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Physics 198-730B: Quantum Field Theory

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Physics 198-730B: Quantum Field Theory Preprint MCGILL-98/NN hep-ph/yymmnnn Physics 198-730B: Quantum Field Theory James M. Cline1 Dept. of Physics, McGill University 3600 University St. Montreal, PQ H3A 2T8 Canada This course builds on the introduction to QFT you received in 198-610A. We will start with the loop expansion in scalar field theory to illustrate the procedure of renormalization, and then extend this to QED and other gauge theories. My goal is to introduce all of the most important concepts and developments in QFT. We cannot treat them all in depth, but you will learn the basic ideas. The topics to be covered (time permitting) will include: Perturbation theory: the loop expansion; regularization; dimensional regularization; • Wick rotation; momentum cutoff; λφ4 theory; renormalization; renormalization group equation; Wilsonian viewpoint; the epsilon expansion; relevant, irrelevant and marginal operators; Callan-Symanzik equation; running couplings; beta function; anomalous dimensions; IR and UV fixed points; asymptotic freedom; triviality; Landau pole The effective action: generating functional; connected diagrams; one-particle-irreducible • diagrams; Legendre transform Gauge theories: QED; QCD; anomalies; gauge invariance and unitarity; gauge fixing; • Faddeev-Popov procedure; ghosts; unitary gauge; covariant gauges; Ward Identities; BRS transformation; vacuum structure of QCD; instantons; tunneling; theta vacuua; superselection sectors. This is as far as the course was able to go in the time that we had. Topics for a continuation course with more topics beyond perturbation theory could include: the strong CP problem; axions; lattice gauge theory; lattice fermion doubling; confine- • ment; the strong coupling expansion; the large N expansion. Effective field theories: chiral lagrangians; the standard model; spontaneous symmetry • breaking; Goldstone’s theorem; the Higgs mechanism; supersymmetry; the minimal supersymmetric standard model; Grand Unified Theories; convergence of gauge cou- plings 1e-mail: [email protected] 1 Field theory at finite temperature and density • 2 1 Introduction Physics, like all sciences, is based upon experimental observations. It’s therefore a good thing to remind ourselves: what are the major experimental observables relevant for particle physics? These are the masses, lifetimes, and scattering cross sections of particles. Poles of propagators, and scattering and decay amplitudes are the quantities which are related to these observables: 1 pole of (mass)2 (1) p2 m2 ↔ − a bc decay rate (2) T → ↔ ab cd scattering cross section (3) T → ↔ These observables, which are components of the S-matrix (scattering matrix) are the main goals of computation in quantum field theory. To be precise, is the transition matrix, T which is related to the S-matrix by eq. (7.14) of [3]: S = 1 + (2π)4iδ(4)(p p ) (4) fi fi f − i Tfi Toward the end of 198-610A you learned about the connection between Green’s functions and amplitudes. The recipe, known as the LSZ reduction procedure (after Lehmann, Szy- manzik and Zimmerman) [1, 2], is the following. For a physical process involving n incoming and m outgoing particles, compute the corresponding Green’s function. Let’s consider a scalar field theory for simplicity: G (x ,...,x ,y ,...,y )= 0 T ∗[φ(x ) φ(x )φ(y ) ...φ(y )] 0 (5) (n+m) 1 n 1 m h out| 1 ··· n 1 m | ini The blob represents the possibly complicated physics occurring in the scattering region, while the lines represent the free propagation of the particles as they are traveling to or from the scattering region. It is useful to go to Fourier space: (2π)4δ(4)( p q )Γ (p ,...,p , q ,...,q ) G˜ (p ,...,p , q ,...,q )= in+m i − i n+m 1 n 1 m (6) (n+m) 1 n 1 m (p2 m2) . (p2 m2)(q2 m2) . (q2 m2) 1 − P nP− 1 − m − The delta function arises because we have translational invariance in space and time, so mo- mentum and energy are conserved by the process. This equation makes the picture explicit. We see in the denominator the product of all the propagators for the free propagation. In the numerator we have the function Γn+m, called the proper vertex function, which repre- sents the blob in the picture. This function contains all the interesting physics, since we are interested in the interactions between the particles and not the free propagation. n { }m Figure 1.n m scattering process. → 3 Now we can state the LSZ procedure: to convert the Green’s function to a transition amplitude, truncate (omit) all the external line propagators. In other words, the