CP violation from the Electroweak SU(2)$_{weak,flavor}$ by $\int \theta F \wedge F$

Question: Why there is NO Charge-Parity (CP) violation from a potential Theta term in the electroweak SU(2)$_{weak,flavor}$ sector by $\theta_{electroweak} \int F \wedge F$?

(ps. an explicit calculation is required.)

Background:

We know for a non-Abelian gauge theory, the $F \wedge F$ term is nontrivial and breaks $CP$ symmetry (thus break $T$ symmetry by $CPT$ theorem), which is this term: $$\int F \wedge F$$ with a field strength $F=dA+A\wedge A$.

$\bullet$ SU(3)$_{strong,color}$ QCD:

To describe strong interactions of gluons (which couple quarks), we use QCD with gauge fields of non-Abelian SU(3)$_{color}$ symmetry. This extra term in the QCD Lagrangian: $$\theta_{QCD} \int G \wedge G =\theta_{QCD} \int d^4x G_{\mu\nu}^a \wedge \tilde{G}^{\mu\nu,a}$$ which any nonzero $\theta_{QCD}$ breaks $CP$ symmetry. (p.s. and there we have the strong CP problem).

$\bullet$ Compare the strong interactions $\theta_{QCD,strong}$ to U(1)$_{em}$ $\theta_{QED}$: For U(1) electromagnetism, even if we have $\theta_{QED} \int F \wedge F$, we can rotate this term and absorb this into the fermion (which couple to U(1)$_{em}$) masses(?). For SU(3) QCD, unlike U(1) electromagnetism, if the quarks are not massless, this term of $\theta_{QCD}$ cannot be rotated away(?) as a trivial $\theta_{QCD}=0$.

$\bullet$ SU(2)$_{weak,flavor}$ electro-weak:

To describe electroweak interactions, we again have gauge fields of non-Abelian SU(2)$_{weak,flavor}$symmetry. Potentially this extra term in the electroweak Lagrangian can break $CP$ symmetry (thus break $T$ symmetry by $CPT$ theorem): $$\theta_{electroweak} \int F \wedge F =\theta_{electroweak} \int d^4x F_{\mu\nu}^a \wedge \tilde{F}^{\mu\nu,a}$$ here the three components gauge fields $A$ under SU(2) are: ($W^{1}$,$W^{2}$,$W^{3}$) or ($W^{+}$,$W^{-}$,$Z^{0}$) of W and Z bosons.

Question [again as the beginning]: We have only heard of CKM matrix in the weak SU(2) sector to break $CP$ symmetry. Why there is NO CP violation from a potential Theta term of an electroweak SU(2)$_{weak,flavor}$ sector $\theta_{electroweak} \int F \wedge F$? Hint: In other words, how should we rotate the $\theta_{electroweak}$ to be trivial $\theta_{electroweak}=0$? ps. I foresee a reason already, but I wish an explicit calculation is carried out. Thanks a lot!

• See section 2 of arxiv.org/abs/hep-ph/9305271 – Mitchell Porter Dec 28 '13 at 6:35
• @ Mitchell, thanks for the comments, let me take a look. – wonderich Dec 28 '13 at 20:07
• @ Mitchell, it will also be nice if you can summarize your/their viewpoints. – wonderich Dec 28 '13 at 20:16

• "The CP-odd term can be removed by a vector baryonic transformation, under which the Dirac masses of the quarks are invariant." I thought CP odd term can only be removed by the chiral or axial transformation (for Dirac notations, we need to have $\gamma^5$ involved)... Could you confirm please? Who misunderstood? – annie marie heart Dec 28 '20 at 0:37