Discrepancy between Dekens/Stoffer and Wilson

[tarendo]

When studying SMEFT influence on bs->mumu with @dvandyk we realized that the one-loop matching of certain four-fermion operators behaves differently in Wilson than in the Dekens/Stoffer mathematica code. Let us for example look at how the the SMEFT operator (in Warsaw Up basis) qq1 with indices (ii33) and (33ii) matches onto the WET operator (in JMS basis) VedLL with indices (2223).

According to Dekens/Stoffer the one-loop matching condition has the following form:

where stands for a constant factor that does not depend on the indices, is the CKM matrix and the index t stands for the top quark. So when setting for all i and when we sum over i=1,2,3 the whole contribution should cancel due to the unitarity of the CKM matrix.

We now repeat this exercise in Wilson where I understand that 'qq1_1133' is in fact qq1_1133+qq1_3311 (at least for real coefficients):

import wilson
wilson.Wilson.set_default_option('smeft_matching_order',1)
qq1_1133=wilson.Wilson({'qq1_1133':1},scale=91.1876,eft='SMEFT',basis='Warsaw up')
qq1_2233=wilson.Wilson({'qq1_2233':1},scale=91.1876,eft='SMEFT',basis='Warsaw up')
qq1_3333=wilson.Wilson({'qq1_3333':1},scale=91.1876,eft='SMEFT',basis='Warsaw up')

qq1_1133matched=qq1_1133.match_run(scale=91.1876,eft='WET',basis='JMS')['VedLL_2223']
qq1_2233matched=qq1_2233.match_run(scale=91.1876,eft='WET',basis='JMS')['VedLL_2223']
qq1_3333matched=qq1_3333.match_run(scale=91.1876,eft='WET',basis='JMS')['VedLL_2223']
print(qq1_1133matched)
print(qq1_2233matched)
print(qq1_3333matched,'\n')
print(qq1_1133matched+qq1_2233matched+qq1_3333matched)
print(qq1_1133matched+qq1_2233matched+qq1_3333matched/2.)

We see that here it is not actually the sum that vanishes but we have to devide qq1_3333matched by 2 in order for the contributions to cancel out.

[DavidMStraub]

Without looking at this in detail, I almost bet it is just a convention issue. Can you please look at appendix A.2 of https://arxiv.org/pdf/1810.07698.pdf and tell me if that solves your issue.

[dvandyk]

Dear @DavidMStraub, @peterstangl,

this is just to inform you that we are still looking into this. It's a bit puzzling to me. The appendix that David provided clears up that in matching and running you switch from the non-redundant basis to the redundant basis. This seems to be done in a 'symmetrize' function.

However, our problem appears to come from the operator with index 3333. In Dekens/Stoffer, this operators is seemingly contributing with a weight 2, since it appear as 'ii33 + 33ii', with i=3. In wilson it contributes with a weight 1, apparently. This is my hypothesis so far, and Sebastian and I will try to confirm it Tuesday.

[DavidMStraub]

The qq1_1133 and qq1_2233 coefficients differ by a factor 2 between the redundant and non-redundant conventions, while qq1_3333 doesn't, so it makes total sense.

You should start with

qq1_1133_nonredundant=wilson.Wilson({'qq1_1133':2},scale=91.1876,eft='SMEFT',basis='Warsaw up')
qq1_2233_nonredundant=wilson.Wilson({'qq1_2233':2},scale=91.1876,eft='SMEFT',basis='Warsaw up')
qq1_3333_nonredundant=wilson.Wilson({'qq1_3333':1},scale=91.1876,eft='SMEFT',basis='Warsaw up')

[dvandyk]

Thanks David, I had misunderstood the appendix. I had understood Wilson applied the symmetry factor internally before the matching (and was looking for it in the symmetrization functions), and removed it again after.

[DavidMStraub]

I had understood Wilson applied the symmetry factor internally before the matching

Yes, it is.

WCxf input: non-redundant basis. Running: redundant (symmetrized) basis WCxf output: non-redudant basis

My point was that the GIM cancellation mentioned by @tarendo holds in the redundant (symmetrized) basis, so to see it in Wilson, one has to start with 2-2-1 (in the non-redundant basis) rather than 1-1-1.

[dvandyk]

@DavidMStraub Thanks for your help, much appreciated!