Mass term in the Lagrangian

I have read that the mass term appearing in the electroweak Lagrangian stops it (the Lagrangian) from becoming gauge invariance. Can someone explain where and why this term is creating the problem?

Answer

Let's assume a typical fermionic mass-term (interacting leptons and quarks are spin 1/2-particles):

$$\tag 1 \bar{\Psi}\Psi = \bar{\Psi}\left(\frac{1 + \gamma_{5}}{2} + \frac{1 - \gamma_{5}}{2}\right)\Psi = \left| \bar{\Psi}\left( 1 \pm \gamma_{5} \right) = \left( (1 \mp \gamma_{5})\Psi\right)^{\dagger}\gamma_{0} \right| =$$ $$=\bar{\Psi}_{L}\Psi_{R} + \bar{\Psi}_{R}\Psi_{L}.$$ Then let's assume $SU(2)\otimes U(1)$ gauge-invariant, realistic theory (the electroweak part of the SM). According to this theory, the left representation $\Psi_{L}$ transforms as the doublet part under the gauge transformations, while $\Psi_{R}$ transforms as the singlet. So of course, the mass term isn't gauge invariant.

But if we assume only $U(1)$ gauge theory, there isn't doublets, so the mass term is indeed gauge invariant (except Majorana case, when $\Psi = \hat{C} \Psi$, where $\hat{C}$ refers to the charge conjugation).

This is the reason why we must include (gauge-invariant) interaction of the Yukawa-type with scalar doublets. For example, I will illustrate my statement by describing the mechanism of appearance of mass of charged leptons into the Standard model. We "replace" the mass term $(1)$ by

$$L_{int} = -G\bar{\Phi}_{L}\varphi \Psi_{R} + h.c.$$ Here $\varphi = \begin{pmatrix} \varphi_{1} & \varphi_{2} \end{pmatrix}^{T}$ refers to the doublet of the scalar complex field, and $\Phi_{L} = \begin{pmatrix} \nu_{L} & \Psi_{L}\end{pmatrix}^{T}$. After using unitary gauge (\varphi \to $\begin{pmatrix} 0 & \sigma \end{pmatrix}^{T}$) and shifting the vacuum ($\sigma \to \sigma + \eta$) we will give the mass term and interaction with Higgs boson:

$$L_{\int} = -G\eta (\bar{\Psi}_{L}\Psi_{R} + h.c.) - G\sigma (\bar{\Psi}_{L}\Psi_{R} + h.c.).$$ So we have the gauge-invariant mass term. But the payment for this is the appearance of Yukawa-interaction with massive real scalar field.