- Scalar Quantum Electrodynamics B This Diagram Is Not Gauge Invariant Independent Of E By Itself What Is The M Mal S 1 (185.61 KiB) Viewed 35 times
Scalar quantum electrodynamics (b) This diagram is not gauge invariant (independent of E) by itself. What is the m mal set of diagrams you need to add to this one for the sum to be gauge invarian Why should the other diagrams cancel on their own? 9.3 In this problem you will prove the uniqueness of non-Abelian gauge theories by com sidering the soft limit when there are multiple scalar fields o. Suppose these fields Moto) have a mass matrix M (i.e. the mass term in the Lagrangian is C sap there are N massless spin-1 particles A, a = 1... N we will call gluons. Then the generic interaction between Aa, o, and o, can be written as I (p, q) as in Eq. (957) (a) Show that in the soft limit, q< p, the charges are now described by a matrix To = Tg. = (b) For N = 1, show that only if [M, T] = 0 can the theory be consistent. Conclude that gluons (or the photon) can only couple between particles of the same mass. (c) Consider Compton scattering, (p) 4 (q) → ;(p) 4 (q), in the soft limit qª, q< p,p'. Evaluate the two diagrams for this process and then show that by setting e qo, the interactions are consistent with Lorentz invariance only if [Ta, Tb] = 0, assuming nothing else is added. (d) Show that one can modify this theory with a contact interaction involving 10 AA of the generic form rabu (p, q, q) so that Lorentz invariance is pre- served in the soft limit. How must rab relate to Ta? Show also that rab have a pole, for example as (qª + qb)² → 0. quand e = ij ij must ij (e) Such a pole indicates a massless particle being exchanged, naturally identified as a gluon. In this case, the rabu interaction in part (d) can be resolved into a 3-point interaction among gluons, of the form rabo (qª, qb, q) and the ra (p, q) vertex. Show that if rabe itself has no poles, then in the soft limit it can be written Show that if and only if [Ta, Tb] = ifabeTe can the Compton .) for some constants fabe and μια μνα uniquely as rabe (qa, qb, q) = fabc (guv qa+ μνα work out the scattering amplitude be Lorentz invariant in the soft limit. This implies that the gluons transform in the adjoint representation of a Lie group, as will be discussed in Chapter 25. The soft limit also im 1