georgi h. lecture notes on weak interactions

The corresponding Noether currents are Ja =?i@TTa: 1.2.10The symmetry, 1.2.8, is the largest internal rotation symmetry that a set ofN real spinlessbosons can have, because the kinetic e

1 | Classical Symmetries

The concept of symmetry will play a crucial role in nearly all aspects of our discussion of weakinteractions At the level of the dynamics, the fundamental interactions (or at least that subset ofthe fundamental interactions that we understand) are associated with \gauge symmetries" Butmore than that, the underlying mathematical language of relativistic quantum mechanics | quan-tum eld theory | is much easier to understand if you make use of all the symmetry informationthat is available In this course, we will make extensive use of symmetry as a mathematical tool tohelp us understand the physics In particular, we make use of the language of representations ofLie algebras

1.1 Noether's Theorem { Classical

At the classical level, symmetries of an action which is an integral of a local Lagrangian density areassociated with conserved currents Consider a set of elds,j(x) wherej= 1 toN, and an action

S[] =Z

d4xL((x);@(x)) (1.1.1)where L is the local Lagrangian density The index, j

L(+;@+@)? L(;@) = L

 + L

(@)@; (1.1.5)1

because @ = @ Note that (1.1.5) is a single equation with no j index The terms on thecolumn vector on the right From (1.1.2), (1.1.4) and (1.1.5), we have

where theTafora= 1 to mare a set ofNN

the a are a set of in nitesimal parameters We can (and sometimes will) exponentiate (1.1.8) toget a nite transformation:

Ta fora= 1 to 3 with the commutation relations of SU(2),

[Ta;Tb] =iabcTc; (1.1.11)and which commute with all the other generators Then this is anSU(2) factor of the algebra Thealgebra can always be decomposed into factors like this, called \simple" subalgebras, and a set ofgenerators which commute with everything, called U(1)'s

The normalization of theU(1) generators must be set by some arbitrary convention However,the normalization of the generators of each simple subgroup is related to the normalization of thestructure constants It is important to normalize them so that in each simple subalgebra,

X

c;d facdfbcd =k ab: (1.1.12)Then for every representation,

for thendimensional representation Then k=n.

the space-time dependence of the elds is called an \internal" symmetry The familiar Poincaresymmetry of relativistic actions is notan internal symmetry

In this book we will distinguish two kinds of internal symmetry If the parameters, a, areindependent of space and time, the symmetry is called a \global" symmetry Global symmetriesinvolve the rotation of 

This is not a very physically appealing idea, because it is hard to imagine doing it, but we willsee that the concept of global symmetry is enormously useful in organizing our knowledge of eldtheory and physics

Later on, we will study what happens if thea depend on x Then the symmetry is a \local"

or \gauge" symmetry As we will see, a local symmetry is not just an organizing principle, but isintimately related to dynamics

For now, consider (1.1.7) for a global internal symmetry of L Because the symmetry is asymmetry ofL, the second term,V is zero

Thus we can write the conserved current as

N= L

(@)iaTa: (1.1.15)Because the in nitesimal parameters are arbitrary, (1.1.15) actually de nes mconserved currents,

This is the canonical form for the Lagrangian for a set of N massless free scalar elds Under an

in rst written down

as a model of leptons, simply because at the time the strong interactions the weak interactions ofthe hadrons were not completely understood

interactions conserve electron number,  number and  number separately Formally, this meansthat the theory has global symmetries:

Further, the interactions of the  and  are exact copies of those of the electron, so we candiscuss only the elds in the electron family (2.2.1)

The gauge group isSU(2)U(1), which means that there are four vector elds, three of whichare associated with the SU(2) group that we will call Wa, where a = 1, 2 or 3, and one X

associated with theU(1)

The structure of the gauge theory is determined by the form of the covariant derivative:

D=@+igWaTa+ig0XS (2.2.5)whereTaand S are matrices acting on the elds, called the generators ofSU(2) and U(1), respec-tively Notice that the coupling constants fall into two groups: there is one g for all of the SU(2)couplings but a dierent one for the U(1) couplings The SU(2) couplings must all be the same(if theTa's are normalized in the same way tr(TaTb) = ab) because they mix with one anotherunder global SU(2) rotations But the U(1) coupling g0 can be dierent because the generatorS

never appears as a commutator of SU(2) generators Even if we did start with equal g and g0,

we would be unable to maintain the equality in the quantum theory in any natural way The twocouplings are renormalized dierently and so require dierent in nite rede nitions in each order ofperturbation theory Since we need dierent counterterms, it would be rather silly to relate thecouplings

To specify the gauge structure completely, we must de ne the action ofTaandS on the fermion elds De ne the doublet

R eld, which is called anSU(2) singlet, (2.2.8) is satis ed in a rather trivial way

In order to incorporate QED into the theory we are building, we must make certain that somelinear combination of the generators is the electric-charge matrixQ The matrixT3is clearly related

to the charge because the dierence between the T3 values of each multiplet (the doublet L and,trivially, the singlet e?

R) us the same as the charge dierence Thus, we de ne

which de nesS We have done this very carefully so thatS will be proportional to the unit matrix

W

= W1

iW2 p

.

Figure 2-1:

There are two things wrong with Figure 2-1 as a picture of the weak interactions The? and

e? are massless, and the W are massless The SU(2)U(1) gauge symmetry does not allow alepton mass term e?

Le?

R or a W mass term The leptons must obviously get mass somehow forthe theory to be sensible The W must also be very heavy in order for the theory to agree withdata A masslessWwould give rise to a long-range weak force In fact, the force has a very shortrange

Despite these shortcomings, we will press on and consider the neutral sector If the theory is toincorporate QED, one linear combination of the W

3 and X elds must be the photon eld A.Thus, we write

A= sin W3+ cos X (2.2.14)Then the orthogonal linear combination is another ... is proportional to the identity TheU(1) is justfermion number

2, four-component fermionelds, , by projection with the projection operators

fermions:

... fermion number is then determined bythe degeneracy of the diagonal mass matrix If all the masses are dierent, then the only symmetry

If fermion number is not conserved, so that the fermions... regularization and renormalization sometimes makes things moreinteresting In this chapter, we discuss an approach to quantum eld theory that is particularlyuseful for the discussion of weak interactions