D f ∂t xx ∂p pp ∂p p2 (1.6) ∂ h p i 1 1 − pf˙ − (∇·~ V~ )f − f − f + Q(p, ~r, t). ∂p 3 τf τr

11 CHAPTER 1. BACKGROUND

Here f(p, ~r, t) is the CR density per unit of total particle momentum p at a position ~r. The term Q(p, ~r, t) represents the sources of CRs including primary, and secondary from hadronic collisions, spallation or decay contributions. In general propagation models consider a simple geometrical configuration for the Galaxy, a cylinder of radius r with absorbing boundaries di- vided into two regions, an inner disk with thickness 2h contained within a halo cylinder of height 2H (see Fig. 1.7). Primary CRs are inserted in the inner disk following SNRs distributions in space, and the spectra are commonly based on experimental data.

Figure 1.7: Galactic geometry considered in the propagation model. Figure taken from [50].

The first term in Eq (1.6) represents the spatial diffusion process of CRs with Dxx as the spatial diffusion tensor. Dxx is, in general, a function of the position (r), the velocity (β) and the rigidity (R) of the particle. The diffusion coefficient is estimated from the microscopical 27 1/3 2 −1 8 theory as D ≈ 2 × 10 βRGV cm s for all CR particles with rigidities R < 10 GV, which is in agreement with the empirical approximation. From this result, it is observed that the index in rigidity is determined by the exponent form of the spectral energy density of interstellar turbulence, characterized by a Kolgomorov type spectrum (δ = 1/3), or a Kraichnan type spectrum (δ = 1/2) [45]. The second term accounts for the convective movement carried by galactic winds transporting CRs with velocity V~ . Convection is usually implemented in two types of models: 1-zone and 2-zone. A 1-zone model considers convective and diffusion motion in all the geometry, meanwhile a 2-zone model establishes diffusion only in the central region of the Galaxy and diffusion plus convection in the outer region. The next term in Eq (1.6) is the re-acceleration part. This expression describes the stochastic acceleration suffered by CRs with the MHD waves as a consequence of the movement of the medium that causes a change in the momentum of the particles. This process contributes less than the main acceleration mechanisms in SNRs, but it could account for the peak observed in secondary-to-primary ratios at a few GeV. As can be seen in Eq (1.6) the re-acceleration is interpreted as diffusion in momentum space with a diffusion coefficient Dpp (subscript pp means 2 2 in momentum space). The coefficient is estimated as Dpp = p Va /9D, where Va is the Alfv´en velocity [45, 50]. The fourth term (p ˙) represents continuous energy losses in the galactic disk from coulomb collisions in an ionized media and ionization losses. The fifth term (∇·~ V~ ) describes adiabatic energy gains or losses resulting from non-uniform convection velocities whose inhomogeneities scatter CRs. τf is the time scale for losses by fragmentation, or other processes like annihilation when are propagated. τr is the time scale for [45, 50].

12 1.1. GALACTIC COSMIC-RAYS (CRS)

Since Eq (1.6) is time-dependent, a stationary density of CRs or a steady-state solution is assumed. This assumption is justified because the propagation time scale is generally smaller than the time scale where galactic propagation conditions change. The solution using an analytic approximation is commonly obtained setting ∂f/∂t = 0, meanwhile, in numerical approaches it is preferred to follow the time dependence until a steady state is reached. The boundary conditions are set for CRs to escape freely into the intergalactic space at the borders of the halo. At the boundary between the disk and the halo, the density and the diffusion flux follow the continuity condition.

Leaky-box model A simpler but meaningful approximation of Eq (1.6), is to neglect the energy loss-and-gain by high-energy CRs in propagation as well as re-acceleration and convection contributions. Assuming a uniform and constant density of CRs in the Galaxy, the diffusion term could be replaced by the escape time  f ∇·~ Dxx(p, ~r)∇~ f → − . (1.7) τesc This approximation, known as the Leaky-box model, describes the Galaxy as a box of reflect- ing walls with free particles within it. A leakage allows CRs to escape with a certain probability. Eq (1.6) now reads f 1 1 = − f − f + Q(p). (1.8) τesc τf τr Let’s assume the nuclei are stable and only fragmentation losses are present. Then the CRs density is  −1 2  −1 τesc QH X(R) f(p) = Qτesc 1 + ∝ 1 + , (1.9) τf D(p) Xr where the escape time is now expressed in terms of the diffusion coefficient D(p), and the grammage in the ISM (X(R)) was introduced as a function of rigidity R. Xr is the critical grammage necessary for a nucleus to interact. For high rigidities, the grammage decreases −γ X(R) << Xr as expected from observations. Thus, for an injection function of the form Q ∝ p and a diffusion coefficient D(p) ∝ pδ, the cosmic-ray spectrum is determined by a power law in −γ−δ momentum defined by the two indexes f ∝ p . If X(R) . Xr, then the observed spectrum gets harder, i.e., similar to the injection one. The relation above is valid for primary nuclei, for secondary nuclei the diffusion equation must include the spallation chain of heavier nuclei into the secondary species. For simplicity let’s assume a secondary nucleus S, as the only result from the nuclear interaction of a nucleus P with the ISM. The process of spallation conserves the kinetic energy per nucleon Ek. Resolving for the secondary nucleus, the expression for the secondary-to-primary ratio is

X(R) XP →S fS/fP = . (1.10) 1 + X(R) XS

Here, XP →S = mp/σP →S and XS = mp/σS where σP →S is the fragmentation cross-section. From Eq (1.10), it is observed that for high rigidities (R > 100 GV), where X(R) << XS, the ratio is proportional to X(R) and the grammage can be estimated from data. The condition X(R) << XS is also met when the secondary nucleus has a long interaction mean free path, as in the case of light nuclei fragmentation like helium. Furthermore, the re-acceleration effects are stronger for light elements. Thus, the secondary-to-primary ratio of light components like -to-helium provides additional relevant information on the propagation process [51].

13 CHAPTER 1. BACKGROUND

1.2. Search for Dark Matter (DM)

Before further reviewing about antimatter in cosmic-rays, it is essential to recapitulate what is known about dark matter. This summary is based on the dedicated works by [52] and [53]. For an extensive review in dark matter see the mentioned references. Evidence about missing matter in our Galaxy started to show up in the early 1930s when J. H. Oort concluded, after a study about the motion of stars, that the galactic mass should be far more than the observed from to explain stars velocities [54]. At around the same time, the Swiss F. Zwicky reached a similar conclusion observing galactic velocities in the Coma cluster employing a different approach [55, 56]. Some decades later, after a continuous development on the topic, the studies of the rotation curves of galaxies provided convincing evidence that the major percent of the matter in galaxies is non-luminous [57, 58]. The movement of stars in the Galaxy was supposed to be close to the motion of the √ around the sun, following a Keplerian behavior v(r) ∝ 1/ r. However, what measurements showed was that the velocities of stars increase with the distance to the galactic center until they reach a limit (see Fig. 1.8). This unexpected behavior is explained if the distribution of mass in the Galaxy is different from the distribution of light. Thus, two solutions were purposed to explain the rotational anomaly, a correction in the gravitational theory and a “dark matter” distribution. The dark matter distribution is considering most of the times spherically symmetric, in what is called a halo, that extends beyond the visible border of the Galaxy.

Figure 1.8: Rotation curve as a function of distance to the center of the spiral galaxy NGC 3198. The disk line is the expected rotation curve when only mass from visible stars is considered. The halo line is the extra dark matter needed to reproduce the observed rotation curve. Figure taken from [58].

Gravitational lensing is the phenomenon of light bending around a massive object, like a galaxy or a galaxy cluster, as a consequence of the space-time deformation caused by it. The phenomenon was discovered around the 70s, and soon it started to contribute to the idea of dark matter. The mass measured in gravitational lensing is independent of the dynamics of the object, reason why this technique provides an independent and unique probe of the distribution of all the matter, including luminous and dark. Several observations comparing the results of the mass obtained with gravitational lensing and the amount of luminous matter calculated with different techniques have been made. From these results, it was determined that a major part of matter in galaxies and galaxy clusters is dark. A recent example of these measurements is the gravitational lensing from the galaxy cluster CL0024+17, detected by the Hubble Space Telescope and shown in Fig. 1.9 [59]. The left image shows the cluster galaxies, most of them in

14 1.2. SEARCH FOR DARK MATTER (DM) brown (color online), together with a repeated galaxy in blue around the center of the cluster, because of the gravitational lensing effect. On the right of Fig. 1.9, the same image is presented but this time with the expected dark matter distribution superimposed in diffused blue (color online). A dark matter distribution with a saturated center and a ring band around it explains the observed gravitational lensing.

Figure 1.9: (Color online) (Left) Image of the gravitational lensing effect in the galaxy cluster CL0024+17, captured by the Hubble Space Telescope. (Right) Same image with the expected dark matter distribution super- imposed in diffused blue. Figures obtained from [59].

One of the explanations for dark matter was the mass contribution from astrophysical objects made of non-luminous baryonic matter, for example: brown dwarfs, neutron stars, black holes, etc. These objects are classified as MAssive Compact Halo Objects (MACHOs). To identify MACHOs a gravitational lensing technique at a very reduced scale is necessary, i.e. the micro- lensing. Despite the challenging of this technique, some experiments such as MACHO, EROS- 2, and OGLE have carried out intensive searches for these objects in the Milky Way. The results reveal that the contribution from MACHOs to the total bulk of dark matter is not significant [52, 53]. In recent years with the development of new and improved experimental techniques, the evidence about the existence of dark matter has become more robust. Combining different observational methods it has been possible to analyze a collision of clusters. Perhaps the most famous event on the topic is the Bullet cluster of galaxies (1E 0657-56), where the process after the collision is captured [60] (see Fig. 1.10). Using gravitational lensing to determine the distribution of dark matter and X-rays to detect the region where most of the baryonic matter resides, it was found that the distribution of baryonic matter is different from that of the dark matter. The baryonic matter (pink in Fig. 1.10) interacts significantly in comparison with dark matter (blue in Fig. 1.10) which is only sensitive to the gravitational effect. Thus, baryonic matter is left behind by a weakly interacting dark matter forming an object like the presented in Fig. 1.10. The proportions of the total mass of the Bullet are 70:10:1 for dark matter, gas, and stars. Other collision clusters have been observed with similar results [61], including the recent event with a simultaneous collision of four galaxies in the Abell 3827 galaxy cluster [62]. The evidence presented above was collected from the observations of astronomical objects, and they were mostly based on their gravitational interaction. Nevertheless, additional data from cosmological observations support the existence of non-. (BBN) is the model that describes the process triggered a few minutes after

15 CHAPTER 1. BACKGROUND

Figure 1.10: (Color online) Mass distribution after the collision of two clusters of galaxies. The image combines the contribution from two different observation techniques. (Pink) X-ray: NASA/CXC/CfA/M.Markevitch et al. [63] ; Optical: NASA/STScI; Magellan/U.Arizona/D.Clowe et al. [60]; (Blue) Lensing Map: NASA/STScI; ESO WFI; Magellan/U.Arizona/D.Clowe et al. [60]

the Big Bang, where protons and neutrons fused to form light nuclei such as deuteron (D), helium (3He and 4He) , and lithium isotopes (6Li and 7Li). BBN has successfully predicted the abundances of light nuclei produced after the Big Bang, specifically the deuterium to hydrogen ratio (D/H). However, this ratio depends on the overall baryonic density, since if the universe were composed mostly of protons and neutrons, then deuterium would have been fused into helium reducing the D/H ratio dramatically. Thus, the primordial abundance of deuterium limits the baryonic matter to lie between 13.3 % to 17 % of the total matter of the universe.

Lastly, the cosmic microwave background (CMB) gives important information about the composition of the universe. CMB is the relic radiation of about 2.73 K left 3.8 × 105 years after the Big Bang and detected in 1964 by Penzias and Wilson. After dedicated observations of the CMB by important experiments like COBE, WMAP, and Planck, it was determined that the CMB temperature is not homogeneous and isotropic, but it presents fluctu- ations of a few parts in 1 × 105. These fluctuations are induced by the gravitational interactions of all the matter in the early universe at the time of decoupling. Since baryonic matter interacts significantly with radiation while dark matter does not, the effect on the anisotropy is different and observable. The mapped temperature anisotropy can be expressed as a in terms of the spherical harmonics to obtain the angular power spectrum shown in Fig. 1.11. Each peak contains information about the structure of the universe. The first peak indicates the universe is spatially flat Ωtotal = 1, the second peak represents the bary- onic dark matter Ωc, and the third determines the physical density of dark matter in the early universe. Based on the results obtained by the Planck experiment in the framework of the 2 standard ΛCDM model, the baryon density is Ωbh = 0.02230 ± 0.00014, and the dark matter 2 density is Ωch = 0.1188 ± 0.0010. This result is interpreted as the total mass-energy of the universe contains 4.9% of ordinary matter and energy, 26.8% is dark matter, and 68.3% is dark energy [22].

16 1.2. SEARCH FOR DARK MATTER (DM)

Figure 1.11: Temperature fluctuations in the CMB detected by Planck at different angular scales on the sky. Dots are the measurements, while the curve represents the best fit of the standard model of cosmology to the Planck data. Figure taken from [64]

1.2.1. DM candidates From the evidence reviewed above, it can be concluded dark matter exists, and its importance is essential to understand our universe. Nonetheless, to determine its nature is an enormous challenge. The pursuit of a solution to such a problem requires to integrate different fields of study, from astrophysics to particle physics. Well-motivated theoretical ideas have shown that particles with specific characteristics, might explain dark matter properties and give answers to other problems related to particle physics. A candidate particle to dark matter has to satisfy some constraints [53]:

It has to be non-baryonic matter.

It has to be stable on the cosmic time scale to ensure it exist today.

It has to have no strong or electromagnetic interactions.

The candidate has to have the required relic density, i.e., the density before the thermal equilibrium breaking in the early universe.

The standard model (SM) of particle physics is a successful but incomplete theory about the composition and interaction of matter. The main frame with the particles from the SM is shown in Fig. 1.12. None of these particles are acceptable candidates for dark matter, either for their short lifetimes or their natural interactions. The only SM particle weakly interacting, electrically neutral, and stable is the neutrino; however, neutrino relic density is much lower than dark matter density Ων . 0.012 [18, 65]. Thus, new particles predicted by theories beyond SM are required to be viable dark matter candidates. Dark matter candidates referred as WIMPs (Weakly Interactive Massive Particles) are highly preferred, because they have masses in the range from 10 GeV to 1 TeV, and they are produced

17 CHAPTER 1. BACKGROUND

Figure 1.12: Standard Model particles. Areas of the circles are proportional to the masses. Figure taken from [18].

with the correct relic density. This last feature arises as a natural consequence of their weak interaction since the contribution to the dark matter relic density depends on its annihilation cross-section −27 3 −1 2 3 × 10 cm s ΩX h = . (1.11) hσX vi The weak mass range that characterizes WIMPs is a consequence derived from the solution to the gauge hierarchy problem in the SM [18], usually referred to as the “WIMP miracle”. Additionally, WIMPs could be detected using different techniques. Some outstanding WIMPs examples are presented below. The neutralino (χ) is a particle predicted by supersymmetry (SUSY), a model which is an extension to the Standard Model and that considers an additional symmetry between and bosons. In the SUSY model, every SM particle has a new, not discovered partner with the same quantum numbers but differs in spin by 1/2. SUSY predicts the existence of four neutral ˜ ˜ 0 ˜ 0 fields with spin 1/2: the gaugino (Z), photino (˜γ), and Higgsinos (H1 and H2 ). These neutral fields mix to form four Majorana fermionic mass eigenstates called neutralinos (χ1, χ2, χ3, and χ4). In the Minimal Supersymmetric Standard Model (MSSM) an additional discrete symmetry named <-parity is necessary to prevent the proton decay. < is a quantum number equal to -1 for the SUSY particles and +1 for the SM particles, defined as

< = (−1)3(B−L)+2s, where s is the spin of the particle, B is the baryonic number, and L is the leptonic number. If <-parity is multiplicatively conserved, SUSY particles can only be produced in pairs, and the Lightest SUSY Particle (LSP) is necessarily stable. The neutralino is usually the LSP of the group and therefore a WIMP dark matter candidate. Furthermore, SUSY establishes the existence of more neutral particles, for example, the gravitino (G˜), a fermion with spin 3/2 and the sneutrinos (˜νe,ν ˜µ andν ˜τ ), scalars with spin 0. Supersymmetry neutrinos have large scattering and annihilation cross-sections, but they are not good dark matter candidates because they have not been observed in collider experiments at the LHC energies. The gravitino, on the other hand, is not a WIMP but it is a viable dark matter candidate [18, 53]. Dark matter candidates also emerge from other models different to supersymmetry, like the Kaluza-Klein theory of extra spatial dimensions [66]. In this model, all standard model particles

18 1.2. SEARCH FOR DARK MATTER (DM) propagate in flat, compact extra dimensions of the order 10−18 m (universal extra dimensions UED) [67]. As in the case of SUSY, SM particles have their partners in the form of Kaluza- Klein (KK) states, with the lightest of such states (LKP) as a dark matter candidate. The mass range of the LKP is between 400 GeV and 1.2 TeV, comparable to the neutralino mass. Although UED does not resolve the gauge hierarchy problem by itself, the model is considered a low-energy approximation to a complete theory that handles this issue [18, 53]. Dark matter candidates are not necessarily restricted to interact weakly. Feasible options denoted as superweakly-interactive massive particles (superWIMPs) have correct relic densities and possibilities to be detected. SuperWIMPs can be produced in late decays of WIMPs or reheating in the inflation era. Some relevant examples are the weak-scale gravitino [68] and the axino [69], the supersymmetric partner of the axion described below. The weak-scale gravitino is expected to have a mass in the range of GeV to a few TeV, which differs from light gravitinos with a mass on the order of eV to keV. The hypothetical elementary particle Axion is a pseudo-Nambu-Goldstone postulated in 1977 by the Peccei-Quinn theory [70]. It solves the strong CP problem in QCD and arises as the result of the breaking of the Peccei-Qinn symmetry. It is believed axions were produced non-thermally in the early universe as a boson condensate from QCD phase transitions. Unlike WIMPs and superWIMPs, axions only predict the correct relic density under limited conditions. Sterile neutrinos are right-handed neutrino fields introduced to extend the standard model [71]. They may get mass following the same mechanisms that generate masses for quarks and charged leptons. Since their standard quantum numbers vanish, they do not interact with SM particles. Sterile neutrinos may be produced by oscillations at temperatures around 100 MeV, or at higher temperatures in the decay of heavy particles. In Table 1.1 are summarized the most important dark matter candidates, together with their principal properties and the experimental methods that favor their detection.

1.2.2. DM detection Regardless the variety in dark matter candidates, or how convenient their properties are in order to solve SM problems, or whether they are compatible with actual cosmological observa- tions, dark matter candidates have to be detectable. Fortunately, the characteristics of WIMPs interactions promises to obtain measurable signals using different approaches. In the framework of particle physics experiments, i.e., not relying on their gravitational effects, dark matter might be detected by three methods: indirect detection, direct detection, and particle .

Direct detection In the direct detection scheme, a dark matter particle such as a WIMP from the halo in our Galaxy scatters off the nucleus of ordinary matter in the detector. In the interaction, an amount of energy on the order of 1 to 100 KeV for WIMP masses around hundreds of GeV is transferred to the environment (see Fig. 1.13 left). The deposited energy is measured as heat, light or ionization, and the expected scattering rate is approximately ρ R = hσvX i, (1.12) MX where ρ is the WIMP density near the Earth, MX is the WIMP mass, σ is the elastic-scattering cross-section, and vX is the WIMP velocity relative to the detector. The density of dark matter is estimated around 0.4 GeV/cm3, and the initial velocity of the WIMP is dominated by the galactic rotation of the solar system which is about 220 km/s. Thus, the scattering rate mainly depends

19 CHAPTER 1. BACKGROUND

WIMPs SuperWIMPs Light G Hidden DM Sterile ν Axions Motivation GHP GHP GHP, NPFP GHP, NPFP ν Mass Strong CP

Naturally Yes Yes No Possible No No Corrected Ω

Production Freeze Out Decay Thermal Various Various Various Mechanism

Mass range GeV-TeV GeV-TeV eV-keV GeV-TeV keV µeV-meV

Temperature Cold Cold/Warm Cold/Warm Cold/Warm Warm Cold √ Collisional √√ √ Early Universe (BBN, CMB) √√ √ √√ Direct Detection √√ √ √ √√ Indirect Detection √√ √√ √√ √ Particle Colliders

Table 1.1: Dark matter particle candidates. Particle physics motivations have been abbreviated as GHP (gauge √√ hierarchy problem) and NPFP (new physics flavor problem). For more details see [18]. The symbol is for a √ generic detection signal while is for a possible signal. Dark matter detection techniques are introduced in the next section. Table taken from [18].

on the mass of the WIMP, and the cross-section, which could be either spin-independent or spin- dependent. The spin-independent cross-section has a coherent enhancement with the of the target nucleus by A2, while the spin-dependent cross-section is proportional to the spin average of the and there is no coherent enhancement effect. Experiments are generally designed to exploit one of both possibilities. In the case of spin-independent coupling, targets with a nucleus mass from Ge to Xe are used in the detection, whereas in the case of spin-dependent coupling, targets such as 19F e and 127I are preferred [22, 53]. WIMPs interaction cross-section is expected to be very low, underneath picobarns, and as a consequence, the scattering rate is highly reduced. Some calculations suggest rates of at most one event per day per kilogram of a detector. A small WIMP mass could increase the scattering rate, but then the energy deposited in the target nucleus would be smaller. Even in the keV energy range of a dark matter signal, natural radioactivity emitting in MeV energies has to be isolated. Thus, WIMPs detection represents a significant technological problem, where underground laboratories are needed to avoid background signals. The number of experiments related to dark matter detection is growing fast. In the right part of Fig. 1.13 can be observed the different detection techniques used by every experiment. In Figs 1.14 and 1.15 the excluded and allowed limits of various detectors for WIMP cross- section as a function of the mass range in GeVs are shown. The areas above the various curves are the excluded regions of the corresponding experiments. One of the most relevant result from Fig. 1.14 is the possible observation of a dark matter signal by the collaborations DAMA and LIBRA using NaI . The interpretations of this particular result in the framework

20 1.2. SEARCH FOR DARK MATTER (DM)

Figure 1.13: (Left) Diagram of the between a WIMP and detector nuclei. (Right) Energy deposited from the WIMP interaction is measured as heat, charge or light. The figure shows the numerous experiments currently active and what they measure.

of the standard halo model, imply a WIMP with a mass of around 50 GeV and a cross-section of around 7 × 10−6 pb, or a WIMP with a low mass in the 6 to 16 GeV range and a cross-section of around 2 × 10−4 pb. However many experiments have rejected these interpretations, some of them using similar technologies like the Korean collaboration KIMS [72] and DM-ICE [73] within the Ice Cube neutrino Telescope. In Figs 1.14 and 1.15 can be observed the liquid noble gas (Xe, Ar) detectors have the highest sensitivity for high mass WIMPs (>10 GeV). The last upgrade of detectors operated in the Gran Sasso Laboratory is the XENON 1T [74]. It has set the best limit on the cross-section for spin-independent interactions at 7.7 × 10−11 pb for a WIMP mass of 35 GeV (see Fig. 1.14). For spin-dependent coupling, the best limit has been reached by LUX experiment, with a cross-section of 1.1 × 10−10 pb for a WIMP mass of 50 GeV [75] (see Fig. 1.15). The competing Xenon detector is PandaX in the Chinese Jinping Laboratory [76], and based detectors already operating such as ArDM-1t, DarkSide50, and DEAP-3600, and some others planned like DEAP-50T and DarkSide-20k. Among other detector technologies are the semiconductors at mK temperature as CDMS, superCDMS and EDELWEISS, which are used to measure recoil energies down to a few keV, and metastable liquids or gels detectors such as PICO and COUPP, oriented to measure spin dependent interactions. For more details about the detectors see [22] and references therein.

Indirect detection Indirect detection is based on the possible annihilation or decay of dark matter in the galactic halo into SM particles. The particles produced to become a component of cosmic-rays, modi- fying in some cases the expected spectrum. Indirect detection methods strongly depend on the candidate to be studied, and they complement the direct search. WIMPs candidates such as neutralinos are classified as Majorana particles, i.e., they are their own antiparticles, and they annihilate in particle-antiparticle pairs e.g. (see Fig. 1.16)

χχ → qq,¯ W +W −,ZZ → γγ, νν,¯ e+e−, pp,¯ dd,¯ The event rate of the annihilation process is

21 CHAPTER 1. BACKGROUND

Figure 1.14: WIMP cross-section per nucleon for spin-independent coupling as a function of WIMP mass. Exclu- sion regions set by different experiments are presented. The enclosed areas are regions of possible signal events explored by DAMA/LIBRA [77] and CDMS-Si, for more details see Reference [22].

Z 2 Rann ∝ nX hvX σannidV. (1.13)

As observed, the event rate depends on the annihilation cross-section and the square of the dark matter particle density. For this reason, natural places to search for indirect signals are those with high dark matter density as the sun, the Earth or the galactic center. Final state particles are expected to have energies below the WIMP mass because dark matter is in the non-relativistic regime. The most interesting final products from the detection point of view are gamma-rays, neutrinos, and antiparticles. However, their observation is challenging since they are competing with the standard CR flux from astrophysical processes that represents a strong background signal. Gamma-rays can be produced from the hadronization jet after the annihilation, or by direct decay. In the case of direct decay, gamma-rays with specific energy proportional to the mass of the WIMP may be generated, which is highly attractive since it would be a direct indication for dark matter annihilation. The detection mechanism is usually based on the pair production e+e− as a result of the interaction between the gamma-ray and a heavy nuclei material. The positron- electron tracks are reconstructed, and its energy is measured with a calorimeter. Examples of gamma-ray experiments are EGRET, Fermi-LAT, H.E.S.S., MAGIC, VERITAS, and HAWC. Neutrinos are good final-state candidates to detect dark matter annihilation in the regions with high dark matter densities. In these areas like the sun, the gravitational field is strong, trapping the final products except for neutrinos. They interact so weakly that most escape from the dense body and can reach the Earth surface. Neutrinos are commonly detected by telescopes that receive the light signal produced by their interaction with the material surrounding the detector, such as air, ice or . Some examples of neutrino telescopes are IceCube, Super-K,

22 1.2. SEARCH FOR DARK MATTER (DM)

Figure 1.15: WIMP cross-section per nucleon for spin-dependent coupling as a function of mass, for more details see Reference [22].

ANTARES, and Borexino. As a result of the existent matter-antimatter asymmetry, antiparticles in cosmic-rays are relatively rare to observe. This asymmetry opens a new opportunity window, where an excess in the antiparticles signals including positrons, antiprotons, antideuterons or antihelium can be an indication of dark matter annihilation. Magnetic are commonly used to detect antimatter in CRs, some examples of these experiments are BESS, PAMELA, and AMS. Nonetheless, different approaches are under development to detect antiparticles using alternative technologies, and an example is the GAPS experiment (see Section 1.3.1). In the next section, an extended review of the latest and most important results of antimatter detection in cosmic-rays is presented.

Particle colliders

The remaining possibility to search for dark matter candidates is their production in particle collisions. The latter arises from the idea that dark matter could be produced in high-energy particle accelerators as they were produced in the high-temperature environment of the early universe. The most powerful accelerator is currently running, and hypothetically the only ma- chine capable of generating dark matter particles is the LHC, a hadron collider experiment planned to reach a total center of mass energy of 14 TeV. Direct WIMP observations of mono-jet (XXj) and mono-photon (XXγ) signals are not possible at the LHC, where the multiple hadrons production obscures those events completely. Instead, dark matter detection in the LHC relies on the measurement of the final products

23 CHAPTER 1. BACKGROUND

Figure 1.16: Schematic diagram for the annihilation of neutralinos into particle-antiparticle pairs. Figure taken from [53].

generated by the specific dark matter candidate. In the context of supersymmetric dark matter particles, for example, pairs of squarks or gluinos produced in the collision will decay into 0 0 neutralinos (p + p → χ1 +χ ¯1 + jets) that escape leaving a missing energy track in the detector, known as missing transverse energy. However, the observation of a missing particle does not imply it is a dark matter candidate. This last conclusion has to be validated using the model parameters to calculate the relic dark matter density and to compare it with the results from the Planck experiment. The detection of dark matter in the LHC is challenging, the expected event rate for the neutralino production is about 105 per fb−1 integrated luminosity, meanwhile the event rate for SM particles is about 108 − 109 per fb−1 [53].

1.3. Antimatter in CRs as a signature for DM

The observation of antimatter has been one of the most important discoveries in the history of science, and it remains as a top field of research to this date, not to mention that the particle- antiparticle symmetry is an established paradigm nowadays. Major unsolved problems in modern physics as the matter-antimatter asymmetry observed in our universe and the nature of dark matter, involve the study of antimatter. The first detection of antimatter was carried out by who observed the positron using a in 1933 [78]. The antiproton was observed until 1955, in the accelerator Bevatron by Chamberlain, Segr`e,Wiegand, and Ypsilantis [79] as the result of hadron collisions. Today, the production of antimatter in collider experiments is very common. However, the measurements of antimatter in cosmic-rays are somewhat scarce and limited because of the reduced flux of antiparticles reaching the Earth. It was not until the late 70’s that antiprotons were observed in CRs by Golden et al. [4] and Bogomolov et al. [5] using spectrometers. The result was unexpected; they found a higher antiproton flux than the one anticipated from hadronic collisions of primary CRs with the ISM. This anomaly resulted in an explosion of new ideas about the origin of the antimatter excess [80, 81]. Subsequent measurements showed that the flux in the energy region observed was in agreement with secondary production. Despite more precise measurements carried out in recent years by experiments like BESS and PAMELA, including the new results from AMS-02, the question whether if antiprotons are exclusively from a secondary origin or exotic sources remains inconclusive.

