implement exact relations between (v,Mh2) and (m2,Lambda)

[peterstangl]

This PR implements the exact relations between the physical Higgs VEV and mass and the parameters of the Higgs potential.

@jasonaebischerGIT, maybe you can check the relations. What I did was:

• Take the Lagrangian and compute the minimum of the Higgs potential. This gives an exact expression for v2 in terms of Lambda, Cphi, and m2.
• Solve this expression for m2 to get an exact expression for m2 in terms of Lambda, Cphi, and v2
• Compute the physical Higgs mass Mh2 and eliminate m2 by using the exact expression for m2 from above to get an exact expression for Mh2 in terms of Lambda, Cphi, Ckin, and v2
• Solve this expression for Lambda to get an exact expression for Lambda in terms of Mh2, Cphi, Ckin, and v2

[coveralls]

Pull Request Test Coverage Report for Build 219

• 177 of 177 (100.0%) changed or added relevant lines in 2 files are covered.
• No unchanged relevant lines lost coverage.
• Overall coverage increased (+0.08%) to 89.979%

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Totals
Change from base Build 215:	0.08%
Covered Lines:	5648
Relevant Lines:	6277

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💛 - Coveralls

[peterstangl]

The check with Python 3.7 failes because:

ERROR: Failure: ImportError (this version of pandas is incompatible with numpy < 1.16.5
your numpy version is 1.16.4.
Please upgrade numpy to >= 1.16.5 to use this pandas version)

[JoseEliel]

I hope it is Ok that I chime in.
I do think that the solution in this PR is probably the best compromise for a general code, but I wanted to point something out.
When tackling this issue on my own I decided to use an improved numerical routine instead. The reason was that in some cases (at the moment I'm thinking about tiny lambda values), it might not be necessary that m2 and lambda give v = 246 GeV and Mh2 = 125 GeV spot-on (In particular, I think there is some freedom for v != 246 GeV as e.g. gauge boson masses depend on the Wilson coefficients as well). In that case, a numerical minimization restricted to some region of the lambda-m2 parameter space becomes interesting (another example is when people want to restrict their studies to negative lambda).

[peterstangl]

@JoseEliel thank you for your comment! I'm not sure if I understand completely, why this PR would be a problem for what you want to do.

First of all, at the moment, as far as I see, wilson is meant basically to run and match dimension six Wilson coefficients in SMEFT and WET. For the SMEFT running, the dimension <=4 SM parameters are required and wilson determines them automatically. But at the moment, the users are not meant to specify the dim <= 4 SM parameters themselves. In fact, there are interesting use cases where the user might want to specify all SM parameters at a high scale Lambda and use those for the running (see e.g. issue #38). In such a case, one would get predictions for the EW-scale SM parameters and one would have to compare these predictions to experimental data (where one can allow for a deviation of v = 246 GeV and Mh2 = 125 GeV). But something like that is not implemented in wilson at the moment. So all the functions modified by this PR are currently not meant to be used by the user directly.

This being said, I do not understand how the modifications of this PR would change anything about the use cases you describe. One can still use the updated m2Lambda_to_vMh2 function and do some numerical minimization with it just as before. As far as I see, this PR is completely backward compatible except for the removal of the now unnecessary _vMh2_to_m2Lambda_SM function.

(Concerning the freedom of v != 246 GeV, this does actually not depend on gauge boson masses if the Fermi constant GF is used as input parameter for fixing v (as currently done in wilson). And GF is measured with high precision, so there is not too much freedom. It is true that dim 6 Wilson coefficients, namely the coefficients Cll and Cphil3, modify the relationship between GF and v. This should be included in wilson, but at the moment is still marked as TODO.)

[peterstangl]

The check with Python 3.7 failes because:

ERROR: Failure: ImportError (this version of pandas is incompatible with numpy < 1.16.5
your numpy version is 1.16.4.
Please upgrade numpy to >= 1.16.5 to use this pandas version)

@jasonaebischerGIT, this problem is fixed by PR #60. Please merge #60 first.

[JoseEliel]

Thanks for the detailed reply. This is very helpful.

In such a case, one would get predictions for the EW-scale SM parameters and one would have to compare these predictions to experimental data (where one can allow for a deviation of v = 246 GeV and Mh2 = 125 GeV). But something like that is not implemented in wilson at the moment.

I think this is what I meant. I originally faced this issue when using smelli actually, and I was thinking about both codes when writing. Apologies for the confusion.

I have tried (though admittedly in a 'hacky' way) to implement something like what you mention with my numerical procedure. My implementation goes like this: When lambda and m2 are found by vMh2_to_m2Lambda, what is found are the "best possible" values numerically within a given region of the lambda-m2 parameter space (in my case, for example, lambda < some value).

I've been assuming that then those are the values used in the running (and checked that the smpars dictionary has stored the values found by the minimization). But this might be wrong though. I then just let Smelli do its thing and calculate likelihoods for EW precision tests and Higgs physics. I've gotten good likelihoods for points with a small lambda that would have come out with a larger lambda (but spot-on Higgs mass) otherwise.

I'm not sure all of that is consistent, and I would really appreciate your input. If you think this is beyond what we should discuss here, I would be happy (and thankful) if you want to continue the discussion by mail or a short call.

[peterstangl]

@JoseEliel I'm happy to discuss this further with you, I'm just not sure if our discussion fits here in this PR. I would suggest you open a new issue or we can also continue the discussion by email: stangl@itp.unibe.ch