Problem with Signs @ RGEs for Cuartic Triplet Couplings

[Nicolas]

We have found a problem while implementing a model in SARAH, we hope you could help us to deal with it

We are implementing the non supersymmetric version of Type II seesaw on SARAH. We have finished the writing of the model, the particle, and the parameter files but despite the implementation runs correctly (after a primary execution in mathematica), we have found that there are some misleading signs in the outcome for the RGEs, after comparing with the literature.

In particular. we defined the contractions for the triplet quartic couplings in the scalar potential (which couple the triplet only, not the Higgs), and after computing the RGEs we found that the numerical factors are correct but signs after comparing with different sources (for instance http://arxiv.org/pdf/1508.02323.pdf although you can compare with you favorite source for T-II-SS). Those signs affect just the couplings before mentioned. Moreover, regardless if you change one those quartics by 'minus' the same coupling in the Lagrangian, you will always get the same RGE with the same problem with the signs. On the contrary, if one change the signs of the SM quartic coupling lambda, we will see that the calculation of the RGE changes accordingly with this convention. We long thought this issue was a matter of the contractions we had to define inside our SARAH model, but even though we have changed them, it seems not to work.

As we said above, your help will be very valuable for us.
Best regards.

PS:

• - SARAH version: 4.8.5
  - Mathematica version: 9.0
  - The model file we have coded is provided in the tarball in the attachments.
  - We also provided the .pdf we generated for this model in order to do a better comparison with the referenced articles.

[FStaub]

Hi,

the origin of this problem is a bit tricky:
SARAH calculates the RGEs for the unbroken gauge groups. Therefore, it assumes that non-fundamental irreps not written as tensor product as this is the case for broken groups, but of vectors of appropriate length. Because of that, it has to re-calculate the contractions internally, and makes use of routines of Susyno of that. However, Susyno returns the contractions with an arbitrary phase which can even change from Mathematica version to Mathematica version, i.e. this can sometimes by rather randomly. If you encounter such a problem, it's best to make use of the option to fix the contractions in the model. That's done via

Code: Select all

ContractionRGE[COUPLING]=CONTRACTION

where COUPLING is the name of the coupling for which you want to 'hard-code' the contraction and CONTRACTION is the given contraction. I guess, it should be possible to define the correct contractions using Kronecker Deltas (see for instance SM+Triplets/Real) for an example. However, what I just did was:
1) I run your file, and extracted the contractions which are used by SARAH via

Code: Select all

?SA`SSSSlist4
?InvMat

In the first line, I check how the invariant matrices are called for the necessary couplings, in the second line I get the expressions.
2) I copied the contractions just to your model and added an additional sign:

Code: Select all

ContractionRGE[lambdaD]=InvMat[100][lef1,lef2,lef3, lef4];
ContractionRGE[lambDP]=InvMat[200][lef1,lef2,lef3, lef4];

InvMat[100][a___Integer]:=-{{{{...}}}[[a]];
InvMat[200][a___Integer]:=-{{{{...}}}}[[a]];

where the dots represent a long list of numbers which I copied. I attach the file, and as first glance it seems that signs are now correct .

[Nicolas]

Hi once more!

the origin of this problem is a bit tricky:
SARAH calculates the RGEs for the unbroken gauge groups. Therefore, it assumes that non-fundamental irreps not written as tensor product as this is the case for broken groups, but of vectors of appropriate length. Because of that, it has to re-calculate the contractions internally, and makes use of routines of Susyno of that. However, Susyno returns the contractions with an arbitrary phase which can even change from Mathematica version to Mathematica version, i.e. this can sometimes by rather randomly. If you encounter such a problem, it's best to make use of the option to fix the contractions in the model. That's done via

Code: Select all

ContractionRGE[COUPLING]=CONTRACTION

So i understood that the conversion amongst "matrices" and "vectors" for the irreps one uses in the model makes use of conversion of the contractions that might carry some signs/phases that Mathematica does not know how to deal with. That messed up all the outcome. Didn't I? On the other hand, well, unfortunately, we did use of that option and it didn't work either. We saw that option included on the model file for SM+Triplet, if i remember well.

In the first line, I check how the invariant matrices are called for the necessary couplings, in the second line I get the expressions.
2) I copied the contractions just to your model and added an additional sign:

Code: Select all

ContractionRGE[lambdaD]=InvMat[100][lef1,lef2,lef3, lef4];
ContractionRGE[lambDP]=InvMat[200][lef1,lef2,lef3, lef4];

InvMat[100][a___Integer]:=-{{{{...}}}[[a]];
InvMat[200][a___Integer]:=-{{{{...}}}}[[a]];

where the dots represent a long list of numbers which I copied. I attach the file, and as first glance it seems that signs are now correct .

Ok. Since the convention of mathematica is a little bit arbitrary, one has to implement the solution by hand with the signs. Good!! I will test it and afterwards i will tell you how it worked in my machine with my mathematica version (I suppose it will work correctly as you said, but it is better to test it first).

Cheers and Thanks for the answer!!