24 1.3. ANTIMATTER IN CRS AS A SIGNATURE FOR DM

The standard mechanism for antiproton production in cosmic-rays is expected to be domi- nated by hadron collisions between primary CRs and the ISM.

p + H → p¯ + X and p + He → p¯ + X

Furthermore as mentioned in the last section, antimatter in cosmic-rays represents a potential indirect dark matter signal. An excess in the flux of CRs antiparticles might be interpreted as the consequence of the annihilation process of dark matter. However, the discrimination between dark matter annihilation and the standard production is not easy at all, neither for antiprotons nor for positrons. In the case of antiprotons, large uncertainties are present in the antiproton production cross- section, when CRs collide with the hydrogen and helium of the Galaxy, and uncertainties are also part of the propagation parameters within the cosmic-ray transport models. The best way to reduce the uncertainties in the production cross-section is to measure with high precision and within a wide energy range, the antiproton production in collider experiments. Fortunately, modern experiments in proton-proton and proton-nucleus collisions have reported new data in the last years [82, 83, 84, 85, 86], which contribute to improving antiproton production models. On the other hand, the recent and more precise measurements on the boron-to-carbon ratio and nuclei fluxes are helping to set tight limits in the parameters of propagation models. To demonstrate the importance of reducing the mentioned uncertainties, in Fig. 1.17 it is shown the antiproton data released by AMS-02 [15] compared with two different interpretations. On the left side, data are compared with previous calculations where new information aboutp ¯ cross-sections and B/C ratio were not available [87, 88, 89]. On the right side, the same data are compared with updated models [90, 91, 92, 93]. As can be seen in the left figure, the secondary antiproton-to-proton ratio calculated by [89] was in disagreement with the AMS-02 data, while a dark matter contribution appropriately fit the results [94]. In the right side of Fig. 1.17, a recent reevaluation of the secondary antiproton flux using new data from accelerator experiments such as NA49, NA61, ALICE and CMS along with a recalibration of the transport parameters to the B/C data is compared to the same data spectrum [93]. Note that the theoretical results from the right figure are in fair agreement with AMS data.

Figure 1.17: (Left) Antiproton to proton ratio measurements from AMS-02 [15], compared to a secondary pro- duction model, and a dark-matter-origin model [94]. Figure taken from [95]. (Right) Antiproton flux reevaluated by [93] compared to AMS-02 data [15].

AMS-02 has opened the precision era in cosmic-rays measurements, reducing experimental uncertainties to the few percent levels, this leads to high sensitive spectra where structures are present and possibly new physics. Due to this high sensitivity, the results presented in Fig. 1.17

25 CHAPTER 1. BACKGROUND do not exclude a contribution from dark matter annihilation in the sub-dominant regime to the antiproton flux.

Figure 1.18: (Left) Best fit to the antiproton-to-proton ratio AMS-02 data [15] using a hypothetical dark matter component. (Right) Best fit regions for a dark matter component of the antiproton flux. The best-fit region for the dark matter interpretation of the Galactic center gamma-ray excess (black region) is also shown. Figures are taken from [96].

The left side in Fig. 1.18 shows the antiproton data measured by AMS-02 compared with a recent work [96]. This study suggests dark matter annihilation of a WIMP with mass ≈ 80 GeV is a minor contributor to the antiproton flux around 18 GV in rigidity. The right side of Fig. 1.18 presents the best regions where a dark matter component fit the antiproton data flux [96]. Each region was calculated with different antiproton cross-section models: Tan and Ng [97], di Mauro et al. [98], and Kachelriess, Moskalenko, and Ostapchenko [99]. Additional studies about the dark matter interpretation of the antiproton flux are presented in Refer- ences [94, 100, 101, 102, 103, 104] Secondary positrons are produced in high-energy CRs collisions via π+, K+ decay:

+ + + + p + p, p + He → K , π + X → µ + νµ + X → e + νe +ν ¯µ + νµ + X

Secondary electrons are also generated through the process above, in addition to the pri- mary production in SNRs and pulsars. Assuming that a source injects primary electrons with a −γ −(γ +δ) spectrum E 0 , the flux observed at Earth would have the form N0E 0 , where the index δ describes the effect of the diffusion and energy losses, and N0 is the normalization coefficient. Secondary electrons, on the other hand, are created with a spectrum E−γ1 and would be ob- −(γ1+δ) served as Ne− E . Thus, considering only electrons are produced in primary sources, the electron+positron flux is:

−(γ0+δ) −(γ1+δ) −(γ1+δ) Ne++e− = N0E + Ne− E + Ne+ E (1.14)

The positron fraction is expressed as:

+ φ(e ) N + R = = e (1.15) + − (γ −γ ) φ(e ) + φ(e ) Ne− + N0E 1 0

26 1.3. ANTIMATTER IN CRS AS A SIGNATURE FOR DM

(γ0−γ1) Since N0 >> Ne− the positron fraction is expected to be proportional to E , a de- creasing function of energy. However, when dedicated antimatter experiments as HEAT (High Energy Antimatter Telescope) started to measure the positron fraction, they found a different behavior in energy, the positron fraction seemed to increase around 10 GeV [105]. Higher pre- cision experiments as PAMELA [106], made evident the data deviation from Eq 1.15 as shown in the left side of Fig. 1.19, but the limitation of the instrument did not allow to extend the measurements to higher energies. With the arrival of AMS-02, the most accurate measurement of the positron fraction to this date has been obtained [107, 108, 109], and it is plotted on the right side of Fig. 1.19. From the AMS-02 latest results, it is evident the positron fraction has an excess and it no longer exhibits an increase with energy above ∼ 200 GeV. The cosmic-ray all-electron spectrum has also been measured in recent years by the -borne experiments DAMPE [110] and CALET [111], in an important effort to reveal nearby CRs sources that justify the positron excess.

Figure 1.19: (Left) Positron fraction measured by previous experiments to AMS-02 such as HEAT [105] and PAMELA [106]. (Right) Latest AMS-02 results on the positron fraction presented in [108, 109].

The three main options to explain an excess in the positron cosmic-ray flux are the annihila- tion or decay of dark matter particles [19, 104, 112, 113, 114, 115, 116, 117, 118, 119], the acceler- ation of secondary positrons within cosmic-ray sources (i.e., SNRs) [88, 120, 121, 122, 123, 124], and nearby primary sources of high-energy positrons such as pulsars [119, 125, 126]. In the left side of Fig. 1.20, the positron fraction data are compared with an extension of the model proposed by [19], where a dark matter particle with a mass of the order of TeV annihilates preferably into leptons to finally produce electrons and positrons. From the figure, it can be inferred that the dark matter explanation is in good agreement with the latest results from AMS-02. Nevertheless, a dark matter interpretation is not as simple as it seems, there are important arguments against it. For instance, the energy loss rate for positrons with energy above 100 GeV requires the source to be close to Earth, where dark matter density is expected to be low and insufficient to account for the observed excess. This discrepancy leads to enhance the positron production rate by a factor of one thousand to match the measurements, far above the thermal annihilation cross-section inferred from the relic density. Besides, the absence of a similar increase in the antiproton production as demonstrated by the results of PAMELA and AMS-02 shows what might be considered as an unusual preference of WIMPs to annihilate into charged leptons, which is not compatible with regular supersymmetric features. Finally, the

27 CHAPTER 1. BACKGROUND lack of evidence from electromagnetic radiation observations and neutrino detection about dark matter annihilation undermines this possibility. These observations set restrictive limits on dark matter annihilation cross-section and masses, that sometimes exclude the regions compatible with the positron anomaly.

Figure 1.20: (Left) Latest AMS-02 results [108, 109] for the positron fraction compared to a dark matter model [19]. (Right) AMS-02 data [14] for the positron fraction compared to a model of pulsars as sources of high-energy positrons [125].

In the right side of Fig. 1.20, the results obtained by [125] in 2013 based on the production of positrons from pulsars is plotted against AMS-02 data. The authors in Reference [125] claim that two well-known nearby pulsars Geminga and Monogem can produce the extra positrons AMS-02 is detecting, emitting e± pairs in a power law spectrum with an index close to 2. They assert the detection of in the cosmic-ray electron flux would be the strongest observational evidence to rule out the dark matter interpretation. The current satellite experiments AMS- 02 and Fermi-LAT set constraints that are one order of magnitude above the anisotropy these pulsars may produce, reason why the authors suggest atmospheric Cherenkov telescopes like H.E.S.S. or the forthcoming CTA, as promising options to reveal an anisotropy. Just last year the High-Altitude Water Cherenkov Observatory (HAWC) collaboration re- ported the detection of an extended tera-electron volt gamma-ray emission from two nearby middle-age pulsars: Geminga and PSR B0656+14 [127]. The collaboration concluded the two pulsars are local, isotropic and homogeneous diffusion sources of accelerated leptons, with a diffusion coefficient 100 times smaller than previous models had suggested. This result leads to a slower emission of electrons and positrons which turns out to be insufficient to explain the positron excess, as can be observed in Fig. 1.21. Models based on constant-velocity convective winds as the dominant effect over diffusion, present an alternative prediction about the lep- ton production in the mentioned pulsars [126]. They consider the positron contribution from Geminga and B0656+14 is significant, and assuming other pulsars in the Milky Way behave similarly, they believe it is very likely the entire positron excess is from pulsars. Moreover as mentioned above, measurements indicate the origin of the high-energy positron flux must be explained by other processes. As indicated above, the known process to generate cosmic positrons is the interaction of CRs protons with the ISM. Although it has been investigated for a long time, many questions remain open about the initial production and propagation of positrons in the Galaxy, which might justify the fraction positron anomaly. Studies like the one presented in Reference [123] suggest a different configuration in the sources, randomly distributed in the Galaxy and surrounded by

28 1.3. ANTIMATTER IN CRS AS A SIGNATURE FOR DM

Figure 1.21: (Left) Estimated positron flux at Earth from Geminga (blue solid line), compared to AMS-02 data (green dots). The shaded blue region indicates a 3σ (99.5% confidence) statistical uncertainty from simula- tions [127]. (Right) Total electron flux and positron fraction predicted by the production model [123], compared to AMS-02 data [14].

a cocoon-like region where spallation takes place but no reacceleration. This model allows high- energy protons to escape and produce high-energy positrons by collisions with the ISM, while low energy CRs are reduced in the cocoon-like region. The positron fraction predicted by this model is shown on the right side of Fig. 1.21. However, these considerations should affect the boron- to-carbon ratio and produce an effect that has not been observed in the latest AMS-02 data. Other works [124], have obtained good results re-analyzing the secondary positron production using updated data, but including an extra secondary source such as pulsars. Thus, even when the secondary positron production has important uncertainties in the injection and propagation process, the excess or anomaly in the positron fraction seems to require an additional source.

1.3.1. Experimental search for CR antinuclei Although a few experiments have measured antiprotons and positrons in CRs, none have reported heavier antinuclei. The reason is understandable when the orders of magnitude of the expected fluxes for these antiparticles are considered. Even with the possible contribution from exotic sources, the sensitivity levels required for detection are challenging. Fortunately, we are entering a precision era in CRs measurements with a highly sophisticated magnetic like AMS-02, which might be able to detect antinuclei in cosmic-rays. Furthermore, the attractiveness of cosmic-rays antideuterons as a possible indirect signal from dark matter has led to developing novel detector designs without the need of a magnet and with increasing sensitivity capable of measuring this flux. Such is the case of the balloon-borne GAPS experiment [128, 129] planned to fly in 2020/2021.

AMS-02 AMS-02 is a general purpose-detector designed to study CRs in an energy range between 0.5 and 2000 GeV. AMS-02 was launch to space and installed on the International Space Station (ISS) in May 2011, and it will continue operating until the end of the space station lifespan. AMS-02 weights 6900 kg, and it has a volume of 64 m3. A total of 56 institutions from 16

29 CHAPTER 1. BACKGROUND

Figure 1.22: AMS-02 detector. (Left) Structure diagram of the detector. (Right) Real event reconstructed.

countries including Mexico are part of the AMS-02 collaboration. Among the objectives of the experiment are the measurement of antimatter, the origin of cosmic rays, the search for dark matter, and the exploration of new astrophysical phenomena. AMS-02 has a powerful magnet and five dedicated sub-detectors described below [14, 130, 131] (see Fig. 1.22). AMS-02 has nine planes of precision silicon tracker, a transition radiation detector (TRD), four planes of time of flight counters (TOF), a permanent magnet, an array of anti-coincidence counters (ACC), a ring imaging Cherenkov detector (RICH), and an electromagnetic calorimeter (ECAL). The silicon tracker is composed of nine planes of silicon sensors supported by an aluminum honeycomb with carbon fiber skins. Three single-layers are located at the top of the TRD (plane 1), above the magnet (plane 2) and between the RICH and ECAL (plane 9). Meanwhile, planes from 3 to 8 are arranged in three double layers inside the magnet (see Fig. 1.22). pc A particle with charge Z traveling through the detector has a rigidity R = Ze = Bρ, which value is measured by the tracker using the known magnetic field intensity (B) and the trajectory of the particle (ρ). The absolute charge is obtained from the deposited ionization energy which is proportional to the square of the particle charge (I ∝ Z2). The detector has a coordinate resolution of 10 µm, and the charge resolution is ∆Z ≈ 0.06 at Z = 1. The TRD has a conical octagon form supported by carbon fiber and an aluminum honeycomb structure to minimize weight and maximize acceptance at the top of AMS-02. The TRD is composed of 20 horizontal layers of 20 mm thick fiber fleece radiator (LRP375) with a density of 0.06 g/cm3. Each layer is interleaved with modules containing 16 proportional tubes of 6 mm diameter and 2 m length, filled with a 90:10 Xe:CO2 mixture. Inside the tubes, a wire is attached parallel to the center carrying the current produced by the interaction between the radiation and the gas. In total there are 328 modules and 5248 proportional tubes. Antimatter search in CRs includes a precise measurement of positrons in an energy range from 10 to 300 GeV. Thus, the detector has to be capable of discriminating e± from the over- whelming flux of protons and nuclei in a factor of 106. The TDR achieves such level of iden- tification using the physical principle of the transition radiation (TR) i.e., the electromagnetic radiation emitted when a charged particle traverses the boundary between two media with dif-

30 1.3. ANTIMATTER IN CRS AS A SIGNATURE FOR DM ferent dielectric constants. In this case, the media is the fiber fleece radiator. The TR intensity is proportional to the particle relativistic Lorentz factor γ = E/m. Hence, for large mass dif- ferences, for example between protons and electrons, a corresponding large difference in the γ factor is expected, and therefore the probability of emitting transition radiation is larger for electrons than for protons. The TOF detector is formed by four layers of 12 cm wide, 1 cm thick, and 117 to 134 cm long plastic stripes, with an effective active area of about 1.2 m2. The four layers are clustered in two planes positioned in orthogonal directions to form an X-Y arrange synchronized with the magnetic field direction. The upper TOF (UTOF) is placed between the TRD and the magnet, and the lower TOF (LTOF) is located below the magnet. At the end of each scintillator paddle photomultipliers (PMTs) are connected, receiving the light signal from the interaction between the particle and the plastic. From the time difference measured by the two planes (with a time resolution of about 180 ps for Z = 1) and the trajectory obtained from the tracker, the detector can determine the velocity with a resolution of the order of 4% for protons, and 1% for . Furthermore, the TOF can determine the charge (Z) of CRs using the energy loss by particles in the plastic. The ACC detector is also composed of plastic scintillator paddles with dimensions 826 x 230 x 8 mm placed in the inner bore of the magnet surrounding the inner tracker. Using the light produced by the interaction of a particle with the plastic, the ACC is capable of detecting CRs from the sides of AMS-02. The principal purpose of the detector is to avoid CRs coming from lateral directions that interfere in the top-bottom signal. Additionally, the ACC provides information about secondary tracks produced by the interaction of CRs with the detector and the back-scattering against the electromagnetic calorimeter. The RICH sub-detector is placed between the LTOF and the electromagnetic calorimeter. It has a truncated conical form with an empty rectangular area in the center of 64 x 64 cm2, that matches with the area of the ECAL just beneath it. The top-plane has a radius of 60 cm and it holds two different radiators: in the central area a 5 mm thick squares of sodium-fluorine (NaF) crystals with refractive index n =1.33, and in the borders a 3 cm thick tiles of silica aerogel with refractive index n =1.05. In the low-plane, a total of 680 light guides are connected to multi-pixel photomultipliers which detect the Cherenkov light produced by the CRs in the interaction with the radiator. The separation between planes is 46 cm, and the inner lateral surface is covered by high reflectivity mirrors to increase the geometrical acceptance. The RICH was designed to measure with high resolution the velocity of the particles in a wide energy range, and to obtain an independent charge measurement of nuclei up to Fe. The velocity is evaluated from the Cherenkov light cone (θ = arcos(1/nβ)), and the charge is deduced from the number of photons emitted in the light cone. The ECAL is a square parallelepiped detector placed at the bottom of AMS-02. It has nine superlayers composed of 1 mm thick lead interleaved with 1 mm thick layers of scintillator fibers. The light signal is transmitted by the fibers and read by the 324 photo-multiplier tubes connected to them. The calorimeter was designed to discriminate between leptons and hadrons based on a precise 3-D image reconstruction of the shower produced by the particles when they go through the detector. For electrons and photons with energies below TeV, it is possible to measure the particle energy, since it is proportional to the energy deposited in the calorimeter. One of the most important pieces of AMS-02 is the cylindrical shape magnet with inner di- ameter of 110 cm, and a height of 80 cm. It is a permanent magnet reutilized from the previous experiment AMS-01 composed by 64 high-grade Nd-Fe-B sectors. The intensity of the field is 1.4 kG uniform in the X direction at the center of the magnet and negligible dipole moment out- side the magnet. Together with the tracker, the magnet provides a maximum rigidity detection

31 CHAPTER 1. BACKGROUND of 2 TV.

GAPS

The General Antiparticle Spectrometer (GAPS) detector [128, 129] is a balloon-borne instru- ment designed to measure antiprotons and antideuterons in a low-energy range <1 GeV. Unlike conventional spectrometers, GAPS does not have a magnet; instead, it uses a different principle to discriminate matter from antimatter. The detection mechanism is based on the production and decay of short-lived exotic atoms by means of the antimatter interaction process with the detector material. The antiparticles entering the GAPS detector will lose energy until they stop in the target material, creating an exotic in an . When the exotic atom decays, it will emit Auger electrons and characteristic atomic X-rays. After the de-excitation, the antiparticle annihilates with the target nucleus producing pions and baryons.

Figure 1.23: GAPS detector. (Left) Structure diagram of the detector. (Right) Antiproton, antideuteron inter- action picture with the detector. Figures taken from [128, 129].

X-rays emitted by an antideuteron exotic atom formed in Si will produce particular energies of 30 keV, 44 keV, and 67 keV. The main background in the search for antideuterons with GAPS is the antiproton signal. Antiprotons produce a similar reaction in the detector as antideuterons. However, they can be separated using the features in annihilation products multiplicity, energy loss, and stopping depth. An antiproton, for example, has approximately the half stopping range than an antideuteron. The X-rays emitted by an exotic atom with an antiproton interaction have different energies than those produced by an antideuteron, the energies of X-rays are 35 keV, 58 keV, and 107 keV. The products from the annihilation of an antiproton are less than those generated with an antideuteron. It is expected GAPS to reach a very high power identification of antideuterons, applying all these selection methods. The GAPS detector is formed by ten planes of lithium-drifted silicon (Si(Li)) detectors with a diameter of 10 cm and a thickness of 2.5 mm, providing a total active area of 10.5 m2. The (Si(Li)) detectors have three main goals: serve as a target to form the exotic atoms, as a tracker for incident particles and annihilation products, and as a detector for atomic X-rays. The planes are surrounding by a cube of plastic scintillator with dimensions 1.6 m×1.6 m×2 m. This cube is the inner time-of-flight detector (TOF). The outer TOF system has dimensions 3.6 m×3.6 m×2 m, and it is separated 1 m from the inner TOF (see Fig. 1.23). The time-of-flight system measures the velocity and direction of an incoming antiparticle.

32 1.4. HIGH-ENERGY HADRON COLLISIONS

GAPS collaboration foresee an acceptance of 18 m2sr in the energy range from 0.05 to 0.25 GeV/n, and a sensitivity of about 2 × 10−6m−2s−1sr−1(GeV/n)−1 over the above energy range for 105 days of flight [128, 129].

1.4. High-energy hadron collisions

1.4.1. Definitions and notation Hadron or nuclear experiments are performed in fixed-target (also called laboratory system) or center-of-mass colliders. If the collision happens in the laboratory system between a beam A with four-momenta pAL = (EL, ~pL) and a target B with four-momenta pBL = (mB,~0), the Lorentz-invariant energy available is

2 2 2 2 s = (pA + pB) = (EL + mB, ~pL) = mA + mB + 2mAEL. (1.16) In the case of a center-of-mass collisions, the Lorentz-invariant of a beam A with four- momenta pA = (EA, ~p) and a beam B with four-momenta pB = (EB, −~p) is

2 2 s = (EA + EB, ~p − ~p) = (EA + EB) . (1.17) √ Thus, in center-of-mass experiments the total energy is s = (EA + EB). Throughout this thesis data from fixed-target and center-of-mass experiments are going to be used, and the √ energies will be expressed in terms of Elab or plab and s. The resulting products from the collision have momentum, energy and angular distribution which are measured in the detectors. To interpret data in high-energy experiments some kine- matical variables are conveniently defined, generally in a cylindrical symmetry. The momentum of the produced particle is commonly decomposed in the transverse (pT ) and the longitudinal (pL) components with respect to the collision axis. A usual quantity defined in high-energy physics is rapidity (y). As its name implies, rapidity is related to velocity, it is dimensionless and describes the rate at which a particle is moving with respect to a chosen reference point situated on the line of motion. It is defined as     −1 1 1 + βz 1 1 + βcosθ y = tanh βz = ln = ln , (1.18) 2 1 − βz 2 1 − βcosθ where β = v/c, v is the velocity of the particle, and θ is the polar angle. Rapidity is often expresed as function of the total energy and momentum:

p 1 E + p  y = tanh−1 z = ln z . (1.19) E 2 E − pz For ultrarelativistic particles the rapidity is determined only by the emission angle θ, since β ≈ 1. This is known as the pseudorapidity, written as

1 1 + cosθ  η = ln . (1.20) 2 1 − cosθ An example of the center-of-mass rapidity distribution in p+p collisions is shown in Fig. 1.24 (Left). The distribution exhibits a central plateau at small y, and falling cross-sections in the fragmentation regions where y → ±ymax. Other useful dimensionless variables to introduce are the Feynman scaling variable (xF ) and the radial scale variable (xR), defined as

33 CHAPTER 1. BACKGROUND

2pL pL xF = √ ≈ , (1.21) s pL,max

E xR = , (1.22) Emax where pL and E are the longitudinal momentum and the energy in the center-of-mass frame re- spectively. pL,max is the maximum possible longitudinal momentum and Emax is the maximum energy. These variables measure, in the center-of-mass frame, the fraction of the beam’s mo- mentum or energy which is contained in the longitudinal momentum or energy of the detected particle. The distribution of xF and xR vary from -1 to 1, with a maximum near 0 due to a large number of slow particles produced. The distribution decreases rapidly to 0 as xF → 1, like n (1 − xF ) . An example of a typical xF distribution is shown in Fig. 1.24 (Right). In general, experiments are only able to measure rapidity, pseudorapidity or xF in limited regions. These variables together with transverse momentum usually describe the available phase space of the produced particles [132].

Figure 1.24: (Left) Rapidity distribution in p+p collisions at 15.4 GeV/c (circles) 26.7 GeV/c (triangles) [132]. (Right) xF distribution of antiprotons in p+C collisions at 158 GeV/c.

Particle and nuclear reactions are characterized by the cross-section, an “effective area” that determines the probability of interaction. Cross-section has units of barns (b), and it is represented generally by the Greek letter σ. Particle interactions can be elastic (σel), where the initial and final states are the same, for example, the initial colliding particles A and B lead to

A + B → A + B. (1.23)

Or can be inelastic (σinel), where new particles with or without those from the initial state are generated:

34 1.4. HIGH-ENERGY HADRON COLLISIONS

A + B → (A + B) + C + D + F + ... (1.24) In any case, the corresponding quantities such as energy, momentum, baryonic number, etc. are conserved. As the energy increase, the number and type of particles produced in the collision are also growing, where different kind of reactions are possible. This process is described in particles physics as the opening of different channels. Inelastic interactions can be decomposed, for processes at low transverse momentum, into single-diffractive, double-diffractive and non- diffractive events as shown in Fig. 1.25. The total cross-section is the sum of the elastic and inelastic cross-sections (σtot = σel + σinel).

Figure 1.25: Diagrams of the different types of inelastic interactions. P indicates Pomeron exchange. (a) and (b) are single-diffractive events, while (c) is a double-diffractive collision, and (d) represents a central collision or non-diffractive. Figure taken from [133].

In a reaction of the form A + B → B + C + F + ..., A and B constitute the initial state i, and B + C + F + ... is the final state f. Let nA be the density of incident particles and vi the relative velocity of A and B. If the target has nB particles per unit of area, then the number of per unit of time per unit volume (dnS/dt) is directly proportional to nA and nB. The overall constant of proportionality is the cross-section σ:

dn S = σφn , (1.25) dt B where φ = nAvi is the incident flux of particle A. In high-energy experimental physics it is common to use the term luminosity, which is no more than the product of the incident flux and the number of target particles (L = φNB). Luminosity represents the figure of merit of a , the bigger the luminosity, the better the accelerator, since rarer events may be seen. dnS/dt is also known as the transition rate. dnS/dt is determined by Fermi’s Golden Rule and depends directly on matrix elements Mi→f , that contains the amplitudes of the quantum transitions of the problem, including coupling strength, energy dependence, angular distribution, spins, etc. and also depends on the phase space factor of the final state ρf : dn Z S = (2π)4 |M |2 × ρ . (1.26) dt i→f f The phase space factor is defined as the Lorentz-invariant expresion:

N 3 N Y d pj X ρ = × δ4( p − p − p ). (1.27) f (2π)32E j A B j=1 j j=1

The individual particle densities are nA = 2EA and nB = 2EB, thus the incident flux is written as φ = 2EAvi. Replacing the transition rate, the incident flux, and the target density in Eq 1.25, and resolving for the total cross-section, the resulting expression is

35 CHAPTER 1. BACKGROUND

  4 Z N 3 N (2π) Y d pj X σ = δ4( p − p − p )|M |2 . (1.28) tot 4E E v  (2π)32E j A B i→f  A B i j=1 j j=1 Eq 1.28 is a Lorentz scalar. In particle collision experiments the products from the reaction are usually detected in a certain region of momentum between pC and pC + dpC , and so for all the particles. Thus, it is common to measure the differential cross-section:

 N  N dσ (2π)4 Y 1 X = δ4( p − p − p )|M |2 (1.29) d3p d3p ....d3p 4E E v  (2π)32E  j A B i→f C D N A B i j=1 j j=1

Furthermore, the number of particles produced in high-energy collisions is enormous and to detect all these particles represents a nonpractical instrumental situation. For this reason, experiments measure inclusive reactions, where one of the products is of importance:

A + B → C + X (1.30) In Eq 1.30, C is the targeted production particle, while X denotes all other particles produced but not detected. An exclusive reaction is of the form of Eq 1.24. The differential invariant single-particle cross-section in cylindrical coordinates is expresed as

2 2 √ dσ EC d σ 1 d σ f(pC , s) = EC 3 = 2 = 2 . (1.31) dpC π dpLdpT π dydpT where the identity dy = dpL/E was used. The differential invariant single-particle cross-section in spherical coordinates is defined as

2 2 √ dσ EC d σ EC d σ f(pC , s) = EC 3 = 2 = 2 . (1.32) dpC p dpdΩ 2πp dpdcosθ The differential cross-section is the probability per unit incident flux to detect particle C 3 within the phase-space volume element d pC . An important additional quantity commonly mentioned in high-energy experiments is the multiplicity (hnchi), i.e., the number of charged particles produced in the collision. Only a small fraction of the incident energy is used to generate new particles, for example at energies on the order of 50 GeV in the center-of-mass system, an average of 12 charged particles are produced, from which 90 % are pions. The behavior of multiplicity with energy is approximated to a logarithmic function (hnchi = 2log(s) − 4).

1.4.2. General features of hadron collisions Quarks and gluons possess a property called colour, which can take three possible values, red (R), green (G) and blue (B). Thus, a with the valence quark structure q1q¯2 is written as 1 M = √ (qRq¯R + qBq¯B + qGq¯G), (1.33) 3 1 2 1 2 1 2 while a baryon with the valence quark q1q2q3 becomes

1 α β γ B = √ (αβγq q q ), (1.34) 6 1 2 3

36 1.4. HIGH-ENERGY HADRON COLLISIONS

where α, β, γ = R, G, B and αβγ is the antisymmetric permutation tensor. The quarks are regarded as the fundamental triplets of an SU(3)C color gauge symmetry group. The Quantum Chromodynamics (QCD) description of the strong interaction establishes this force is the result of the exchange of colored massless vector gluons between colored quarks [132]. α The quark-gluon vertex coupling is expressed as 1/2gsλij, where gs is the strong coupling, and α the λij are Gell-Mann’s SU(3) representation matrices, with α = 1, ..., 8 for the gluon colors, while i, j = 1, 2, 3 for the quark colors.

q i j i j i j

α q g

q

k (a) l k (b) l k (c) l (d)

Figure 1.26: (a) Lowest-order qq¯ interaction (b) and (c) Lowest-order corrections to the quark-gluon coupling. (d) qq¯ colour field with V (r) ∼ λr.

