Some simple questions regarding leptonic masses in the Standard Model (SM):

  1. Why there is not an explicit mass term in addition to the effective mass term that arises from the Yukawa terms after spontaneous symmetry breaking (SSB).

  2. The lepton mass term is: $$m_{l_i}\sim y_i\cdot v \sim y_i\cdot \frac{m_H}{\sqrt{\lambda}}$$ with $m_{l_i}$ the lepton mass, $y_i$ the Yukawa coupling, $v$ the vacuum expectation value of the Higgs field, $m_H$ the Higgs mass, and $\lambda$ the self-coupling of the Higgs.
    So the running of leptonic masses in the SM are given by the running of $y_i$, $m_H$, and $\lambda$. Thus the lepton masses are not protected to radiative corrections $\beta(m_{l_i})\neq 0$ for $m_{l_i} = 0$. This is in contrast with explicit lepton mass terms in theories without SSB where Chiral symmetry protect fermions' masses from radiative corrections. Is this correct? If not, why?


2 Answers 2

  1. An explicit mass term violates Gauge invariance, because left and right particles belong to different representations.

  2. At one loop, the lepton mass is given by

$m_{1L} = M_{bare} + \Delta M_{1L}(\mu = m_{1L})$.

This condition uniquely defines the bare mass.

The correction $\Delta M(\mu)$ is proportional to some power of the yukawa, in any case, and thus is very small.


Let me answer the first question. There is not any explicit mass term which be a singlet under $SU(3)\times SU(2)\times U(1)$ transformations.


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