Is the expression for CP asymmetry wrong in Wikipedia?

In the wikipedia page about neutrino oscillation (or ate least in the current revision), the CP-asymmetry is written as $$A^{(\alpha\beta)}_{\text{CP}}=P(\nu_\alpha \rightarrow\nu_\beta)-P(\bar{\nu}_{\alpha}\rightarrow\bar{\nu}_{\beta}) =4\sum_{i>j}\mathrm{Im}\big(U^*_{\alpha i} U_{\beta i}U_{\alpha j} U^*_{\beta j}\big) \sin\Big(\tfrac{\Delta m^2_{ij}L}{2E}\Big)\tag{1}$$ where where $U$ is the neutrino mixing matrix with elements $U_{\alpha i}$. Here, $\alpha$ labels neutrino flavours ($e,\mu$ or $\tau$) and $i$ labels neutrino mass eigenstates such that $$|\nu_\alpha\rangle=\sum\limits_{i=1}^{3}U^*_{\alpha i}|\nu_i\rangle.$$

Now, the probability of oscillation from neutrino flavour $\alpha$ to flavour $\beta$ is given by\begin{aligned}P_{\alpha\beta}=P(\nu_\alpha \rightarrow\nu_\beta)&=\delta_{\alpha\beta}- 4\sum\limits_{i>j} {\text{Re}}(U_{\alpha i}^{*} U_{\beta i} U_{\alpha j} U_{\beta j}^{*}) \sin^2(\Delta m_{ij}^2 L/4E)\\&+ 2\sum_{i>j}{\text{Im}}(U_{\alpha i}^{*}U_{\beta i}U_{\alpha j} U_{\beta j}^{*}) \sin(2\Delta m_{ij}^2 L/4E ),\end{aligned}\tag{2} And the probability of oscillation from an antineutrino flavour $\alpha$ to flavour $\beta$ is given by \begin{aligned}P(\bar{\nu}_{\alpha}\rightarrow\bar{\nu}_{\beta})&=P_{\alpha\beta}(U_{\alpha i}\to U^*_{\alpha i})\\&= \delta_{\alpha\beta}- 4\sum\limits_{i>j} {\text{Re}}(U_{\alpha i} U^*_{\beta i} U^*_{\alpha j} U_{\beta j}) \sin^2(\Delta m_{ij}^2 L/4E) \\&+ 2\sum_{i>j}{\text{Im}}(U_{\alpha i}U^*_{\beta i}U^*_{\alpha j} U_{\beta j}) \sin(2\Delta m_{ij}^2 L/4E),\end{aligned}\tag{3}. Simple subtraction of (3) from (2), does not lead to Eq. (1). How is the formula Eq.(1) derived?

• In WP the formula for (2) given is quite consistent with (1). What makes your think they are not consistent? Are you hiding a bogus discrepancy in your undefined $\Delta_{ij}$? Are you flipping the sign of the imaginary part of the conjugate argument? – Cosmas Zachos Oct 29 '17 at 0:46
• @CosmasZachos As fas fas the notation $\Delta_{ij}$ is concerned, I have corrected it. I don't understand your other confusions related to the question. I just ask subtracting (3) from (2), does not lead to the expression (1) in the question. This expression is quoted from Wikipedia. – SRS Oct 30 '17 at 12:43
• In eqn (3), the argument of Re and Im is the complex conjugate of the argument of Re and Im in eqn (2), isn't it? – user154997 Oct 30 '17 at 13:17
• WP is right. Unless you show your work, (2)-(3) and what confuses you, one cannot figure out your sticking point. – Cosmas Zachos Oct 30 '17 at 15:57
• @LucJ.Bourhis Oh yes! Thanks . The real parts cancel, and the imaginary parts add up. – SRS Oct 30 '17 at 20:41

1 Answer

CP-asymmetry occurs due to phase $\delta$ (I would call this cp-violating phase $\delta$), which makes the elements of mixing matrix complex because of the presence of $e^{-I \delta}$ term. When we deal with anti-neutrinos, the sign of this phase $\delta$ gets changed (as you have done by using the complex conjugation in Eq.(3)). Hence, the $\Re$ part remains the same as is in Eq.(2) but the $\Im$ part obtains the same value as that in Eq.(2) with a (-) sign. This makes Eq.(1) correct, in my opinion.