Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2018 / 3 / b

The quadratic part of the covariant-derivative term contains $g^2{v}^2{A}_{\mu }{A}^{\mu }/2-gv{A}_{\mu }{\partial }^{\mu }b$. Its mixing term cancels the $+gv{A}_{\mu }{\partial }^{\mu }b$ in the stated gauge fixing term. Thus the vector mass and diagonal quadratic action areFor nonzero $p^2$ introduce the transverse and longitudinal transverse and longitudinal momentum projectors $P_T^{\mu \nu }=g^{\mu \nu }-p^{\mu }{p}^{\nu }/p^2$ and $P_L^{\mu \nu }=p^{\mu }{p}^{\nu }/p^2$. After integration by parts, the inverse quadratic kernel isTheir orthogonality makes the inversion immediate. The gauge-boson propagator isEquivalently,The projector formula explains the physical massive pole and the gauge-dependent longitudinal pole. The apparent $1/p^2$ in the projectors cancels in the equivalent expression; it is not an extra massless physical state. At $\xi =1$ the numerator reduces to $g^{\mu \nu }$. At fixed momentum the $\xi \to \infty$ limit gives the unitary gauge propagator $-i(g^{\mu \nu }-p^{\mu }{p}^{\nu }/M_A^2)/(p^2-M_A^2+i0)$. The relation between this limit and ultraviolet power counting is not uniform in momentum.^[ref]

At a stationary symmetry-breaking vacuum of a gauge-invariant scalar potential, the angular field has zero potential curvature. Its free kinetic term is $(\partial b{)}^2/2$, while the displayed gauge-fixing term adds $-\xi g^2{v}^2{b}^2/2$. HenceThis is the would-be Goldstone boson, whose gauge-dependent pole accompanies the longitudinal pole in i; it is not an additional physical scalar particle of that mass. For $M_A\ne 0$, fixed finite $p$ and $\xi \to +\infty$,The angular field becomes infinitely heavy in this gauge and is absent from the unitary gauge propagating description. The qualification about fixed momentum matters: loop integrations probe arbitrarily large $p$, so one cannot automatically interchange the $\xi \to \infty$ limit with ultraviolet integration. If $gv=0$, there is no massive Higgs phase and the asserted heavy-field limit does not follow. A separate radial Higgs boson may remain; its mass depends on the unspecified potential and is not determined by this gauge-fixing term.