[...]... we can tell, the weak and electromagnetic interactions act on the and exactly as they act on the electron Because they are heavier, they decay, by weak interactions The electron, the lightest charged particle, must be absolutely stable unless electromagnetic gauge invariance and global charge conservation are violated The SU (2) U (1) theory of the weak and electromagnetic interactions was rst written... the functional integral formulation of eld theory, one tries to write the Green functions as functional integrals: ~ (1b.2.3) h0 j T g(G; ) j 0i = hgi ; ~i h1 23 Weak Interactions | Howard Georgi | draft - February 10, 1998 | where hgi = ~ Z dG] d ] g(G; ) eiS(G; ) : (1b.2.4) In the functional integral description, the di culty with gauge invariance shows up in the independence of the action on gauge... with your new sources, one for each new dimension 4 object you can build These will be required to absorb dependence on the unknown short distance physics Weak Interactions | Howard Georgi | draft - February 10, 1998 | 18 3 Once this is done, the generating functional, W , depends on the new sources, and we can nd the e ect of an insertion of a composite operator in any Green function simply by di erentiating... the weak and electromagnetic interactions was rst written down as a model of leptons, simply because at the time the strong interactions the weak interactions of the hadrons were not completely understood 2.2 SU (2) U (1) In its simplest and original version, the SU (2) U (1) model describes the weak interactions of the leptons Nine elds needed to describe the e, , and and their neutrinos eL ; L; e? ;... dimension, , and the function, First of all, the sign in front of the anomalous dimension term depends on the fact we are studying 1PI functions, ?n , rather than connected Green functions, Gn , which satisfy @ + @ @ +n @ Gn = 0 : (1a.1.21) Secondly, because W (s) is the generating function for the connected Green functions, we can obtain (1a.1.21) for all n from the single equation @ + @ @ Z 4 + d x s(x)... commenting on some minus signs that frequently cause confusion Note that the dependence of ( ) is opposite to that of ?4 Because the wave function renormalization vanishes, ?4 is a directly physical scattering amplitude, to this order Thus when changes, ?4 remains constant That is a direct translation of (1a.1.14), with = 0 On the other hand, the \solution", to the renormalization group equation, is ?4... dependence on the external momenta, pi for i = 1 to n, on the parameter , and on the scale If we change , we must change the wave function renormalization of each of the external lines and also the coupling to maintain the same physics This is the conveniently summarized in the renormalization group equation: @ ?n = 0 : (1a.1.14) + @@ ? n @ In addition to , (1a.1.14) depends on an \anomalous dimension",... not need a R eld The neutrinos are known to be very light They may be massless, but there are some (still confusing) indications that they may have small masses If the neutrinos are massless, the weak 27 Weak Interactions | Howard Georgi | draft - February 10, 1998 | 28 interactions conserve electron number, number and number separately Formally, this means that the theory has global symmetries: eL !... charged-current weak interactions based on a four-fermion interaction For the decay interaction, the form was as follows: GF J J p e 2 where and (1 + 5) ? J = (2.3.1) (2.3.2) (1 + 5 )e? (2.3.3) (2.3.1) is called a current-current interaction for obvious reasons The constant GF (F is for Fermi) is determined from the decay rate It has units of 1/mass2 Je = e m2 GF ' 10?5 p (2.3.4) where mp is the proton mass... draft - February 10, 1998 | 35 2.6 The -Model Before discussing the spontaneous breakdown of symmetry in gauge theories, we will work out in detail one example of global symmetry breakdown in the strong interactions that will be useful when we discuss hadrons | and will turn out to have a curious connection with the weak interactions as well The example is the -model of Gell-Mann and Levy (Nuovo Cimento