Physics 198-730B: Quantum Field Theory

1 Preprint MCGILL-98/NNhep-ph/yymmnnnPhysics 198-730B: Quantum Field TheoryJames M. Cline1 Dept. of Physics , McGill University3600 University , PQ H3A 2T8 CanadaThis course builds on the introduction to QFT you received in198-610A. We will startwith 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 themost important concepts and developments in QFT. We cannot treat them all in depth, butyou will learn the basic ideas. The topics to be covered (timepermitting) will include: Perturbation Theory : the loop expansion; regularization;dimensional regularization;Wick rotation; momentum cutoff; 4theory; renormalization; renormalization groupequation; Wilsonian viewpoint; the epsilon expansion; relevant, irrelevant and marginaloperators; Callan-Symanzik equation; running couplings;beta function; anomalousdimensions; IR and UV fixed points; asymptotic freedom; triviality; Landau pole The effective action: generating functional; connected diagrams; one-particle-irreduciblediagrams; 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.

2 Superselection is as far as the course was able to go in the time that we for a continuation course with more topics beyond perturbation Theory couldinclude: 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 symmetrybreaking; Goldstone s theorem; the Higgs mechanism; supersymmetry; the minimalsupersymmetric standard model; Grand Unified Theories; convergence of gauge cou-plings1e-mail: Field Theory at finite temperature and density21 IntroductionPhysics, like all sciences, is based upon experimental observations. It s therefore a goodthing to remind ourselves: what are the major experimental observables relevant for particlephysics? These are the masses, lifetimes, and scattering cross sections of particles. Polesof propagators, and scattering and decay amplitudes are thequantities which are related tothese observables:pole of1p2 m2 (mass)2(1)Ta bc decay rate(2)Tab cd scattering cross section(3)These observables, which are components of theS-matrix(scattering matrix) are the maingoals of computation in Quantum Field Theory .

3 To be precise,Tis the transition matrix,which is related to the S-matrix by eq. ( ) of [3]:Sfi=1fi+ (2 )4i (4)(pf pi)Tfi(4)Toward the end of 198-610A you learned about the connection between Green s functionsand amplitudes. The recipe, known as the LSZ reduction procedure (after Lehmann, Szy-manzik and Zimmerman) [1, 2], is the following. For a physical process involvingnincomingandmoutgoing particles, compute the corresponding Green s function. Let s consider ascalar Field Theory for simplicity:G(n+m)(x1,..,xn,y1,..,ym) =h0out|T [ (x1) (xn) (y1).. (ym)]|0ini(5)The blob represents the possibly complicated Physics occurring in the scattering region, whilethe lines represent the free propagation of the particles asthey are traveling to or from thescattering region. It is useful to go to Fourier space: G(n+m)(p1,..,pn,q1,..,qm) =in+m(2 )4 (4)(Ppi Pqi) n+m(p1,..,pn,q1,..,qm)(p21 m2)..(p2n m2)(q21 m2).

4 (q2m m2)(6)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 see in the denominator the product of all the propagators for the free propagation. Inthe numerator we have the function n+m, called theproper vertex function, which repre-sents the blob in the picture. This function contains all theinteresting Physics , since we areinterested in the interactions between the particles and not the free propagation.{}mnFigure mscattering we can state the LSZ procedure: to convert the Green s function to a transitionamplitude, truncate (omit) all the external line propagators. In other words, the propervertex n+mis the T-matrix element we are interested in. I will come backto the nontrivialproof of this statement later. First, we would like to use it to get some concrete results.

5 Theproblem is that, in general, there is no analytic way to compute n+mif it is nontrivial. In freefield Theory , the only nonvanishing vertex function is 2=i(p2 m2), the inverse we introduce interactions, we get all the vertex functions, but they can t be computedexactly. We have to resort to some kind of approximation. Since we know how to computefor free fields, the most straightforward approximation is that in where the interactions areweak and can be treated perturbatively. In nature this is a good approximation for QEDand the weak interactions, and also for the strong interaction at sufficiently high , these realistic theories are a bit complicated to start with. It is easier to learnthe basics using a toy model Field Theory . The simplest theorywith interactions which has astable vacuum is 4. We simply add this term to the Klein-Gordon Lagrangian for arealscalar Field :2L=12 (m2 i ) 2 4!

