Implementing a flavour-changing neutral current

[Anders]

Dear experts,

I am working to implement a flavour-changing neutral current (FCNC) between the b and s quarks at tree level in SARAH. To do this, I am trying to figure out the best way to "flavour-decompose" the quark fields in SM.m so that I can eventually make two flavours oppositely charged under a new U(1)' gauge group. I am considering three approaches to this. Could you help me by pointing out which approach, if any, seems viable?

The simplest way to flavour-decompose the quark fields seems to be as outlined in https://gitlab.in2p3.fr/goodsell/sarah/ ... omposition:

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DEFINITION[EWSB][Flavors]= { ...
 {Field, {Name 1, Name 2, ... }},
 ... }

However, if I do this it isn't clear to me how to assign different charges under U(1)' to different flavours, as the assignment of charges is something usually done when defining the quark fields. Is there a way to assign U(1)' charges after this step?


As an second approach, I am decomposing the fields upon definition. First, I define the quark fields as such:

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FermionFields[[1]] = {q01,1, {u01L, d01L},     1/6, 2,  3};
FermionFields[[2]] = {q02,1, {u02L, d02L},     1/6, 2,  3};
FermionFields[[3]] = {q03,1, {u03L, d03L},     1/6, 2,  3};
FermionFields[[4]] = {d01,1, conj[d01R],     1/3, 1, -3};
FermionFields[[5]] = {d02,1, conj[d02R],     1/3, 1, -3};
FermionFields[[6]] = {d03,1, conj[d03R],     1/3, 1, -3};
FermionFields[[7]] = {u01, 1, conj[u01R],    -2/3, 1, -3};
FermionFields[[8]] = {u02, 1, conj[u02R],    -2/3, 1, -3};
FermionFields[[9]] = {u03,1, conj[u03R],    -2/3, 1, -3};

Then, I rewrite the Lagrangian to account for this decomposition:

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LagHC =  -(Yd1 conj[H].d01.q01 + Yd2 conj[H].d02.q02 + Yd3 conj[H].d03.q03 + Ye conj[H].e.l + Yu1 u01.q01.H + Yu2 u02.q02.H + Yu3 u03.q03.H);

with the Yukawa couplings defined in parameters.m as scalars rather than 3x3 matrices.

Finally, I rewrite the mixing in the matter sector as follows:

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DEFINITION[EWSB][MatterSector]=   
  {{{{d01L,d02L,d03L}, {conj[d01R],conj[d02R],conj[d03R]}}, {{DL,Vd}, {DR,Ud}}},
   {{{u01L,u02L,u03L}, {conj[u01R],conj[u02R],conj[u03R]}}, {{UL,Vu}, {UR,Uu}}},
   {{{eL}, {conj[eR]}}, {{EL,Ve}, {ER,Ue}}}};  

With this approach, I run into errors related to the matching of Yukawas in SPheno. If I write the matching conditions in SPheno.m as Yd11 = YdSM[1,1] etc. I get errors like this when running makeSPheno:

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Transpose::nmtx: "The first two levels of {{v,vSM},{Ye,YeSM},{YdSM[1,1]},{YdSM[1,2]},{YdSM[1,3]},{YdSM[2,1]},{YdSM[2,2]},{YdSM[2,3]},{YdSM[3,1]},{YdSM[3,2]},{YdSM[3,3]},{YuSM[1,1]},{YuSM[1,2]},{YuSM[1,3]},{YuSM[2,1]},{YuSM[2,2]},{YuSM[2,3]},{YuSM[3,1]},{YuSM[3,2]},{YuSM[3,3]},{g1,g1SM},{g2,g2SM},{g3,g3SM}} cannot be transposed.

