Precision Calculations

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====== Andreas t-channel Simple Models ======

**Fawzi's analytical cross section/implementation for e+e- to nu_e nu_e. Thermal mass vs cut-off**
Write-up, see {{ :micromegas:tchannel:tchannelmain.pdf |here}}

**Andreas' checks in practice**
Write-up, see  {{ :uvfitests.pdf |}}

**Andreas' SM single-generation-no-QCD FeynRules implementation**
{{ :smforuvbehaviourfr.zip |}}

**Andreas' SM single-generation-no-QCD-zero-temperature CalcHEP implementation**
{{ :smforuvbehaviournothermal2.zip |}}

**Andreas' SM single-generation-no-QCD-T-dependent-vev CalcHEP implementation**
{{ :smforuvbehaviourt-dependentvev2.zip |}}

**Andreas' SM single-generation-no-QCD-zero-temperature CalcHEP implementation**
{{ :SMforUVbehaviourNoThermalNoZ2.zip |}}

**Andreas' SM single-generation-no-QCD-T-dependent-vev CalcHEP implementation**
{{ :SMforUVbehaviourNoZ2.zip |}}



I want to do two tests:

1) W boson t-channel exchange: Work in a simple model which consists of 
the SM with only one generation of leptons and nothing else (I might 
have to add a RH neutrino as well, for the moment I haven't reached this
point yet). The role of the dark matter will be played by the neutrino 
(gauge-charged, but I don't care if it's a viable freeze-in model, I'm 
interested in the general behaviour of the cross-section and that of the
yields). This model contains DM (identified as the SM neutrino) 
production through e+ e- -> nu nubar, which also proceeds through W 

//FB In your example, take nu_e//

//AG: Yes, I'll only be taking a single lepton generation, so there's only nu_e//

//FB Ok, I just wanted to make sure the t-channel W exchange was operative, otherwise only s-channel with the other nu's//

exchange. Then I want to:

- Reproduce the pathologic behaviour (large contributions from high 
temperatures).

- Figure out around which temperature the cross-section obtains an 
asymptotic value.

- Compute the relic with T-dependent masses (my idea is to do that by 
introducing a T-dependent vev) and no cut on the integration (I don't 
think this is doable with MO, so I'll have to do it externally by myself).

//FB For GI this may do as discussed but check that (since you are most probably going to give nu_e a Temp. dep. mass
as if the latter are in the bath ...what will you be assuming for v(T) the scale has to match the one introduced by the cut. Of course this should have no impact if the Yukawa of the neutrino =0 // 

//AG: I'm not sure I understand the comment. I think that I'll have to add a RH neutrino (otherwise the neutrino will be massless, at least from the point of view of a Higgs-induced mass) with a small Yukawa. For v(T), I'm sure people have already computed its expression in the SM and I'll use that. As for the temperature, I was thinking of just integrating the Boltzman equation up to T_RH and compute the yield. This is the point of the whole exercise: compare the cut method with a more "proper" calculation in which thermal masses are taken into account. Or am I misunderstanding your comment?// 


// FB: this point is not important after all. you can leave the neutrino massless and no need for a RH. We want a different mass for the W, this is the important mass which will have to run with T. Before integrating over the temp (for the relic) you need to produce a T dependent cross section. The T dependent X sec will come from MW(T), in your case (via v(T)). which is not the same as the induced thermal masses. Say the t-channel were a photon, then you would not have a mass to regularise with, through v(T). but what expression are you going to give for v(T)?. It would be wiser that you give no mass and check at this step that the cross section is  sensitive to the constant cross regime, or for what value of "TRH" which I call the integration limit parameter you are sensitive to the large cross section."//


- Compute the relic with T-independent masses and the cut Sasha has 
implemented, and compare the results.

2) Exchange of a fermion in the t-channel: Here I think that I can work 
with the 3rd model that we had in the micromegas paper and repeat the 
steps described previously (it has the advantage that several masses are
not vev-induced, so they can be changed independently).

My goal is to check the cut method in two different models, involving 
exchange of different particles in the t-channel.

//FB (a reminder) Of course the limitations of MO5 is that a model is not defined through the different stages of SB, ie SB phase, Symmetric phase,...the temperature implementation is just a "limit"of integration. Yes?//

//AG: Yes, this is one of the limitations, and this also applies to the previous model. I see three options: i) Just use the broken model ii) Just use the unbroken model iii) Write down two models, one broken and one unbroken, run the code with the unbroken model down to T_EWSB, get a result, continue with the broken model below T_EWSB, get another result and add the two (here it would also be interesting to compare the results obtained through the three options). In passing, I'm wondering if this is actually already doable with MO (of course with no T-dependent masses), one just needs to stop integrating at T_EWSB (or whatever SB temperature), then run the code with a different model below T_EWSB and (externally) add up the two results. If this is a correct procedure, then even using the current MO version one should be able to handle both phases.//

//FB let's discuss this later. You also have to be weary about how to calculate with calchep when you are in the Sym. phase//

//STATUS OF DISCUSSION ABOUT TCUT DEC. 20//

1- We all agree that we cannot do a complete thermal treatment in micromegas and that we are just trying to find a reasonable solution that allows to suppress the behaviour of t-channel diagrams based on the fact that we know that they should be strongly suppressed by thermal masses.

2- I propose to drop the topic of the case where we would have  many t-channel diagrams for the same process. It is clear that the FI module is not as universal as the FO case and since for all models considered so far this situation does not occur no point trying to find a solution for that. In the wiki the list of model included in micromegas is given so anyone can see that for themselves.

3- We still have to decide on what to do in practice, the following solutions are proposed
A) For processes that have a t-channel contribution, use tcut as defined in relation with the thermal mass. I think we all agree on how to implement tcut as well as how to implement the thermal mass - by default the SM thermal mass. Here what is not clear is how this will affect the s-channel contribution and/or interference. Andreas promised to do some more test on that point so let’s wait for his input.
B) For processes with only a s-channel contribution, we should not use a cut (I am not sure everyone agrees on this). There are two opinions, a) not to put tcut on processes without t-channel (is there a practical disadvantage in implementing this in micromegas?) b) to treat the s-channel resonance as a decay process which should take care of the cut issue since the decay is not subject to tCut. The added advantage of solution b (which could be implemented irrespective of tcut) is that the calculation is faster. The disadvantage is that there is some uncertainty on how to cut the resonance 3 widths- 5 widths etc… Moreover solution b) from Sasha proposes to use tcut on everything but the decay. This raises some issues as was illustrated in the test I have done with only s-channel. I do not know what to think of this test but I thought  it would  be good  to have a coherent answer to that  as it might explain some issues in the problem  about the interference s and t channel.
Precision Calculations

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