Subatomic Physics: the Notes1

Prepared for submission to JHEP Subatomic Physics: the Notes1 C.P. Burgess Department of Physics & Astronomy, McMaster University and Perimeter Institute for Theoretical Physics 1 c Cliff Burgess, for Physics 4E03 Winter Term 2016 Contents 1 The story so far...2 1.1 Prequel: substructure and atoms2 1.2 Units and scales 19 1.3 Lies, Damn Lies, and Measurement Errors 23 1.4 Relativistic kinematics 30 2 Calculational tools I 39 2.1 Conserved quantities 39 2.2 Decays: general properties 41 2.3 Scattering: general properties 55 3 Calculational tools II 65 3.1 Classical two-body scattering 65 3.2 Quantum potential scattering 75 3.3 Perturbation theory and the Born approximation 92 4 Nucleon substructure 100 4.1 Electrons, nucleons and quarks 101 4.2 Elastic scattering 106 4.3 Inelastic ep scattering 111 5 Nuclear structure 117 5.1 Nuclear binding energies and nucleon forces 118 5.2 Nuclear models 126 5.3 Isospin and meson exchange 139 5.4 Radioactivity 147 6 Quantum Field Theory 161 6.1 Heisenberg's harmonic oscillator 161 6.2 Creation and annihilation operators 163 6.3 Interactions and fields 167 6.4 Relativistic quantum field theory 171 6.5 Bosons and forces 174 7 The Standard Model 176 7.1 Fermions and the generation puzzle 176 7.2 Bosons and the four forces 179 { 1 { 7.3 Where the Standard Model fails 182 A major theme of 20th Century physics is that we are surrounded by substructure: what we see around us is built from smaller (often initially invisible) constituents and much of the diversity we see can be efficiently understood as consequences of the properties of these constituents. Furthermore this is a recursive process, with the constituents themselves often built from still-smaller pieces: matter is made of molecules; molecules are made of atoms; atoms made of nuclei and electrons; nuclei are built from nucleons (i.e. protons and neutrons); nucleons are made of quarks and gluons; and so on. Subatomic physics is the part of this story starting with nuclei and continuing on to the smallest constituents known. We call particles èlementary' if they have no substructure so far as we can tell, and at present the list of such particles contains around 20 entries. The theory of these particles and their mutual interactions is called the Standard Model and works extremely well (with a few noteworthy exceptions). But history teaches us that this classification of the elementary is at best provisional and may be changed in light of later evidence with finer resolution. These notes summarize the evidence for the present picture, as well as the flaws it is known to have, at a level appropriate for upper-year physics undergraduates. The reader is assumed to be familiar with non-relativistic quantum mechanics, electromagnetism and the rudiments of special relativity. 1 The story so far... This section contains some preliminary background information needed to tell this story, and starts by summarizing the first indications that there might be an interesting story to tell. 1.1 Prequel: substructure and atoms The evidence that many of the properties of macroscopic things are best understood if those things are regarded as being built of numerous much smaller constituents { atoms { started to accumulate convincingly in the 19th and early 20th Centuries. Partly this came about as the rules governing chemical reactions became clearer, with the emergence of a pattern of sys- tematic properties for the elements, summarized by the periodic table (see Figure9). Partly it emerged with the realization that the thermal properties of fluids (and thermodynamics in general) could be understood in terms of the random motion of their constituent atoms. It was clinched by the development of quantum mechanics and the ability this brought to compute the properties of simple atoms from first principles, including an understanding of the patterns of the periodic table. { 2 { Starting with Newton In retrospect, the possibility that substructure could be useful was already implicit in the recursiveness of Newton's Laws. To see what this means, suppose that a macroscopic object, O, is made up of a collection of N point-like atoms that mutually interact through forces Fij (which describe the force acting on particle ì' due to particle `j'), with the atoms labelled by an index i; j = 1; ··· ;N. Then Newton's 2nd law for the motion of each atom is given by m1 x¨1 = + F12 + F13 + ··· + F1N + Fext 1 m2 x¨2 = F21 + F23 + ··· + F2N + Fext 2 m2 x¨3 = F31 + F32 + ··· + F3N + Fext 1 (1.1) . = mN x¨N = FN1 + FN2 + FN3 + ··· + Fext N ; where over-dots denote differentiation with respect to time { i.e. x_ := dx=dt and x¨ := d2x=dt2 { while Fext i denotes any external forces (e.g. attraction by the Earth's gravity etc.) acting on atom number ì'. The laws of motion for the entire macroscopic object must follow as consequences of eqs. (1.1), and at first sight it seems remarkable that any simple laws should be possible at all for macroscopic objects if this is so. A wonderful thing happens if all of these equations are added together, however, since then Newton's third law (which states that Fij = −Fji for all i and j) implies that all of the Fij cancel in the sum, leaving m1 x¨1 + m2 x¨2 + ··· + mN x¨N = Fext 1 + ··· + Fext N : (1.2) This takes the same form as did Newton's law for each atom: M X¨ = Fext ; (1.3) with total mass and net external force given by N N X X M := mi ; Fext := Fext i ; (1.4) i=1 i=1 provided one defines N 1 X X := m x : (1.5) M i i i=1 This shows that Newton's law applies in the same way to the entire macroscopic object provided the acceleration that appears in it is chosen to be the acceleration of the object's centre of mass | defined by (1.5). Furthermore, this shows that Newton's 2nd law is recursive in the sense that it also applies equally well to various macroscopic subsets of macroscopic objects. For example suppose the { 3 { Figure 1. A sketch (not to scale) of atoms in a macroscopic object, illustrating the difference between the atomic position xi and its position, yi = xi − X, relative to the object's centre of mass, X. object described above can be regarded as the union of two pieces, denoted A and B, so O = A [ B. (Maybe the macroscopic object considered above was the Earth-Moon system and A is the Earth while B is the Moon.) Then all sums over i in the above argument can be broken up into sums separately over A and B: N X X X M = mi = mi + mi =: MA + MB ; (1.6) i=1 i2A i2B and similarly N X X X Fext = Fext i = Fext i + Fext i =: Fext A + Fext B : (1.7) i=1 i2A i2B So if we define 1 X 1 X X := m x and X := m x ; (1.8) A M i i B M i i A i2A B i2B then N ¨ X X X ¨ ¨ M X = mi xï = mi xï + mi xi = MA XA + MB XB ; (1.9) i=1 i2A i2B where the last equality uses (1.8). Repeating the arguments leading to (1.3) separately for each of objects A and B then implies ¨ ¨ MA XA = Fext A and MB XA = Fext B ; (1.10) and (1.9) shows that these are consistent with (1.3). { 4 { Taken together, the above arguments show that the relationship between Newton's law for the whole system and Newton's law for its two subsystems is identical to the relationship derived earlier with Newton's laws for the N atoms, specialized to the case N = 2. (That is, it is conceptually as if each of A and B were themselves to be considered to be àtoms'.) This recursive nature of Newton's laws shows that the laws themselves cannot tell what the fundamental smallest objects are, and apply equally well at all levels of substructure. If tomorrow evidence were to emerge that all of our atoms in eq. (1.1) turn out to contain still- smaller teeny-weeny atoms, each of which themselves satisfy Newtons 2nd law, then nothing in the above arguments need change at all (provided we assume the position xi to be an appropriately defined centre-of-mass coordinate). Of course just because the internal forces cancel when Newton's 2nd law is summed over atoms doesn't mean that internal dynamics has no physical effect. For instance the average kinetic energy of motion of the constituent atoms relative to their centre of mass is a source of internal energy, since N N X 1 1 X 1 1 E = m x_ 2 = M X_ 2 + m y_ 2 =: M X_ 2 + E ; (1.11) kin 2 i i 2 2 i 2 int i=1 i=1 shows that it is an energy that survives even in the absence of motion of the object's overall centre of mass. In this expression yi := xi − X and no term linear in y_ i arises due to the easily proven identity N X mi yi = 0 ; (1.12) i=1 that follows directly from the definition (1.5) of X. At this point Eint as defined above could equally well describe the energy of an overall rigid rotation of the macroscopic body, or the kinetic energy associated with altering its shape, or the random motion of its constituent atoms for an object whose macroscopic orientation and shape do not change.

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