Physics 215B: Particles and Fields Winter 2017 Lecturer: McGreevy These lecture notes live here and are currently growing. Please email corrections to mcgreevy at physics dot ucsd dot edu. Last updated: 2017/03/15, 23:19:56 1 Contents 0.1 Sources....................................3 0.2 Conventions.................................4 6 To infinity and beyond5 6.1 A parable from quantum mechanics on the breaking of scale invariance5 6.2 A simple example of perturbative renormalization in QFT....... 13 6.3 Classical interlude: Mott formula..................... 15 6.4 Electron self-energy in QED........................ 20 6.5 Big picture interlude............................ 25 6.6 Vertex correction in QED......................... 34 6.7 Vacuum polarization............................ 45 7 Consequences of unitarity 54 7.1 Spectral density............................... 54 7.2 Cutting rules and optical theorem..................... 61 7.3 How to study hadrons with perturbative QCD.............. 67 8 A parable on integrating out degrees of freedom 69 9 The Wilsonian perspective on renormalization 78 9.1 Where do field theories come from?.................... 78 9.2 The continuum version of blocking.................... 86 9.3 An extended example: XY model..................... 89 9.4 Which bits of the beta function are universal?.............. 110 10 Gauge theory 116 10.1 Massive vector fields as gauge fields.................... 116 10.2 Festival of gauge invariance........................ 118 10.3 Lattice gauge theory............................ 122 2 0.1 Sources The material in these notes is collected from many places, among which I should mention in particular the following: Peskin and Schroeder, An introduction to quantum field theory (Wiley) Zee, Quantum Field Theory (Princeton, 2d Edition) Banks, Modern Quantum Field Theory: A Concise Introduction (Cambridge) Schwartz, Quantum field theory and the standard model (Cambridge) David Tong's lecture notes Many other bits of wisdom come from the Berkeley QFT courses of Prof. L. Hall and Prof. M. Halpern. 3 0.2 Conventions Following most QFT books, I am going to use the + − −− signature convention for the Minkowski metric. I am used to the other convention, where time is the weird one, so I'll need your help checking my signs. More explicitly, denoting a small spacetime displacement as dxµ ≡ (dt; d~x)µ, the Lorentz-invariant distance is: 0 1 +1 0 0 0 B 0 −1 0 0 C 2 2 µ ν µν B C ds = +dt − d~x · d~x = ηµνdx dx with η = ηµν = B C : @ 0 0 −1 0 A 0 0 0 −1 µν µ @ ~ µ µν (spacelike is negative). We will also write @µ ≡ @xµ = @t; rx , and @ ≡ η @ν. I'll use µ, ν::: for Lorentz indices, and i; k; ::: for spatial indices. The convention that repeated indices are summed is always in effect unless otherwise indicated. A consequence of the fact that english and math are written from left to right is that time goes to the left. h A useful generalization of the shorthand ~ ≡ 2π is dk ¯dk ≡ : 2π I will also write =δd(q) ≡ (2π)dδ(d)(q). I will try to be consistent about writing Fourier transforms as Z ddk Z eikxf~(k) ≡ ¯ddk eikxf~(k) ≡ f(x): (2π)d IFF ≡ if and only if. RHS ≡ right-hand side. LHS ≡ left-hand side. BHS ≡ both-hand side. IBP ≡ integration by parts. WLOG ≡ without loss of generality. +O(xn) ≡ plus terms which go like xn (and higher powers) when x is small. +h:c: ≡ plus hermitian conjugate. We work in units where ~ and the speed of light, c, are equal to one unless otherwise noted. When I say `Peskin' I usually mean `Peskin & Schroeder'. Please tell me if you find typos or errors or violations of the rules above. 4 6 To infinity and beyond Last quarter we ended at a high point, computing the amplitudes and cross-sections for many processes using QED. More precisely, we studied the leading-order-in-α am- plitudes, using Feynman diagrams which were trees { no loops. The natural next step is to look at the next terms in the perturbation expansion in α, which come from dia- grams with one loop. When we do that we're going to encounter some confusing stuff, in fact some of the same confusing stuff we encountered in thinking about Casimir forces at the beginning of last quarter. We didn't encounter these short-distance issues in studying tree-level diagrams because in a tree-level diagram, the quantum numbers (and in particular the momenta) of the intermediate states are fixed by the external states. In contrast, once there is a loop, there are undetermined momenta which must be summed, and this sum includes, it seems, arbitrarily high momentum modes, about which surely we have