Standard Model and Beyond · Problem 3.26

Gauge Theories

Physics — Particle Physics · sun-left-right-chiral-symmetry-scalar

Problem

Let

\phi_{ab}

a, b = 1 \cdots n

be a complex scalar transforming as

(n, n^*)

under

SU(n)_L \times SU(n)_R

, i.e.,

[T^i_L, \phi_{ab}] = -(L^i_n)_{ac}\phi_{cb}, \quad [T^i_R, \phi_{ab}] = +\phi_{ac}(L^i_n)_{cb},

where

T^i_L

and

T^i_R

are, respectively, the generators of

SU(n)_L

and

SU(n)_R

. (a) Calculate a finite transformation of

\phi

. (b) Show that the Yukawa interaction

\mathcal{L}_Y = h\bar{\psi}_{aL}\phi_{ab}\psi_{bR} + h^*\bar{\psi}_{bR}(\phi_{ab})^\dagger \psi_{aL} \equiv h\bar{\psi}_L \phi \psi_R + h^* \bar{\psi}_R \phi^\dagger \psi_L

and the mass term

\mathcal{L}_m = -\mu^2(\phi_{ab})^\dagger \phi_{ab} \equiv -\mu^2 \mathrm{Tr}(\phi^\dagger \phi)

are invariant under

U(n)_L \times U(n)_R

. In the second forms,

\phi

represents an

n \times n

matrix with components

\phi_{ab}

, and

(\phi^\dagger)_{ba} \equiv (\phi_{ab})^\dagger

. (c) Construct the most general

U(n)_L \times U(n)_R

invariant renormalizable Lagrangian density for

\psi

and

\phi

. (d) Display the

U(n)_L \times U(n)_R

Noether currents for the Lagrangian density in (c). Hint: both

\phi

and

\phi^\dagger

must be included in the sum over fields in (3.46).