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I was reading a paper https://arxiv.org/abs/1707.03094 considering the SM extension with $SU(2)$ singlet and doublet fermions. After EWSB, the mass eigenstates are the linear combinations of gauge eigenstates. However, when I tried to diagonalize the mass matrix shown in Eq.7 of this paper, I found that eigenvalues of non-diagonal mass matrix contains negative values. Does it mean that the particle has negative mass? What does this mean?

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The sign of fermion mass is just a human convention.

Let's say you have a Dirac Lagrangian with positive mass $m$: $$ L = i\bar{\psi}\not{\partial}\psi - m\bar{\psi}\psi $$

If you redefine the fermion as $$ \psi' = i\gamma_5\psi $$ it can be easily verified that the Dirac Lagrangian turns into: $$ L = i\bar{\psi'}\not{\partial}\psi' + m\bar{\psi'}\psi' $$ The mass term changes sign, therefore fermion $\psi'$ has negative mass $-m$. Does it change any thing? No, the sign of fermion mass is just a human convention, which usually does not matter.

One caveat: one measurable effect of the sign/phase of the fermion mass is the phase of the strong CP violating topological term. But it is far removed from our usual daily concern.

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  • $\begingroup$ Very clear explanation, thanks a lot! $\endgroup$
    – Tsukishima
    May 26, 2023 at 20:00

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