Hadron Physics Lectures for the 19th UK Nuclear Physics Summer School, Queen’s University Belfast D. G. Ireland (University of Glasgow) 30 August, 31 August and 1 September, 2017 2 d g ireland 1 Introduction Hadrons are the particles that feel the strong nuclear force. This force is described by the theory of Quantum ChromoDynamics (QCD), a field theory whose constituents are quarks (the particles) and gluons (the force carriers). QCD forms the strongly interacting sector of the standard model of particle physics, but has a unique character compared to the other forces of nature. One of the main features of the strong interaction is that the constituents have never been observed in isolation. Hadrons are composite particles, made from quarks and bound by gluons. They are the only physical manifestations of QCD that we can study. Nuclei are built from protons and neutrons (and very occasionally hyperons!) and are held together by pions, all of which are hadrons. An understanding of nuclear physics therefore rests on an under- standing of hadrons, even if the details of hadronic interactions at high energies are not relevant for understanding collective phenom- ena of heavy nuclei. With only three lectures on the topic, there is barely time to scratch the surface. However, rather than give a superficial overview of as much as possible, beyond a general introduction, I have cho- sen a few topics to look at in sufficient detail so as to stretch the participants during their time at the summer school. This document contains material presented on the lecture slides. As you can see, it does not look exactly the same, but these notes contain all the information on the slides, plus more. There are occa- sional extra comments (like this one) in these printed notes that do 1 1 There are also notes placed in the mar- not appear on the slides. To help you follow the lectures, the slide gins, like this one. The side notes are number is written, underlined, in the margin. used to highlight important informa- tion or provide some background. Please contact me ([email protected]) if you spot any mistakes, or would like further explanation. hadron physics 3 The Big Picture 2 The study of Hadron Physics is part of a wide spectrum of re- search that aims to be able to describe the nature of the matter that we observe in the universe. It sits at the interface between particle-, or high-energy physics, and nuclear physics. From particle physics it shares a “reductionist” philosophy - a desire to understand ev- erything from basic constituents. On the other hand it involves the study of the structure of composite particles, and thus shares a great deal of common ground with nuclear structure physics, such as a study of effects that are “emergent properties” due to the interaction of several constituents. In hadron physics, we study the hadrons, the particles in nature that feel the strong nuclear force. These are categorised in to mesons (bosons, with integer spin quantum numbers) and baryons (fermions, with half-integer spin). We want to know how many hadrons there are, how they interact with each other, how they decay, etc.. We also want to know about their properties: static quantities such as mass, shape, size, magnetic moment; dynamic quantities such as the distribution of charge and current that are measured with structure functions. 4 d g ireland The Nuclear Chart 3 Part of the reason for studying hadrons is that the basic con- stituents of nuclei (protons, neutrons, pions,...) are such particles. Without a full understanding of hadrons, therefore, we cannot really say that we understand nuclear matter. Quantum ChromoDynamics 4 The theory underlying the strong nuclear reaction is quantum chromodynamics (QCD), a quantum field theory that describes fields of “matter” particles, quarks, that interact via the exchange of force- carrying bosons called gluons. It is somewhat like quantum electro- dynamics (QED), the theory underlying electromagnetism, but there are some important differences that lead to quite unique phenomena. Due to the mathematical structure of the interaction, gluons carry “colour” charge, and can therefore interact with each other. This is quite different to QED, where the force carriers, the photons, do not hadron physics 5 interact with each other. In QCD this leads to many of the features that we are quite used to, such as superposition of photon states (e.g. lasers). QCD Running Coupling Constant Figure 1: Nucleons become more com- plex, the closer one looks at them. Figure 2: Models that retain features of QCD are needed for practical calcula- tions. 