Color summation from 4-scalar vertex

[AlexanderVoigt]

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

[AlexanderVoigt]

For completeness, please find attached the SARAH model file.

[FStaub]

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