- #1

Antarres

- 200

- 100

Covariant derivative is defined with:

$$D_\mu\phi_a = \partial\phi_a + gf_{abc}A^b_\mu\phi_c$$

The model considered is of a multiplet of real scalar fields coupled to gauge bosons. He's looking for mass matrix of bosons, which he gets from the term in Lagrangian given in (20.33):

$$\Delta\mathcal{L} = \frac{g^2}{2}(f_{abc}A^b_\mu\phi_c)^2$$

Now he defines a variable ##\Phi = \phi_c t^c##, where ##t^c## are the generators of SU(3) which are traceless 3x3 Hermitian matrices.

He says (equation 20.35), that we can then rewrite this term in form:

$$\Delta\mathcal{L} = -g^2tr([t^a,\Phi][t^b,\Phi])A^a_\mu A^{b\mu}$$

However, checking this trace, and substituting the definition of commutators of generators, there is a residual term that is the trace of two generators which arise from those two commutators. But that trace equals to the invariant C(r) multiplied by a Kronecker delta over the gauge indices. The invariant C(r) is equal to quadratic Casimir operator in adjoint representation of SU(N), ##C(G) = C_2(G) = N##. Therefore, a factor of 3 should arise there, and so I don't see how that factor transfers into the factor of one half in the mass term given in the first equation.

Any help would be appreciated. I suspect maybe I'm missing some kind of symmetrization step, but I'm not sure, I can't see a mistake in my calculations.