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Yes it must be zero. The topological susceptibility is an observable and in the case of $SU_L(2)$ interactions, the chiral rotations can "absorb" the theta as you mentioned. Therefore it cannot possibly be something physical.

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I don't know where that "we" came from, but the SM "unification" is partial, more of a Weinberg-angle tilt: It involves EM and WI inextricably linked and mixed. The different masses of W and the Z mirror the two couplings. For vanishing Weinberg angle, there would be no mixing and e=g'. The question then is just a matter of terminology, ...

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First order phase transitions are characterized by discontinuous first derivatives of the free energy. Specifically, the derivative of Gibbs free energy with respect to pressure determines the density of the gas ($\Delta N/\Delta V$) and this may be discontinuous. The Early Universe was nearly but not quite homogeneous, and near the phase transition ...

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