6 4.(7)The factor of 1/4! is merely for convenience, as will become apparent. Thereare severalthings to notice. (1) Thei is to remind us of how to define the pole in the propagatorso as to get physical (Feynman) boundary conditions. We always takei 0 finally, sothat (2) the Lagrangian is real-valued. The latter is necessary in order fore iS/ hto be apure phase. Violation of this condition will lead to loss of unitarity, , probability willnot be conserved. (3) The 4interaction comes with a sign: the Lagrangian is kineticminus potential energy. The sign is necessary so that the potential energy is boundedfrom below. This is the reason we consider 4rather than 3as the simplest realisticscalar Field potential. Although 3would be simpler, simpler, it does not have a stableminimum the Field would like to run off to .The above Lagrangian does not describe any real particles known in nature, but it issimilar to that of the Higgs Field which we shall study later onwhen we get to the standardmodel.

7 The coupling constant is a dimensionless number since has dimensions of will be able to treat the interaction as a perturbation if is sufficiently small; to determinehow small, we should compute the first few terms in the perturbation series and see whenthe corrections start to become as important as the leading tool which I find most convenient for developing perturbation Theory is the Feynmanpath integral for the Green s functions. Let s consider thegenerating function for Green sfunctions:Z[J] =ZD eiS[J]/ h(8)whereS[J] =Zd4x(L+J(x) (x))(9)Recall theraison d etreofZ[j]: the Green s functions can be derived from it by taking2I use the metric conventionp p =E2 ~ derivatives. For example, the four-point function isG4(w,x,y,z) =1i4Z 4Z[J] J(x1) J(x2 J(y1) J(y2) J=0(10)=1Z[0]ZD eiS[0]/ h (w) (x) (y) (z)(11)If we had no interactions, this Green s function could be computed using Wick s theorem tomake contractions of all possible pairs of s as shown in figure 2.)

8 This is not very interest-ing: it just describes the free propagation of two independent particles. The correspondingFeynman diagram is calleddisconnectedsince the lines remain separate. The disconnectedprocess is not very interesting experimentally. It corresponds to two particles in a collisionmissing each other and going down the beam pipe without any deflection. These events arenot observed (since the detector is not placed in the path of the beam).xyzw+z xwxzw+yyFigure 2. 4-point function in the absence of when we include the interaction we get scattering between the particles. This canbe seen by expanding the exponential to first order in :G4(w,x,y,z) =1Z[0]ZD eiS[0]/ h (w) (x) (y) (z)Zd4x i h 4! 4!.(12)Before evaluating this path integral, I would like to digress for a moment to discuss itsmuch simpler analog, the ordinary integralZ=Z dx 2 ei2ax2(13)This is related to the well-known trigonometric ones, the Fresnel integrals, and they can becomputed using contour integration [4] along the contour shown in fig.

9 :Z= 2Z 0dx 2 ei2ax2= 2ei /4Z0 dy 2 e 12ay2=ei /4 a(14)The second equality is obtained by using the fact that the integral around the full contourvanishes, as well as that along the circular arc (at ). Therefore we have shown that theoscillatory Gaussian integral is related to the real one. The integralZis analogous to the field5theory generating functionalZ[0]. And the analogy to the 2-point function (the propagator)is1ZZ dx 2 ei2ax2x2=1Z2idZda=ia(15)If we carry out the analogous procedure in Field Theory , we obtain the momentum-spacepropagator G2=i/(p2 m2). This little exercise shows you where the factor ofiis comingfrom. It also belies the statement you will sometimes hear, that the Feynman path integralonly rigorously exists in Euclidean Complex contour for evaluating complex Gaussian let s return to the 4-point function. When we do the contractions, we have thepossibility of contracting the external fields with the fields from the interaction vertex.

10 Thisnewconnectedcontribution (figure 3) comesin additionto the disconnected one shown infigure 4. In the latter, we contract the s within the interaction only with results in aloopdiagram, in fact a two-loop diagram. There is also another kind ofdisconnected diagram where one of the particles is freely propagating, while the other feelsthe effect of the interaction, figure 5. We are going to focus onthe connected contribution,fig. 3, for right now. This kind of diagram is called atree diagrambecause of its stick-likeconstruction, to distinguish it from loop diagrams, such asthe figure-eight appearing in yxzFigure 3. Connected 4-point function at linear order in : a tree +z xwxzw+yy()xx Figure 4. Contribution to disconnected 4-point function atlinear order in .wxyzx + permutationsFigure 5. Another contribution to the disconnected 4-pointfunction at linear order in .In evaluating the connected diagram, we have to take into account all the possible waysof Wick-contracting.