Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 9.3

Scalar quantum electrodynamics — Schwartz M.D. z-lib.org

Physics — QFT · non-abelian-gauge-theories-soft-limit

Problem

In this problem you will prove the uniqueness of non-Abelian gauge theories by considering the soft limit when there are multiple scalar fields ϕi\phi_i. Suppose these fields have a mass matrix MM (i.e. the mass term in the Lagrangian is L=Mijϕiϕj\mathcal{L} = M_{ij}\phi^i\phi_j) and there are NN massless spin-1 particles AμaA^a_\mu, a=1…Na = 1\ldots N we will call gluons. Then the generic interaction between AμaA^a_\mu, ϕi\phi_i and ϕj\phi_j can be written as Γijaμ(p,q)\Gamma^{a\mu}_{ij}(p,q) as in Eq. (9.57). ((a)) Show that in the soft limit, q≪pq \ll p, the charges are now described by a matrix Ta=TijaT^a = T^a_{ij}. ((b)) For N=1N=1, show that only if [M,T]=0[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, ϕi(p)Aμa(qa)→ϕj(p′)Aνb(qb)\phi_i(p)A^a_\mu(q^a) \to \phi_j(p')A^b_\nu(q^b), in the soft limit qa,qb≪p,p′q^a, q^b \ll p, p'. Evaluate the two diagrams for this process and then show that, by setting ϵμa=qμa\epsilon^a_\mu = q^a_\mu and ϵνb=qνb\epsilon^b_\nu = q^b_\nu, the interactions are consistent with Lorentz invariance only if [Ta,Tb]=0[T^a, T^b] = 0, assuming nothing else is added. ((d)) Show that one can modify this theory with a contact interaction involving ϕiϕjAμaAνb\phi_i\phi_j A^a_\mu A^b_\nu of the generic form Γijabμν ⁣(p,qa,qb)\Gamma^{ab\mu\nu}_{ij}\!\left(p, q^a, q^b\right) so that Lorentz invariance is preserved in the soft limit. How must Γijabμν\Gamma^{ab\mu\nu}_{ij} relate to TijaT^a_{ij}? Show also that Γijabμν\Gamma^{ab\mu\nu}_{ij} must have a pole, for example as (qa+qb)2→0(q^a + q^b)^2 \to 0. ((e)) Such a pole indicates a massless particle being exchanged, naturally identified as a gluon. In this case, the Γijabμν\Gamma^{ab\mu\nu}_{ij} interaction in part (d) can be resolved into a 3-point interaction among gluons, of the form Γμναabc ⁣(qa,qb,qc)\Gamma^{abc}_{\mu\nu\alpha}\!\left(q^a, q^b, q^c\right) and the Γijaμ(p,q)\Gamma^{a\mu}_{ij}(p,q) vertex. Show that if Γμναabc\Gamma^{abc}_{\mu\nu\alpha} itself has no poles, then in the soft limit it can be written uniquely as Γμναabc ⁣(qa,qb,qc)=fabc(gμνqcα+⋯ )\Gamma^{abc}_{\mu\nu\alpha}\!\left(q^a, q^b, q^c\right) = f^{abc}(g_{\mu\nu}q^\alpha_c + \cdots) for some constants fabcf^{abc} and work out the ⋯\cdots. Show that if and only if [Ta,Tb]=ifabcTc[T^a, T^b] = if^{abc}T^c can the Compton 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.

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