In Fig. 1.26(a) it is shown a diagram of a single gluon exchange between a quark and an antiquark. The potential for this case is expressed as

8 X 1 V (r) = − λα λα (α /r), (1.35) ijkl 4 ij lk s α=1 2 where αs = gs /4π. For colorless initial and final states the result is V (r) = −4/3(αs/r). However, the effect of higher-order diagrams like those in Fig. 1.26(b) and (c) gives

(  2    2 2 ) 2 αsb0 Q αsb0 Q 1 αs(Q ) ≈ αs 1 − log 2 + log 2 + ... = 2 2 , (1.36) 4π µ 4π µ (b0/4π)log(Q /Λ )

2 2 2 2 2 where Λ = µ exp(−4π/αsb0), Q is the four-momentum of the gluon, µ is the value of Q at which αs is measured, and b0 = 11/3Nc − 2/3Nf , where Nc is the number of colors (3) and Nf is the number of flavors of quarks (6). 2 Eq 1.36 has two important consequences it is necessary to point out. First, αs(Q ) → 0 as Q2 → ∞, which means that quarks and gluons are almost-free at very high-energy collisions, 2 and they reach a regime called “asymptotic freedom” [134, 135]. Second, αs(Q ) → ∞ as Q2 → Λ2 and the perturbation series breaks down at small Q2. This means the coupling becomes stronger as the separation between the q andq ¯ increases, because of the gluon self-coupling. The exchanged gluons attract each other constraining the color lines force to a tube-like region between the quarks (see Fig. 1.26(d)). Thus, the potential energy of the interaction increases with the separation between the pair of quarks (V (r) ∼ λr), and they can never escape from the hadron. The latter phenomenon is called “color confinement” [136, 137], and has important

37 CHAPTER 1. BACKGROUND effects in hadron collisions. For example, in a proton-proton collision, quarks from the parent protons may scatter each other increasing the force between them until the potential energy of the color field is large enough to produce qq¯ pairs as in Fig. 1.26(d). The first qq¯ pair then breaks up, becoming the endpoints of two new qq¯ pairs with lower energy. The process continues until the quark energy has degraded into clusters of quarks with zero net color and low internal momentum. The clusters of quarks are confined into hadrons, and that is why this mechanism is known as “hadronization” [132]. Quantum Chromodynamics (QCD) has successfully described the strong interaction in the perturbative regime where αs < 1. However, these perturbative methods are not applicable to the bound-state with αs > 1. This is the case, for example, of the hadrons composed of light quarks (u, d, s) where the probability of creating additional virtual gluons and qq¯ pairs within it becomes very high. It is also the case of elastic and diffractive scatterings between hadrons which are non-perturbative and therefore cannot be described by QCD. Fortunately, the parton model alongside with experimental information allow to understand and explain many features of the hadron structure and its collisions [132]. Partons was the name given to the constituents of the proton observed by electron scattering experiments in the late 1960s. Today partons are no more than quarks and gluons, confined in clusters called hadrons. From this perspective, the interaction of hadrons in high-energy collisions is reduced to the scattering and exchange of partons. As in molecular physics, where the is the residual electromagnetic polarisation effect between neutral atoms, the can be considered as the residual color polarisa- tion force between colorless hadrons. This interpretation involves the exchange of colorled quarks and gluons, i.e. partons between the hadrons. The idea is similar to that proposed by Yukawa in 1935, but with the difference that the exchange particle now is considered non-elementary [132].

C

A a c fA DC

a c

b d B b fB

Figure 1.27: (Left) Hadronic reaction A + B → C + X at large pT , in terms of the parton sub-process a + b → c + d [132]. (Right) Proton structure function given at two Q2 values (6.5 GeV2 and 90 GeV2). For more details see [22].

A diagram of the parton interaction process is shown on the left side of Fig. 1.27, where the inclusive reaction in Eq 1.30 is represented. The incoming particles A and B contain partons a and b with momenta qa and qb respectively, which scatter producing partons c and d with a momenta qc and qd. Hadron C is formed from the confinement of parton c. Let fA(xa) be the

38 1.4. HIGH-ENERGY HADRON COLLISIONS

“structure function”, defined as the probability the hadron A contains a parton a carrying a fraction of its momentum xa = qa/pA (0 6 xa 6 1). Similarly, the “fragmentation function” C Dc (zc) is introduced, representing the probability that the outgoing parton c produces an hadron C carrying a momentum fraction zc = pC /qc (0 66 zc 6 1). Neglecting the masses of hadrons and partons and the transverse momentum of the a parton, and assuming that C is produced collinearly with c, the invariant variables of the reaction are 2 s = (pA + pB) ≈ 2pApB, (1.37)

2 t = (pA − pB) ≈ −2pApC , (1.38) √ where −t is the invariant momentum transfer from A to C. The corresponding variables for the parton sub-process are

2 s¯ = (qa + qb) ≈ 2qaqb = 2xaxbpApB ≈ xaxbs, (1.39)

2 t¯= (qa − qb) ≈ −2qaqc = −2xapApC /zc ≈ xat/zc. (1.40) The invariant cross-section of the reaction can be expressed as the weighted sum of the differential cross-sections, dσ/dt¯, of all possible parton scatterings that can contribute:

Z 1 Z 1 dσ X a b 1 dσ C EC 3 = dxa dxbfA(xa)fB(xb) (ab → cd)Dc (zc). (1.41) d pC πzc dt¯ abcd 0 0 From Eq 1.41 is observed that according to the parton model, if the structure functions, the fragmentation functions, and the cross-section for all the parton sub-processes are known, then the invariant cross-section of the hadron collision can be calculated [132].

Structure functions Structure functions are determined by deep inelastic -nucleon scattering (DIS) and some additional processes such as single jet inclusive production in nucleon-nucleon interac- tions, dilepton production in the virtual Drell-Yan process, and electroweak Z and W boson production. A parametrization originated in analytical assumptions or using computational tools is fitted to various sets of experimental data, mainly deep inelastic scattering data. Two variables conveniently describe this inelastic interaction: the four-momentum transfer squared Q2 = −q2 and the Bjorken scaling variable x = Q2/2qP . Here q is the four-momentum trans- ferred by the lepton to the parton and P is the four-momentum of the target nucleon. The structure function describes the distribution in Q2 and x of the constituents (partons), and it is generally expressed as 2 X 2 2 F2(x, Q ) = ei xifi(x, Q ), (1.42) i=u,d,... 2 where ei is the charge of each flavor, and xifi(x, Q ) are “parton distribution functions” (PDFs). If PDFs are understood, then it is possible to know hadron structure. In the right side of Fig. 1.27, a structure function is shown as a function of x for two values of Q or two resolutions. 2 As can be observed F2 increases as Q rises in the region where x < 0.1. This behavior is explained in the framework of QCD dynamics because of the massive production of pairs quark- antiquark (sea quarks) by a growing density of gluons. In the regime where x > 0.1, F2 decreases 2 as Q increases, due to the formation of gluons by the valence quarks. At x around 0.1 F2 is independent of Q2 because there is no dependence of the electron-proton scattering process on a distance scale [138].

39 CHAPTER 1. BACKGROUND

Fragmentation functions

h 2 Fragmentation functions (Di (z, Q )) describe the formation of colorless bound final states after the interaction. They represent the probability a parton of type i fragments into a hadron of type h with a fraction of the parton momentum z. In the context of QCD, fragmenta- tion functions are a consequence of color confinement, a regime dominated by non-perturbative processes, which explains the difficulty to be calculated analytically. Thus, fragmentation func- tions are obtained from scattering data, principally from single-inclusive hadron production in electron-positron annihilation (e+ + e− → h + X), semi-inclusive deep-inelastic lepton-nucleon scattering (l+N → l+h+X) and single-inclusive hadron production in proton-proton collisions (p + p → h + X) [22, 139].

Parton-parton cross-section The third ingredient to obtain the invariant differential cross-section of an inclusive reaction is the differential cross-section of the parton interaction. This value can be derived from first-order (or leading-order) perturbation theory in QCD. The parton-parton differential cross-section has the form dσ πα2|A2| = s , (1.43) dt s2 where |A| is the amplitude of the contributions to the sub-process ab → cd calculated from standard QCD. Since the parton differential cross-section has a structure of the type dσ 1 (a + b → c + d) ∼ . (1.44) dt¯ s¯2 The differential cross-section of the hadronic process A + B → C + D can be anticipated:

3 d σ −4 EC 3 (s, pT ) = F (xT )pT , (1.45) dpC √ where xT = 2pT / s and F is a scale-invariant function. A more general parametrization has demonstrated to describe the data on large pT scattering, and it is used to estimate antiproton production in proton-proton collisions:

3 d σ −n m EC 3 ∼ pT (1 − xT ) , (1.46) dpC √ where xT = 2pT / s. An important characteristic of hadronic collisions is described in Eq 1.46, where can be noted the cross-section decreases as a function of pT because most of the produced particles have low transverse momentum and they are emitted close to the direction of the incident beam [132].

1.5. Monte Carlo (MC) generators

MC generators are software packages created to simulate in detail high-energy collisions. MC generators play an essential role in theoretical and experimental particle physics where they are used to make predictions about new physics processes or modeling QCD and aspects beyond the perturbative regime as well as to analyze data, estimate signals and backgrounds in particle detectors, and planning new experiments. The importance of MC is exemplified in the Higgs discovery, which relies strongly on MC predictions. In this case, MC generators were used to set limits on Higgses in specific parameter space regions that helped to its finding [22, 140, 141, 142].

40 1.5. MONTE CARLO (MC) GENERATORS

The mechanism how MC generators describe a proton-proton collision is based on QCD and QCD-inspired phenomenological models. The parton model explained in the last section (see Section 1.4) is used altogether with hadronization models and experimental data, to repro- duce and predict the result from the hadronic collision. Although each MC model has specific characteristics to simulate a p+p collision, most of them follow a general structure born in the factorization tool, which allows separating the processes according to the momentum transfer scale. A diagram of a p+p collision event simulation by an MC generator is shown in Fig. 1.28.

Figure 1.28: Event structure in a Monte Carlo generator of the p+p collision.

The simulation starts from a very short distance scale, where QDC coupling is weak, and perturbation theory can be applied to calculate the probability distribution of a particular hard process. As exposed in the last section, initially the colliding protons have a partonic flux given by the Parton Distribution Functions (PDFs) or structure functions (see Section 1.4). The collision between the partons is the hard process, characterized by the matrix elements obtained from the leading order (LO) or next-to-LO (NLO) perturbation calculations. The cross-section is obtained from a collinear factorization computation of the PDFs and the parton cross-section (see Eq 1.41). If short-lived resonances are produced in the stage (W ±, Z0, H0) their decay are viewed as part of this process itself. Nevertheless, in the hard parton production process, an indefinite sequence of collinear soft radiation (gluons) is produced, as in the case of accelerated electromagnetic charges (), with the difference gluons are carrying color charge themselves and hence they can emit more gluons or pairs qq¯. This effect cannot be described by perturbation theory, and in order to solve it, parton-shower algorithms were developed. The parton-shower is simulated as a sequential step-by-step process where momentum is transferred from the high scale of the hard-process to lower scales of the order of 1 GeV, associated with confinement of the partons that are going to be packed into the final state hadrons. In the final state radiation (FSR) approach, all the final states that can be built the q andq ¯ partons with an indefinite number of splitting processes are considered. By recursively clustering together final state parton pairs with the smallest relative momentum transfer, two trees rooted at q andq ¯ can be constructed from each final configuration. The momenta of all the intermediate states of the tree are determined from the final-state momenta [22]. Initial-state radiation (ISR) is a different algorithm to develop a parton shower, which starts with the hard process and ends with the incoming parton of the hadron. This approach arises because incoming particles may undergo collinear radiation before entering the hard-scattering

41 CHAPTER 1. BACKGROUND process, acquiring a low transverse momentum [22]. When the momentum scale is reduced to the non-perturbative limit, a different approach must be used to determine the distribution of the final colorless state hadrons from the already produced colored partons. This process is called hadronization and despite not being derived directly from QCD, it is to a good approximation independent of how a specific system produced the final partons. Therefore, although it has various free parameters that are tuned to data, hadronization models have predictive power for other collisions types and energies. There are two main models of hadronization: the String Model, implemented in Phytia [143], and the Cluster Model implemented in Herwing [144] and Sherpa [145]. The string picture is based on the space-time diagram of a meson, where the pair quark- antiquark evolves with each particle flying apart from the other. The separation of the pair is represented as a color flux tube being stretched (see Fig. 1.26(d)) to a “string”, and with the potential energy increasing with distance (V (r) = kr). When the distance between the quark-antiquark reaches a limit from 1 to 5 fm, the string breaks generating a new pair qq¯, emulating the intense chromomagnetic field produced between the pair. The new quarks are accelerated away from each other before they annihilate generating two strings. The breaking process is repeated until a definite number of hadrons is obtained, each hadron formed by the quark from one break and the antiquark from the adjacent break (qq¯1, q1q¯2, q2q¯3,...qn−1q¯) as shown in Fig. 1.29. The probability that a break occurs is proportional to h π i exp − (m2 + p2 ) . (1.47) k q T As a consequence of the mass dependence, strange quarks will be produced to a lesser extent than light quarks. The fragmentation function for hadrons is expressed as

 b(m2 + p2 ) f(z) ∝ zaα−aβ −1(1 − z)aβ exp − h T , (1.48) z where z is the fraction of the remaining momentum the new hadron takes, and aα, aβ, b are free parameters to be adjusted by comparison with data.

Figure 1.29: (Left) Motion and breakup of a string system with two transverse degrees of freedom suppressed. (Right) Color structure of a parton shower in the framework of the cluster model.

The cluster hadronization model is based on the preconfinement property of parton showers. This mechanism can be illustrated considering a large number of colors in the parton shower, with gluons represented as pairs of color-anticolor lines that are connected at vertices (see Fig. 1.29). Thus, the structure of the shower favors the formation of color singlets because color-anticolor partners are adjacent. Non-adjacent partners have a vanishing probability as the number of

42 1.5. MONTE CARLO (MC) GENERATORS colors goes to infinity. The adjacency condition limits the phase space suppressing large masses and leading to an asymptotically mass distribution of adjacent objects. The color-singlet combinations of partons are called “clusters” and after being formed from the asymptotically mass distribution, they can be treated as a spectrum of excited mesons that decay to lighter well-known resonances and stable hadrons. Because of the characteristics of the parton shower production, most of the energy is carried by localized collinear bunches of partons, called jets. The jet structure is preserved in the hadronization process, and they are experimentally observed predominantly in high-momentum- transfer hadronic collisions. Finally, many of the hadrons that are produced during hadronization are unstable resonances. Advanced models are used to simulate their decay to the lighter hadrons that are long-lived enough to be considered stable on the timescales of particle physics detectors. Since many of the particles involved with all stages of the simulation are charged, QED radiation effects can also be inserted into the event chain at various stages [22, 140, 141]. Throughout the above description of the event generator, special attention has been paid to the interaction of one of the partons within the incoming hadron. However, hadrons contain many other colored partons, that interact with each other and have to hadronize. These multiple interactions produce additional partons in the event, which may contribute to any observable, in addition to those from the hard process and associated parton showers considered at the beginning. This part of the event structure is known as the underlying event. The underlying event is modeled generally as multiple parton interactions (MPI) distributed across the hadron in space and momentum with PDFs, each one producing their hard processes and parton showers. These MPIs produce additional back-to-back jet pairs with low transverse momentum, and soft interactions such as color exchange and small momentum transfers, af- fecting the final state activity through the increase of the hadron-multiplicity, redistribution of transverse energy, and the break-up of the beam remnants in the forward direction. MPIs are regulated using momentum conservation, color screening controlled by an effective cutoff parameter, flavor sum rules, etc. that introduce correlations among the MPI. Additional effects have to be included in the simulation depending on the energy regime of the collision, such as Bose-Einstein correlations and color reconnections [22, 140, 141].

43 CHAPTER 1. BACKGROUND

44 CHAPTER 2 Deuteron and antideuteron production model in high-energy collisions and space

In the previous chapter, basic concepts related to cosmic-rays properties, dark matter evi- dence and detection, as well as high-energy hadron collisions and simulations were reviewed. An introduction to such topics was necessary to explain the interest to investigate CR deuterons and antideuterons, which is the principal aim of this thesis. However, before any explanation related to deuterons and antideuterons produced in space, the subjacent mechanism by which deuterons and antideuterons are created, has to be un- derstood. Experimental observations in laboratory colliders have demonstrated that d and d¯ production in high-energy hadron collisions is successfully described through the coalescence model. Thus, a detailed description of this model is presented in Section 2.1. The possible production of antideuterons in CRs as a result of dark matter annihilation motivates their research. Especially because of the advantage antideuterons offer respect to other indirect observations of a hypothetical DM signal, as it will be detailed in Section 2.2. On the other hand, deuteron study is motivated by the information about the propagation of CRs in the Galaxy that can be obtained from the fraction of deuterons created by helium fragmentation in the ISM. This approach will be exposed in Section 2.3.

2.1. Coalescence model

To describe (anti)deuteron formation the coalescence model is used [21, 146, 147]. This model postulates that proton-neutron (pn) or antiproton- pairs (¯p¯n)that are close enough in phase space could result in the formation of deuterons (d) or antideuterons (d),¯ respectively. In the remaining of this section, the antinucleon notation will be used, although the equations are equally valid for nucleons. √ Antideuteron formation occurs with a probability C( s,~kp¯,~kn¯), known as the coalescence function. C depends on the momentum difference 2∆~k = ~k − ~k and the total energy avail- √ p¯ n¯ able ( s). Following the derivation presented in [148, 149], the momentum distribution of an- tideuterons produced in the coalescence scheme can be expressed as

45 CHAPTER 2. (ANTI)DEUTERON PRODUCTION MODEL IN HIGH-ENERGY COLLISIONS

! ! √ Z √ √ dNd¯ ~ 3~ 3~ dNp¯n¯ ~ ~ ~ ~ ( s, kd¯) = d kp¯d kn¯ × ( s, kp¯, kn¯) C( s, kp¯, kn¯). (2.1) d~k3 d~k3d~k3 d¯ p¯ n¯ As a first approximation, it is assumed that the coalescence function does not depend on √ collision energy, resulting in C( s,~kp¯,~kn¯) = C(~kp¯,~kn¯). Momentum conservation leads to express Eq.2.1 as

! ! √ Z √ dNd¯ ~ 3~ 3~ dNp¯n¯ ~ ~ ~ ~ ~ ~ ~ ( s, kd¯) = d kp¯d kn¯ × ( s, kp¯, kn¯) C(kp¯, kn¯)δ(kd¯ − kp¯ − kn¯), (2.2) d~k3 d~k3d~k3 d¯ p¯ n¯

3 3 where dNd¯ = d σd¯/σtot, with σtot and d σd¯ being the total and differential cross-sections and 6 dNp¯n¯ = d σp¯n¯/σtot the number of pairs (¯p¯n)produced in the collision. Transforming the integra- ~ ~ ~ ~ ~ ~ tion variables to the total momentum kp¯ + kn¯ = kd¯ and the relative momentum 2∆k = kp¯ − kn¯, the result is ! ! √ Z √ dNd¯ ~ 3 ~ 3~ dNp¯n¯ ~ ∗ ~ ∗ ~ ~ ~ ~ ( s, kd¯) = d ∆kd kd¯ × ( s, kp¯, kn¯) C(∆k)δ(kd¯ − kp¯ − kn¯), (2.3) d~k3 d~k3d~k3 d¯ p¯ n¯

~ ∗ ~ ~ ~ ∗ ~ ~ ¯ where kp¯ = kd¯/2 + ∆k and kn¯ = kd¯/2 − ∆k. Integrating on the d momenta:

! ! √ Z √ dNd¯ ~ 3 ~ dNp¯n¯ ~ ∗ ~ ∗ ~ ( s, kd¯) = d ∆k × ( s, kp¯, kn¯) C(∆k). (2.4) d~k3 d~k3d~k3 d¯ p¯ n¯ The coalescence idea is based on the very small relative momentum of the pair, reason why ~ ~ is justified to consider that |∆k|  |kd¯|. Under this approximation Eq. (2.4) becomes ! ! √  Z  √ dNd¯ ~ γd¯ 3 ~ ~ dNp¯n¯ ~ ~ γd¯ ( s, kd¯) ≈ d ∆k C(∆k) × γp¯γn¯ ( s, kd¯/2, kn¯/2) , (2.5) d~k3 γp¯γn¯ d~k3d~k3 d¯ p¯ n¯ where the γ factor was introduced to show the result in a Lorentz-invariant form. Next, C is 2 2 approximated by a step function Θ(∆k −p0), where p0 is a free parameter called the coalescence momentum, representing the magnitude of the maximal radius in momentum space that allows antideuteron formation. Under this approximation, the probability changes from zero when |∆~k| > p0 to one if |∆~k| < p0. Evaluating the term within brackets in the rest frame of the pair, the result is

! ! √  3  √ dNd¯ ~ 4πp0 dNp¯n¯ ~ ~ ~ ~ γd¯ ( s, kd¯) ≈ × γp¯γn¯ ( s, kp¯ = kd¯/2, kn¯ = kd¯/2) . (2.6) d~k3 3 d~k3d~k3 d¯ p¯ n¯

Eq. (2.6) indicates that antiproton and antineutron momentum distributions, as well as the coalescence momentum, are the necessary ingredients to estimate the antideuteron cross-section. The coalescence momentum is commonly obtained from the comparison of the model with experimental data. However, the antiproton and antineutron momentum distributions can be considered following an analytical approximation used historically, or the most recent method of the Monte Carlo generators. Both procedures are explained below.

46 2.1. COALESCENCE MODEL

2.1.1. Analytical approach The analytical solution expresses the momentum distribution of the pair antiproton-antineutron as the product of two independent (uncorrelated) isotropic distributions:

dN dN dN p¯n¯ = p¯ × n¯ . (2.7) ~ 3 ~ 3 ~ 3 ~ 3 dkp¯dkn¯ dkp¯ dkn¯ This assumption, however, is overly simplistic since correlations have an important effect on deuteron and antideuteron formation [150, 151, 152]. Replacing Eq. (2.7) in Eq. (2.6) leds to:

 3  ! ! dNd¯ 4πp0 md¯ dNp¯ dNn¯ Ed¯ ≈ Ep¯ En¯ . (2.8) d~k3 3 mp¯mn¯ d~k3 d~k3 d¯ p¯ n¯ Eq 2.8 is known as the analytical coalescence model. The approximation can be taken even further and to consider total symmetry between antiproton and antineutron production, which is derived in !2 dNd¯ dNp¯ Ed¯ ≈ B2 Ep¯ (2.9) d~k3 d~k3 d¯ p¯ 3 4πp0 md¯ B2 = 2 (2.10) 3 mp¯ where B2 is called the coalescence parameter, and it is usually measured by the collision exper- iments. Nonetheless, the assumption about the symmetry has been recently questioned based on the data by the NA49 experiment [153]. There, it was found that a higher antiproton mul- tiplicity was produced in p+n collisions in comparison with p+p collisions [154]. According to [154], this suggests that at this energy: “The measured yield asymmetry between p and n projectiles corresponds to the preference of the positively charged pn¯ combination over the pn¯ with a proton projectile and the negatively charged combination pn¯ over the pn¯ with a neutron projectile”. Such asymmetry seems to be a consequence of isospin effects, where the conservation in the collision process may cause the preferred final production of pn¯ or pn¯ depending on the initial isospin states. At higher energies, the isospin effects disappear due to efficient charge exchange reactions, as can be observed from the antiproton-to-proton ratio measured at the LHC energies [155]. Assumptions of independent (uncorrelated) production of antiprotons and have been used in several analytical calculations of the antideuteron production in CRs [148, 156, 157, 158]. The differential antiproton production cross-section is parameterized using data from collider experiments and theoretical models. Traditional parametrizations [159, 160] were based on experimental data from the 70s and 80s, and they considered the last approximation presented in Eq 2.9. Earlier and most elaborated parametrizations [93, 98, 161], use recent data from NA49 and ALICE experiments, and to consider other important effects into Eq 2.8 as the antiproton- antineutron asymmetry, hyperon decay and scaling violation at high energies.

2.1.2. MC approach Monte Carlo generators, on the other hand, take into account the correlations or anticor- relations involved in the production, with the caveat that there can be uncertainties in the description of these effects. The reason why these effects are included in MC generators comes from the kinematical detail in which the antiproton and antineutron are produced into these

47 CHAPTER 2. (ANTI)DEUTERON PRODUCTION MODEL IN HIGH-ENERGY COLLISIONS programs. This characteristic allows mutual correlations being included when coalescence con- ditions are implemented on an event-by-event basis. Such correlations or anticorrelations may be related to phase space availability, spin alignments, energy conservation, and antiproton- antineutron production asymmetry. These effects are absorbed in the coalescence momentum p0. From the perspective of the coalescence model formulation, MC models solve Eq 2.1 without any assumption of independence or isotropy about the antiproton or antineutron momentum distribution and considering the coalescence function as a step function. Studies of the an- tideuteron production in CRs using MC generators have been developed, showing significant differences in comparison with analytical approaches, as the enhancement found in [150, 152] of the antideuteron production in dark matter annihilation process, and the reduction of an- tideuterons by means of the resonance decay of Upsilon(nS) mesons predicted in [151]. However, most of them were based on old data and assumed a coalescence function independent of the collision energy. In this thesis, the coalescence mechanism is reproduced using MC generators updated to the most recent data of the LHC experiments and a possible collision energy depen- dence of the coalescence function is not discarded. The details about the implementation of the simulation for the coalescence model are described in the next chapter.

2.2. CR antideuterons as a signature for DM

Antideuterons as the result of an accelerator experiment at CERN, colliding proton and beryllium at about 19 GeV were observed for the first time in 1965 [162]. The difficulty to produce antideuterons from a hadron collision lies in the energy threshold production. In a proton-proton collision, the threshold energy required in the center-of-mass system to generate √ an antideuteron is s ∼ 6 GeV, considering the minimum reaction:

p + p → p + p + p + n + d¯

. In the laboratory system projectile energy is around 17 GeV. As a consequence of the high- energy incident particle, the products of the collision are boosted relative to the center-of-mass frame in the direction of the projectile. Hence, the antideuterons produced will have an en- ergy distribution away from the rest system. In the context of cosmic-rays, it is expected an- tideuterons to be produced in the dominant p+p and p+He collisions with an energy distribution (> 1 GeV) elsewhere to the rest galactic frame. A precise evaluation of the antideuteron production cross-section would not be so relevant if an additional signal were not expected. As mentioned in Section 1.3 of the previous chapter, dark matter WIMPs annihilation might lead to the production of light nuclei and antinuclei such as antideuterons. This prediction represents an indirect method for dark matter detection with significant advantages in comparison to other antimatter particles. Because of the very low-velocity distribution of dark matter in the Galaxy, antiprotons and antideuterons created in the annihilation or decay processes involved are at rest with respect to the Galaxy, generating a flux highly populated at low energies (< 1 GeV). Indirect searches exploit the kinematic differences between the production of cosmic-rays through dark matter and standard astrophysical processes to identify dark matter signals. Unfortunately, ionization losses, as well as inelastic non-annihilation scatterings on the inter- stellar medium, produce a redistribution in the energy of antiprotons increasing the low-energy tail of the flux with antiprotons from higher energies. Additionally, solar modulation shifts the

48 2.2. CR ANTIDEUTERONS AS A SIGNATURE FOR DM energy spectrum toward lower energies. The result of these effects is a secondary antiproton contribution that covers a possible exotic signal making it much more difficult to disentangle. The secondary low-energy antideuteron production is further suppressed in comparison with antiprotons as a consequence of the higher energy production threshold, which causes a reduced population of low-energy antinucleons to coalesce into an antideuteron. Besides, the energy loss effects listed above are less efficient in the antideuteron flux. On the other hand, the abundant low-energy population of antinucleons from dark matter annihilation or decay, enhance the antideuteron production because pairs antiproton-antineutron are close in momentum. These features cause that predicted antideuteron fluxes from different dark matter candidates to be a few orders of magnitude above the secondary flux (see Fig. 2.1). The disadvantage with the antideuteron option, as an indirect search for dark matter, is the experimental challenge that represents to detect an antideuteron, because of the highly reduced magnitude of the flux. The BESS experiment set the limit detection in 1.9 × 10−4m−2s−1sr−1(GeV/n)−1 for the d¯ kinetic energy range 0.17 ≤ Ekin ≤ 1.15 GeV/n, and two experiments AMS-02 (currently in operation) and GAPS are expected to reach a sensitivity of 1.0 × 10−6m−2s−1sr−1(GeV/n)−1 and 1.2 × 10−6m−2s−1sr−1(GeV/n)−1 respectively. At the time this document is being written no antideuteron detection has been reported in cosmic-rays. Taking as an example the annihilation of a neutralino in the Galaxy [148], the source term for the SUSY antideuteron production is  2 dNd¯ ρχ q ¯(χ + χ → d¯+ X) = hσ vi , (2.11) d ann d¯ m dEkin χ where hσannvi is the mean value over the galactic velocity distribution of the neutralino pair annihilation cross-section multiplied by the relative velocity. The value of hσannvi is defined considering the relic density range established by cosmological observations, and it is close to −26 3 −1 d¯ 3 × 10 cm s . dNd¯/dEkin is the antideuteron differential energy spectrum, mχ is the mass of the neutralino and ρχ is the mass distribution function of neutralinos inside the galactic halo. The antideuteron differential energy spectrum or multiplicity is calculated using the coales- cence model and may be expressed as

 3      dNd¯ 4p0 md¯ X (F ) dNp¯ dNn¯ ¯ = Bχh × . (2.12) d 3k ¯ mp¯mn¯ dEp¯ dEn¯ dEkin d F,h

As it was detailed in Section 2.1, p0 is the coalescence momentum to be defined from experi- (F ) mental measurements. Bχh are the branching ratios for all annihilation final-states F which may produce antiprotons and antineutrons. After annihilation, quarks and gluons may be directly produced or may be generated through the intermediate states like Higgs bosons, gauge bosons and t quarks creating jets whose hadronization yield the antiproton and antineutron energy spectra. The hadronization process is commonly calculated with Monte Carlo generators. The neutralino mass distribution is assumed to be the same as the dark matter density distri- bution in the Galaxy. Most of the commonly used dark matter density profiles are parametrized as ρ ρ (r) = 0 , (2.13) DM (r/a)γ[1 + (r/a)α](β−γ)/α where a and ρ0 are a characteristic length and a characteristic density, and α, β, γ are dimen- sionless parameters. In the case of the Navarro-Frenk-White (NFW) profile the values are α = 1, β = 3, and γ = 1. As seen before, the source antideuteron term from neutralino annihilation contains astrophys- ical factors such as the dark matter density profile and particle physics aspects like neutralino

49 CHAPTER 2. (ANTI)DEUTERON PRODUCTION MODEL IN HIGH-ENERGY COLLISIONS self-annihilation cross-section. This source term should be inserted in Eq 1.6 to propagate in the Galaxy and, thus, to obtain the flux observed at the Earth. Although the description above was made for a specific WIMP SUSY particle, the procedure is similar to other dark matter candidates, considering their respective cross-sections.

Figure 2.1: (Left) Expected antideuteron flux from three different dark matter candidates [148, 163, 164] com- pared to expected secondary production [151] and the corresponding detector sensitivity limits. (Right) Expected antideuteron flux from heavy dark matter candidates [165, 166].

