Wilson Coefficient Values for b->s l l in the Standard Model: What Do We Know?

[effrit]

TL;DR

Wilson Coefficient values

Hi, does anyone happen to know a paper somewhere with a list of the standard model wilson coefficients for effective field theory?

I see the same basis everywhere and I can find the new physics corrections, but nowhere seems to have an actual number for the standard model value? Specifically this is C9 and C10, the two operators for b>s l l

 

[Orodruin]

$b\to s\ell\ell$ should be GIM-suppressed in the standard model.

 

[effrit]

$b\to s\ell\ell$ should be GIM-suppressed in the standard model.

So this decay should only be able to happen via the penguin feynman diagram right, but isn't the point of the effective field theory to replace the loop diagrams with a single vertex, which can be represented using a wilson coefficient?
Surely this coefficient has a specific value at the ~10s of GeV scale? Because clearly this decay does occour in the standard model

 

[Orodruin]

Right, that is a charged current with a top loop so no GIM.

At 10s of GeV you are not really in the EFT regime for the SM itself since you are getting close to the masses of the SM particles you would integrate out to get the effective operators.

When it comes to new physics, the associated scales are typically higher. The point of effective field theory is to integrate out high energy degrees of freedom from your theory and to have higher dimensional operators suppressed by the heavy scale.

 

[Reggid]

I see the same basis everywhere and I can find the new physics corrections, but nowhere seems to have an actual number for the standard model value?

Aren't they all zero by construction?

The Wilson coefficients of an effective theory are calculated as the matching coefficients between the full (high energy) theory and the effective (low energy) theory at a given matching scale.

So to calculate the Wilson coefficients to a given order in perturbation theory you calculate all the relevant loop diagrams in the full theory (i.e. light degrees of freedom + heavy degrees of freedom) and in the effective theory, (i.e. only light degrees of freedom) and fix the Wilson coefficient by the condition that at some matching scale the results have to agree.

In case of SMEFT the standard model + higher dimensional operators of standard model fields ist the low energy effective theory, of an unknown high energy theory with some unknown heavy degrees of freedom (new heavy particles or whatever). So when you calculate the Wilson coefficients the pure standard model diagrams will exactly cancel out because they contribute both in the high energy theory above the matching scale and in the low energy theory below the matching scale.

Assuming there is really no new physics beyond the standard model, you would just match two identical theories (the standard model above and below the matching scale) onto each other, which leads to the Wilson coefficients of the higher dimensional operators all being zero.

What you seem to have in mind, that you shrink the standard model diagrams to a point interaction, would mean that you are integrating out heavy standard model particles. But that is not the point in SMEFT. You do that for example in Fermi theory of weak interaction when integrating out the W-boson, or when you introduce an effective gluon-Higgs coupling when integrating out the top loop, but in SMEFT the full standard model IS the low energy effective theory, and you integrate out only new physics.

 

[Orodruin]

What you seem to have in mind, that you shrink the standard model diagrams to a point interaction, would mean that you are integrating out heavy standard model particles.

So this was kind of the point in my previous post. At the energies mentioned by OP, no standard model fields cannot really be considered to be heavy and integrated out of the theory. Of course, if you go to lower energy processes, such as beta decays, you can comfortably integrate out the electroweak gauge bosons and be left with a 4-fermi operator essentially resulting in Fermi theory.

That the Wilson coefficients of those operators are zero in the standard model of course also does not mean that the process does not occur in the standard model as there are diagrams involving the SM fields that contribute.

 

[effrit]

So essentially the issue is that loop diagrams are too computationally expensive for me to compute, which is why I'm using effective field theory to stand in for the b > s l l vertex at low enough energy while calculating everything at 1st order. Is this a reasonable thing to do?

And is there somewhere I can find what coefficients I should use for this vertex, and what energy range they would actually be valid for?

 

[Orodruin]

What do you assume is ”low enough” energy here?

 

[effrit]

I found this presentation (http://v17flavour.in2p3.fr/MondayMorning/Jager.pdf) which uses 4 for C9 and -4 for C10 at 4.6 GeV on page 8, which is about the region I'm interested in. It's very vague though, so I think just a more precise version of this figure is exactly what I'm looking for

 

[ohwilleke]

Specifically this is C9 and C10, the two operators for b>s l l

The papers I've seen state that three Wilson coefficients are implicated in b -> s l l. They are C7, C9 and C10.

The master's thesis at http://essay.utwente.nl/80992/1/Master_Thesis_BAKortmanv1.0.pdf has a pretty good discussion of the question, but doesn't have a nice clean simple table reciting what you are looking for.

A discussion more focused on the decay you are looking for can be found at https://www.slac.stanford.edu/pubs/slacpubs/7500/slac-pub-7702.pdf But this also doesn't give you a straight answer, although it, in turn, relies heavily on the following references for its SM baseline: J.L. Hewett, Phys. Rev. D53, 4964 (1996); J.L. Hewett and J. Wells, Phys. Rev. D55, 55 (1997); See also A. Ali, G.F. Giudice and T. Mannel, Z. Phys. C67, 417 (1995).

Values are listed at page 15 of this power point presentation: https://avvicente.files.wordpress.com/2018/07/fpcp18_lecture2.pdf but the reference source for those numbers in the power point could be more clearly indicated. The values are as follows:

This further implies Chiral combinations of C9 and C10 of CL=4.2 and CR=-0.1.

The relevant excerpts from the Jager power point presentation at page 8, cited at #9 in this thread, is:

This is also less clear than it might be about the authority it is relying upon for these numbers.

Q=m_b (ca. 4.2 GeV) of Avvincente would seem to make more sense than Q=4.6 GeV in Jager for purposes of these decays. The Avvincente numbers are also at least to two significant digits instead of the one in Jager.