If I instead leave the matching conditions blank, I get errors like the one below when I run make Model=... in SPheno:

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Yu1 = YuSM
Error: Incompatible ranks 0 and 2 in assignment at (1)

Finally, as a third approach I am trying to decompose the quark fields and then "recomposing" them into the format used in SM.m to avoid any issues downstream. For this, I would like to define the quark generations separately as

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FermionFields[[1]] = {q01,1, {u01L, d01L},     1/6, 2,  3};
FermionFields[[2]] = {q02,1, {u02L, d02L},     1/6, 2,  3};
FermionFields[[3]] = {q03,1, {u03L, d03L},     1/6, 2,  3};
FermionFields[[4]] = {d01,1, conj[d01R],     1/3, 1, -3};
FermionFields[[5]] = {d02,1, conj[d02R],     1/3, 1, -3};
FermionFields[[6]] = {d03,1, conj[d03R],     1/3, 1, -3};
FermionFields[[7]] = {u01, 1, conj[u01R],    -2/3, 1, -3};
FermionFields[[8]] = {u02, 1, conj[u02R],    -2/3, 1, -3};
FermionFields[[9]] = {u03,1, conj[u03R],    -2/3, 1, -3};

and then collect them in the definitions of q, d, and u as

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FermionFields[[10]] = {q, 3, ..., 1/6, 2, 3}; (* ... placeholder *)
FermionFields[[11]] = {d, 3, ..., 1/3, 1,-3};
FermionFields[[12]] = {u, 3, ...,-2/3, 1,-3};

where I somehow make q01, q02, q03 the three generations of q and similarly for d and u. My problem with this approach is that it isn't clear to me how to make the three generations correspond to my previously defined fields.


Thank you for your time and best regards,
Anders

[Anders]

Update: I made quite some headway using the second approach above. After I included all intergenerational Yukawa couplings in the Lagrangian, i.e. non-diagonal ones in addition to the ones in the post above, the model runs all the way through SARAH and SPheno and reproduces SM results IF I add the SM Yukawa coupling values explicitly in the LesHouches file. I'm writing out my current solution here and attaching my model file in case anyone else needs to flavour-decompose the quark sector in this way.

First, I define the quark fields as such:

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FermionFields[[1]] = {q01,1, {u01L, d01L},     1/6, 2,  3};
FermionFields[[2]] = {q02,1, {u02L, d02L},     1/6, 2,  3};
FermionFields[[3]] = {q03,1, {u03L, d03L},     1/6, 2,  3};
FermionFields[[4]] = {d01,1, conj[d01R],     1/3, 1, -3};
FermionFields[[5]] = {d02,1, conj[d02R],     1/3, 1, -3};
FermionFields[[6]] = {d03,1, conj[d03R],     1/3, 1, -3};
FermionFields[[7]] = {u01, 1, conj[u01R],    -2/3, 1, -3};
FermionFields[[8]] = {u02, 1, conj[u02R],    -2/3, 1, -3};
FermionFields[[9]] = {u03,1, conj[u03R],    -2/3, 1, -3};

Then, I rewrite the Lagrangian to account for this decomposition:

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LagHC =  -(Yd11 conj[H].d01.q01 + Yd12 conj[H].d01.q02 + Yd13 conj[H].d01.q03 + Yd21 conj[H].d02.q01 + Yd22 conj[H].d02.q02 + Yd23 conj[H].d02.q03 + Yd31 conj[H].d03.q01 + Yd32 conj[H].d03.q02 + Yd33 conj[H].d03.q03 + Ye conj[H].e.l + Yu11 u01.q01.H + Yu12 u01.q02.H + Yu13 u01.q03.H + Yu21 u02.q01.H + Yu22 u02.q02.H + Yu23 u02.q03.H + Yu31 u03.q01.H + Yu32 u03.q02.H + Yu33 u03.q03.H);

and I define the Yukawa couplings Yd11 etc. in parameters.m as scalars rather than as the matrices they are in the SM.

Finally, I rewrite the mixing in the quark sector:

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  {{{{d01L,d02L,d03L}, {conj[d01R],conj[d02R],conj[d03R]}}, {{DL,Vd}, {DR,Ud}}},
   {{{u01L,u02L,u03L}, {conj[u01R],conj[u02R],conj[u03R]}}, {{UL,Vu}, {UR,Uu}}},

I leave the matching conditions for the up-and-down-type Yukawa couplings in SPheno.m blank. Now, the fields are flavour-decomposed and the different generations can be defined to carry different charges under a new U(1)' gauge group.

The only problem left is how to get SARAH to calculate the Yukawa couplings Yd11, Yd12, etc. rather than needing to input them explicitly in the LesHouches file.