implementing extended SUSY breaking terms

[gilbertmoultaka]

Hi Florian,

we have an NMSSM like model (with at least two extra singlets) but, more importantly, with hard SUSY breaking terms on top of the usual soft ones.
We wonder what is the best way to implement this model:
1) if we implement it as an MSSM extension with two singlets, giving the superpotential, etc., how can we add extra (had breaking) terms that are not generated automatically by SARAH, AND have access to the corresponding RGEs.
2) If we implement the model as a non-susy model, can we avoid the awsome task of coding the full lagrangian, by using the SARAH generated largrangian of the susy case to build in the supersymmetric and soft breaking parts, and then add the missing terms, thus obtaining a non-susy
model with the proper RGEs.

Thanks a lot for your help,

Gilbert & Michel

[FStaub]

Dear Gilbert and Michel,

I need to think a bit further about that, but right now I worry that I don't have good news:
1) The SUSY RGEs make, of course, explicitly use of SUSY properties. For instance that quartic couplings are given by D- and F-terms and that superpotential terms only receive wave-function contribution. Both might be spoilt by hard-breaking terms
2) I have doubts that this will work in practice. Even if one finds a way to export the Lagrangian from SARAH (might be possible), you need to drop all SUSY relations among parameters. Thus, each quartic coupling and Yukawa-like interaction will be an independent parameter. See for instance 0712.2858 for a related discussion in the MSSM which, however, doesn't include the quartics because it's hardly feasible.
The computational difficulty grows quickly with the number of quartic interaction. Already in models with 10 quartics, I sometimes get problems with the memory. Your model might have 50 quartics or even more.

Sorry that I don't have more positive things to say...

Florian

[gilbertmoultaka]

Dear Florian,

Thanks a lot for your prompt reply. We agree with point 1). Concerning point 2) maybe we were not clear enough.
Actually our model is a low energy limit of a supergravity theory where spontaneous susy breaking generates
soft and hard breaking; This is unusual and due to new forms of the superpotential and Kahler potential.
So the susy relations among the parameters are preserved. In that sense we would be satisfied if we could export
the Lagrangian from SARAH. Then use the non-SUSY RGEs but with the susy relations when they apply; for
instance quartic couplings from D- and F-terms are the usual ones, plus new quartic contributions.
We thought the coded non-SUSY RGEs would perfectly fit to this case.
Or are we perhaps still missing something in your argument?

Thanks again for your help!
Cheers,
Gilbert & Michel

[FStaub]

Hi,

I still see two problems:
1) from the technical point of view you need still to write down the full potential including all quartic terms. Independent of the question if they are now related or not, all possible loop diagrams need to be generated. That will be a huge number. I'm not sure of Mathematica could handle that.
2) Honestly, I'm not convinced that Susy relations survive the RGE running. For instance, consider the Higgs-fermion and Higgsino-fermion-sfermion coupling. Those are identical due to Susy and also the beta functions are the same. If you do the calculation in a non-Susy fashion you will find that this is only the case because of subtle cancellations between contributions from quartics as well as from gauginos/higgsinos. Of course, the reason is that these couplings are related by F- and D-terms.
If you now modify the quartics, the chances are high that the running of these two couplings (and many others) is no longer the same because you spoil these cancellations, or?
In order to check that, it might be a good start first with a toy model to see how the non-Susy rges behave once you add your hard breaking. I have set up such a model some time. If you want to try, I can give you the files.

Cheers
Florian

[gilbertmoultaka]

Hi Florian,

1) in our model all the hard breaking terms involve necessarily the two singlet fields. So they do not lead to the most generic quartic terms, even though higgs, squarks, sleptons fields are present too in these terms.
So perhaps the number of loop diagrams remains relatively moderate(?) In any case, one could perhaps proceed
partially "by hand" by identifying the types of diagrams with SARAH and adding all the contributions with
mathematica outside SARAH.
We should try with a toy model first. If you have one we are interested.
Moreover, if you think we can export the SARAH generated Lagrangian of an N2MSSM,
we will have only to add the hard terms which we have already generated in mathematica.

2) yes we agree. we probably misunderstood your initial comment about having to modify the susy relations.
what we meant is to start from the tree-level susy relations + new hard terms and we get what we get from the
non-SUSY RGE. If we get deviations in the runnings we can always trace them back to the contributions of
the singlets running in the loops.

