# Fermions in Euclidean vs Lorentzian Signature

We know that in Lorentzian signature, fermions are representations of $$$$Spin(3,1)\cong SL(2,\mathbb{C})\cong SU(2)\times SU(2)^*$$$$ where crucially left/right handed fermions are exchanged under complex conjugation. This is what one means by the $$SU(2)^*$$ above. From an operational perspective, we write undotted and dotted indices, $$\psi^\alpha$$ and $$\bar{\chi}_{\dot{\alpha}}$$ to keep track of the left/right handed Weyl spinors. The statement that left/right handed spinors are related by conjugation is then captured by $$(\psi_\alpha)^\dagger=\bar{\psi}_{\dot{\alpha}}$$. An important consequence here is that one can’t write a $$U(1)$$ invariant mass term such as $$\psi \bar{\psi}$$.

Now if we move to Euclidean signature, fermions transform under $$$$Spin(4)\cong SU(2)\times SU(2)$$$$ and this time we have two independent copies of $$SU(2)$$. This means that left/right handed spinors now do not transform into each other under conjugation. This leads me to the following two questions:

1. Why can I not write a $$U(1)$$ invariant mass term $$\psi \psi^\dagger$$ in Euclidean signature? This to me seems fine as it is just the contraction of a fundamental/anti-fundamental $$SU(2)$$ index?

2. How does this conjugation difference between the two signatures get resolved under Wick rotation? Are there any issues with degrees of freedom on the two sides?