In Fig. 2.1, the antideuteron fluxes predicted by different dark matter candidates are com- pared. The experimental sensitivity limits for the relevant detectors and the secondary an- tideuteron production evaluated by [151] are also shown. The left side of Fig. 2.1 shows the an- tideuteron flux expected from three dark matter candidates with masses in the tenths of GeVs region. The lightest supersymmetric particle (LSP) neutralino [148], a 5D warped GUT Dirac neutrino (LZP) [163], and the LSP gravitino [164]. As can be seen in the figure, antideuteron expected signals from dark matter are two orders of magnitude over the secondary production for kinetic energy below 0.2 GeV/n. In the right side of Fig. 2.1 heavy dark matter candidates with masses between 0.5 and 20 TeV are presented. The analysis of dark matter candidates in this high mass range was motivated by the interpretation of the positron excess through a heavy dark matter particle (see Section 1.3). However, the multi-TeV candidates have important problems already mentioned in Section 1.3. They need a very large annihilation cross-section in order to match the positron flux, which is not consistent with the relic density. To solve this problem, some models propose an enhancement mechanism effective today but not in the early universe (Sommerfeld enhancement [167]). The heavy dark matter signals shown in Fig. 2.1 are taken from Reference [165], where the model used was the Minimal Dark Matter (MDM) [168], a minimalistic approach not inspired by Beyond- the-Standard-Model ideas such as supersymmetry or extra dimensions. As observed in the Fig. 2.1, the resulting fluxes are promising even when they now extend to the multi-GeV range, above the astrophysical background. The additional case illustrated in Fig. 2.1 is the expected antideuteron signal from the heavy supersymmetric pure-Wino particle as a candidate for dark matter [166], which spreads all over the high kinetic energy region. Whatever the model involved, detectors have to be sensitive enough to discriminate a possible dark matter signal correctly from the astrophysical background. Hence, it necessary to calculate with better precision the secondary antideuteron flux, reducing the production as well as the

50 2.3. DEUTERON AS A PROBE FOR CR PROPAGATION propagation uncertainties. Furthermore, the experiments capable of detecting antideuterons in the near future have a limited detection power, which means that a misinterpretation is likely if all the possible contaminating sources are not appropriately considered.

2.3. Deuteron as a probe for CR propagation

Deuterons are destroyed rather than formed in the stellar formation, and for that reason, a primary source of ejected deuterons from SNRs is highly unlikely. Deuterons in CRs are rare, and they are expected to be produced mainly in CRs nuclear collisions, specifically the fragmentation of He4 and He3 with the ISM. A significant contribution is also expected from the resonant reaction p + p → d + π+, in which deuterons are produced in a narrow energy distribution (FWHM ≈ 320 MeV) with the maximum around ∼ 600 MeV [169]. This last reaction is only significant for energies below 1 GeV, meanwhile fragmentation is the main origin for deuterons at higher energies.

Figure 2.2: (Left) Deuteron flux measurements by [170, 171, 172, 173, 174]. Figure taken from [174]. (Right) Measurements of the ratio 2H/He by [170, 174, 175, 176, 177]. Figure taken from [177].

Besides the two processes described above, CR deuterons are produced in high energy hadron collisions of primary CRs with the ISM. The generation mechanism is the same as the explained in Section 2.1, the coalescence model. At sufficiently high energies, p + p and p + nuclei interac- tions can create multiple nucleon-antinucleon pairs, generating conditions for the formation of deuterons through coalescence, not incorporated yet in the standard calculation of the secondary deuteron CR flux. As noted in Section 1.3 from Chapter 1, the ratio deuteron-to-helium provides important additional information about the diffusive process of the CRs in the ISM, that complements the observations from the B/C ratio. Furthermore, most relevant results are obtained from the high-energy region where solar modulation effect is no longer significant. However, cosmic-ray deuterons at high energies have been poorly explored experimentally, except for two detectors, the balloon-borne experiment CAPRICE98 [174] and the satellite mission SOKOL. Indeed, the identification of deuteron at high energies is not trivial and sometimes requires innovative anal- ysis techniques like that proposed in [177], where the data from SOKOL was analyzed using neural networks. Although the results need to be validated by more data, the authors in [177]

51 CHAPTER 2. (ANTI)DEUTERON PRODUCTION MODEL IN HIGH-ENERGY COLLISIONS found a significant increase of the deuterium to helium ratio for an energy range of 0.5 to 2 TeV (see Fig. 2.2). The most relevant measurements about the deuteron flux and the 2H/He ratio are shown in Fig. 2.2. One of the principal purposes of this thesis is to reevaluate the deuteron flux and the deuteron-to-helium ratio in cosmic-rays, using the expected contribution from the coalescence model and the new propagation parameters extracted from updated data. This reevaluation will allow obtaining a new prediction about the secondary production in CRs with reduced uncertainties that may be compared with data to come from AMS-02.

52 CHAPTER 3 Deuteron and antideuteron production simulation

In the previous chapter, the motivation for this thesis was established as well as the mecha- nism by which deuterons and antideuterons are produced in hadron collisions, and therefore in CRs interactions. The next step is to describe the methodology developed to calculate d and d¯ production cross-section and to corroborate the results with experimental data. In this chapter, results from simulations about deuteron and antideuteron production in hadron collisions are presented. The first section is dedicated to explaining the simulation pro- cess, including the implementation of deuteron and antideuteron production in MC generators, and how the results are compared and presented. In the second section, the results from the simulation are shown including the corresponding analysis.

3.1. Simulation

The simulation of the deuteron and antideuteron production process starts using high-energy MC models to generate hadron and nuclear collisions. The type of collisions to be simulated depends on the experimental data with which model is going to be compared and the phenomena to study. Here, p+p, p+Be, p+Al, p+C, He+p, andp ¯+ p collisions are produced following the data characteristics in Table 3.1 and the most important CRs interactions.

3.1.1. MC generators In the present study, high-energy MC generators have been preferred over their counterparts at low energy. Our choice is based on the conclusions presented in Reference [99], where the authors showed that MC models used in low energy have strong deviations (up to an order of magnitude) from the measured ¯pspectra, while they demonstrate that advanced high- energy MC generators like EPOS-LHC [178] predict the antiproton yield reliably. Furthermore, these generators have been tuned to experimental results in a wide energy range, and they are extensively and consistently used in simulating CRs interactions. Specifically, the Cosmic Ray Monte Carlo package (CRMC) [179] was used to simulate the collisions. Inside CRMC are available various MC generators widely used in high-energy CRs. In this thesis, it was decided to employ EPOS-LHC [178], QGSJETII-04 [180], and SIBYLL2.1 [181]. Besides the CRMC package, simulations were also produced with PYTHIA-8.205 [143] and two

53 CHAPTER 3. DEUTERON AND ANTIDEUTERON PRODUCTION SIMULATION

GEANT4 (version:10.02.p02) [182] hadronic models: FTFP-BERT (based on the Fritiof descrip- tion of string fragmentation [183] with the Bertini intra-nuclear cascade model) and QGSP-BERT (quark-gluon string based model [184] with the Bertini intra-nuclear cascade model). The main features of the MC models used in this thesis are described below.

EPOS-LHC EPOS-LHC [178] is an event generator tuned version of the EPOS 1.99 to reproduce LHC data from various experiments, with an improved implementation of the collective flow and hadronization processes. EPOS [185] is among the MC generators based on the Parton-Based Gribov-Regge theory. EPOS is a combination of previous models, it uses model [186] for soft interactions and the QGSJET model [187] for the semi-hard scattering. EPOS 1.99 and EPOS-LHC are extensively used in cosmic-ray and ultra-high-energy cosmic-ray physics. EPOS model follows the general structure of a Monte Carlo generator described in Sec- tion 1.4, with important additional features. As other MCs, EPOS simulates hadronization employing the string model, however it also contains a statistical hadronization model using a microcanonical scheme. Once strings have been produced and before the hadrons are formed, high and low-density areas of string segments are identified in order to hadronize the high-density regions via the microcanonical procedure. A collective flow in the form of a Bjorken-fluid be- havior is considered and simulated as the final stage of the hadronization process. Although collective effects are more notorious in Heavy- collisions, they are also present in p+p and p+A collisions inserting significant correlations in the final states.

QGSJETII-04 and QGS As in the case of EPOS, QGSJET and QGS are MC generators developed in the framework of the Reggeon Field Theory, which considers the contribution for the soft and hard parton dynamics. Soft interactions (i.e., non-perturbative regime) are dominant in hadronic collisions, and Quark-Gluon String Models (QGSM), as well as Dual-Parton Models (DPM), combine Regge theory and parton model to calculate cross-sections. The principal object in this approach is the Pomeron, and the interactions are mediated by the exchange of one or more Pomerons [187] (see Section 1.3). Since the Regge-Grivob approximation assumes the reaction probability can be factorized, the cross-section is calculated as the sum of all the Pomerons exchanged between any pair of participants partons. Strings are formed using the parton exchange mechanism by the sampling of parton densities and ordering pairs of partons into color coupled entities [188]. Each Pomeron is considered as a pair of color triplet strings, where the strings ends are attached to partons in the interacting hadrons. Strings are latterly decayed by fragmentation models as those described in Section 1.4.

FTF The Fritiof model is also a Regge-Grivob approximation that acts in the diffractive compo- nent of inelastic collisions. It assumes that all hadron-hadron interactions are binary reactions A + B → A0 + B0, where A0 + B0 are excited states of the hadrons. The reaction is classified as single-diffractive or double-diffractive if just one of the hadrons is excited or both hadrons are excited respectively. The string masses are sampled in a region limited by the kinematic condi- 0 0 tions of the reaction M1 + M2 = s, where M1 and M2 are the masses of the A , B hadrons. The excited hadrons are considered as QCD-strings, and the corresponding fragmentation models are adopted to simulate the hadronization process [189].

54 3.1. SIMULATION

Sybill is a Monte Carlo generator widely used in extensive air showers. It is based on the Dual Parton Model, where a nucleon consists of a quark (q, color triplet) and a diquark (qq, color antitriplet). The projectile quark (diquark) combines with the target diquark (quark) to form two strings. Each string fragments separately following the string fragmentation model [181]. Pythia, on the other hand, is a Monte Carlo generator extensively used high-energy hadron collisions with a general structure as the presented in Section 1.4. In version 8, Pythia migrated from Fortran to C++ programming language. Pythia has the additional advantage to include numerous processes related to physics beyond the Standard Model, like supersymmetry [143]. As explained in Section 2.1, to produce (anti)deuterons using MC generators, it is necessary to have a correct prediction of the (anti)proton production. For this reason, once collisions were generated using the MC models described above, the nucleons and antinucleons from the output stack were collected and saved in a new stack file to subsequent analysis. Both outputs were stored in a tree configuration file of the ROOT analysis framework [190], with the crucial kinematical information (momentum and energy) in each branch (see Fig. 3.2). Then, using a typical ROOT macro (analysis.C ), a new stack files were analyzed obtaining the results to be compared to data. In the analysis process, the desired particle to study was selected, and the convenient variables were calculated. Due to the diversity of data presentation, different kinds of variables were calculated, including invariant differential cross-sections or differential cross-sections as a function of rapidity (y), transverse momentum (pT ), Feynman variable (xF ), etc. To contain all these options a C++ class named “Kinematics” was constructed and it was called in the analysis macro to calculate the required quantity. The output was stored once again in a ROOT file but in the form of a graph. With a different ROOT macro, graphs from simulation and data were compared in final plots as the presented in Fig. 3.1, where several MC models were tested and compared to (anti)proton data. )] )] 3 3 /c /c

EPOS•LHC 2 b) 2 SIBYLL 2.1 a) 1 p =0.5 GeV/c T FTFP•BERT PYTHIA 8.205 10 QGSP•BERT NA49 QGSJETII•04 NA61 [mb/(GeV [mb/(GeV 3 3 10•1 /dp /dp σ σ 3 3 p =0.9 GeV/c × 0.3 Ed Ed p =0.5 GeV/c T 1 T

10•2

EPOS•LHC PYTHIA 8.205 •1 10 FTFP•BERT Duperray et al.(2003) 10•3 QGSP•BERT Korsmeier et al. (2018) p =0.9 GeV/c × 0.5 T QGSJETII•04 NA49 SIBYLL 2.1 NA61 •0.5 0 0.5 1 1.5 2 2.5 •0.5 0 0.5 1 1.5 2 y y

Figure 3.1: Invariant differential cross-sections as a function of rapidity (y) are calculated with different MC models for protons a), and antiprotons b) in p+p collisions at 158 GeV/c. Results for two bins of transverse momentum (pT ) are compared with data from experiments NA49 [82] and NA61 [85].

In Table 3.1 a list of the experimental data considered in this thesis is shown along with their collision characteristics. The selection of these experimental data was based on their relevance to the most abundant cosmic ray species, as well as to the energy range in which deuterons

55 CHAPTER 3. DEUTERON AND ANTIDEUTERON PRODUCTION SIMULATION and antideuterons are produced in CRs collisions. Since part of the available experimental data is old enough to lack the precision tracking and vertex determination techniques available today, this might have introduced inherent systematic uncertainties. For example, feed-down contribution to protons and antiprotons (from decays of heavier baryons) were not handled well in some of these data, contributing to the mismatch between data and MC production. The detected fraction of protons and antiprotons produced by this mechanism depends on the energy boost generated by the parent hyperons decay, as well as the details of the detector. These characteristics make it difficult to estimate, a posteriori, the proper correction [91, 93, 191]. For the case of experiments at CERN-ISR, where p+p collisions with center of mass energy from 23 to 53 GeV were studied, a correction was possible. According to [82], the detector design of this experiment allowed nearly all baryonic decay products to be included in the measured cross-section. Thus, here the corresponding correction factors were extracted from simulations and applied to this group of data. This was not the case for other data sets, as indicated in Table 3.1. √ Experiment or Reference Collision Final states plab s Phase Space Laboratory (GeV/c) (GeV) ITEP 1 [192] p+Be p 10.1 4.5 1≤ p ≤7.5 GeV/c; θ = 3.5 deg CERN 1 [193, 194] p+p p, ¯p 19.2 6.1 2≤ p ≤19 GeV/c; p+Be p, ¯p 0.72 ≤ θ ≤ 6.6 deg CERN 1 [194] p+p p 24 6.8 2≤ p ≤9 GeV/c; θ = 6.6 deg NA61/SHINE [195] p+C p 31 7.7 0≤ p ≤25 GeV/c; 0≤ θ ≤ 20.6 deg [85] p+p p, ¯p pT ≤ 1.5 GeV/c; 0.1≤ y ≤2.0 NA61/SHINE [85] p+p p, ¯p 40 8.8 pT ≤ 1.5 GeV/c; 0.1≤ y ≤2.0 1 Serpukhov [196, 197] p+p p, ¯p 70 11.5 0.48≤ pT ≤ 4.22 GeV/c; θlab = 9.2 deg [198] p+Be p, ¯p [199] p+Al p, ¯p NA61/SHINE [85] p+p p, ¯p 80 12.3 pT ≤ 1.5 GeV/c; 0.1≤ y ≤2.0 CERN-NA49 [82] p+p p, ¯p 158 17.5 pT ≤ 1.9 GeV/c; xF ≤1.0 [83] p+C p, ¯p CERN-NA61 [85] p+p p, ¯p pT ≤ 1.5 GeV/c; 0.1≤ y ≤2.0 CERN-SPS 1 [200, 201] p+Be p, ¯p 200 19.4 23≤ p ≤197 GeV/c p+Al p, ¯p θlab = 3.6 mr, θlab = 0 1 Fermilab [202, 203] p+p p, ¯p 300 23.8 0.77 ≤ pT ≤ 6.91 GeV/c; p+Be p, ¯p θlab = 4.4 deg, θcm = 90 deg 1 Fermilab [202, 203] p+p p, ¯p 400 27.4 0.77 ≤ pT ≤ 6.91 GeV/c; θlab = 4.4 deg p+Be p, ¯p CERN-ISR [204] p+p p, ¯p 1078 45.0 0.1< pT <4.8 GeV/c; 0.0≤ y ≤1.0 CERN-ISR [204] p+p p, ¯p 1498 53.0 0.1< pT <4.8 GeV/c; 0.0≤ y ≤1.0 3 CERN-LHCb [86] p+He ¯p 6.5× 10 110 0.0≤ pT ≤4.0 GeV/c; 12≤ p ≤110 5 CERN-ALICE [84] p+p p, ¯p 4.3× 10 900 0.0≤ pT ≤2.0 GeV/c; -0.5≤ y ≤0.5 7 CERN-ALICE [84] p+p p, ¯p 2.6× 10 7000 0.0≤ pT ≤2.0 GeV/c; -0.5≤ y ≤0.5 1 No feed-down correction. Table 3.1: Experimental data list on proton and antiproton production in p+p and p+A collisions considered in this thesis to be compared to simulations.

3.1.2. Afterburner To generate (anti)deuterons emulating the coalescence process, an afterburner [155] was cre- ated to be coupled to the MC generators output. An afterburner is a name given to routines com- monly used in MC codes to modify the particle distribution produced by the generator according to a model. The afterburner performed an iterative operation for every event, by identifying all proton-neutron and antiproton-antineutron pairs from the stack of particles created by the generator and calculating the difference in momenta of each pair in their center-of-mass frame. Half of the magnitude of this difference (∆k = |~kp¯ − ~kn¯|/2) was compared to the coalescence ~ ~ ~ momentum p0. If ∆k was lower than p0, (an)a (anti)deuteron with momentum kd = kp + kn (or

56 3.2. RESULTS

q q ~k = ~k +~k ) and energy E = ~k2 + m2 (or E = ~k2 + m2) was included in the stack, while d¯ p¯ n¯ d d d d¯ d¯ d¯ the corresponding nucleons were deleted from it (see Fig. 3.2). (Anti)protons and (anti)neutrons from weak decays were excluded from the simulations.

Monte Carlo Generator

STACK i+1 P0, E0 P1, E1 ¼. j+1 Pi, Ei NO

YES NEW STACK AFTERBURNER Nucleons Ppi, Epi Pnj, Enj Coalescence conditions

NO YES

PUT IN STACK Pdi = Ppi+Pnj Edi = Epi+Enj

i+1 DELETE j+1 Ppi, Epi Pnj, Enj

Figure 3.2: Diagram of the afterburner implementation for the production of deuterons and antideuterons.

The afterburner functions were written in a ROOT class (AntiNucGen, see AppendixA) which is programmed to read the output stack that contains the nucleons and antinucleons produced by the MC generator. The code receives as an input parameter the test value of the coalescence momentum p0. Then, all the iterative operations are run, and the coalescence condition is set considering the test value entered. Finally, the program saves a new ROOT output file with the deuterons and antideuterons who fulfilled the conditions. The value of the coalescence momentum was varied in steps of 5 MeV/c, and the (anti)deuteron spectra corresponding to each of these values were compared with the experimental data in Table 3.2. As can be observed, data include ALICE experiment measurements. The p0 that produced the lowest χ2 fit was thus selected as the optimal value to reproduce experimental results. Coalescence parameters at different energies were estimated with the intention to produce a global parametrization that allows predicting the deuteron and antideuteron production in the characteristic energy range for CRs. Results of the complete process are presented in the next section.

3.2. Results

In this section, the results from the simulation process described above are presented. First, the proton and antiproton production obtained with the MC models is compared to data in

57 CHAPTER 3. DEUTERON AND ANTIDEUTERON PRODUCTION SIMULATION

√ Experiment or Reference Collision plab s No. of points Phase Space Laboratory (GeV/c) (GeV) d dbar CERN [194] p+p 19 6.15 6 0 0≤ p ≤9 GeV; θ = 6.6 deg CERN [194] p+p 24 6.8 4 0 0≤ p ≤9 GeV; θ = 6.6 deg Serpukhov [198] p+p 70 11.5 7 2 0.48≤ pT ≤ 2.4 GeV; θlab = 9.2 deg p+Be 6 3 CERN-SPS [200, 205] p+Be 200 19.4 3 5 15≤ plab ≤ 40 GeV; θlab = 0 deg p+Al 3 3 Fermilab [203] p+Be 300 23.8 4 1 0.77 ≤ pT ≤ 6.91 GeV; θlab = 4.4 deg CERN-ISR [206, 207, 208] p+p 1497.8 53 3 8 0.0≤ pT ≤1.0 ; θcm = 90 deg 5 CERN-ALICE [155, 209] p+p 4.3× 10 900 3 3 0.0≤ pT ≤2.0 ; -0.5≤ y ≤0.5 7 CERN-ALICE [155, 209, 210] p+p 2.6× 10 7000 21 20 0.0≤ pT ≤2.0 ; -0.5≤ y ≤0.5

Table 3.2: Experimental data list on deuteron and antideuteron production in p+p and p+A collisions considered in this thesis to be compared to simulations. order to select the most appropiate MC generators. Next, the afterburner is coupled to the selected MC models, and the deuteron and antideuteron production is estimated.

3.2.1. p and p¯ production simulation and MC selection

250 250 EPOS•LHC QGSP•BERT EPOS•LHC SIBYLL Mean = 0.6 Mean = •2.4 Mean = 0.8 Entries Mean = 0.8 Entries RMS = 13.0 200 RMS = 6.0 200 RMS = 6.0 RMS = 5.7

FTFP•BERT PYTHIA8 QGSJETII•04 Mean = 3.7 Mean = •1.6 Mean = 4.2 150 RMS = 9.3 RMS = 10.4 150 RMS = 10.6

100 100

50 50

0 0 •15 •10 •5 0 5 10 15 •15 •10 •5 0 5 10 15 ∆ ∈ / ∆ ∆/∈∆

Figure 3.3: Distributions of difference between measurements and the MC generators divided by the uncertainty (see Eq. 3.1) for proton production in p+p and p+A collisions.

To determine which MC is describing (anti)proton measurements most reliably in the energy range considered, a quantitative comparison between MC models, parametrizations and data is made with the help of Eq. 3.1.

 d3σ sim d3σ data ∆ E dp3 − E dp3 = (3.1) p 2 2 ∆ (sim) + (data) This equation allows calculating the difference (∆) between measurement and simulated 3 3 differential cross-sections (Ed σ/dp ). Then ∆ is divided by the total uncertainty (∆). The

58 3.2. RESULTS

resulting quantity (∆/∆) is evaluated for every data set listed in Table 3.1, and their distri- butions for a choice of models are illustrated in Figs. 3.3 and 3.4 for protons and antiprotons, respectively. Ideally, these distributions should be centered at zero, with the RMS value close to 1 when the measurement and the theoretical value are compatible on an absolute scale.

180 180 EPOS•LHC Duperray et al QGSJETII•04 EPOS•LHC 160 Mean = •0.7 Mean = 2.1 160 Mean = 3.6 Mean = •0.7 Entries Entries RMS = 4.1 RMS = 4.6 RMS = 5.2 RMS = 4.1 140 140 FTFP•BERT Korsmeier et al Mean = 0.0 Mean = •0.5 SIBYLL Korsmeier et al 120 RMS = 5.9 RMS = 5.0 120 Mean = •1.9 Mean = •0.5 RMS = 5.1 RMS = 5.0 100 QGSP•BERT PYTHIA8 100 Mean = 4.0 Mean = 7.3 RMS = 7.6 RMS = 6.4 80 80

60 60

40 40

20 20

0 0 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15 ∆ ∈ / ∆ ∆/∈∆

Figure 3.4: Distributions of difference between measurements and the MC generators divided by the uncertainty (see Eq. 3.1) for antiproton production in p+p and p+A collisions.

Fig. 3.3 illustrates how proton production in p+p and p+A collisions is, in general, better described by EPOS-LHC. A similar analysis for antiprotons is presented in Fig. 3.4, but in this case, the parameterization of Duperray et al. [160] and the parametrization presented by Winkler [93] which was updated by Korsmeier et al. [161] to the latest NA61 and LHCb data are added. As in the case of protons, the antiproton prediction from EPOS-LHC provides better results than other MC models, while being comparable to the parametrizations. The momenta dependence corresponding to the EPOS-LHC simulation of Fig. 3.3 and Fig. 3.4 are shown on the left side of Fig. 3.5 for protons, and on the right side for antiprotons. In these plots, the distribution was divided into two momentum regions, low (from 10 to 100 GeV/c) and high (> 100 GeV/c). For protons (Fig. 3.5 left), the low momentum distribution (solid red line) is shifted to positive values, accounting for the positive value tail in Fig. 3.3. In the high momentum region (dashed red line) the distribution is more symmetric but broader. For antiprotons, the resulting distributions from Korsmeier et al. parametrization have also been included in Fig. 3.5 right. As can be observed the low momentum distribution of EPOS-LHC is shifted to positive values indicating an overestimation of antiprotons. However, it also shows a lower RMS value compared to the parametrization. The high-energy distribution for EPOS- LHC under-predicts antiproton production, revealing that both distributions contribute to the positive and negative value tails in Fig. 3.4 left. From the results shown above, EPOS-LHC is the best option, among MC generators, to estimate proton and antiproton production in p+p and p+A high-energy collisions. Furthermore, because the GEANT4 framework is broadly used in simulations of particle interactions with detectors, the GEANT4 hadronic model FTFP-BERT is also included here. Note, however, the

59 CHAPTER 3. DEUTERON AND ANTIDEUTERON PRODUCTION SIMULATION

140 120 EPOS•LHC EPOS•LHC Korsmeier et al 10•100 GeV 10•100 GeV

10•100 GeV Entries Entries 120 Mean = 1.9 100 Mean = 1.0 Mean = •0.4 RMS = 3.6 RMS = 4.4 RMS = 5.5 100 >100 GeV >100 GeV >100 GeV 80 Mean = •1.7 Mean = •0.5 Mean = •0.7 RMS = 4.1 RMS = 5.3 80 RMS = 6.3 60 60

40 40

20 20

0 0 •15 •10 •5 0 5 10 15 −15 −10 −5 0 5 10 15 ∆/∈ ∆ ∆/∈∆

Figure 3.5: Distributions, in two different energy regions, of the difference between measurements and EPOS- LHC divided by the uncertainty (see Eq. 3.1) for proton (left) and antiproton (right) production in p+p and p+A collisions.

use of this last MC model is limited to kinetic energy collisions having T < 10 TeV.

Total p¯ production cross-section Finally, the total antiproton productions in p+p and p+He collisions obtained with EPOS- LHC are compared to the parametrizations by Duperray et al. and Korsmeier et al. In Fig. 3.6 the results for these three models as a function of the collision kinetic energy T in the laboratory system are presented. From Fig. 3.6 can be observed EPOS-LHC has notable differences respect to parametrizations. In both interactions, EPOS-LHC predicts a higher antiproton production than Korsmeier et al. in the region from about 30 GeV to 200 GeV, and a lower cross-section from about 200 GeV to 2×106. In contrast, EPOS-LHC produces a lower number of antiprotons in the whole energy range for p+p collisions compared to Duperray et al. (see Fig. 3.6 left). The above agrees with the analysis of Fig. 3.5, where it was shown EPOS-LHC overpredict the production of antiprotons at low energies, while at high energies the Monte Carlo tends to underpredict it. For p+He collisions, EPOS-LHC and Korsmeier, with updated p+A reaction cross-sections based in recent data, prognosticate a largerp ¯ production cross-section production than Duperray (see Fig. 3.6 right). Results for He+p and He+He collisions from EPOS-LHC are also included in Fig. 3.6 right. Note that the total antiproton cross-section presented in Fig. 3.6 was evaluated for antipro- tons produced in short time-scales and originated from the fragmentation of the colliding par- tons, not from the decay of intermediate strange baryons. However, to determine the antiproton flux in the Galaxy, it is necessary to account for antiprotons produced by the late decays of antineutrons, and hyperons. Λ¯ In recent works using parametrizations, the ratio of hyperon-induced (fpp→p¯) to promptly- 0 ¯ ¯ produced (fpp→p¯) antiprotons is determined by considering the contribution of Λ and Σ as well as some symmetry assumptions and the limited data available [91], obtaining

60 3.2. RESULTS

102 [mb] [mb] 2 p p 10 σ σ

10 10

1 1

EPOS•LHC −1 −1 10 10 Duperray et al. p+He EPOS•LHC Duperray et al. p+He Korsmeier et al. Korsmeier et al. He+p × 2 −2 10−2 10 He+He

102 103 104 105 106 107 102 103 104 T [GeV] T [GeV]

Figure 3.6: EPOS-LHC total antiproton production cross-section comparted to parametrizations from Duperray et al. and Korsmeier et al. in p+p collisions (left) and p+He, He+p, and He+He (right).

Λ¯ fpp→p¯ 0 = 0.2-0.3. (3.2) fpp→p¯ In EPOS-LHC the decay-length is set to a long distance (100 cm) to assure a major fraction of antiprotons from hyperons decay are included without sacrifice a reasonable computing time. The ratio of the hyperon-induced to promptly-produced antiprotons in EPOS-LHC is between 0.2 to 0.3, which means EPOS-LHC describes the antiproton multiplicity from hyperon decays in agreement with [91]. As pointed in [91], recent data analysis in proton collisions at different center-of-mass energies have shown discreapancies with the general symmetry assumption betweenp ¯ andn ¯ production. The results in [153, 154] indicate a possible preference for the production of pn¯ pairs compared to np¯ pairs in p+p collisions, which is parametrized by an isospin factor defined as

0 fn¯ ∆IS = 0 − 1. (3.3) fp¯ This isospin factor measures the enhancement of antineutron over antiproton production, and it has been conservatively estimated to be in the range ∆IS = 0−0.43 [91]. In Reference [93], the isospin factor is considered energy-dependent based on the argument isospin effects disappear at high energies due to very efficient charge exchange reactions, which permute protons and neutrons. This argument is supported by measurements of the antiproton-to-proton ratio at LHC energies. Surprisingly, EPOS-LHC, unlike other Monte Carlo generators, has implemented in its hadronization model, an asymmetry in the antiproton-antineutron production that favors the antineutron production in p+p collisions. This asymmetry arises as a consequence of the tuned parameters in string fragmentation, since it is easier to produce a dd diquak than a uu accord- ing to LEP data on particle production [211]. EPOS-LHC predicts an antineutron-to-antiproton

61 CHAPTER 3. DEUTERON AND ANTIDEUTERON PRODUCTION SIMULATION ratio that varies between 1.2 and 2.0, or comparing with the parametrization, an isospin factor in the range from 0.2 to 1.0.