proper vertex Γn+m is the T-matrix element we are interested in. I will come back to the nontrivial proof of this statement later. First, we would like to use it to get some concrete results. The problem is that, in general, there is no analytic way to compute Γn+m if it is nontrivial. In free 2 2 field theory, the only nonvanishing vertex function is Γ2 = i(p m ), the inverse propagator. Once we introduce interactions, we get all the vertex functions,− but they can’t be computed exactly. We have to resort to some kind of approximation. Since we know how to compute for free fields, the most straightforward approximation is that in where the interactions are weak and can be treated perturbatively. In nature this is a good approximation for QED and the weak interactions, and also for the strong interaction at sufficiently high energies. However, these realistic theories are a bit complicated to start with. It is easier to learn the basics using a toy model field theory. The simplest theory with interactions which has a stable vacuum is λφ4. We simply add this term to the Klein-Gordon Lagrangian for a real scalar field:2 1 λ = ∂ φ∂µφ (m2 iε)φ2 φ4. (7) L 2 µ − − − 4! The factor of 1/4! is merely for convenience, as will become apparent. There are several things to notice. (1) The iε is to remind us of how to define the pole in the propagator so as to get physical (Feynman) boundary conditions. We always take iε 0 finally, so → iS/¯h that (2) the Lagrangian is real-valued. The latter is necessary in order for e− to be a pure phase. Violation of this condition will lead to loss of unitarity, i.e., probability will not be conserved. (3) The λφ4 interaction comes with a sign: the Lagrangian is kinetic − minus potential energy. The sign is necessary so that the potential energy is bounded from below. This is the reason− we consider λφ4 rather than µφ3 as the simplest realistic scalar field potential. Although µφ3 would be simpler, simpler, it does not have a stable minimum—the field would like to run off to . The above Lagrangian does not describe−∞ any real particles known in nature, but it is similar to that of the Higgs field which we shall study later on when we get to the standard model. The coupling constant λ is a dimensionless number since φ has dimensions of mass. We will be able to treat the interaction as a perturbation if λ is sufficiently small; to determine how small, we should compute the first few terms in the perturbation series and see when the corrections start to become as important as the leading term. The tool which I find most convenient for developing perturbation theory is the Feynman path integral for the Green’s functions. Let’s consider the generating function for Green’s functions: Z[J]= φ eiS[J]/¯h (8) Z D where S[J]= d 4x ( + J(x)φ(x)) (9) Z L Recall the raison d’ˆetre of Z[j]: the Green’s functions can be derived from it by taking 2 µ 2 2 I use the metric convention pµp = E ~p . − 4 functional derivatives. For example, the four-point function is 1 δ4Z[J] G4(w,x,y,z) = (10) i4Z δJ(x )δJ(x δJ(y )δJ(y ) 1 2 1 2 J=0 1 iS[0]/¯h = φ e φ(w)φ(x)φ(y) φ(z) (11) Z[0] Z D If we had no interactions, this Green’s function could be computed using Wick’s theorem to make contractions of all possible pairs of φ’s as shown in figure 2. This is not very interest- ing: it just describes the free propagation of two independent particles. The corresponding Feynman diagram is called disconnected since the lines remain separate. The disconnected process is not very interesting experimentally. It corresponds to two particles in a collision missing each other and going down the beam pipe without any deflection. These events are not observed (since the detector is not placed in the path of the beam). w y w x w y + + x z y z z x Figure 2. 4-point function in the absence of interactions. But when we include the interaction we get scattering between the particles. This can be seen by expanding the exponential to first order in λ: 1 iS[0]/¯h 4 i λ 4 G4(w,x,y,z) ∼= φ e φ(w)φ(x)φ(y)φ(z) d x′ φ . (12) Z[0] Z D Z −h¯ 4! ! Before evaluating this path integral, I would like to digress for a moment to discuss its much simpler analog, the ordinary integral ∞ dx i ax2 Z = e 2 (13) √ Z−∞ 2π This is related to the well-known trigonometric ones, the Fresnel integrals, and they can be computed using contour integration [4] along the contour shown in fig. 2.5: 0 iπ/4 ∞ dx i ax2 iπ/4 dy 1 ay2 e Z =2 e 2 = 2e e− 2 = (14) 0 √ − √ √a Z 2π Z∞ 2π The second equality is obtained by using the fact that the integral around the full contour vanishes, as well as that along the circular arc (at ).
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