Higgs and Particle Production in Nucleus-Nucleus Collisions Zhe Liu Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences Columbia University 2016 c 2015 Zhe Liu All Rights Reserved Abstract Higgs and Particle Production in Nucleus-Nucleus Collisions Zhe Liu We apply a diagrammatic approach to study Higgs boson, a color-neutral heavy particle, pro- duction in nucleus-nucleus collisions in the saturation framework without quantum evolution. We assume the strong coupling constant much smaller than one. Due to the heavy mass and colorless nature of Higgs particle, final state interactions are absent in our calculation. In order to treat the two nuclei dynamically symmetric, we use the Coulomb gauge which gives the appropriate light cone gauge for each nucleus. To further eliminate initial state interactions we choose specific prescriptions in the light cone propagators. We start the calculation from only two nucleons in each nucleus and then demonstrate how to generalize the calculation to higher orders diagrammatically. We simplify the diagrams by the Slavnov-Taylor-Ward identities. The resulting cross section is factorized into a product of two Weizsäcker-Williams gluon distributions of the two nuclei when the transverse momentum of the produced scalar particle is around the saturation momentum. To our knowledge this is the first process where an exact analytic formula has been formed for a physical process, involving momenta on the order of the saturation momentum, in nucleus-nucleus collisions in the quasi-classical approximation. Since we have performed the calculation in an unconventional gauge choice, we further confirm our results in Feynman gauge where the Weizsäcker-Williams gluon distribution is interpreted as a transverse momentum broadening of a hard gluons traversing a nuclear medium. The transverse momentum factorization manifests itself in light cone gauge but not so clearly in Feynman gauge. In saturation physics there are two different unintegrated gluon distributions usually encountered in the literature: the Weizsäcker-Williams gluon distribution and the dipole gluon distribution. The first gluon distribution is constructed by solving classical Yang-Mills equation of motion in the McLerran-Venugopalan model, while the second gluon distribution is related to the dipole scattering amplitude. So far, the quantum structure of the dipole gluon distribution has not been thoroughly investigated. Applying the same diagrammatic techniques, we carry out a detail study of the quantum structure of the color dipole gluon distribution, and then compare it with that of the Weizsäcker-Williams gluon distribution. Table of Contents List of Figures . iv List of Tables . ix 1 Introduction1 2 Theory of Strong Interactions6 2.1 A Brief History of Strong Interactions . .6 2.1.1 The Quark Model . .6 2.1.2 The Theory of Strong Interactions . .9 2.2 Quantum Chromodynamics . 11 2.3 Light Cone Variables . 13 2.4 Quantize QCD on the Light Cone . 17 2.4.1 Light Cone Propagator . 18 2.4.2 Perturbation Theory on the Light Cone . 18 3 High Energy Evolution Equations in QCD 22 3.1 Deep Inelastic Scattering . 23 3.2 DGLAP Equation . 27 3.3 Regge Theory . 30 3.4 BFKL Evolution Equation . 32 3.4.1 Quark-quark Scattering . 33 3.4.2 DIS at Small-x .............................. 35 i 3.4.3 Dipole Formalism . 37 3.4.4 Problems in BFKL Evolution . 38 3.5 Saturation of the Gluon Density . 39 3.6 A Map of Parton Evolution in QCD . 41 Low Density Region . 43 High Density Region . 43 Saturation Region . 44 4 Parton Saturation in Large Nuclei 45 4.1 Saturation in Large Nuclei . 45 4.2 Gluon Distribution in Large Nuclei . 48 4.2.1 Weizsäcker-Williams Gluon Distribution . 48 4.2.2 Dipole Gluon Distribution . 49 4.3 Non-linear Evolution Equations . 53 4.3.1 JIMWLK Evolution . 53 4.3.2 BK Evolution . 54 4.4 Quantum Structure of the Weizsäcker-Williams Gluon Distribution . 57 4.4.1 STW Identities . 58 4.4.2 Quantum Structure of the Weizsäcker-Williams Gluon Distribution . 61 i Prescriptions in Light Cone Propagator . 63 Diagrammatic Calculation . 64 5 Particle Production in Nucleus-Nucleus Collisions 71 5.1 Parton Distribution and Factorization . 72 5.1.1 Collinear Factorization . 72 5.1.2 Transverse Momentum Dependent Factorization . 76 5.2 Color-Neutral Heavy particle Production in Nucleus-Nucleus Collisions . 78 5.3 Coulomb and Light Cone Gauges . 81 5.4 Light Cone Calculation of Scalar Particle Production . 84 ii 5.5 Feynman Gauge Calculation . 94 6 Quantum Structure of Dipole Gluon Distribution 100 6.1 Introduction . 101 6.2 Gluon Production in Proton-Nucleus Collisions . 105 6.2.1 From Proton-Proton Collisions to Proton-Nucleus Collisions . 