• Color summation from 4-scalar vertex

Hi Florian,

I have a question concerning a specific vertex with 4 colored scalar fields. Tomas Gonzalo created a SARAH model by extending the SM model with the new field

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ScalarFields[[2]] = {s1, 1, s11, 0, 1, 3};

The interesting part of the new Lagrangian reads

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Lags1 = - 1/2 ls1 conj[s1].s1.conj[s1].s1;

When I start this model in SARAH and ask for the 4-scalar vertex I get

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In[]:= Vertex[{conj[s11],s11,conj[s11],s11}] // InputForm                     

Out[]//InputForm= 
{{conj[s11[{ct1}]], s11[{ct2}], conj[s11[{ct3}]], s11[{ct4}]}, 
 {(-I/2)*ls1*(CG[SU[3], {{0, 1}, {1, 0}, {0, 1}, {1, 0}}][ct1, ct2, ct3, ct4] + 
    CG[SU[3], {{0, 1}, {1, 0}, {0, 1}, {1, 0}}][ct1, ct4, ct3, ct2] + 
    CG[SU[3], {{0, 1}, {1, 0}, {0, 1}, {1, 0}}][ct3, ct2, ct1, ct4] + 
    CG[SU[3], {{0, 1}, {1, 0}, {0, 1}, {1, 0}}][ct3, ct4, ct1, ct2]), 1}}

You said in another thread, that these CGs are a just a product of two Kronecker deltas.
So, this vertex would be equivalent to

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In[]:= Vertex[{conj[s11],s11,conj[s11],s11}] /.
  CG[SU[3], {{0, 1}, {1, 0}, {0, 1}, {1, 0}}][ct1_, ct2_, ct3_, ct4_] :>
  Delta[ct1,ct2] Delta[ct3,ct4] // Simplify // InputForm

Out[]//InputForm=
{{conj[s11[{ct1}]], s11[{ct2}], conj[s11[{ct3}]], s11[{ct4}]}, 
 {(-I)*ls1*(Delta[ct1, ct4]*Delta[ct2, ct3] + Delta[ct1, ct2]*Delta[ct3, ct4]), 
  1}}

Now, when I run MakeSPheno[], SARAH generates the following vertices in Fortran (I've left out parts of the routine for brevity here):

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Subroutine Couplings11s11cs11cs11T(ls1,res)
res = res-2._dp*(ls1)
End Subroutine Couplings11s11cs11cs11T

Subroutine Couplings11s11cs11cs11L(ls1,res)
res = res-6._dp*(ls1)
End Subroutine Couplings11s11cs11cs11L

I'm assuming SARAH/SPheno uses the latter of these two vertices to calculate the 1-loop contribution to s1. There are 2 diagrams that contribute to the 1-loop self energy: One with a gluon and one with another s1 field. The latter is a tadpole-like diagram where the 4-scalar s1 vertex appears. So, the contribution from this diagram would be in SARAH/SPheno (-6*ls1*A0[ms1^2]).

However, this final expression (-6*ls1*A0[ms1^2]) for the self energy contribution looks suspicious to me. As far as I can see, there are 4 possible contractions of external and internal fields for this diagram: 2 of them have a structure where there is a sum over the internal colors, so their overall contribution is 2*3 = 6. The other 2 contractions have no color sum, because in the sum appears delta_{ik}, so the contribution from these 2 diagrams is 2. So, in total I would expect that the self energy contribution has a factor (6 + 2)/2 = 4 as prefactor. (The factor 1/2 comes from the Lagrangian.) So, I would assume the contribution to the self energy is (-4*ls1*A0[ms1^2]). My question is, why is the SARAH/SPheno vertex 6?

Btw, when I explicitly add the Deltas to the Lagrangian as

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Lags1 = - 1/2 Delta[col1,col2] Delta[col3,col4] ls1 conj[s1].s1.conj[s1].s1;

I get immediately

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In[]:= Vertex[{conj[s11],s11,conj[s11],s11}] // InputForm                     

Out[]//InputForm= 
{{conj[s11[{ct1}]], s11[{ct2}], conj[s11[{ct3}]], s11[{ct4}]}, 
 {(-I)*ls1*(Delta[ct1, ct4]*Delta[ct2, ct3] + Delta[ct1, ct2]*Delta[ct3, ct4]), 
  1}}

which is the same expression as above after replacing the CG symbols by Deltas. But if I then call MakeSPheno[] with this Lagrangian, it produces the vertices

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Subroutine Couplings11s11cs11cs11T(ls1,res)
res = res-2._dp*(ls1)
End Subroutine Couplings11s11cs11cs11T  

Subroutine Couplings11s11cs11cs11L(ls1,res)
res = res-4._dp*(ls1)
End Subroutine Couplings11s11cs11cs11L

So, with the Deltas written explicitly in the Lagrangian SARAH/SPheno produces different vertices in the Fortran files. Although, I think this generated vertex leads to the correct contribution (-4*ls1*A0[ms1^2]).

Could you please explain what is going on? Am I missing something?

Many thanks and best regards,
Alex

For completeness, please find attached the SARAH model file.

As we discussed in the other thread, this CG is not correctly initialized (there are no numerical values associated with it). Therefore, I'm not surprised that the color-sum is wrong. That's all what I can so far. More information (and hopefully a patch) will follow when I'm back at work (end of January).

Cheers,
Florian

Color summation from 4-scalar vertex

Color summation from 4-scalar vertex

Re: Color summation from 4-scalar vertex

Re: Color summation from 4-scalar vertex