Problem with matching of C_qe from Warsaw to JMS basis

[micha-a-schmidt]

There is a problem with the matching of the Wilson coefficient C_{qe}. When matching to weak effective theory, the mixing matrices for up and down-type quarks drop out, since (Qbar \gamma^\mu QL) (ebar \gamma\mu e_R) = (ubar \gamma^\mu uL) (ebar \gamma\mu e_R) + (dbar \gamma^\mu dL) (ebar \gamma\mu e_R) , but there are off-diagonal entries in the result of Wilson. See Matching_Cqe_Warsaw_JMS.pdf for a simple example.

Starting with (q_2 q_2) with WC = 1, the (u_1 u1) operator is generated with a coefficient ~ V{us} V^_{us} ~ 0.05. Similarly for (q_3 q_3), Wilson returns (u_1 u1) ~ V{ub} V^_{ub} ~ 1.3e-5, and (u_2 u2) ~ V{cb} V^*_{cb} ~ 1.7e-3.

[DavidMStraub]

Can you try with Warsaw up as input basis?

[micha-a-schmidt]

Thanks for the quick reply! There are also off-diagonal entries. This time for down-type quarks. Matching_Cqe_Warsaw_up_JMS.pdf.

[micha-a-schmidt]

The result in Wilson is correct. The CKM matrix only drops out if the SMEFT Wilson coefficient is proportional to the identity in the quark flavours.