5 The Strong Coupling constant, which measures the strength of the interaction, varies with energy and at the scale of hadron masses, its value approaches unity. Any scheme that relies on a small param- eter to use perturbation theory for calculations is therefore bound to fail for hadrons. 6 Confinement Figure 3: If mesons are “pulled” apart, the energy stored in the gluon string is enough to create a quark-antiquark pair if the string breaks. 7 Since gluons interact with each other, this gives rise to an observed phenomenon know as confinement: quarks and gluons have 2 2 never been observed in isolation. In relativistic heavy ion collisions the quark-gluon plasma has been shown to form as a new state of matter. This is sometimes referred to as a de-confined phase. However at the energy scales relevant to ordinary hadronic matter, all hadronic states are confined. 6 d g ireland The Origin of Mass 8 The Higgs Mechanism is only responsible for about 1-2% of the mass of nucleons, though the generation of quark mass. That means that around 98% of the visible mass in the universe must be gener- ated by an additional mechanism. Recent calculations have shown that even if quarks were to have no mass from the interaction with the Higgs field, it is possible via a mechanism called dynamical chiral symmetry breaking (DCSB) for QCD to give mass to quarks. The simple picture is that quarks moving through the “glue” acquire inertia due to the interaction with the “stickiness”. The situation is a little more complicated than that, but it has been proven that DCSB is a feature of QCD, and is thought to be related to confinement. However it has not yet been proved that confinement is a predictable consequence of QCD. Light and Heavy Particles 9 In these lectures we will concentrate on what is referred to as the light quark sector, which generally encompasses hadrons that contain hadron physics 7 combinations of up, down or strange quarks. The heavy quark sector (charm, bottom and top) is still very much the purview of high- energy physics, although the boundary between the two is bot fuzzy and arbitrary. Scattering Experiments Figure 4: Conceptually, most mea- surements are identical to Rutherford’s original experiment 10 As with other areas of nuclear physics, most of the infor- mation about hadrons comes from scattering experiments. We will primarily discuss experiments that involve beams of photons and electrons, although will note in passing that a large amount of infor- mation has been derived from hadron beam experiments. Cross Sections • Select beam (type and current), target (type and thickness) • Measure number of particles in detector, beam charge Reaction a + b ! X We imagine an experiment where a beam of particles a hits a target containing particles b. We want to know how a reacts with b. 8 d g ireland • Flux: 1 dN F = a = n v a A dt a a where na is beam particle density, A is the illuminated target area and va is the beam velocity. • Number of target particles Nb = nb Ad where nb is target particle density, A is the illuminated target area and d is the target thickness. • Luminosity: dN L = F N = a n d a b dt b which has dimensions L2T−1. 11 Differential Cross Section Consider a detector, area A, angle q and distance r from beam A DW = r2 Number of reactions seen by detector in time Dt is ds (E, q) N (E, q) = L DWDt dW where the piece ds (E, q) dW is the differential cross section ds (E, q) 12 An angular distribution is the variation of with angle q. dW If the detector determines energy E0 of scattered particles d2s (E, E0, q) dWdE0 is the double differential cross section The total cross section is the integral ZZ d2s (E, E0, q) s = dWdE0 tot dWdE0 13 hadron physics 9 Scattering Theory Assumptions Wavefunction of beam represented by plane waves. i.e. ∝ yi exp (iki.r) 14 The momentum transfer is given by q = h¯ (ki − k f ) Using quantum collision theory, it can be shown that the differ- ential cross-section is equal to the square magnitude of a function called the scattering amplitude: ds = j f (q)j2 dW where • m Z iq.r f (q) = − V(r) exp dr 2ph¯ 2 h¯ • m = mass of electron • V(r) = potential between target and electron 15 Now q.r = qr cos q0 10 d g ireland dr = r2 sin q0drdq0df so the expression for the scattering amplitude becomes 2m Z ¥ f (q) = V (r) r sin (qr/¯h) dr hq¯ 0 16 Coulomb Scattering As an example of how to calculate scattering amplitudes, we start with a point charge scattering from a point charge (Rutherford scat- tering). Screened Coulomb potential: Ze2 V (r) = exp (−r/a) r the scattering amplitude becomes: 2mZe2 Z ¥ qr −r f (q) = sin exp dr hq¯ 0 h¯ a [N.B.