107 6.3 Quantum Structure of the Dipole Gluon Distribution . 111 6.4 Comparison . 119 Bibliography 120 Appendix 125 A.1 Diagrammatic Representation of STW Identities in Non-Abelian Gauge Field 125 A.2 Other Choices of i’s ............................... 127 A.3 Lipatov Vertex in Feynman Gauge . 130 iii List of Figures 2 2.1 A summary of measurements of the coupling αs(Q ), which is taken from [55]. Q is the momentum scale at which the measurement was made. The data clearly indicate that as the energy increases the strength of the QCD coupling constant decreases. 14 2.2 Light cone coordinates and the usual space-time coordinates. 15 2.3 A bare quark emits a soft gluon in the infinite momentum frame. The plus sign indicates that the quark carries a large plus momentum component. The dashed vertical line at t = 0 indicates that the quark-gluon state is measured. r and r0 label the quark helicity. 19 3.1 Deep inelastic scattering. l is the momentum of the incoming lepton and q is the momentum transfer between the lepton and the proton. γ∗ is a virtual photon. 23 3.2 The handbag diagram in DIS. γ∗ represents the virtual photon and it scatters with a quark from the proton. The filled circle represents the proton. The vertical dashed line represents the final state cut. 24 2 3.3 Structure function F2 as a function of Q from a combination of modern experimental data [63]. For moderate values of x, there is a good agreement between the data and Bjorken scaling. 26 3.4 QCD corrections to the γ∗q vertex in the parton model. The wavy line not labeled by γ∗ are gluons. 27 iv 3.5 Ladder diagram for gluon cascade in the DGLAP evolution. The proton, carrying longitudinal momentum p+ and no transverse momentum, splits into gluons with smaller and smaller plus momentum components but bigger and bigger transverse momentum components. 29 3.6 The imaginary part of the quark-quark elastic scattering amplitude with exchange of a gluon ladder. The filled circles represent the Lipatov vertices. The double wavy line represents the reggeized gluons. The gluon ladder is often called BFKL ladder. 32 3.7 Deep inelastic scattering at low x with BFKL evolution in the Bjorken frame. The circle at the center represents the full BFKL ladder in Fig. 3.6...... 35 3.8 BFKL evolution in DIS in the dipole frame. The virtual photon first splits into a quark-antiquark pair, a dipole. Then, gluon cascades are developed in the dipole wavefunction. The density of the dipoles increases according to the dipole form of the BFKL equation. 37 3.9 A phase diagram of parton evolution in QCD. A filled circle represents a parton with transverse size 1=Q2 and rapidity y. The solid line represents ∼ the boundary between a dilute parton system and a dense parton system. 42 4.1 A quark scatters on two nucleons in Feynman gauge. The circles represent the nucleons. The vertical line on the third diagram denotes the fact that the quark is put on-shell between the two scatterings. Thus, successively scatterings become independent in Feynman gauge. 49 4.2 Quark-quark and Quark-dipole scatterings in Feynman gauge. The transverse coordinates of the quark and antiquark in the dipole are x? and 0?, respectively. The arrows on the line indicate momentum flows. 51 v 4.3 A graphical representation of the S-matrix of a dipole scattering on a nucleus. The S-matrix can be equivalently represented as the quark-nucleus scattering amplitude squared or the dipole-nucleus scattering amplitude. The vertical line on the left side of the identity indicates a final state cut. 0? and x? are the transverse coordinates of the quark in the amplitude and the complex conjugate amplitude. 53 4.4 Figs. 4.4(a) to 4.4(c) represent the amplitude squared of a dipole emitting a gluon. Their sum can be denoted by Fig. 4.4(d) in the large Nc limit, where the emitted gluon is replaced by a double quark line. x1? and x0? are the transverse coordinates of the quark and the anti-quark. The transverse coordinate of the emitted gluon at the final state is x2?............ 55 4.5 A graphical representation of the two terms in the Balitsky-Kovchegov equation, Eq. (4.29). A dipole of size x01? is splitted into two separate dipoles of sizes x02? and x21?, respectively. The interaction with the nucleus can happen before and after the dipole splitting. Fig. 4.5(a) is the real gluon emission term, while Figs. 4.5(b) and 4.5(c) are the virtual terms. 56 4.6 A generalized Ward identity in non-Abelian gauge theory. The solid lines represent on-shell particle states, the dashed line represents a longitudinally polarized gluon. 58 4.7 STW identities for fermion and quark propagators. An external gluon state when multiplied by its four momentum becomes a longitudinally polarized gluon.