3.2.2. d and d¯ Production Simulation Estimation of coalescence momentum As described in Section 3.1.2, the coalescence momentum derived from the joint simulation of MC and afterburner is compared to data in Table 3.2. An example of the results from this analysis is presented in Fig. 3.7, where the p+p at 70 GeV/c case is shown. As observed, the best values of p0 at this particular energy were 25 MeV/c for EPOS-LHC and 50 MeV/c for FTFP- BERT. In the Korsmeier et al. parametrization case, p0 was evaluated using the analytical expression in Eq. 2.8 assuming antiproton-antineutron independence and symmetry, i.e., the analytical coalescence model. The result from fitting Eq. 2.8 to data was a value p0 = 32 MeV/c (dashed cyan line in Fig. 3.7). Duperray et al. proposed a constant p0 = 79 MeV/c over the whole energy range, also shown in Fig. 3.7 (solid magenta line). )] 3

/c −1

2 10 data p at y = 0 10−2 data d at y = 0 10−3

[mb/(GeV −4

3 10

/dp 10−5 σ 3 −6 EPOS•LHC (p = 25 MeV/c) Ed 10 0 FTFP•BERT (p = 50 MeV/c) 10−7 0 Duperray (p = 79 MeV/c) 10−8 0 Korsmeier (p = 32 MeV/c) 0 10−9

1.5 1 0.5 Model/Data 1.5

1

0.5 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 p /nucleon (GeV/c) T

Figure 3.7: Antiproton and antideuteron invariant differential cross-sections in p+p collisions at 70 GeV/c as function of transverse momentum (pT ) calculated with EPOS-LHC, FTFP-BERT and parametrizations [160, 161]. The results are compared to data [196, 197, 198] (see text for details).

The differential cross-sections computed with the resulting p0 values for EPOS-LHC, FTFP- BERT, as well as the parameterizations [160, 161] are compared with the data in Appendix B.1. The values of p0 extracted from the comparison to data are shown in Fig. 3.8 (a) for deuterons and in Fig. 3.8 (b) for antideuterons, as a function of the collision kinetic energy (T) in the laboratory system. Although the trend of the p0 values obtained with different MC models as a function of T is similar, their magnitudes differ from one simulator to the other and also with respect to parametrizations. Differences between MC models and parametrizations result from the correlations (or anticorrelations) in the antinucleon pairs only present in the MC generators [149, 150, 151, 152]. Disparities in the corresponding MC model assumptions, lead to deviations of their predictions for nucleon and antinucleon production, causing differences in the extracted p0 among MC generators. To compare the coalescence momentum among MC models it is useful to factorize the (anti)nucleon mismatch assuming uncorrelated and

62 3.2. RESULTS

DEUTERONS ANTIDEUTERONS 300 220 FTFP•BERT EPOS•LHC QGSJETII•04 EPOS•LHC 200 p+p p+p 250 p+p p+p [MeV/c] p+Be p+Be [MeV/c] 0 0 180 p

p p+Be p+Be p+Al p+Al 160 200 FTFP•BERT p+p 140 p+Be 120 150 100

100 80 60 Duperray et al. 50 40 Korsmeier et al. (a) + Coalescence (b) 20

102 103 104 105 106 107 102 103 104 105 106 107 T [GeV] T [GeV]

Figure 3.8: Extracted coalescence momentum p0 (symbols) for deuterons (a) and antideuterons (b) as function of the collision kinetic energy (T). Fit functions [Eqs. (3.4) and (3.5)] for EPOS-LHC (long-dashed red line) and FTFP-BERT (dashed blue line) are shown. Additionally, the p0 values obtained from the analytic coalescence model and the parametrization of Korsmeier et al. are included (dashed cyan line and dots). Also, the constant value of p0 = 79 MeV/c estimated by Duperray et al. is plotted (solid magenta line).

symmetric production, hence treating the p0 difference as due to antiproton mismatch. The details and results of this process are shown in Appendix B.2. As shown in the next subsection this factorization does not affect the deuteron and antideuteron cross-section calculations. Note that in the low collision-energy region (T < 100 GeV) shown in Fig. 3.8 (a) the p0 for deuterons decreases reaching a saturation value for T > 100 GeV. The measurements reported in Table 3.2 show that the deuteron production cross-section is larger at T ≈ 19-24 GeV than for higher energies. The increase in production seems to be induced by the contribution of opening inelastic channels, not related to coalescence. However, this increase is reproduced in the simulation through the rise in p0 near that particular energy region. Below 19 GeV no further comparisons in deuteron production were made, due to limitations of the MC models used. Down at 1−3 GeV, the coalescence model is no longer valid. In this low energy region, deuteron production is determined by direct reactions correlated to the initial state as p + p → d + π+ and is independent of similar processes where protons and neutrons are created as p + p → p + n + π+ [212]. In the case of antideuterons, p0 increases beyond the production threshold (T ≈ 17 GeV) until it saturates at high energies (see Fig. 3.8 (b)). Keep in mind that this energy dependence appears in the MC simulations, as well as in the Korsmeier et al. parametrization shown in Fig. 3.8 because they reflect best fits to the characteristic trend of the data. However, the gradual growth of p0 beyond the antideuteron production threshold is expected due to phase space [151, 213]. To generate an energy-dependent p0 parameterization that can be used with MC codes, the p0 points shown in Fig. 3.8 have been fitted using Eq. 3.4 for deuterons and Eq. 3.5 for antideuterons. The resulting parameters are given in Table 3.3. Since in Fig. 3.8 the p0 obtained at specific energy shows no significant differences among p+p, p+Be, and p+Al, Eq. 3.4 and Eq. 3.5 are used to produce a common (target independent) parameterization for deuterons and

63 CHAPTER 3. DEUTERON AND ANTIDEUTERON PRODUCTION SIMULATION antideuterons respectively. Fit function for deuterons:

  ln(T/GeV) p = A 1 + exp B − (3.4) 0 C Fit function for antideuterons:

A p = (3.5) 0 1 + exp(B − ln(T/GeV)/C)

Model A (MeV/c) BC

Deuterons EPOS-LHC 80.6 ± 2.39 4.02 ± 0.62 0.71 ± 0.11 FTFP-BERT 118.1 ± 2.42 5.53 ± 2.28 0.43 ± 0.14 QGSJETII-04 93.1 ± 2.89 3.20 ± 0.12 1.19 ± 0.07

Antideuterons EPOS-LHC 89.6 ± 3.0 6.6 ± 0.88 0.73 ± 0.10 FTFP-BERT 170.2 ± 10.5 5.8 ± 0.47 0.85 ± 0.08 Korsmeier et al.2 153.6 ± 3.7 4.5 ± 0.36 1.47 ± 0.14 2 Used with the analytical coalescence model.

Table 3.3: Values of the parameters for the fitting functions 3.4 and 3.5.

Total d and d¯ production cross-section in p+p and p+He collisions Based on the coalescence momentum parametrizations of Eq. 3.4 and 3.5, the total deuteron and antideuteron cross-sections (σd,d¯ = σp+p(p+A) ×nd,d¯/Nevt) were estimated using the MC sim- ulations to extract the total inelastic cross-section (σp+p(p+A)), as well as the number of events ¯ with at least one d or d (nd,d¯), for a given total number of events (Nevt). In the Korsmeier et al. parametrization case, Eq. 2.2 (with antiproton-antineutron independence and symmetry) was integrated using Eq. 3.5 and parameters in Table 3.3. The results in p+p and p+He colli- sions as a function of the collision kinetic energy are plotted in Fig. 3.9, together with available measurements. The left panels of Fig. 3.9 show the results in p+p collisions. The data extracted from Meyer, J. P. [169] show the reaction p+p → d + π+, while the other data [212] and the simulations represent the inclusive reaction p+p → d + X. Fig. 3.9 (a) shows how deuteron cross-section starts to decrease with energy until it reaches the point-of-inflection of about 100 GeV which marks the change of slope in the p0 parametrization. From this point, thanks to the constant p0, the cross-section starts to grow continuously. The antideuteron cross-section on the other hand (Fig. 3.9 (b)), emerges from the production threshold and rapidly grows until it changes of slope around T˜1000 GeV, where the coalescence momentum changes to a constant value. The total antideuteron cross-section increases to meet the deuteron one at a very high-energy. On the right side of Fig. 3.9, the results for p+He collisions are plotted along with data at lower energy from Meyer, J. P. [169]. Data only include the reactions: p + He4 → He3 + d and p + He4 → d + n + 2p (see Fig. 3.9 (c)). Simulations have higher values because they include the

64 3.2. RESULTS coalescence contribution and the fragmentation reactions. However, MC estimation is not far from Meyer extrapolation. The cross-section for antideuterons has a similar behavior in p+He as for p+p collisions (see Fig. 3.9 (d)), because antinucleons are formed in nucleon-nucleon collisions. In the lower panels of Figs. 3.9 (b) and (d), the ratios of the antideuteron cross-section between the Duperray et al. parametrization and the results from EPOS-LHC, FTFP-BERT and Korsmeier et al. were plotted. As can be observed, the estimations from this thesis are significantly lower at T<100 GeV than the prediction from Duperray et al. This reduction is a direct consequence of the behavior of p0 in this energy region, where instead of having a constant value the coalescence momentum grows gradually.

p+p collisions p+He collisions 10 103 p+p → d+X p+p → d+π+ 1 Holt, et al Meyer, J.P. [mb] [mb] 2 d Bugg et al. d 10

σ D. E. Pellet σ 10−1 Zorn 10 − Meyer J. P. Deuteron 10 2 Deuteron EPOS•LHC Data EPOS•LHC − 1 Extrapolation FTFP•BERT 10 3 (a) FTFP•BERT (c)

1 10−1 1 (b) − −2 1 (d) [mb] 10 [mb] 10 d d −3 −2 σ 10 σ 10 Antideuteron − Antideuteron 10−4 10 3 − EPOS•LHC EPOS•LHC 10 5 FTFP•BERT 10−4 −6 − FTFP•BERT 10 10 5 − Duperray et al 7 − Duperray et al 10 Korsmeier et al 10 6 −8 Korsmeier et al 10 10−7 300 10 200 5 100 0

Duperray/MC 0 Duperray/MC − − 10 1 1 10 102 103 104 105 106 107 10 1 1 10 102 103 104 105 106 107 T [GeV] T [GeV]

Figure 3.9: Deuteron (a) and antideuteron (b) total production cross-section in p+p collisions. Deuteron (c) and antideuteron (d) total production cross-section in p+He collisions. The expected antideuteron cross-section from Duperray’s parametrization has been added. In the lower panels Duperray to MC predictions in antideuteron are compared. Vertical broken lines represent the antideuteron production threshold.

Total d¯ production cross-section in p¯ + p, p¯ + He and He+He collisions Because of the kinematical characteristics of the antideuteron production process, antipro- tons colliding with the interstellar medium have become, as shown by [157], a significant con- tribution to the antideuteron cosmic-ray flux. Although antiproton cosmic-ray flux is around 4 to 5 orders of magnitude lower than proton CR flux, the threshold energy production is only about 4 GeV in the center-of-mass frame. The threshold energy is lower because baryon conser- vation number requires two additional baryons in the final state. This contribution stops being important as energy goes beyond threshold because, despite cross-section production increases, antiproton CR flux decreases hardly. In previous works [157], the calculation of the antideuteron cross-section inp ¯+p collisions was made under certain assumptions. Since they consider an independent production of antiprotons and antineutrons, and the differential cross-sections for the independent processesp ¯ + p → pX¯ andp ¯ + p → n¯ + X are unknown, then the differential cross-section for the former process was approximated to that in p + p → pX¯ , and the differential cross-section for the later was taken

65 CHAPTER 3. DEUTERON AND ANTIDEUTERON PRODUCTION SIMULATION as in p + p → p + X, i.e.,

dσn¯ dσp¯ En¯ (¯p + p → n¯ + X) ≈ Ep¯ (p + p → p¯ + X), (3.6) ~ 3 ~ 3 dkn¯ dkp¯

dσp¯ dσp Ep¯ (¯p + p → p¯ + X) ≈ Ep (p + p → p + X). (3.7) ~ 3 ~ 3 dkp¯ dkp However, as pointed by [151] this approximation can be questioned because baryon number conservation varies between both reactions in Eq. 3.6. On the left side, an additional baryon is required, while on the right side at least three additional baryons have to be produced in the final state. Hence, the reaction on the right-hand side is suppressed, and they should not be comparable.

10 2 T p

/d EPOS•LHC

σ 2

10 FTFP•BERT [mb] d

d 1 QGSJETII•04 PYTHIA 8.205 σ Duperray et al.(2003) 10 data, p+p (32GeV/c) 10−1

10−2 1

−3 10 EPOS•LHC 10−1 p+He × 102 2 10−4 p+p × 10 2.2 He+He × 10 2 1.8 He+p 1.6 10−5 1.4 He+p 1.2 Model/Data 1 Duperray et al. −6 0.8 10 Korsmeier et al. 0.6 0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 102 103 104 p2 (GeV/c) T T [GeV]

2 Figure 3.10: (Left) Antiproton differential cross-section as a function of pT in ¯p+ p collisions at 32 GeV. Results from simulations with Monte Carlo generators are compared to data from [214]. (Right) Total antideuteron production cross-section in ¯p+ p, ¯p+ He, He+p and He+He collisions as a function of kinetic energy of the collision T.

In this thesis, the antideuteron production in ¯p+ p collisions is calculated using the Monte Carlo generators studied before. To validate the adequate predictability of these generators, the results for antiproton production in ¯p+ p at 32 GeV/c collisions are compared to data from the Mirabelle bubble chamber at IHEP [214], as presented on the left side of Fig. 3.10. In the figure can be observed how PYTHIA 8.2 reproduces reasonably well the data while EPOS-LHC and QGSJET-II-04 underpredict the antiproton production at around 20%. The parametrization from Duperray et al. on the other hand widely overpredicts antiprotons in this process. Assuming the same value of the coalescence momentum for the ¯p+ p collision as in the case of p + p, the complete event-by-event analysis as described in the last subsection is performed to the EPOS-LHC output. As before, the antiprotons and antineutrons produced by the Monte Carlo are clustered in the afterburner by means of the coalescence model conditions. Resulting antideuterons are counted at each bin of energy collision and normalized to the total number of events to calculate the energy differential cross-section and the total antideuteron cross-section

66 3.2. RESULTS for that specific channel. The result of the total antideuteron cross-section for ¯p+ p and ¯p+ He as a function of the collision kinetic energy is shown on the right side of Fig. 3.10. In addition to the antiproton collision channels the right side of Fig. 3.10 includes the results for the total antideuteron production in He+p and He+He collisions, which represent a sizeable contribution to the total antideuteron flux. To calculate the antideuteron production, the same coalescence momentum as in p+p collisions was considered in both interactions. For the case of He+p, such consideration is based on the results obtained from the analysis to p+A collision data performed above, where it was concluded a similar p0 value could describe experimental results. However, in He+He collisions such supposition is not that obvious, mainly because data in heavy-ion interactions show the coalescence parameter (B2 in Eq. 2.10) is smaller in heavy-ion collisions than in hadron collisions [215]. This result is a consequence of the source volume inmediatly after the collision, which is bigger in heavy-ion interactions than in hadron interactions. Thus, the produced nucleons are far from each other decreasing the probability to coalesce. Nevertheless, the reaction under study is a light-nuclei interaction closer to the results obtained for p+A collisions and therefore with a similar coalescence momentum.

Total d production cross-section in He+p collisions Unlike antideuterons, the production of the deuteron in cosmic-rays is mainly due to collisions of helium nuclei with the interstellar medium. Although contributions from heavier nuclei are also considered to calculate the final deuteron flux. Thus, the deuteron production from the coalescence mechanism estimated in the previous subsections does not play a primary role in the deuteron cosmic-ray production. As observed in Fig. 3.9, the coalescence influence is important at high energies (>1 GeV), where the proton flux has decreased enough to make its contribution negligible. On the contrary, the low energy deuteron production due to the resonant reaction p+p → d+π+ represents around a 40% of the total contribution at ∼ 1 GeV/c energy where the peak of the cross-section is located. The critical quantity to analyze in this subsection is the total inclusive production cross- section ((3,4)He + p → d), not the cross-section of the different channels that lead to particular final states. Since the deuteron energy distribution follows a peaked gaussian around the energy of the projectile, the final energy of the deuteron is approximated to that of the projectile, this is known as the straight-ahead approximation, and it has been shown to be adequate for the level of precision attained by the CR data [216]. Measurements about fragmentation reaction are available only below 10 GeV as shown in Fig. 3.11. The sets of data taken into account, as the parametrizations used in this thesis are extracted from the work by Coste et al. [216] and references therein. For the deuteron produc- tion through the fragmentation of 4He, they consider the break-up and stripping cross-sections separately (see Fig. 3.11). Break-up reaction (dotted red line) corresponds to the case where the helium nucleus breaks up leading to coalescence of free nucleons into a new nucleus. The stripping on the other hand (dashed blue line), happens via the pickup reaction where the in- cident proton tears a neutron or a proton off the helium nucleus. From Fig. 3.11 is observed the break-up process is dominant at high energies but parametrization only can be extended until 3 GeV per nucleon where the limit of data is reached. Beyond that energy there is no information about the cross-section, thus in this thesis, a calculation using the hadronic Monte Carlo generators QGSJET-II-04 and EPOS-LHC was performed in the energy region >10 GeV. On the left side of Fig. 3.11 is shown the data and the parametrization of the reaction 4He + p → d together with the results from the Monte Carlo simulation. The dotted red line below 0.2 GeV is the stripping parametrization fitted to data from Tannenwald et al. [169], Griffiths & Hairbison [217], Rogers et al. [218], Jung et al. [219], while the dashed blue line represents the

67 CHAPTER 3. DEUTERON AND ANTIDEUTERON PRODUCTION SIMULATION break-up process extending up to around 10 GeV fitted to data from Innes and Caims et al. [169], Jung et al. [219], Webber [220], Aladashvill et al. [221], Glagolev et al. [222], Abdullin et al. [223] (for details see [216]). Above this point, the dotted-dashed green line shows the prediction from QGSJET-II-04 and the dotted-dashed red line is the result from EPOS-LHC. As can be observed in Fig. 3.11, EPOS-LHC produces an erratic behavior in the cross-section, probably originated in a careless tunning to this specific process [211]. However, the prediction from QGSJET-II- 04 is satisfactory, producing a curve with a cross-section value near to the expected from the parametrization to data. In the right side of Fig. 3.11, the deuteron total inclusive production cross-section generated by the process 3He + p → d is shown. The parametrization, respresented by the solid black line, is fitted to data from Griffiths & Hairbison [217], Meyer [169], Blinov et al. [224], and Glagolev et al. [222]. As in the case of 4He, 3He fragmentation is also the addition of two contributions: stripping and break-up. Despite the similar magnitude of the production cross-section in 3He fragmentation compared to 4He fragmentation, 3He cosmic-rays are less abundant and the contribution to the deuteron flux is expected to be about 20% at 1 GeV, decreasing at higher energies.

Deuteron production in 4He+p → d+X Deuteron production in 3He+p → d+X 120 120

QGSJETII•04 EPOS•LHC [mb] [mb]

d EPOS•LHC d 100 + coalescence + coalescence

σ 100 σ + coalescence Alone Alone Alone

80 QGSJETII•04 80 + coalescence Alone 60 60

40 40

20 20

0 0 − − 10−2 10−1 1 10 102 103 104 10 2 10 1 1 10 102 103 104 T [GeV] T [GeV]

Figure 3.11: Parametrization and data for deuteron production in the reaction 4He+p → d (Left) and 3He+p → d. Stripping (dotted red line), break-up (dashed blue line) and total (solid black line) parametrizations fitted to data are shown togheter with simulations results from QGSJET-II-04 (dotted-dash and long-dash green lines) and EPOS-LHC (dotted-dash and long-dash red lines) (see text for details). Data and parametrizations taken from [216].

The unexpected results from [177] show an overproduction of deuterons at very high-energy (see Section 2.3). This excess is not explained by the expected production cross-sections modeled with Monte Carlo generators, and as mentioned, no data has been measured in this energy region. Thus, even when more data is necessary to probe any anomaly in cosmic-ray deuterons, it may be assumed that some additional mechanism is creating more deuterons than expected. The hypothesis claimed in this thesis is that coalescence, after fragmentation processes, might be a contributor to the deuteron production at high energies. This supposition is not that far from what is observed since the break-up process is indeed a coalescence action at low energies. Hence to determine the magnitud of the cross-section in (3,4)He + p → d reactions, the after-

68 3.2. RESULTS burner togheter with the output from the Monte Carlo generators QGSJET-II-04 and EPOS- LHC is used. To evaluate the deuteron production, a constant value of the coalescence parameter (p0) is adopted, specifically that given by the parameter A presented in Table 3.3 obtained by fitting Eq. 3.4 to deuteron coalescence momentum. The results for QGSJET-II-04 and EPOS- LHC are shown in Fig. 3.11 as a long-dashed green line and a long-dashed red line respectively. EPOS-LHC shows now a steadily growing cross-section that reaches values near 100 mb. The observed curve indicates that the lack of deuterons in the region >400 GeV is compensated by forming more deuterons through coalescing the fragmented protons and neutrons. This effect indicates that the break-up process has not been appropriately tuned and when an external tool as the afterburner is used to coalesce proton-neutron pairs, then an adequated behavior is obtained. QGSJET-II-04 predicts lower cross-section values reaching a maximum of 80 mb at 104 GeV, and it shows a faster-growing tendency as in the case without coalescence.

69 CHAPTER 3. DEUTERON AND ANTIDEUTERON PRODUCTION SIMULATION

70 CHAPTER 4 Galactic secondary deuteron and antideuteron flux

Once the deuteron and antideuteron production cross-sections are known, they can be used along with transport models to determine the flux in different media. In this chapter, the propagation of deuterons and antideuterons through the Galaxy is calculated employing one of the most advanced CRs transport codes: GALPROP. In the first section, a description of GAL- PROP and the propagation process is presented. In the second section, the source term obtained using the production cross-section is explained and analyzed. The interaction of deuterons and antideuterons with the interstellar medium is described in the third section. Finally, the results of this study are shown and compared with previous works in the fourth section, significant discrepancies are underlined.

4.1. GALPROP

GALPROP [45, 225, 226] is a numerical code for calculating the propagation of relativistic charged particles and the diffuse emissions produced during their propagation. The GALPROP code incorporates as much realistic astrophysical input as possible, together with the latest theoretical developments. The code calculates the propagation of cosmic-ray nuclei, antiprotons, electrons, and positrons, and computes diffuse γ-rays and synchrotron emission in the same framework. GALPROP code is public and flexible since it can be updated introducing new data or processes. Besides, it has an entirely physical approach because a real propagation environment and all the known propagation effects are included. Finally, it can reproduce simultaneously almost all the data from space missions. GALPROP allows the user to define the main parameters of the simulation through a file called galdef. The galdef file used in this thesis is shown in Appendix C, and the essential values of the parameters considered for our simulation are explained below.

4.1.1. Galactic structure GALPROP is able to simulate the propagation of cosmic rays in 2D and 3D models of the galaxy, using information from different fields of and astrophysics. In this thesis a 2D model is used to simulate the propagation of cosmic-rays, assuming a cylindrical symmetry (r, z) with isotropy in momentum space. In this model, the Galaxy is considered as a dense disk

71 CHAPTER 4. GALACTIC SECONDARY DEUTERON AND ANTIDEUTERON FLUX of thickness 2h with h = 100 pc, and it is immersed in a bigger cylindrical halo centered on the inner disk, where cosmic rays are trapped by the galactic magnetic field. The half-height of the halo is defined by the user, employing the z min and z max in the galdef file, as well as the radius of the halo named r min and r max. The value of the half-height parameter usually is between 1 kpc and 15 kpc as suggested by previous studies on radioactive nuclei [227] and distributions of synchrotron radiation [45]. The radius of the halo usually runs from 10 kpc to 30 kpc.

4.1.2. Galactic source distribution As it was explained in Section 1.1, galactic cosmic-rays are produced mostly by supernovae. These sources are located only in the inner disk of the Galaxy, and its distribution is parame- terized in GALPROP using the equation

−1 −γi qi(r, z, R) = fi(r, z)β R δ(r − rmax), (4.1) where R is the rigidity, γi is the spectral index for every species (γ ≡ γe(R), γp(R), γnuclei(R)) and fi(r, z) is the spatial source distribution defined as

r r−r0 z α −β r − z fi = q0( ) e 0 scale . (4.2) r0

Here r0 =8.5 kpc represents the radial distance from the center of the Galaxy to the Solar System, and q0 is a normalization factor. The parameters to define in the galdef file are: zscale =0.2 kpc, which is a modulation factor that takes into account the confinement of the sources into the inner disk, and corresponds to source parameter 0 ; α =1.5, is source parameter - 1 ; and β =3.5 represents the parameter (not the velocity) source parameter 2. Eq 4.2 is chosen because it reproduces (after propagation) the cosmic-ray distribution determined by analysis of EGRET gamma-ray data [43]. GALPROP has a number of options both to change the parameters of the parametrization, as to introduce other distributions. In this thesis, the default values presented above are used.

4.1.3. Galactic magnetic field The default Galactic magnetic model implemented in GALPROP is based in the work by [228], where a large scale data sets on polarization with nearly 7000 stars were analyzed and modeled. The resulting parametrization of the uniform magnetic field is

r−r0 z r − z B = B0e scale scale , (4.3) where parameters B0 = 5 µ G, rscale =10 kpc, and zscale =2 kpc. The total magnetic field distribution is adjusted to match the 408 MHz synchrotron and distributions.

4.1.4. Interstellar gas distribution As mentioned in Section 1.1 the most important components of the interstellar medium are hydrogen and helium. Hydrogen is present in three forms, atomic hydrogen (HI), molecular hydrogen (H2), and ionized hydrogen (HII). The atomic hydrogen spatial distribution is param- eterized as an exponential function of the halo height

2 −ln2(z/z0) nHI (r, z) = nHI (r)e , (4.4)

72 4.1. GALPROP giving an exponential increase in the width if the HI layer outside the solar circle: ( 0.25 kpc if r ≤ 10 kpc z0(r) = (4.5) 0.083e0.11r kpc if r > 10 kpc

The distribution of the molecular hydrogen is modeled as

−ln2(z/70pc)2 nH2 (r, z) = nH2 (r)e . (4.6)

The adopted radial distribution of nHI (r) and nH2 (r) is represented in Fig. 4.1. For the ion- ized gas, a two-component model is used, with the first term representing an extensive warm ion- ized gas added to a second component that represents the concentration of HII around r = 4 kpc.

2 2 |z|  r  |z|  r  − 1 kpc − 20 kpc − 0.15 kpc − 2 kpc −2 −3 nHII (r, z) = 0.025e + 0.2e cm (4.7) The He/H ratio for the interstellar gas is taken as 0.11, consistent with photospheric deter- minations [43].

Figure 4.1: Adopted radial distribution of atomic (HI), molecular (H2), and ionized hydrogen (HII) at z = 0. Figure taken from [43].

4.1.5. Isotopic abundances

1 64 GALPROP allows propagating a large number of nuclear cosmic-rays, from 1H to 28Ni as well as elementary particles: e, e+, and γ. The selection of the particles to propagate is set using the galdef parameters max Z which determines the maximum number of nuclei and use Z i, which indicates if the nucleus of number i is going to be used. Additionally to the number and species of nuclear cosmic-ray to be included in the simulation, in the galdef file it is possible to specify the various isotopes abundances at the source for each nuclei setting the parameter iso abundance xx yy (see Appendix C).

73 CHAPTER 4. GALACTIC SECONDARY DEUTERON AND ANTIDEUTERON FLUX

4.1.6. Transport equation GALPROP solves the transport equation considering a source distribution and boundary conditions for all cosmic-ray species. The equation includes Galactic wind (convection), diffusive re-acceleration in the interstellar medium, energy losses, nuclear fragmentation, and nuclear de- cay. The numerical solution of the transport equation is based on a second-order implicit scheme. The spatial boundary conditions assume free particle escape. The details about GALPROP code can be found in [225]; here the most relevant properties are outlined. The transport equation is written as ∂f(p, ~r, t) ∂ ∂ 1 = ∇·~ D (p, ~r)∇~ f − V~ f + p2D f ∂t xx ∂p pp ∂p p2 (4.8) ∂ h p i 1 1 − pf˙ − (∇·~ V~ )f − f − f + q(p, ~r, t), ∂p 3 τf τr where f(p, ~r, t) is the number density of particles in the ISM per unit of total momentum. Dxx is the spatial diffusion coefficient, V~ is the convection velocity, Dpp is the diffusion coefficient in momentum space describing re-acceleration,p ˙ is the momentum loss rate, τf is the timescale for fragmentation, and τr is the timescale for the radioactive decay. The spatial diffusion coefficient is η δ taken as Dxx = β D0(R/R0) and depends on rigidity R with a break below/above R0. δ values are commonly between 0.3 and 0.6, which correspond to Kolmogorov and Kraichnan spectra of interstellar turbulence. The convection velocity V (z) is assumed to increase linearly with distance from the plane (dV/dz > 0 for all z). The physical interpretation of these quantities was described in Section 1.1, where the transport equation was introduced. Here, it will be explained how these parameters are introduced in the GALPROP model. As mentioned above, GALPROP initial conditions and settings are listed in the galdef file (see Appendix C). The normalization factor of the spatial difussion coefficient D0 is represented by a parameter called D0 xx and the corresponding labels for δ are D g 1 and D g 2 indicating the exponent value before and after the rigidity break R0, represented in the galdef file as D - rigid br. Diffusive reacceleration and convection are activated through the parameters diff reacc = 1 and convection = 1, meanwhile initial convection velocity and gradient are inserted as v0 conv and dvdz conv. Re-acceleration is determined by means of the Alfv´envelocity, which corresponding parameter is v Alfven. From the analysis in [229] to fit the AMS-02 proton data correctly at low energies and to maintain a good agreement above 200 MV with data, an exponent η in the diffusion coefficient was introduced. The values of the parameters were taken from the same work [229] since they were in excellent agreement with data. The values are shown in Table 4.1.