no information yet. In order to put ourselves in the right frame of mind to think about that stuff, let's make a brief retreat to systems with finitely many degrees of freedom. Then we'll apply some of these lessons to a toy field theory example (scalar field theory). Then we'll come back to perturbation theory in QED. Reading assignment for this chapter: Zee xIII. 6.1 A parable from quantum mechanics on the breaking of scale invariance Recall that the coupling constant in φ4 theory in D = 3 + 1 spacetime dimensions is dimensionless, and the same is true of the electromagnetic coupling e in QED in D = 3+1 spacetime dimensions. In fact, the mass parameters are the only dimensionful quantities in those theories, at least in their classical avatars. This means that if we ignore the masses, for example because we are interested in physics at much higher energies, then these models seem to possess scale invariance: the physics is unchanged under zooming in. Here we will study a simple quantum mechanical example (that is: an example with a finite number of degrees of freedom)1 with such (classical) scale invariance. It exhibits many interesting features that can happen in strongly interacting quantum field theory { asymptotic freedom, dimensional transmutation. Because the model is simple, we can understand these phenomena without resort to perturbation theory. 1I learned this example from Marty Halpern. 5 They will nevertheless illuminate some ways of thinking which we'll need in examples where perturbating is our only option. Consider the following (`bare') action: Z 1 Z 1 S[q] = dt ~q_2 + g δ(2)(~q) ≡ dt ~q_2 − V (~q) 2 0 2 where ~q = (x; y) are two coordinates of a quantum particle, and the potential involves (2) δ (~q) ≡ δ(x)δ(y), a Dirac delta function. I chose the sign so that g0 > 0 is attractive. (Notice that I have absorbed the inertial mass m in 1 mv2 into a redefinition of the p 2 variable q, q ! mq.) First, let's do dimensional analysis (always a good idea). Since ~ = c = 1, all dimensionful quantites are some power of a length. Let −[X] denote the number of powers of length in the units of the quantity X; that is, if X ∼ (length)ν(X) then we have [X] = −ν(X), a number. We have: [t] = [length=c] = −1 =) [dt] = −1: The action appears in exponents and is therefore dimensionless (it has units of ~), so we had better have: 0 = [S] = [~] and this applies to each term in the action. We begin with the kinetic term: Z 0 = [ dt~q_2] =) 1 1 [~q_2] = +1 =) [~q_] = + =) [~q] = − : 2 2 Since 1 = R dqδ(q), we have 0 = [dq] + [δ(q)] and D [δD(~q)] = −[q]D = ; and in particular [δ2(~q)] = 1: 2 This implies that the naive (\engineering") dimensions of the coupling constant g0 are [g0] = 0 { it is dimensionless. Classically, the theory does not have a special length scale; it is scale invariant. The Hamiltonian associated with the Lagrangian above is 1 H = p2 + p2 + V (~q): 2 x y 6 Now we treat this as a quantum system. Acting in the position basis, the quantum Hamiltonian operator is 2 H = −~ @2 + @2 − g δ(2)(~q) 2 x y 0 2 ~ 2 So in the Schr¨odingerequation H = − 2 r + V (~q) = E , the second term on the LHS is (2) V (~q) (~q) = −g0δ (~q) (0): To make it look more like we are doing QFT, let's solve it in momentum space: Z d2p i~p·~q=~ (~q) ≡ 2 e '(~p) (2π~) The delta function is Z d2p (2) i~p·~q=~ δ (q) = 2 e : (2π~) So the Schr¨odingerequation says 1 − r2 − E (q) = −V (q) (q) 2 Z p2 ¯d2peip·q − E '(p) = +g δ2(q) (0) 2 0 Z 2 ip·q = +g0 ¯d pe (0) (6.1) which (integrating the both-hand side of (6.1) over q: R d2qeip·q ((6.1)) ) says 2 Z 2 0 ~p d p 0 − E '(~p) = +g0 2 '(~p ) 2 (2π~) | {z } = (0) There are two cases to consider: • (~q = 0) = R ¯d2p'(~p) = 0. Then this case is the same as a free theory, with the constraint that (0) = 0, ~p2 − E '(~p) = 0 2 i.e. plane waves which vanish at the origin, e.g. / sin pxx e±ipyy=~. These scat- ~ tering solutions don't see the delta-function potential at all. 7 • (0) ≡ α 6= 0, some constant to be determined. This means ~p2=2 − E 6= 0, so we can divide by it : g Z g '(~p) = 0 ¯d2p0'(~p0) = 0 α: ~p2 ~p2 2 − E 2 − E The integral on the RHS (for (0) = α) is a little problematic if E > 0, since then there is some value of p where p2 = 2E. Avoid this singularity by going to the boundstate region: consider E = −B < 0. So: g '(~p) = 0 α: ~p2 2 + B What happens if we integrate this R ¯d2p to check self-consistency { the LHS should give α again: ! Z Z g 0=! ¯d2p'(~p) 1 − ¯d2p 0 ~p2 + B | {z } 2 = (0)=α6=0 Z g =) ¯d2p 0 = 1 ~p2 2 + B is a condition on the energy B of possible boundstates.