To summarize, we are interested in

a) trying your toy model

b) forgetting about the hard terms in a first step, trying the exercise of generating the N2MSSM model as a
non-susy model by exporting the SARAH generated Lagrangian of a (susy) N2MSSM,
if this is technically feasible.

cheers,
Gilbert & Michel

[FStaub]

Hi,

1) I attach the toy model which is the non-SUSY version of the model

Code: Select all

W=\lambda S H_u H_d + \mu H_u H_d

with hypercharge only. You can see that already the non-SUSY RGEs for this toy model turn out to be very lengthy.

2) Unfortunately, I was too optimistic. While looking for a way how to export the Lagrangian, I recognised that I decompose SU(2) before generating the Lagrangian from the Superpotential, ie. I don't have the Lagrangian in terms of the SU(3)xSU(2)xU(1) representations. Anyways, there is some code snippet which shows a possibility how to get the different terms in the Lagrangian. Maybe, it's still of some help for you:

Code: Select all

Start["MSSM"]

(* Routine to generate independent names of all couplings *)
cF = 0;
cS = 0;
cY = 0;
cA = 0;
cL = 0;
MakeCoup[in_] := Block[{},
   Switch[Length[in],
     2,
     If[getType[in[[1]]] === F,
       cF++; Return[ToExpression["mF" <> ToString[cF]]];,
       cS++; Return[ToExpression["mS" <> ToString[cS]]];
       ];,
     3,
     If[getType[in[[1]]] === F || getType[in[[2]]] === F,
       cY++; Return[ToExpression["Y" <> ToString[cY]]];,
       cA++; Return[ToExpression["A" <> ToString[cA]]];
       ];,
     4,
     cL++; Return[ToExpression["L" <> ToString[cL]]];
     ];
   ];
   
Expand[LagFFS[GaugeES]] /. sum[a__] -> 1 /. Delta[a__] -> 1 /. 
          x_?(FreeQ[parameters, #] == False &) -> 1 /. Sig[__] -> 1 /. 
        Power[a_, 2] -> a.a /. Lam[__] -> 1 /. 1[___] -> 1 //. 
     A_[1] -> A /. Power[a_, 2] -> a.a //. x_?NumericQ -> 1 /. 
  Times -> Dot /. a_Dot :> MakeCoup[a] a   
 

The output looks like

Code: Select all

mF1 fB.fB + mF2 fG.fG + mF3 FHd0.FHu0 + mF4 FHdm.FHup + mF5 fWB.fWB + 
 mS1 SdL.conj[SdL] + mS2 SdR.conj[SdR] + mS3 SeL.conj[SeL] + 
 mS4 SeR.conj[SeR] + mS5 SHd0.SHu0 + mS6 SHd0.conj[SHd0] + 
 mS7 SHdm.SHup + mS8 SHdm.conj[SHdm] + mS9 SHu0.conj[SHu0] + 
 mS10 SHup.conj[SHup] + mS11 SuL.conj[SuL] + mS12 SuR.conj[SuR] + 
 mS13 SvL.conj[SvL] + mF6 conj[fB].conj[fB] + mF7 conj[fG].conj[fG] + 
 mF8 conj[FHd0].conj[FHu0] + mF9 conj[FHdm].conj[FHup] + 
 mF10 conj[fWB].conj[fWB] + mS14 conj[SHd0].conj[SHu0] + 
 mS15 conj[SHdm].conj[SHup] + Y1 fB.FdL.conj[SdL] + 
 Y2 fB.FeL.conj[SeL] + Y3 fB.FHd0.conj[SHd0] + Y4 fB.FHdm.conj[SHdm] +
  Y5 fB.FHu0.conj[SHu0] + Y6 fB.FHup.conj[SHup] + 
 Y7 fB.FuL.conj[SuL] + Y8 fB.FvL.conj[SvL] .
... +L77 SuR.SvL.conj[SuR].conj[SvL] + L78 SvL.SvL.conj[SvL].conj[SvL]

Note, I haven't check in much detail that the result is totally correct. So, please use it careful.

Cheers,
Florian

[gilbertmoultaka]

Hi Florian,

Thanks a lot!
We'll play with all this and let you know.

Cheers,
Gilbert & Michel