Parameter Best Value Units z 4.0 kpc 28 2 −1 D0/10 4.3 cm s δ 0.395 - −1 Valf 28.6 kms V 12.4 kms−1 dV/dz 10.2 kms−1kpc−1 η 0.91 -

Table 4.1: Propagation parameter values and spectral parameters obtained from Reference [229].

The nucleon injection spectrum is assumed to be a power law in momentum p−γ, as explained

74 4.1. GALPROP in Section 1.1.1 due to the acceleration mechanisms. Since rigidity is the parameter governing the propagation in the galactic magnetic field, it is preferable to replace momentum per rigidity. Thus in GALPROP, the injection spectrum has the form

∂s  R −γ ∝ , (4.9) ∂p R0 where R0 is, as usual, a reference rigidity where the injection spectrum is breaking. In order to fit the latest results from AMS-02, the injection spectrum is extended to have three different exponents γ1, γ2, and γ3 with two reference rigidities R1 and R2, their values are presented in Table 4.2. In the galdef file the parameters γ1, γ2, and γ3 correspond to nuc g 1, nuc g 2, and nuc g 3 while the parameters R1 and R2 correspond to nuc rigid br1 and nuc rigid br2.

Parameter Best Value Units

R1 7 GV R2 360 GV γ1 1.69 - γ2 2.44 - γ3 2.28 -

Table 4.2: Spectral parameters obtained from Reference [229].

Energy losses for nucleons by ionization and Coulomb interactions are included as described in [43]. The total magnetic field distribution is adjusted to match the 408 MHz synchrotron longitude and latitude distributions. The secondary source term q(p) in Eq. 4.8 for deuterons and antideuterons will be described and calculated in Section 4.2. The propagation process starts reading the galdef file and assigning the initial parameter values. Then a spatial grid is created, defining the gas distribution and magnetic field in every point according to the equations exposed in the previous subsections. The interstellar radiation field energy density is read from a file, and the skymap parameters for gamma rays and syn- chrotron emission are defined. After galaxy structure has been formed, a loop over every cosmic ray species is executed, calling in each iteration a group of routines with specific functions. The first is the create transport arrays routine which assigns primary source functions, diffusion coefficient, fragmentation rate, momentum loss rate and decay rate to every given species. The fragmentation rate is assigned using the total cross-section calculation on a proton target, and it is extended to a He target by a geometrical scaling. It is calculated in every spatial and energy point as   1 nHe F = = βc(nH2 + nHI + nHII ) σp + σHe . (4.10) τf (nH2 + nHI + nHII ) The cross-sections used to calculate the fragmentation rate are those given in [43] and refer- ences therein. Only in the case of deuteron production, the fragmentation rate of helium was modified according to the cross-section presented in Section 3.2.2, including the prediction from Monte Carlo generators. In this part, it is also implemented the inelastic cross-section reaction of antideuterons as described in Section 4.3. Energy losses for nucleons are produced by ioniza- tion and Coulomb interactions which formulae used in GALPROP are shown in Section 4.3. For electrons, the essential processes are ionization, Coulomb scattering, bremsstrahlung in the neu- tral and ionized medium, and Compton and synchrotron losses. For details about the equations used in each case see [43].

75 CHAPTER 4. GALACTIC SECONDARY DEUTERON AND ANTIDEUTERON FLUX

After fragmentation rates and energy losses are assigned, the next step is the evaluation of the secondary contributions. This process is implemented in the routine gen secondary - source, where all the secondary source functions are combined. It includes the secondary positron, antiproton and proton source terms, and also the tertiary antiproton contribution. As part of this thesis, a new secondary source function for antideuterons was created and added to the main routine. The decay cross-section is also introduced as part of this routine taking into account short-lived intermediate states and the network. Finally, the routine propel propagates the particles resolving the diffusion equation using the Crack-Nicolson method. This method consists in approximating the derivatives employing finite differences. For example, considering the differential equation in one dimension

∂u ∂2u = D , (4.11) ∂t ∂x2 expressed in terms of the CR distribution function u(x, t). The Crank-Nicholson discretization is then: " # un+1 − un D (un+1 − 2un+1 + un+1) + (un − 2un + un ) i i = i+1 i i−1 i+1 i i−1 , (4.12) ∆t 2 (∆x)2 where the index i refers to the spatial grid i∆x, while the indexes n + 1 and n refer to the evaluation of the quantities at time t + ∆t and t. This method was shown to be stable and accurate to second-order in space and time. Note Eq 4.12 is a combination of the forward (n) and backward (n + 1) Euler method. A system of algebraic equations must be solved, to obtain the next u value as a function of time. In this case, the algebraic problem is tridiagonal and may be efficiently solved with the tridiagonal matrix algorithm. For the case of the general Eq 4.8, this can be finite-differenced for each dimension (R, z, p) or (x, y, z, p) in the form

n+1 n n+1 n+1 n+1 n n n ∂f f − f α1f − α2f + α3f α1f − α2f + α3f i = i i = i−1 i i+1 + i−1 i i+1 + q (4.13) ∂t ∆t 2∆t 2∆t i where α1, α2, and α3 are coefficients defined according to the processes including in the calcu- lation (for example re-acceleration, convection, etc) and the defined coordinates [43]. The boundary conditions imposed at each iteration for 2D are

f(R, zh, p) = f(R, −zh, p) = f(Rh, z, p) = 0. (4.14) Typical grid intervals go like ∆R = 1 kpc, ∆z = 0.1 kpc. For 3D, the boundary conditions are f(±xh, y, z, p) = f(x, ±yh, z, p) = f(x, y, ±zh, p) = 0, (4.15) with typical grid intervals ∆x = ∆y = 0.5 kpc, ∆z = 0.1 kpc.

4.1.7. Solar modulation In order to compare the calculations performed with GALPROP and the cross-sections evaluated in the previous chapter with data collected in Earth by cosmic-ray experiments such as AMS-02, it is necessary to take into account the effect of the solar wind. In this thesis, the effective theory of a spherically symmetric and charge-independent force field is adopted as presented in [230, 231]. The model accounts for three key processes: cosmic-ray diffusion through the magnetic field carried by the solar wind; convection by the outward motion of the solar wind; and adiabatic deceleration of the cosmic-rays in this flow.

76 4.2. DEUTERON AND ANTIDEUTERON SOURCE TERMS

The first two processes lead to a rigidity-dependent decrease of the particle flux, while the third one leads to a decrease in the energy of the particles that penetrate the . The solar wind mostly affects rigidities below 20 GV [3]. From the mentioned references it is concluded that at high enough energies (>0.1 GeV), cosmic-ray transport in the solar wind under this approximation can be expressed as ∂f VR ∂f + ≈ 0 (4.16) ∂r 3k ∂R where f is the cosmic-ray distribution function, r is the heliocentric radial distance, and R is the particle rigidity. V represents the solar wind speed, and k is the diffusion coefficient for radial propagation. The final solution only depends on one parameter

1 Z rhs V φ = dr (4.17) 3 r k where r1 is the heliospheric radius of the Earth (1 AU) and rhs is the boundary of the heliosphere usually assumed to be 50 AU. A particle that has a total energy EIS in the interstellar space would reach the Earth with energy ETOA = EIS − |Z|φF , where φF = eφ is known as the Fisk potential and it commonly runs from 500 MeV to 1.3 GeV over the eleven years solar cycle. The flux at the top of the Earth´satmosphere ΦTOA is related to the interstellar flux ΦIS in terms of the kinetic energy through

p2 2mAETOA + (AETOA)2  Φ (ETOA) = TOA Φ (EIS ) = kin kin Φ (EIS ) (4.18) TOA kin 2 IS kin IS IS 2 IS kin pIS 2mAEkin + (AEkin)

TOA IS where the kinetic energy per nucleon is expressed as Ekin = Ekin − (|Z|/A)φF [3, 151]. GALPROP does not consider any solar modulation correction. It only delivers the interstellar flux for every particle in every position of the galaxy into a .gz file which contains the FITS file. A program in ROOT was created to extract the flux at the top of the Earth. This program includes some python routines to read the FITS file and to get the specific particle flux at the solar system position. Once the interstellar flux is obtained, Eq. 4.18 is used to calculate the top of the atmosphere flux with a given value of Fisk potential (see Appendix C).

4.2. Deuteron and antideuteron source terms

4.2.1. Deuterons The secondary deuteron source is introduced in GALPROP as a fragmentation rate in the gen secondary source routine. From Section 3.2.2 it is known that deuteron production at high energies is dominated by the fragmentation of 4He and 3He isotopes, and at low ener- gies, it is mainly due to the fusion reaction p + p → π + d. Spallation of heavier nuclei (C, O, Fe) gives a minor contribution, less than around 10% of their fractional abundance. For f all these processes it is assumed that the fragment is ejected (Ekin) with the same kinetic en- p ergy per nucleon as the projectile Ekin. This straight-ahead approximation is expressed by f p f f p σ(Ekin,Ekin) ≈ σ(Ekin)δ(Ekin − Ekin). The energy of the fragments roughly follows a Gaus- sian distribution, with an insignificant impact (< 10%) in the secondary flux compared to the straight-ahead approximation [216, 232]. Thus, the results for the fragmentation cross-section in Fig. 3.11 found combining data parametrization for Ekin <10 GeV and Monte Carlo simulations to predict the result in the energy region Ekin >10 GeV, are inserted in Eq. 4.19.

77 CHAPTER 4. GALACTIC SECONDARY DEUTERON AND ANTIDEUTERON FLUX

  1 nHe Fd = = βc(nH2 + nHI + nHII ) σHe+p→d + σHe+He→d (4.19) τd (nH2 + nHI + nHII )

For nuclear reactions involving heavier nuclei (Z >2), the default cross-section parametriza- tion of GALPROP was used. The fusion reaction contribution was already added in the version of GALPROP adopted in this thesis, into the class gen secondary source, where the differential cross-section was evaluated based on the data and parametrizations by Meyer [169] and Coste [216].

4.2.2. Antideuterons

The secondary source function for antideuterons (qd¯) generated in CRs collisions with the ISM is defined as the antideuteron production rate per unit of volume, time, and kinetic energy d¯ per nucleon (Ekin):

∞ Z ! sec d¯ X X dσ q¯ (Ekin) = 4πnj dTi Φi(Ti) (4.20) d dEd¯ i=p,He,¯p j=p,He kin ij Tmin

Here, the index i represents all the incident cosmic rays species with flux Φi and kinetic energy per nucleon Ti. The index j represents the ISM components with number densities −3 −3 np = 0.9 cm and nHe = 0.1 cm , which were distributed over the galactic disk according to Section 4.1.4. The threshold energy (Tmin) is ≈ 17 GeV for p+p collisions and ≈ 7 for ¯p+p collisions. The proton, helium and antiproton fluxes have been measured recently, with an unprecedented precision by the AMS-02 collaboration [15, 233, 234]. The results indicate a spectral hardening at high rigidity (R ∼ 400 GV) for proton and helium spectra, described by a double power law function. These new spectra are considered as the primary fluxes Φi in the antideuteron source function. To obtain the secondary source term, the antideuteron differential production cross-section (dσ/dTd¯)ij was calculated from the collision simulations with EPOS-LHC (see Chapter 3 and Reference [235]). The resulting (dσ/dTd¯)ij were inserted in Eq. 4.20 for 44 different collision ener- gies between 17 and 1×107 GeV. To introduce EPOS-LHC results into GALPROP propagation model a new routine called gen secondary antideuteron was created. In this routine, a loop over the energy and position grid is executed calling in each iteration the corresponding values of the cross-section obtained with EPOS-LHC. At the end of the loop, the source term matrix is assigned to the secondary source function, which is called by the propel routine responsible for the propagation. The secondary source term obtained with EPOS-LHC is shown in Fig. 4.2. On the left side are gathered all the CR interactions considered in this thesis that contribute to the antideuteron flux, and whose cross-sections were evaluated in Sections 3.2.2 and 3.2.2. From this study, it is concluded the p+p channel represents around 48% of the total flux, while the antiproton channels account for ∼ 5.2% of the antideuteron flux. However, this low contribution is dominant below ∼1 GeV as can be observed in Fig. 4.2. In this thesis, the contribution from He+He has been included although it constitutes a meager percentage (8.7%) of the total Helium channels which are around 47% of the flux. On the right side of Fig. 4.2, the EPOS-LHC source term is compared with results from previ- ous works [151, 157]. It is evident from the figure that EPOS-LHC predicts a higher antideuteron production in CRs collisions, reaching around twice the value from Ibarra et al. [151] and ∼1.2

78 4.2. DEUTERON AND ANTIDEUTERON SOURCE TERMS times the result from Duperray et al. [157]. Furthermore, the maximum of the source distribu- tion is slightly shifted in the antideuteron kinetic energy in comparison with parametrizations, being located at approximately 5 GeV/n. This shift is a consequence of the competing factors that fold the secondary source term, i.e., the rising production cross-section that dominates at low energies and the rapidly decreasing incident CR flux that governs the high-energy region. Thus, in contrast with previous calculations, the production cross-section from EPOS-LHC is more significant in the high energy range, causing the increase observed in qd¯ and despite the decreasing incident flux it also produces a broader distribution with a flattened shape on the top of the curve and the shifting of its maximum. •1 •1 − Total 10 34 10−34 p+p p+He He+p s GeV/n] s GeV/n] 3

3 He+He 10−35 p+p 10−35 [cm

[cm p+He d d q q

10−36 EPOS•LHC 10−36 Ibarra et al.

10−37 Duperray et al.

10−37

10−38

10−1 1 10 102 103 10−1 1 10 102 Kinetic energy per nucleon E [GeV/n] Kinetic energy per nucleon E [GeV/n] kin kin

Figure 4.2: Secondary source term for antideuterons as function of kinetic energy per nucleon. On the left side the total contribution from the most important collisions is shown. On the right side, the result is compared to previous works [151, 157].

Besides the initial creation of antideuterons in CR collisions, these light nuclei interact with the ISM in a process named non-annihilation. The non-annihilating cross-section corresponds to antideuterons that interact inelastically with the hydrogen of the media, but survive the collision, ¯ 0 ¯ losing a fraction of their initial energy, i.e., the process is represented as d(Td¯) + p → d(Td¯) + X. The emerging contribution from this interaction is called the tertiary term, and its importance was emphasized in Reference [236]. In a medium of constant density np, the tertiary term reads:

 ∞  Z dσnon−ann ! dp¯ ter  0 non−anndp¯  qd¯ (Td¯) = 4πnp dTd¯ Φd¯(Td¯) − σ Φd¯(Td¯) (4.21)  dT¯  d ij Td¯ Therefore the total source term is:

sec ter qd¯ = qd¯ + qd¯ (4.22) The tertiary term is implemented in GALPROP by constructing an additional routine called gen tertiary antideuteron. An additional iteration is requested to the GALPROP main pro- cess in order to secondary antideuterons were produced in the first step. Then, these secondary particles are inserted into the tertiary routine in the second step. At the end of the iteration, the total (secondary and tertiary) contribution is transferred to propel for the propagation.

79 CHAPTER 4. GALACTIC SECONDARY DEUTERON AND ANTIDEUTERON FLUX

4.3. Deuteron and antideuteron interaction with matter

After deuterons and antideuteron are generated in the collisions of CRs with the ISM, they interact with the matter in the Galaxy. They suffer energy losses and absorption because of ionization, Coulomb scattering, and nuclear collisions. The electromagnetic interactions are well understood, and appropriate equations for their description have been developed. The formulae used in GALPROP, for example, are based on the works of Mannheim and Schlikeiser 1994. The energy losses in Coulomb collisions in a completely ionized plasma are given by

  2 dE 2 2 2 β ≈ −4πre cmec Z nelnΛ 3 3 (4.23) dt Coul xm + β where re is the classical electron radius, me is the electron rest mass, c is the velocity of light, Z is the projectile charge, β = v/c is the nucleon speed, ne is the electron number density in plasma, xm is function of the electron temperature, and lnΛ is the cold plasma limit, and depends on the projectile mass and the nucleon velocity [43]. For the ionization losses the general formula based on the Bethe-Bloch equation is

dE  1 X (β > β ) = −2πr2cm c2Z2 n B , (4.24) dt 0 e e β s s s=H,He

2 where ns is the density of the corresponding species in the ISM, β0 = 1.4e /~c is the characteristic velocity determined by the orbital velocity of the electrons in hydrogen, and

  2 2 2   2mec β γ Qmax 2 2Cs Bs = ln 2 − 2β − − δs , (4.25) Is zs where γ is the Lorentz factor of the ion, Qmax is the largest possible energy transfer from the incident particle to the atomic electron, and Is is the geometric mean ionization and excitation potential of the atom for the ISM species [43]. These equations are implemented in GALPROP within the energy losses.cc class.

4.3.1. Antideuteron inelastic cross-section The energy loss induced by the elastic scattering of the transported particles at the energies considered here involves only small momentum transfers, i.e., it is negligible. Inelastic processes, on the other hand, may involve sizeable energy-momentum loss of the scattered particles, jus- tifying the need to include it in any propagation calculation. Here, the analysis presented in References [157, 158] is followed. However, the fact to consider other approximations do not lead to significant changes, since the primary source of uncertainty remains in the production cross-section. The inelastic cross-section is defined as

σinel = σtot − σel. (4.26) No data exist for the total cross-section of the process d+H,¯ but there are measurements for the charge conjugate reaction d+¯pas shown in the left side of Fig. 4.3. Thus, the total cross- dp¯ d¯p section is assumed as σtot = σtot represented by the black solid line in Fig. 4.3 (Left). As the d¯p ¯pp total cross-section σtot is well approximated to 2σtot (see Fig. 4.3 (Left)), a similar assumption pd¯ ¯pp is considered for the elastic cross-section σel ≈ 2σel , where no data exist even for the charge conjugate reaction. The elastic cross-section approximation is shown as the black solid line

80 4.3. DEUTERON AND ANTIDEUTERON INTERACTION WITH MATTER in the right of Fig. 4.3, and as can be observed it is a factor of two larger than the Glauber description (black dashed line) used in [157]. Both parametrizations of the d¯ptotal cross-section and ¯pptotal elastic cross-section have been taken from [22].

Figure 4.3: (Left) Total cross-section data of thepd ¯ reaction and its parametrization (black solid line) [22]. The black dashed line correspond to the Glauber approximation used in [157]. The two dotted curves are estimations of the dp¯ total cross-section by means ofpp ¯ andpn ¯ cross-sections. (Right) Elastic cross-section data for thepp ¯ andpn ¯ reactions and its parametrization (dotted line) [22]. The black line is a model of the pd¯ elastic cross- ¯pp section using 2σel . The dashed-dotted line corresponds to the Glauber cross-section used by [157]. Figures taken from [158].

The inelastic cross-section has two contributions, the non-annihilation and the annihilation cross-section:

non−ann ann σinel = σinel + σinel . (4.27) The total non-annihilating cross-section is observed when antideuterons interact inelastically with protons but survive the collision losing a fraction of their initial energy. The importance of this process is based on experimental evidence and formal grounds. Inelastic cross-section induced by a composite nuclear projectile is expected to be larger than for incident nucleons. Indeed, from the definition at high energies, it is anticipated cross-section increases with the nuclear number as the area “seen” by the particles colliding is bigger. This result is confirmed by experimental data from [237], where a larger cross-section for dn → d(pπ−) than for pn → p(pπ−) was measured. The practical approach to estimate the non-annihilation cross-section is the same as con- sidered in [157, 158]. The energy integrated dp¯ → dX¯ cross-section has been inferred from the experimental values of the cross-section for the symmetric systempd ¯ → Xd. The total inelastic non-annihilation cross-section has been obtained by summing up the ¯p+ d → (nπ)¯pdcross- section which is experimentally available, leading to a peak around 4 mb as shown in the left side of Fig. 4.4. For the energy distribution of the non-annihilation antideuteron cross-section, a similar shape to the differential cross-section in p+p → p+X [238] is considered:

2 2 d σ p γ(E − βpcosθ) 2 h pT i = 610pT exp − , (4.28) dpdΩ 2πpT E 0.166 where γ and β are the usual Lorentz factor and particle velocity, and pT the transverse mo- mentum of the particle. Eq 4.28 is a consequence of the large independence of the inclusive

81 CHAPTER 4. GALACTIC SECONDARY DEUTERON AND ANTIDEUTERON FLUX

Figure 4.4: (Left) Total, inelastic, elastic and inelastic non-annihilation cross-sections for d¯+ H as a function of the antideuteron kinetic energy per nucleon. (Right) Differential redistribution cross-section d¯+ p → q¯ + X as a function of the antideuteron kinetic energy per nucleon. Figures taken from [158].

cross-section respect to the longitudinal momentum of the produced particles. Hence, integrat- ing Eq. 4.28 in the solid angle and normalizing, the differential non-annihilation cross-section reads:

dH¯ →dX¯ pp→pX "Z T 0 pp→pX # dσ non−ann dσ dσ 00 = σinel 00 dT . (4.29) dT dT 0 dT

The inelastic scattering spectra in the laboratory system for 0.55 (black), 1 (blue), and 4 GeV/c (red) antideuteron energies are shown in the right part of Fig. 4.4. For each of these energies, Eq 4.29 efficiently redistributes antideuteron at very low energy. For the total, elastic, and non-annihilation cross-section in d¯ + He collisions, the geometrical approximation is used, multiplying the factor A2/3 by the d¯ + H cross-section. These parametrizations of the antideuteron total inelastic cross-sections are implemented into GALPROP as part of the nucleon cs.cc class, that provides the antiproton-proton and in gen- eral p+p and p+A total inelastic cross-sections. The energy distribution of the non-annihilation cross-section is included as a method into the new created gen secondary antideuteron.cc class.

4.4. Results

4.4.1. GALPROP validation The first step to calculate the propagation of deuterons and antideuteron in the Galaxy correctly is to be sure the transport model is accurately set up. Special care has to be taken with the proton and helium fluxes, since they are the primary incident particles. Additionally, the precise measurement of the Boron-to-Carbon ratio by AMS-02 allows restricting the diffusion coefficient value improving the predictability of the model. Therefore, in Fig. 4.5 a comparison of the fluxes obtained using GALPROP with the param- eters in Tables 4.1 and 4.2 against AMS-02 data is shown. On the left-top side of Fig. 4.5 is the proton flux, where the solid blue line represents the GALPROP prediction using the solar mod- ulation model described in Section 4.1.7, with a Fisk potential of 570 MeV. The dashed blue line is the same GALPROP result but with the solar modulation model HelMod [229], which solves

82 4.4. RESULTS by a Monte Carlo approach the entire CRs transport equation, from the termination shock to the Earth orbit. As observed the results differ by a maximum of 10% from AMS-02 data. On the right-top side of Fig. 4.5 is presented the results for the Boron-to-Carbon ratio, while on the low part are the results for the Helium flux. In both cases, the relative difference between simulation and data is less than 20%, and the solar modulation model used is the force field approximation described in Section 4.1.7, again with a Fisk potential of 570 MeV.

•1 103 0.5 B/C 2 B/C Galprop (φ = 570 MeV) 10 F 0.4 10 data AMS•02 s sr GeV/n] 2 1 0.3 [m p 10−1 Φ 10−2 0.2 Flux 10−3 Proton flux Galprop (φ = 570 MeV) F 0.1 10−4 Proton flux Galprop+HelMod 10−5 data AMS•02 0 0.4 0.4 0.2 0.2 0 0

−0.2 −0.2

−0.4 Relative difference −0.4 Relative difference 10−1 1 10 102 103 10−1 1 10 102 103 Kinetic energy per nucleon E [GeV/n] Kinetic energy per nucleon E [GeV/n] kin kin •1 102 10 1 s sr GeV/n] 2 10−1 [m

He −2

Φ 10 10−3 Flux 10−4 5 Helium flux Galprop (φ = 570 MeV) 10− F 10−6 data AMS•02 0.4 0.2 0

−0.2 −0.4 Relative difference 1 10 102 103 Kinetic energy per nucleon E [GeV/n] kin

Figure 4.5: Results obtained with GALPROP and the parameters in Table 4.1 for proton, Boron-to-Carbon ratio and Helium compared to the most recent data from AMS-02 experiment.

Antiprotons In addition to the most abundant CRs species simulated and validated in Fig. 4.5, antiprotons have an essential role in the antideuteron production as explained in Section 3.2.2, and for that reason, their generation in GALPROP has to be analyzed. Furthermore, antiprotons have

83 CHAPTER 4. GALACTIC SECONDARY DEUTERON AND ANTIDEUTERON FLUX been measured recently by the AMS-02 experiment showing important features that have been studied in some current works (see Section 1.3), and a comparison of these studies with the results obtained with EPOS-LHC is well worth it. In the first part of the antiproton analysis, the aim was to evaluate the correct behavior of the propagation stage with GALPROP and the parameters selected. For this purpose, the antipro- ton secondary source term calculated in Winkler et al. [93] was taken as a probe. The authors in [93] determined the antiproton production cross-section using an improved parametrization, and the propagation was performed within the standard two-zone diffusion model. The Winkler antiproton secondary source term then was inserted into GALPROP through a routine called gen secondary antiproton, and the tertiary term was calculated in gen tertiary antipro- ton (see Section 4.2). The results of such comparison are shown in Fig. 4.6, where the solid line represents the Winkler et al. antiproton source term propagated with GALPROP and the broken-line is the original flux computed by them. This figure shows the propagation with GAL- PROP produces similar and even better results than the propagation in [93]. This result means GALPROP parameters configuration and the simulation performance are working correctly.

•1 10−1

10−2

3 s sr GeV/n] −

2 10 [m p 10−4 Φ

10−5 Antiprotons Flux Winkler et al. −6 10 Winkler et al. + Galprop (φ = 570 MeV) F data AMS•02 10−7

0.4 0.2 0

−0.2 −0.4 Relative difference 1 10 102 103 Kinetic energy per nucleon E [GeV/n] kin

Figure 4.6: Comparison between the antiproton flux obtained by Winkler et al. [93] and the secondary source term in Winkler et al. propagated with GALPROP.

Now, the next step is to determine the antiproton flux using EPOS-LHC and GALPROP to compare with parametrizations. As in the case of antideuterons, to obtain the antiproton sec ter flux, the secondary (qp¯ ) and tertiary (qp¯ ) source terms have to be calculated (see Eqs. 4.20 and 4.21). The dominant channels for the production of the secondary antiprotons are p+p at roughly 50-60% of the total contribution, and p+He, He+p, and He+He at 10-20% each, while the channels involving heavier incoming CRs contribute only up to a few percents. Thus, in this thesis contributions from the reactions p+p, p+He, He+p, and He+He were included and antiprotons created in heavier CRs collisions were ignored. Following Eq 4.20 for antiprotons and the corresponding production cross-sections evaluated in Section 3.2.1, the secondary source term is calculated. The antiproton multiplicity produced in CRs includes antiprotons produced in CRs collisions and antiprotons from hyperon and antineu- tron decay. The result is shown on the left side of Fig. 4.7. As can be observed, the individual interaction contributions together with the total antiproton source are presented. The distribu-

84 4.4. RESULTS •1 •1 EPOS•LHC 10−1 Total 10−29 p+p p+He 10−2 s sr GeV/n] s GeV/n] 2 3 He+p −30 10 He+He [m p −3 [cm Total* Φ 10 p q

10−31 Flux 10−4

Antiprotons (φ = 570 MeV) −32 F 10 −5 10 EPOS•LHC+Galprop Winkler+Galprop −33 10 Duperray et al. 10−6 data AMS•02 Winkler et al. Korsmeier et al. 10−34 10−7

10−1 1 10 102 103 10−1 1 10 102 103 Kinetic energy per nucleon E [GeV/n] Kinetic energy per nucleon E [GeV/n] kin kin

Figure 4.7: (Left) Antiproton secondary source term for EPOS-LHC. (Right) Antiproton flux obtained with EPOS-LHC and GALPROP compared to Winkler et al. [93] and AMS-02 data.

tion shows its maximum at around 2 GeV/n, although it is slightly wider than previous studies. It also predicts a larger antiproton production than Duperray et al. [157], Winkler et al. [93], and Korsmeier et al. [161]. Additionally, an estimation of the source term using EPOS-LHC but considering the antiproton contribution from antineutron decay as in Korsmeier et al. [161] is presented in Fig. 4.7, it is labeled as the solid red line (Total∗). This additional estimation pre- dicts a lower antiproton production conserving the shape of the original EPOS-LHC calculation (solid black line). The latter means the antiproton overproduction generated by EPOS-LHC is related to an excess in the contribution from antineutron decays. On the right side of Fig. 4.7 is the antiproton flux compared with the one obtained in [93] and the data from AMS-02. It is observed from the figure that EPOS-LHC predicts a larger antiproton flux below 10 GeV/n reaching a magnitude 1.5 times larger than Winkler calculation. At high energies the antiproton flux calculated with EPOS-LHC shows similar results as the flux from Winkler et al.

4.4.2. Deuteron flux

Once the propagation parameters have been defined and tested, it is safe to proceed with the transport of deuterons and antideuterons. At this point all the inputs have been defined: the production cross-section defined in Section 3.2.2, the interaction cross-sections, and the propa- gation model. Using all these calculations, along with the numerical solution from GALPROP, the flux of deuterons is obtained and presented in Fig. 4.8. On the left side of the figure, the 2.7 deuteron flux times Ekin is plotted as a function of the deuteron kinetic energy. Besides the deuteron result, the helium flux is included in the same plot to show the good description of data by GALPROP. Data from AMS-1 and AMS-02 are presented in the figure together with data from CAPRICE [174], which is the only deuteron measurement at high energies. The deuteron flux from simulations was obtained using the estimated cross-sections in Section 3.2.2, which are the combination of the parametrization from Coste et al. [216] at low energies and the Monte Carlo generator QGSJET-II-04 at high energies. The solid cyan line represents the deuteron

85 CHAPTER 4. GALACTIC SECONDARY DEUTERON AND ANTIDEUTERON FLUX •1 3 10 With coalescence d/He Without coalescence CAPRICE98 1

s sr GeV/n] 2 SOKOL 2 10 IMAX [m 2.7 kin E ×

d 10 −1 Φ 10 MED propagation φ = 500 MeV F Deuteron 1 Deuteron+Coalescence Helium

−2 data d CAPRICE98 10 10−1 data d AMS data He AMS•02

10−1 1 10 102 103 10−1 1 10 102 103 Kinetic energy per nucleon E [GeV/n] Kinetic energy per nucleon E [GeV/n] kin kin

Figure 4.8: (Left) Deuteron flux produced using QGSJET-II-04 with coalescence (solid cyan line) and with- out coalescence (dotdashed cyan line) propagated with GALPROP and compared to data from CAPRICE [174] and AMS [173]. (Right) Deuteron-to-Helium ratio compared to data from CAPRICE [174], IMAX [175] and SOKOL [177].

calculation considering coalescence, while the dot-dashed cyan line is the result without coales- cence (see Fig. 3.11). As can be observed in the figure, the deuteron flux including coalescence production of deuterons at high energies shows an improved description of data. On the right side of Fig. 4.8, the deuteron-to-helium ratio results are presented and compared to data from CAPRICE and SOKOL. The CAPRICE ratio was reevaluated with the helium data from AMS- 02. Again, as in the left side figure, it is observed deuteron simulation containing coalescence production describes data satisfactorily; however, this simulation is not enough to account for the excess predicted by SOKOL.

4.4.3. Antideuteron flux The final result of the antideuteron flux calculated with EPOS-LHC as event generator to simulate CR collisions, simultaneously with the afterburner to produce antideuterons through the coalescence model, and the propagation of these particles in the Galaxy by means of GAL- PROP is presented in Fig. 4.9. The first notorious feature on the left side of Fig. 4.9 is that the antideuteron flux evaluated in this thesis (solid red line) is the largest one compared with previ- ous works. Indeed, the predicted antideuteron flux is around 30% larger than the flux calculated by Duperray et al. and nearly to 4 times the one inferred by Ibarra et al. The last result is not surprising, since from the source term in Fig. 4.2 it was evident EPOS-LHC+afterburner produce a higher number of antideuterons as a consequence of the larger cross-section. The predicted antideuteron flux obtained with EPOS-LHC, also shows a wider distribution compared with the mentioned studies, due to a broader high energy contribution originated in a high production cross-section. An effect of the energy-dependent coalescence momentum analyzed in Section 3.2.2 and the cross-section derived from that analysis is the slight shift of the maximum in the antideuteron flux that now is around 4-5 GeV/n, when other calculations place it around 3-4 GeV/n. If the

86 4.4. RESULTS

10−5 10−2 •1 •1 GAPS AMS•02 AMS•02 LSP (SUSY) m = 30 GeV 10−3 χ LZP (UED) 6 10− m = 40 GeV GAPS AMS•02 AMS•02 LZP s sr GeV/n] s sr GeV/n] 4 2 − gravitino (decay) 2 10 m = 50 GeV [m [m d d 7 EPOS•LHC+GALPROP

− Φ Φ 10 10−5 Flux Flux 10−6 10−8 MED propagation φ = 500 MeV F EPOS•LHC+GALPROP 10−7 Ibarra et al. 10−9 Donato et al. 8 Duperray et al. 10−

10−1 1 10 102 10−1 1 10 102 Kinetic energy per nucleon E [GeV/n] Kinetic energy per nucleon E [GeV/n] kin kin

Figure 4.9: Antideuteron flux at the top of the atmosphere obtained with EPOS-LHC and GALPROP. (Left) The result is compared to previous works from Ibarra et al. [151], Donato et al. [158] and Duperray et al. [157]. (Right) The result is compared to antideuteron fluxes expected from dark matter annihilation or decay [148, 163, 164]. Sensitivity limits for AMS-02 and GAPS are also included.

coalescence momentum would have been considered constant at all energies, then the shape of the distribution would be the same as Duperray et al., or Ibarra et al. But since the suppression of the coalescence momentum was taken into account near the energy threshold, the low energy region of the flux (<1 GeV) is also suppressed, and therefore the shape of the distribution is modified. On the right side of Fig. 4.9, the resulting antideuteron flux is plotted along with three different expected antideuteron fluxes from dark matter annihilation or decay. These candidates, already shown in Fig. 2.1, continue having a considerable larger flux below 1 GeV/n compared to the secondary, despite its increase using EPOS-LHC.

87 CHAPTER 4. GALACTIC SECONDARY DEUTERON AND ANTIDEUTERON FLUX

88 CHAPTER 5 Secondary antideuterons produced in AMS-02

Recently, antideuterons from DM origin are reaching the expected observable limit in CR experiments. Despite the reduced antideuteron flux, experimental efforts have led to the pos- sibility to measure at least a few candidates. However, this low-rate antideuteron count brings the critical issue of background misidentification. The evaluation of the antideuteron cross-section in p+p and p+A collisions, developed in Chapter 3, may be useful to determine the antideuteron production in the detector materials such as AMS-02. These secondary antideuterons might represent an important contribution to the systematic errors to be considered in this, and future experiments. In this chapter, an estimation of the antideuterons produced in the AMS-02 detector is pre- sented using GEANT4 toolkit and the coalescence model parametrization proposed in Chapter 3

5.1. Afterburner implementation in GEANT4

Beyond only seeking for antideuteron production in one particular detector, this thesis has the intention to be useful for other studies in planned or present detectors. Following this pur- pose, the GEANT4 toolkit [182] was chosen to introduce the antideuteron formation algorithm (Afterburner). GEANT4 is a widely used software developed to simulate the interactions of particles with detectors. GEANT4 is able to reproduce detector responses, covering a large energy region of in- teraction and including an impressive number of reactions. GEANT4 is continuously improving and increasing its action field thanks to the positive results obtained in a large number of areas where it has been employed. From high energy physics, space, and radiation, to medical appli- cations. CRs detectors are not the exception, and most of the simulations are performed with this software. Such universality allows accomplishing the objective of making the antideuteron production accessible for future analysis. The implemetation of the afterburner described in Section 3.1.2 was made using two different approaches. The first method was introduced at the user level, where the simulation program was built to perform an iterative reading of the particle’s stack. In every cycle, the program searches for antideuterons who meet the coalescence conditions. The main class where this process was inserted is StackingAction. Here, the information from the stack for every event (collision) can be manipulated by means of three labels: fUrgent, which gives a free pass to the

89 CHAPTER 5. SECONDARY ANTIDEUTERONS PRODUCED IN AMS-02 particles to continue the next processes; fWaiting, which stops the whole process allowing the user to add actions that change the stack, including to modify the label to fUrgent; and fKill, used to eliminate a particle track from the stack. The simulation starts with a projectile proton fired towards the detector. Once the collision takes place, and all the particles are produced, the process is stopped with fWaiting. In this stage, coalescence conditions are set, and the antideuterons generated are included in the stack, while the antiprotons and antineutrons used in the formation are deleted from it. After coalescence- conditions have been run for all the antinucleon pairs, and all antideuterons have been produced, the process is switched-on with fUrgent and particles continue traveling through the detector (see Fig. 5.1).

Figure 5.1: Diagram of the afterburner implementation at the user level, for antideuteron production in GEANT4.

Although the process described before works correctly, it is inefficient since all the infor- mation from the stack has to be stored momentarily and it has to be accessed several times in a single iteration. The above, and the intention to make the afterburner implementation useful at the long term led to introduce the coalescence parametrization at a developer level in the source code of GEANT4, specifically in the class G4TheoFSGenerator located in the path source/processes/hadronic/models/theo high energy. G4TheoFSGenerator class is provided to organize interaction between G4VHighEnergyGene- rator, G4VIntranuclearTransportModel, G4VPreCompoundModel, and G4VExcitationHandler. This class inherits from G4HadronicInteraction, and hence can be registered as a model for a final state production with a hadronic process. It allows a detailed implementation of G4VIntranuclearTransportModel and G4VHighEnergyGenerator to be registered. Addition- ally, it distributes initial interactions and intra-nuclear transport of the corresponding secon- daries to the respective classes [239] (see Fig. 5.2). G4TheoFSGenerator declares a minimal interface of only one pure virtual method: ApplyY- ourself(const G4Track, G4Nucleus), where a particular implementation usable as a stand- alone model can be included. The value of the coalescence momentum (p0) is set at the beginning of this method by taking the information of the projectile, the target, and the model from the initialization arguments. Then, projectile and target are led to scatter, and the products are led to decay (strong decay) in order to obtain the G4ReactionProductVector, which contains the

90 5.2. SIMULATION

Figure 5.2: Level 3 implementation framework in the hadronic category of GEANT4 [239].

information of the generated particles before any propagation action. A newly added method called GenerateDeuterons takes the object of G4ReactionProductVector and apply the coa- lescence conditions to deliver a new vector finally, but this time with the antideuterons included. At the end of the ApplyYourself method, the information of the recently created vector is in- cluded in the object of the G4HadFinalState class to continue with the transportation process. The results from both approaches were validated for the FTFP model, comparing them with the product obtained in Section 3.2.2.

5.2. Simulation

Once antideuteron production has been implemented in GEANT4, the next step is to simu- late the interaction of the proton flux with the detector of interest and to collect all the relevant information about the antideuterons generated. For that purpose, the geometry of the detector should be constructed within the simulation. The AMS-02 detector geometry modeled in this thesis is described below. A review of the principal characteristics of AMS-02 was presented in Section 1.3.1.

5.2.1. AMS-02 geometry The geometry created for the simulation is based on the original detector dimensions and materials used for its construction. Special attention has been paid to the top part of AMS- 02, above the upper Time of Flight detector (UTOF), where antideuterons produced by CRs protons interactions are more unlikely to be rejected. In Fig. 5.3 it is shown a side sketch of the simulated geometry. The principal elements considered are the tracker planes, the upper and lower TOF, the TRD, and the corresponding support planes and structures. As mentioned in Section 1.3.1 AMS-02 has 9 tracker planes after its reconfiguration with a permanent magnet. Tracker planes are groups of silicon strips with different lengths. However, in this simulation, they are regarded as uniform silicon circular planes of 0.3 mm thickness and radius 112 cm. The plane support 1NS is a 70 mm high aluminum honeycomb, cover with carbon fiber face sheets of 2.5 mm thick and 231 mm diameter, that holds up the first silicon layer. It

91 CHAPTER 5. SECONDARY ANTIDEUTERONS PRODUCED IN AMS-02

0.005 cm 1NS 7.5 cm L1 4.17 cm UTRD 10 cm

TRD 62.1 cm

5

.

LTRD 4 5.6 cm 7

7

1

UTOF 12 cm 6

7

1

5 L2 4.44 cm 5

.

1

9

2

.

8 L3-4 1.3 cm 6

7

8

2

.

5

9 L5-6 1.3 cm 2

2

.

9

2

0 L7-8 1.3 cm 5 Carbon

7

3

112 cm 1 Aluminum LTOF 7.7 cm Xe/CO2(80/20) 140 cm Silicon Radiator L9 4.44 cm PVT

Figure 5.3: Detector geometry model used in the GEANT4 simulation (not to scale).

has an octagonal shape, similar to the TRD detector and besides the provided support, 1NS serve as a low energy particle shield. A second support plane 1N is attached just beneath 1NS; it has a circular shape with a diameter of 112 cm. As in the case of the 1NS plane, 1N is formed by an aluminum honeycomb of 40 mm, covered by two carbon fiber sheets of 0.7 mm. The first single silicon plane L1 is situated below 1N [240] (see Fig. 5.3). Under L1 is the TRD, which has two support planes covering the top and bottom of the detector. These support planes have the same configuration as 1NS and 1N but different di- mensions. The top(bottom) plane is a 90 mm(50.4 mm) thick aluminum honeycomb with a 5 mm(2.8 mm) thick carbon foil in each face. In between, as explained in Section 1.3.1 are the straw tubes modules where each tube wall is formed as a sandwich by a combination of thin films of carbon, Kapton, Polyurethane, and aluminum. Tubes have a diameter of 6 mm and are filled with a gas mixture of Xe/CO2. Every array of tubes in the plane is alternated with a −3 fleece radiator made of Polyethylene/Polypropylene (C2H4/C3H6) of density 0.06 gcm . This arrangement is simulated in the geometry creating a set of multiple planes made by all the ma- terials that constitute the tubes and the radiator. This set with an octagonal shape is repeated 20 times as in the real detector with a variable diameter to adjust the appropriate dimensions (see Fig. 5.3). Additionally, two 50 µm aluminum foils that cover the top side of the TRD and the top side of plane 1NS are also included. The next part of the detector included in the geometry is the upper (UTOF) and lower (LTOF) sides of the Time of Flight. This is a plastic scintillator with the characteristics presented

92 5.2. SIMULATION in Reference [241]. Each plastic pad is 1 cm thick and between 117 and 134 cm long. Although the shape of the detector is not entirely a square, this is considered as such in the simulation with the longest longitud. Both UTOF and LTOF are formed by two plastic planes separated by a special foam to reduce the effect of vibrational modes. The support of the UTOF is a 100 mm thick aluminum honeycomb connected to the lower support of the TRD, and 1.7 mm carbon layers cover it in each face. The LTOF support structure is also an aluminum honeycomb but 50 mm thick, also covered by 1.7 mm carbon layers. Finally, the remaining planes L 2-9 are implemented in the geometry simulation as uniform silicon cylinders with their corresponding support structures as shown in Fig. 5.3. Planes L 3-8 are organized between the upper and lower parts of the TOF, and L9 is below the lower-TOF. Since in this thesis, it is assumed antideuterons are produced only at the top of AMS-02 no additional sub-detectors are included.

5.2.2. Event generation After the geometry was constructed, a representative sample of cosmic-rays flux impinging on the detector was simulated. It was done generating protons distributed uniformly in a plane 80 cm above the detector, with a length and width of 231 cm. The direction of the particles was configurated to swept up the region of geometrical acceptance of the detector. Thus, all the fired protons traverse the top (above the UTOF) of the simulated detector. The enegy spectrum of the proton flux simulated was a power law distribution of the form E−2.7. The particles were generated in an energy range from 16 GeV to 10 TeV. The lower limit was based on the antideuteron production threshold, and the upper limit was set considering the limitations of the event generator. Nevertheless, the CR flux around and beyond 10 TeV is significantly reduced, and therefore, its contribution to the antideuteron production is negligible.

5.2.3. Event selection The Time of Flight (TOF) and the Ring Imaging Cherenkov (RICH) detectors play the principal role identifying antideuterons in AMS-02 (see Section 1.3.1). From the velocity and charge measurements provided by these two detectors, along with the trajectory determined by the Silicon Tracker, the mass of the incoming particles is obtained, and antideuterons may be distinguished from the rest of cosmic-rays. However, the enormous differences in the magnitude of the fluxes between antideuterons and other abundant CRs species like protons, electrons or even antiprotons, make the identification process very challenging. The background rejection power of AMS-02 and its capacity to detect antideuterons is expressed as the acceptance of the detector. It is evaluated by means of a detailed simulation study as the presented in [130]. From the acceptance, the sensitivity limits to detect antideuterons by AMS-02 after five years of op- eration are estimated in 2 × 10−6m−2s−1sr−1(GeV/n)−1 and 1.4 × 10−6m−2s−1sr−1(GeV/n)−1 within the energy ranges of 0.2-0.8 and 2.2-4.4 GeV/n respectively [129]. The resulting accep- tance calculated in [130] is shown in Fig. 5.4. The preselection criteria chosen in the cited work were

at least one reconstructed track in the Tracker,

at least one reconstructed track in the TRD,

a velocity measurement compatible with a downgoing particle (β > 0),

the reconstructed charge is |Z| = 1.

93 CHAPTER 5. SECONDARY ANTIDEUTERONS PRODUCED IN AMS-02

Figure 5.4: Final AMS-02 acceptance for antideuterons signal [130].

Besides the above criteria, additional conditions in the event selection are set depending on detector calibration and response. Among them, there are some related to the suppresion of events with interactions, which affects the selection of events that may produce secondary antideuterons. For example, an event with a signal in the anti-coincidence counter (ACC) is rejected, because it may be caused by secondaries produced at large angles. To guarantee TOF does not count particles produced in the detector, or near the TOF planes, a single hit from a unique paddle is required. The energy deposited in the UTOF or the LTOF should not be greater than 6 MeV. Additionally, to avoid particles produced in the proximity of the Tracker planes, the number of hits in the vicinity of the reconstructed track is limited to be below the channel 60 in the ADC. A similar condition is set in the number of reconstructed segments in the TRD, where it is considered that a single particle would be formed at most by 4 TRD segments, whereas a higher number of segments is likely because secondary particles. For the simulation developed in this thesis, and the analysis of the events selected, various assumptions based on the acceptance evaluation reviewed above have been introduced.

1. It is assumed the antideuterons produced in the detector, due to the interactions of protons with AMS-02 materials are ruled by the acceptance conditions established in [130].

2. Only events with a signal in the UTOF and LTOF are selected.

3. All the events analyzed let a signal in the inner planes (2-8) of the Silicon Tracker.

5.3. Results

A sample of 1.2 × 109 protons was targeted toward the constructed detector, with the spec- ifications pointed in the last section. This is approximately the number of protons expected to be collected during 10 years of operation by AMS-02. The information of every event was saved in ROOT files with a TTree structure. Only events with antideuterons were written in the raw simulation files. The variables contained in each tree were particle code, energy, momentum direction, position, and hits in the volumes of the detector. From the proton sample, only 18 events containing antideuterons survived to the selection conditions established previously. These events are considered as candidates to be misidentified by the detector, hereafter called final events.

94 5.3. RESULTS

5.3.1. Antideuteron selection A first look into the final events and, particularly, into the characteristics of the antideuterons produced led to the results shown in Fig. 5.5. In the left side of the figure, the kinetic energy distribution of the antideuterons selected with the previous conditions is presented. As observed in the figure, most of the antideuterons generated in the detector are in a high energy region (>2 GeV). This distribution is a consequence of the kinematics involved in the reaction (see Section 2.2) and the power law of the proton flux. For the same reason, the number of additional tracks generated in each event is generally high. To produce an antideuteron in a p+p collisions it is necessary to create at least 4 additional hadrons in the same event, three protons and one neutron, to keep the baryonic number conserved. In the right side of Fig. 5.5, the number of tracks accompanying the antideuteron in each event is plotted against the kinetic energy of the antideuteron in that event. Note that the minimum number of tracks is around 100, and it seems that there is no correlation between the energy of the antideuteron and the secondary tracks generated in the event.

5 800

4.5 700 4 18 d candidates 600 3.5

500 3

Number antideuterons 2.5 400

2 Number of tracks per event 300 1.5 200 1

0.5 100

0 0 10−1 1 10 102 0 2 4 6 8 10 12 14 16 18 Kinetic Energy per nucleon (GeV/n) Kinetic Energy per nucleon (GeV/n)

Figure 5.5: (Left) Energy distribution of the antideuterons likely to be misidentified by the detector. (Right) Dispersion of the number of tracks per event vs kinetic energy of the antideuterons in that event.

The results obtained at this point suggest, even though antideuterons are produced in the detector, it is unlikely these candidates could be taken as truly CRs antideuterons. The argument to justify this affirmation is the large number of secondary tracks accompanying the generated antideuteron. However, most of these tracks do not penetrate larger distances into the detector, and they remain close to the primary track. Assuming the only valid secondary tracks per event are those who give a signal in the UTOF and LTOF, then the number of additional particles accompanying the antideuteron is considerably reduced, as shown in the left side of Fig. 5.6. Now as can be observed, the number of secondary tracks is lower than 45, and there are two events with less than 5 accompanying particles. From the antideuteron candidates shown in Fig. 5.5, just a small fraction is in the energy region accepted by the AMS-02 detector (see Fig. 5.4). To be precise, only 7 candidates remain, represented as those within the shaded area and at the border in Fig. 5.6 (Left). The distribution on the right side of Fig. 5.6 describes the number of accompanying tracks of those events in the accepted energy region. As observed the mean number of tracks is around 13, and the minimum value is 4 tracks. In Fig. 5.7, the only event with the minimum number of secondary tracks and within the acceptance region of the detector is displayed inside the geometry of AMS-02 constructed for

95 CHAPTER 5. SECONDARY ANTIDEUTERONS PRODUCED IN AMS-02

45 2.5

40 Entries 2 35

30 1.5 25

20 Number of tracks per event 1 15

10 0.5

5

0 0 0 2 4 6 8 10 12 14 16 18 0 5 10 15 20 25 30 35 40 45 50 Kinetic Energy per nucleon (GeV/n) Number of tracks per event

Figure 5.6: (Left) Dispersion of the number of tracks per event vs kinetic energy of the antideuterons in that event. (Right) Angular distribution of the secondary particles produced in an event, respect to the generated antideuteron.

this simulation. The red line represents the antideuteron produced in the collision, and different colors represent the additional secondary tracks. As can be observed in this example, the secondary tracks leave a signal in the TOF detector and the Silicon Tracker planes, that is clearly distinguished from the secondary antideuteron. Furthermore, in this case, the collision occurs at the top of the TRD, producing multitrack signals in this sub-detector not allowed in a common AMS-02 event selection.

Figure 5.7: Event simulation of an antideuteron (red line) and the additional secondary tracks produced by the collision of a proton with the AMS-02 detector.

96 5.3. RESULTS

5.3.2. Antideuteron misidentification Despite not having a candidate for antideuteron misidentification with the simulated sample, an estimation of the maximum flux produced by these particles in the detector can be devel- oped. Let’s consider two scenarios, in the first scenario, AMS-02 is unable to discard the final event where four additional tracks are accompanying the produced antideuteron (see Fig. 5.7). This scenario is highly improbable as justified before. Here, the efficiency of the antideuteron −10 selection in the corresponding energy bin is d¯ = Nacc/Nevents = 8.3 × 10 , where Nacc = 1. In the second, more reliable scenario, the probability to obtain an antideuteron with no secondary tracks in the inner part of the silicon tracker is evaluated assuming a Poisson continuous distri- bution in the right side of Fig.5.6 (solid red line). The probability is estimated at 0, because that means the antideuteron will go alone through the tracker and it can be misidentified. Thus, in this case, the probability is around 4.6 × 10−15, and it can be assumed as the value of the antideuteron efficiency. The secondary antideuteron background events can be estimated as Z  i p i Nd¯ = Fp(E)AaccT dE × d¯, (5.1)

p where Fp is the proton flux at the top of the atmosphere, Aacc is the detector acceptance to protons, and T is the exposure time. Taking an approximate geometric acceptance of 0.1 m2sr, and an exposure time of 10 years, the result in the first scenario is Nd¯ = 1.62, while for the −6 second scenario Nd¯ ∼ 8.96 × 10 , for an energy range less than 4.4 GeV/n.

•1 10−4

GAPS AMS•02 AMS•02 EPOS•LHC+GALPROP 10−5 d misident. in AMS•02

s sr GeV/n] −6

2 10 [m 10d −7 Φ

−8

Flux 10

10−9

10−10

10−11

10−1 1 10 102 Kinetic energy per nucleon E [GeV/n] kin

Figure 5.8: Upper limit flux for antideuterons produced in the materials of AMS-02 detector, that might be misidentified with Galactic antideuterons.

The flux is estimating using the following relation:

Nd¯(E) F ¯(E) = . (5.2) d d¯ ∆EAaccT −6 −2 −1 −1 −1 The resulting upper limit for the first scenario is Fd¯ < 1.07 × 10 m s sr (GeV/n) . −10 −2 −1 −1 −1 For the second and most reliable scenario, Fd¯ . 5.3 × 10 m s sr (GeV/n) , and it is shown in Fig. 5.8 along with the galactic antideuteron flux and the sensitivity limits for AMS-02 and GAPS. Here, it has been assumed the antideuterons are above the geomagnetic cut-off and

97 CHAPTER 5. SECONDARY ANTIDEUTERONS PRODUCED IN AMS-02 the minimum acceptance value has been taken. As can be observed in Fig. 5.8, the upper limit for an antideuteron misidentification in AMS-02 detector is far below its maximum sensitivity, in fact, it is various orders of magnitud lower than the secondary antideuteron flux. The fluxes estimated show to misidentified an antideuteron produced in the detector with an antideuteron originated in the Galaxy is very unlikely. Based on the kinematics of the process involved, and the characteristics of the detector, all events where collisions generate antideuterons are expected to be accompanying with multi-tracks that give signals in the TOF, the Silicon Tracker and even the TRD. These events are going to be rejected because of the large spatial separation between tracks. Although the upper limit is similar in the energy region from 2-4 GeV/n and in the energy range between 0.2 to 0.8 GeV/n, a lower value is expected in this last region because antideuterons produced in the detector have a similar energy distribution as those produced in the Galaxy.

98 CHAPTER 6 Conclusions

To improve the coalescence formation modeling of light nuclei, deuteron and antideuteron production in p+p and p+Be collisions with energies in the laboratory system from 20 to 2.6× 107 GeV were reevaluated. As no commonly used hadronic MC generator describes (anti)deuteron production, the goal was to create an afterburner based on experimental data to generate d and d¯ in p+p and p+A interactions in a reliable way. After an event-by-event analysis using three of the most relevant MC generators (EPOS-LHC, QGSJET-II-04, and Geant4’s FTFP-BERT), it was found that the coalescence momentum p0 depends on the collision energy (see Fig. 3.8), and it is not constant over the entire energy range as previous works suggested. For deuterons, p0 decreases with energy until it reaches a constant value, and for antideuterons, p0 starts to grow after the production threshold, and then it reaches a constant value. The behavior of p0 seems to be related to the increase in the available phase space as a function of energy [151, 213]; however, more data in this energy region is necessary to verify this dependence. Also, it was found, there is no substantial difference in the p0 values between p+p and p+Be collisions. Based on these results, parameterizations were developed and used in tandem with EPOS- LHC, QGSJET-II-04, and FTFP-BERT. Such parameterizations allowed to estimate the dif- ferential and total production cross-section for deuterons and antideuterons in p+p and p+A collisions (assuming A to be a light nuclei). As an example of the power of this tool, an esti- mation of the total production cross-section of deuterons and antideuterons in p+p and p+He is presented in Fig. 3.9. This new estimation predicts an antideuteron cross-section in p+p col- lisions that can be at least 20 times smaller than the value expected from the parametrization of Duperray et al. [157, 160] in the low kinetic energy (T) region 20-100 GeV, while at high energies (∼ 1000 GeV) the cross-section is 2.4 times larger. A similar result is obtained in p+He collisions, where this thesis estimates a cross-section at least six times smaller than Duperray et al. in the low energy region. In addition to p+p and p+He interactions, other important reactions were simulated using the mentioned Monte Carlo generators and the afterburner. Among them are the helium channel reactions He+p and He+He and the antiproton contributionsp ¯+p andp ¯+He. For all these interactions the same value of the coalescence momentum as in p+p collisions was used. The results for the production cross-sections are shown in Fig. 3.10. For the specific case of deuterons, the interaction that contributes most to the CR deuteron flux is the fragmentation of Helium. Measurements about this reaction are only available below ∼ 5 GeV; thus a Monte Carlo simulation was performed to calculate the deuteron production

99 CHAPTER 6. CONCLUSIONS cross-section at higher energies (see Fig. 3.11). Comparing EPOS-LHC and QGSJET-II-04 re- sults, it was determined that the last generator produces a more reliable description compared to the former, as observed in Fig. 3.11. Furthermore, it was noticed that introducing coalescence deuteron production causes the cross-section magnitude to increase significantly and continu- ously with energy. Then, the calculated cross-sections were used as an input into the propagation process de- veloped with GALPROP, an advanced numerical code that solves the CRs transport equations in the Galaxy. For the sake of consistency, the GALPROP output was validated comparing the simulated fluxes of proton, helium and boron-to-carbon ratio with the most recent data from AMS-02. The results were satisfactory, showing an excellent agreement with measurements and a maximum deviation of around 20% from data. Besides the mentioned species, antiproton prop- agation through the Galaxy was also simulated, both using an existing parametrization of the production cross-section [93] and using EPOS-LHC. The antiproton flux predicted by the combi- nation of EPOS-LHC and GALPROP showed an excess in comparison with the parametrization. This excess is related to the enhanced antineutron production in the Monte Carlo generator that leads to an increment in the CR antiproton multiplicity when antineutrons finally decay. The predicted deuteron flux evaluated with the Monte Carlo generator QGSJET-II-04, and the coalescence deuteron production at high energies (>10 GeV) is consistent with the available data as shown in Fig. 4.8. The deuteron flux also generates an improved description of measure- ments at high energy compared to the result without considering coalescence. The deuteron- to-helium ratio computed in this thesis using QGSJET-II-04, and the coalescence model does not agree with the result from the SOKOL analysis [177], which reports a significant increase of deuterons at around 1 TeV/n. The antideuteron flux obtained in this thesis shows a larger magnitude compared to other studies, and a slight difference in the shape distribution, as a consequence of the energy depen- dence of the coalescence parameter (see Fig. 4.9). Although the resulting flux is 1.3 times larger than the predicted by [157] and four times than [151], it is still below the expected sensitivities of AMS-02 and GAPS. These findings strengthen the idea that if these experiments detect cosmic antideuterons, they are probably from exotic sources.

100 Appendices

101

APPENDIX A Afterburner

The output from a Monte Carlo generator (EPOS-LHC, QGSJET-II, etc.) is configurated to produce a ROOT file with a TTree structure containing the primary information as shown in the next scheme.

TTree *event

Number of particles per event Branch 1

PDG code for every particle Branch 2

Particle’s momentum in X Branch 3

Particle’s momentum in Y Branch 4

Particle’s momentum in Z Branch 5

Particle’s energy Branch 6

Particle’s mass Branch 7

Figure A.1: TTree scheme of the event information saved in a ROOT file.

A.1. Nucleon selection

In the simulation process just after obtaining the standard MC output information in a TTree as the shown in Fig. A.1, a ROOT macro separates the nucleons and antinucleons from the event stack. This macro creates a new file where only the information of the nucleons and antinucleons is saved. The initial file containing all the event information is deleted.

103 APPENDIX A. AFTERBURNER

Into the macro, a new TTree is created: TFile* foutput = new TFile(outputfile.Data(), "recreate");

TTree* nucleons = new TTree("nucleons", "(anti)nucleons"); nucleons->Branch("nPart", &fNpart, "nPart/I"); nucleons->Branch("pdgid", fPdg, "pdgid[nPart]/I"); nucleons->Branch("status", fStatus,"status[nPart]/I"); nucleons->Branch("px", fPx, "px[nPart]/D"); nucleons->Branch("py", fPy, "py[nPart]/D"); nucleons->Branch("pz", fPz, "pz[nPart]/D"); nucleons->Branch("E", fE, "E[nPart]/D"); nucleons->Branch("m", fm, "m[nPart]/D"); .... and only nucleons are saved ... if((TMath::Abs(pdg[j]) != 2212) && (TMath::Abs(pdg[j]) != 2112)) continue; fPdg[fNpart] = pdg[j]; fStatus[fNpart] = status[j]; fPx[fNpart] = px[j]; fPy[fNpart] = py[j]; fPz[fNpart] = pz[j]; fE[fNpart] = E[j]; fm[fNpart] = m[j];

++fNpart; ...

A.2. Coalescence conditions

The class AntiNucGen takes the nucleons and antinucleons filtered previously, and check the coalescence conditions for all the possible pairs in each event. It compares the difference momentum of the pair to the coalescence momentum. ... Bool_t AntiNucGen::Coalescence( Double_t p1x, Double_t p1y, Double_t p1z, Double_t m1, Double_t p2x, Double_t p2y, Double_t p2z, Double_t m2) const { // // returns true if the nucleons are inside of an sphere of radius p0 // Double_t deltaP = this->GetPcm( p1x, p1y, p1z, m1, p2x, p2y, p2z, m2); return (deltaP < fP0); } ...

104 A.3. ANALYSIS where fP0 is the value of the coalescence momentum defined by the user at the beginning of the program, and deltaP is the momentum difference of the pair in the center-of-mass system.

A.3. Analysis

Multiple ROOT files are created with the kinetic information of the deuterons and an- tideuterons created before. These files are read by the macro Analysis, where the relevant information is extracted in order to compute variables that can be compared to data. Below is shown part of the analysis code for the process p+p at 70 GeV/c.

... // loop over nucleons for(Int_t j=0; jSetParticleVar(px[j], py[j], pz[j], E[j], m[j]); pt[j] = kin->GetPt(px[j], py[j]); rapidity[j] = kin->GetY(); T[j] = kin->GetT(E[j], m[j]);

//antiduteron selection if(pdg[j]==-1000010020) {

hAntiDeuteronYPt->Fill(pt[j], rapidity[j]); hAntiDeuteronT->Fill(T[j]/2.); nd++; } } ....

An example of these variables is the invariant differential cross section as function of the transverse momentum, which is calculated in the class Kinematics as follows:

... //______TGraphErrors *Kinematics::GetSpectrumY(const TH1D *hPt, const Double_t sinet, const Int_t neve, const Double_t ymin, const Double_t ymax) const { Int_t nbins = hPt->GetNbinsX(); Double_t dpt = 0; Double_t dY = ymax-ymin; Double_t invyield = 0; Double_t errdN = 0; Double_t norm = 2.0*TMath::Pi()*dY*neve;

TGraphErrors *gr = new TGraphErrors(nbins);

for(Int_t i = 0; i < nbins; i++) {

dpt = hPt->GetBinWidth(i+1);

105 APPENDIX A. AFTERBURNER

Double_t pt = hPt->GetBinCenter(i+1); Double_t dN = hPt->GetBinContent(i+1); invyield = (dN*sinet)/(pt*dpt*norm); errdN = hPt->GetBinError(i+1); gr->SetPoint(i, pt/2., invyield); gr->SetPointError(i, dpt/4., (errdN*sinet)/(pt*dpt*norm)); }

return gr; } //______TGraphErrors *Kinematics::GetInvCrosSecY(const TH2D *hYPt, const Double_t sefine, const Int_t nevents, const Double_t rmin, const Double_t rmax) const { TAxis *axis = hYPt->GetYaxis(); Int_t binmin = axis->FindFixBin(rmin); Int_t binmax = axis->FindFixBin(rmax); TH1D *hPtantiprotonY = hYPt->ProjectionX("antiproton_Pt_Y", binmin, binmax, "e");

TGraphErrors *gr = new TGraphErrors(); gr = this->GetSpectrumY(hPtantiprotonY, sefine, nevents, rmin, rmax);

return gr; } ...

106 APPENDIX B MC simulations vs accelerator data

B.1. Comparison of simulations to accelerator data (p, p,¯ d and d)¯

This appendix is a collection of all comparisons made between accelerator data and MC mod- els. The three MC models studied are plotted in each figure with the same marker and color convention: EPOS-LHC (red circle); FTFP-BERT (blue square); and QGSP-BERT (green tri- angle). Data are presented as black dots or black squares. The comparisons are shown for either the differential cross sections or invariant differential cross sections as a function of laboratory or transverse momentum per nucleon. When possible, (anti)protons and (anti)deuterons are shown in the same figure.

B.1.1. p+p and p+Be at plab = 19.2 GeV/c )] 10 c 10 θ=12.5 mrad × 100 data Protons 1 1 EPOS•LHC θ × •2 [mb/(GeV/c str)] FTFP•BERT −1 =20 mrad 10

[mb/(sr GeV/ 10 Ω −1 QGSP•BERT

10 Ω

/dpd −2 σ 10 2 /dpd d −2 σ 10 2 data Deuterons d θ=30 mrad × 10•4 10−3 EPOS•LHC (p = 155 MeV/c) 0 10−3 FTFP•BERT (p = 150 MeV/c) 0 10−4 2 1.5 10−5 EPOS•LHC 1 FTFP•BERT QGSP•BERT 0.5 −6 Model/Data 10 Duperray et al.(2003) 1.5 Korsmeier et al. (2018) 1 10−7 Data antiprotons 0.5 0 0 1 2 3 4 5 6 7 4 5 6 7 8 9 10 11 12 p /nucleon (GeV/c) p /nucleon (GeV/c) lab lab

Figure B.1: Double differential cross sections from MC Figure B.2: Double differential cross sections from models compared to data of protons and deuterons pro- MC models and Duperray’s parametrization (pink line) duced in p+p collisions at 19 GeV/c [194]. compared to data of antiprotons produced in p+Be col- lisions at 19.2 GeV/c [193]. 107 APPENDIX B. MC SIMULATIONS VS ACCELERATOR DATA

Results from [193] show p and ¯pproduction in p+p, p+Be and p+Al collisions. The nucleons produced cover a laboratory momentum range from 2 to 19 GeV and an angular region from 12.5 to 70 mrad. Another experiment [194] at nearly the same energy (19 GeV/c) reported p, ¯p and d production in p+p collisions for θ = 116 mrad. In Fig. B.1, proton and deuteron production in p+p are shown in comparison to data of [194]. Values of p0 = 155 MeV/c and p0 = 150 MeV/c were determined from the fit to deuteron data with EPOS-LHC and FTFP-BERT, respectively. In Fig. B.2, antiproton production in p+Be collisions is shown for three different angles, alongside with the parameterization of Duperray [160] (magenta continuous line). ] •1 10 10 data Protons 1 1 EPOS•LHC FTFP•BERT EPOS•LHC [(rad GeV/c)

[mb/(GeV/c str)] −1

QGSP•BERT θ 10 FTFP•BERT Ω 10−1 θ=0•10 mrad, × 10 QGSP•BERT

/dpd −2

σ 10

2 data Protons dN/dpd d 10−2 data Deuterons 10−3 EPOS•LHC (p = 145 MeV/c) −3 0 10 FTFP•BERT (p = 145 MeV/c) 0 10−4 2.5 θ=60•100 mrad, × 10•2 2 10−5 1.5 1 •4 0.5 −6 θ=140•180 mrad, × 10 Model/Data 10 1.5 1 10−7 0.5 0 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 p /nucleon (GeV/c) p /nucleon (GeV/c) lab lab

Figure B.3: Double differential cross sections from MC Figure B.4: Double differential momentum distribution models compared to data of protons and deuterons pro- from MC models compared to data of protons produced duced in p+p collisions at 24 GeV/c [194]. in p+C collisions at 31 GeV/c [195].

B.1.2. p+p at plab = 24 GeV/c The same group that measured p, ¯pand d production in p+p collisions at 19 GeV also reported results at 24 GeV [194]. The results are compared with the MC models in Fig. B.3. Best fit values of the coalescence momentum for deuterons are p0 = 145 MeV/c and p0 = 145 MeV/c for EPOS-LHC and FTFP-BERT.

B.1.3. p+C at plab = 31 GeV/c The NA61/SHINE collaboration reported the production of mesons and baryons in p+C collisions at an incoming momentum of 31 GeV/c in 2016 [195]. In Fig. B.4 data at three different angles is plotted in comparison with MC models.

B.1.4. p+p, p+Be and p+Al at plab = 70 GeV/c A series of experiments performed in the Russian Institute for High Energy Physics at Serpukhov measured the production of p, ¯p,d and d¯ in p+p, p+Be and p+Al collisions at 70 GeV/c [196, 197, 198, 199]. Protons and antiprotons were detected in a transverse momentum region from 0.48 to 4.22 GeV/c and deuterons and antideuterons were evaluated until pT ≈

108 B.1. COMPARISON OF SIMULATIONS TO ACCELERATOR DATA (p, p,¯ d AND d)¯ )] 3 10 data Protons /c 2 EPOS•LHC 1 FTFP•BERT 10−1 QGSP•BERT

−2

[mb/(GeV 10 3 −3

/dp 10 σ 3 10−4 Ed 10−5 10−6 data Deuterons EPOS•LHC (p = 75 MeV/c) 10−7 0 FTFP•BERT (p = 105 MeV/c) 0 1

0.5

0 Model/Data 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 p /nucleon (GeV/c) T

Figure B.5: Invariant differential cross section for protons and deuterons produced in p+p collisions at 70 GeV/c. Data taken from [196, 197, 198].

3.8 GeV/c. Both hadrons and nuclei were measured at an angle of θ = 160 mrad or 90◦ in the center-of-mass frame. Figs. B.5, 3.7, B.6 and B.7 present this set of data in comparison with MC generators. The best fit values for p0 are shown in the figures. Despite the fact that some authors like Duperray et al. [157, 160] excluded these data from their analysis, the authors of this study did not find a reason to reject them. Besides, this is the lowest energy at which the spectrum of the invariant antideuteron cross section was measured so far. )] )] 3 3 2 data Protons 10 /c /c 10 2 EPOS•LHC 2 data p at y = 0 1 10 FTFP•BERT data d at y = 0 QGSP•BERT 10•1 1 10•2 [mb/(GeV [mb/(GeV 3 3 10−1 •3 /dp /dp 10 σ σ −2 3 3 10 10•4 Ed Ed EPOS•LHC (p = 35 MeV/c) 10−3 0 10•5 FTFP•BERT (p = 65 MeV/c) 0 −4 Duperray (p = 79 MeV/c) 10 data Deuterons 10•6 0 Korsmeier (p = 30 MeV/c) EPOS•LHC (p = 110 MeV/c) 0 10−5 0 •7 FTFP•BERT (p = 120 MeV/c) 10 0 1.5 1

0.5 1 0.5

Model/Data 0 Model/Data 1.5 1.5

1 1

0.5 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 p /nucleon (GeV/c) p /nucleon (GeV/c) T T

Figure B.6: Invariant differential cross section for pro- Figure B.7: Invariant differential cross section for an- tons and deuteron produced in p+Be collisions at tiprotons and antideuterons produced in p+Be colli- 70 GeV/c. Data taken from [198, 198]. sions at 70 GeV/c. Data taken from [198, 198].

109 APPENDIX B. MC SIMULATIONS VS ACCELERATOR DATA

B.1.5. p+p, p+C at plab = 158 GeV/c NA49 experiment published results on the production of protons, deuterons and antiprotons in p+p and p+C collisions at 158 GeV/c in 2009 and 2012 [82, 83]. These modern data sets are important since they are achieved with up-to-date techniques in hardware and data analysis and have low systematic errors. Figs. B.8 and B.9 show the invariant differential cross sections as function of pT for different values of Feynman xF calculated with MC and compared with data. Only protons from p+p collisions (Fig. B.8) and antiprotons from p+C collisions (Fig. B.9) are displayed, however, the analysis also includes antiprotons from p+p and protons from p+C. )] )] 3 3 10 10 /c

/c 0 EPOS•LHC 2 2 × 10 xF = •0.1 FTFP•BERT 1 1 QGSP•BERT •1 data Protons 10 × 10•2 x = 0 10−1 xF = 0 F [mb/(GeV [mb/(GeV 3 3 10•2 /dp /dp −2 σ

σ 10 3 3 10•3 × 10•4 x = 0.1 Ed F Ed x = 0.1 −3 F 10 10•4

−4 •5 10 10 × 10•6 x = 0.2 x = 0.3 F F 10•6 10−5 10•7 EPOS•LHC −6 FTFP•BERT 10 QGSP•BERT 10•8 xF = 0.7 Duperray et al. Korsmeier et al. 10−7 10•9 data p NA49

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 p /nucleon (GeV/c) p /nucleon (GeV/c) T T

Figure B.8: Invariant differential cross section for pro- Figure B.9: Invariant differential cross section for an- tons produced in p+p collisions at 158 GeV/c. Data tiprotons produced in p+C collisions at 158 GeV/c. taken from [82]. Data taken from [83].

B.1.6. p+Be, p+Al at plab = 200 GeV/c Protons, antiprotons, deuterons, and antideuterons produced in p+Be and p+Al collisions using the CERN-SPS accelerator were measured by [200, 205]. Proton and antiproton produc- tion was also measured at the Fermi National Accelerator Laboratory between 23 GeV/c and 200 GeV/c in p+Be collisions at 3.6 mrad [201]. Data from CERN were reported as ratios of differential cross section with respect to pions. Following the procedure used by [160], the differ- ential cross sections were calculated from the measured ratios. Results in p+Be for protons and deuterons are presented in Fig. B.10 while results for antiprotons and antideuterons are shown in Fig. B.11.

B.1.7. p+p, p+Be at plab = 300 and 400 GeV/c A large group of measurements were conducted at the Fermilab synchrotron with incident momenta of 200, 300 and 400 GeV/c using various targets, such as p, D2, Be, Ti and W. Protons and antiprotons were measured for every type of collision, but deuterons and antideuterons were only extracted at 300 GeV/c and measured at large transverse momentum pT /nucleon > 1 GeV/c. All the particles emitted from collisions were computed at 77 mrad which corresponds

110 B.1. COMPARISON OF SIMULATIONS TO ACCELERATOR DATA (p, p,¯ d AND d)¯ to an angle of ≈ 90◦ in the center-of-mass system [202, 203]. The specific case of p+Be at 300 GeV/c compared to MC models is shown in Figs. B.12 and B.13. )] )] 3 3 /c /c 10 2 2 10 data Protons 1 1 EPOS•LHC data p at y = 0 EPOS•LHC (p = 55 MeV/c)

FTFP•BERT [mb/(GeV 0

[mb/(GeV •1 3 3 10 data d at y = 0 FTFP•BERT (p = 85 MeV/c) −1 QGSP•BERT 0 /dp /dp 10 σ

σ Duperray (p = 79 MeV/c) 3 3 10•2 0 data Deuterons Korsmeier (p = 40 MeV/c) Ed Ed 0 −2 EPOS•LHC (p = 60 MeV/c) 10 0 FTFP•BERT (p = 70 MeV/c) •3 0 10 10−3 10•4

1.5 2 1.5 1 1 0.5 0.5 Model/Data Model/Data 1.5 1.5

1 1

0.5 0.5 5 10 15 20 25 30 35 40 45 5 10 15 20 25 30 p /nucleon (GeV/c) p /nucleon (GeV/c) lab lab

Figure B.10: Invariant differential cross section for Figure B.11: Invariant differential cross section for an- protons and deuteron produced in p+Be collisions at tiprotons and antideuterons produced in p+Be colli- 200 GeV/c. Data taken from [200, 205]. sions at 200 GeV/c. Data taken from [200, 205]. )] )] 2 3 3 10 data Protons 10 /c /c 2 2 10 data p at y = 0 EPOS•LHC 1 1 FTFP•BERT data d at y = 0 •1 10−1 QGSP•BERT 10 10−2 10•2 [mb/(GeV [mb/(GeV −3 3 3 10 −4 •3 /dp /dp 10 10 σ σ 3 3 10−5 EPOS•LHC (p = 90 MeV/c) 10•4 0 Ed Ed −6 FTFP•BERT (p = 140 MeV/c) 10 0 •5 −7 10 Duperray (p = 79 MeV/c) 10 0 −8 data Deuterons Korsmeier (p = 52 MeV/c) 10 10•6 0 10−9 EPOS•LHC (p = 95 MeV/c) 0 •7 −10 10 10 FTFP•BERT (p = 190 MeV/c) 0 1 1.5

0.5 1

0 0.5 Model/Data Model/Data 2 1.5 1.5 1 1 0.5 0.5 0.5 1 1.5 2 2.5 3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 p /nucleon (GeV/c) p /nucleon (GeV/c) T T

Figure B.12: Invariant differential cross section for pro- Figure B.13: Invariant differential cross section for an- tons and deuterons produced in p+Be collisions at tiprotons and antideuterons produced in p+Be colli- 300 GeV/c. Data taken from [202, 203]. sions at 300 GeV/c. Data taken from [202, 203]. √ B.1.8. p+p at s = 45 and 53 GeV The production of pions, , nucleons and antinucleons was measured at the CERN Intersecting Storage Ring in p+p collisions at a variety of energies in the center-of-mass frame √ with s = 23, 31, 45, 53, 63 GeV [204]. Deuterons and antideuterons were only reported for

111 APPENDIX B. MC SIMULATIONS VS ACCELERATOR DATA

45 and 53 GeV [206, 207, 208]. Following the analysis of proton and antiproton production by the NA49 collaboration, a feed down excess of 25% was estimated from simulations and it was applied to the whole sample. This correction significantly reduces the proton production, but leaves antiprotons essentially unchanged because of systematic errors in the nuclear absorption correction of about 30%. Results are shown in Figs. B.14 and B.15. )] )] 3 10 3 /c /c 2 2 1 1 data Protons xF = 0.15 •1 EPOS•LHC 10 10−1 data p at y = 0

[mb/(GeV FTFP•BERT [mb/(GeV 3 3 QGSP•BERT •2 data d at y = 0 /dp

/dp 10 −2 data Deuterons x = 0.18•0.25 σ 10 F σ 3 3 EPOS•LHC (p = 75 MeV/c) EPOS•LHC (p = 85 MeV/c)

Ed 0 Ed 0 •3 −3 FTFP•BERT (p = 105 MeV/c) 10 FTFP•BERT (p = 155 MeV/c) 10 0 0 Duperray (p = 79 MeV/c) 0 10−4 10•4 Korsmeier (p = 95 MeV/c) 0

1.2 1.5 1 0.8 1 0.6 0.5 Model/Data

1.6 Model/Data 1.4 1.5 1.2 1 1 0.8 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 p /nucleon (GeV/c) p /nucleon (GeV/c) T T

Figure B.14: Invariant differential cross section for Figure B.15: Invariant differential cross section for an- protons and deuteron produced in p+p collisions at tiprotons and antideuterons produced in p+p collisions √ √ s = 53 GeV. Data taken from [204, 208]. at s = 53 GeV. Data taken from [204, 206, 207]. )]

2 1 c /

2 0 10−1 × 10 , 12.0

10−2 × 10•2, 27.7.0

[mb/(GeV −3

T 10 p d

p −4 10 × 10•4, 56.7

2 −5

d 10

10−6 × 10•6, 98.7

10−8

−9 EPOS•LHC 10 FTFP•BERT

−10 QGSP•BERT 10 Duperray et al.(2003) Korsmeier et al. (2018) −11 10 data LHCb 0 0.5 1 1.5 2 2.5 3 3.5 p /nucleon (GeV/c) T √ Figure B.16: Differential cross section for antiprotons produced in p+He collisions at sNN = 110 GeV. Data taken from [86].

112 B.2. (ANTI)PROTON MISMATCH FACTORIZATION FOR EPOS-LHC AND FTFP-BERT

√ B.1.9. p+He at sNN = 110 GeV Antiprotons produced in p+He collisions with a 6.5 TeV proton beam were measured recently by the LHCb experiment at CERN. The antiproton momentum range covered was from 12 to 110 GeV/c. The antiprotons collected were produced only by direct collisions or from resonances decaying via strong interaction. In Fig. B.16 the data is compared with the MC models EPOS- LHC, FTFP-BERT, and QGSP-BERT. The parametrizations from Duperray and Korsmeier are also included. √ B.1.10. p+p at s = 900 and 7000 GeV At the LHC, protons and antiprotons as well as deuterons and antideuterons are produced in p+p and Pb+Pb collisions at very high energies. ALICE reported results at 0.9, 2.76 and 7 TeV in the central rapidity region -0.5 < y < 0.5 for a wide range of transverse momentum (pT < 5 GeV/c) [84, 155, 209, 210]. The data are compared with EPOS-LHC and the Duperray parameterization in Figs. B.17 and B.18. FTFP and QGSP were not included, since Geant4 √ models have an energy limit of s ≈ 430 GeV. )] )] 3

3 10 /c /c 2 2 data Protons 10−1 data p at y = 0 1 EPOS•LHC data d at y = 0 10−2 10−1 dy) [1/(GeV [mb/(GeV T

3 −3

−2 dp 10

10 T /dp σ 3

N/(p −4 −3 2 10

Ed 10 EPOS•LHC (p = 100 MeV/c) d 0 ×

ev Duperray (p = 79 MeV/c) −4 −5 0 10 data Deuterons N 10 π Korsmeier (p = 150 MeV/c) EPOS•LHC (p = 100 MeV/c) 0 0 1/2 −5 10 10−6

1.2 2 1 1.5 0.8 1 0.6 0.5 Model/Data Model/Data 1.2 1 1 0.8 0.5 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 p /nucleon (GeV/c) p /nucleon (GeV/c) T T

Figure B.17: Invariant differential cross section for Figure B.18: Invariant differential momentum dis- protons and deuteron produced in p+p collisions at tribution for antiprotons and antideuterons produced √ √ s = 900 GeV. Data taken from [84, 155, 209]. in p+p collisions at s = 900 GeV. Data taken from [84, 155, 209].

B.2. (Anti)proton mismatch factorization for EPOS-LHC and FTFP-BERT

Assuming (anti)proton-(anti)neutron independence and symmetry, Eq. 2.2 can be rewritten as:

2 sim 3 sim ! dNd¯ ~ 4πp0 dNp¯ ~ γd¯ (kd¯) = γp¯ (kp¯) (B.1) d~k3 3 d~k3 d¯ p¯ The proton or antiproton mismatch can be represented by the energy-dependent ratio.

113 APPENDIX B. MC SIMULATIONS VS ACCELERATOR DATA

sim  dNp¯  γd¯ ~ 3  dkp¯  r(T) = data . (B.2)  dNp¯  γp¯ ~ 3 dkp¯ Inserting the r(T) factor in Eq. B.1, the final result is:

2 sim data ! dNd¯ ~ 4π 0 3 dNp¯ ~ γd¯ (kd¯) = (p0) γp¯ (kp¯) (B.3) d~k3 3 d~k3 d¯ p¯ 0 2/3 Where p0 = p0 · r(T) , is the redefined coalescence momentum that is now more specific to the coalescence process rather than scaling the mismatch of the (anti)protons. The values of 0 p0 for EPOS-LHC and FTFP-BERT are shown in Fig. B.19 as function of the collision kinetic 0 energy (T). As observed, after factorizing the mismatch the p0 values of FTFP-BERT are close to the values of EPOS-LHC, showing a similar energy dependence. This, justified the use of 0 Eqs. 3.4 and 3.5 to fit the extracted p0 for both models. Differences in p0 for EPOS-LHC and FTFP-BERT after the mismatch factorization, are related to the intrinsic effects of the models as for example (anti)nucleon production asymmetries.

DEUTERONS ANTIDEUTERONS 200 220 180 FTFP•BERT EPOS•LHC EPOS•LHC 200 p+p p+p [MeV/c] [MeV/c] 0

0 160 p+p p+Be p+Be

p’ 180 p’ p+Be p+Al p+Al 140 160

120 140 120 100 100 80 80 60 FTFP•BERT 60 p+p 40 40 (a) p+Be (b) 20 20

102 103 104 105 106 107 102 103 104 105 106 107 T [GeV] T [GeV]

0 Figure B.19: (Color online) Extracted coalescence momentum p0 (symbols) for deuterons (a) and antideuterons (b) as function of the collision kinetic energy (T). Fit functions [Eqs. (3.4) and (3.5)] for EPOS-LHC (long-dashed red line) and FTFP-BERT (dashed blue line) are shown.

114 APPENDIX C Galdef file

In this appendix, the most important parts of the GALDEF file used in this thesis to calculate the transport of CRs through the Galaxy are shown. This file contains the initial parameters required to solve the propagation equation. In the first part of the file, the parameters related to the geometry considered, and the ranges in kinetic energy and moomentum are defined. Then, the values of the propagation parameters, source distribution and magentic field are outlined. For a detailed example of a GALDEF files see Reference [226].

Title = test_parm0 n_spatial_dimensions = 2 r_min = 0.0 min r r_max = 20.0 max r dr = 1.0 delta r z_min = -04.0 min z z_max = +04.0 max z dz = 0.1 delta z x_min = -20.0 min x x_max = +20.0 max x dx = 1.0 delta x y_min = -20.0 min y y_max = +20.0 max y dy = 1.0 delta y p_Ekin_grid = Ekin p||Ekin alignment p_min = 1000 min momentum (MV) p_max = 4000 max momentum (MV) p_factor = 1.35936 momentum factor

Ekin_min = 1.0e2 min kinetic energy per nucleon (MeV) Ekin_max = 1.0e9 max kinetic energy per nucleon (MeV) Ekin_factor = 1.35936 kinetic energy per nucleon factor

#...propagation equation parameters

D0_xx = 4.3e28 diffusion coefficient at reference rigidity

115 APPENDIX C. GALDEF FILE

D_rigid_br = 4.5e3 reference rigidity for diff. coefficient, MV D_g_1 = 0.395 diff. coeff. index below reference rigidity D_g_2 = 0.395 diff. coeff. index above reference rigidity diff_reacc = 1.0 1=include diffusive reacceleration v_Alfven = 28.6 Alfven speed in km s^{-1} damping_p0 = 1.0e6 some rigidity, MV, (where CR density is low) damping_const_G = 0.02 a const derived from fitting B/C damping_max_path_L = 3.0e21 Lmax~1 kpc, max free path convection = 1 1=include convection v0_conv = 12.4 V0 convection in km s^-1 dvdz_conv = 10.2 dV/dz=grad V in km s^-1 kpc^-1 nuc_rigid_br0 = 7.e3 ref. rig. for primary nucleus inj. index in MV nuc_rigid_br = 360.0e3 nuc_g_0 = 1.69 nucleus injection index below ref. rigidity nuc_g_1 = 2.40 nucleus injection index above ref. rigidity nuc_g_2 = 2.32 inj_spectrum_type = rigidity rigidity||beta_rig||Etot nucleon inj.spec.type electron_g_0 = 1.45 electron inj. index below electron_rigid_br0 electron_rigid_br0 = 6.0e3 ref. rigidity0 for electron inj. index in MV electron_g_1 = 2.75 electron inj. index electron_rigid_br = 1.0e5 reference rig. for electron inj. index in MV electron_g_2 = 2.49 electron inj. index above ref. rig. He_H_ratio = 0.11 He/H of ISM, by number

#...source distribution source_model = 1 0=zero 1=parameterized 2=case-B 3=pulsars source_parameters_0 = 0.2 zscale not used source_parameters_1 = 1.5 model 1:alpha source_parameters_2 = 3.5 model 1: source_parameters_3 = 20.0 model 1:rmax source_parameters_4 = 20.0 model 1:rmax

#...

B_field_name = galprop_original the name of the B-field model n_B_field_parameters = 10 number of B-field parameters B_field_parameters = 0,0,0,0,0,0,0,0,0,0 model param. by B_field_name B_field_model = 050100020 bbrrrzzz bbb=10*B(0) rrr=10*rscale zzz=10*zscale

116 Acronyms

ACC Anti-Coincidence Counters

AGN Active Galactic Nuclei

ALICE A Large Ion Collider

AMS Alpha Magnetic Spectrometer

ANTARES Astronomy with a Neutrino Telescope and Abyss enviromental RESearch

BESS Ballon-borne Experiment with Superconducting Spectrometer

CAPRICE Cosmic AntiParticle Ring Imaging Cherenkov Experiment

CMB Cosmic Microwave Background

CMS Compaq Solenoid

COBE Cosmic Background Explorer

CRMC Cosmic-Ray Monte Carlo

CRs Cosmic-Rays

CTA Cherenkov Telescope Array

DM Dark Matter

ECAL Electromagnetic Calorimeter

EGRET Energetic Experiment Telescope

GAPS General Antiparticle Spectrometer

HAWC High-Altitude Water Cherenkov Observatory

HESS The High Energy Stereoscopic System

ISM Interstellar Medium

MACHOs Massive Compact Halo Objects

117 Acronyms

MC Monte Carlo

PAMELA Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics

PDF Parton Distribution Function

QCD Quantum Chromodynamics

RICH Ring Imaging Cherenkov

SM Standard Model

SNE Supernova Explosions

SNRs Supernova Remnantes

Super-K Super-Kamiokande

SuperCDMS Super Cryogenic Dark Matter Search

TOF Time Of Flight

TRD Transition Radiation Detector

UED Universal Extra Dimensions

UHECR Ultra High Energy Cosmic-Ray

VERITAS The Very Energetic Radiation Imaging Telescope Array System

WIMP Weak Interacting Massive Particle

WMAP Wilkinson Microwave Anisotropy Probe

118 Notation

A : .

β : Particle velocity.

B2 : Coalescence parameter. c : Speed of light.

Ekin : Particle kinetic energy. η = 1/2ln(1 + cosθ/1 − cosθ): Pseudorapidity.

Φ: Flux.

γ = 1/p1 − β2 : Lorentz factor.

∆IS : Isospin factor. ~k : Vector momentum. m : Particle mass. p0 : Coalescence momentum. plab : Momentum in the laboratory reference system. pL : Longitudinal momentum. pT : Transverse momentum. qsec : Secondary source term. qter : Tertiary source term. q = qsec + qter : Total source term. √ s : total energy in the center-of-mass frame

σel : Elastic cross-section.

σinel : Inelastic cross-section.

119 Notation f = Ed3σ/dp3 : Differential invariant cross-section.

φ : Solar modulation.

σtot : Total cross-section. T : Kinetic energy of the collision in the laboratory. √ xF = 2pL/ s : Feynman scale variable. xR = E/Emax : Radial scale variable. y = 1/2ln(E + pz/1 − pz): Rapidity.

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