Lecture 1: EFT basis derivation
SMEFT'2022 Physics School

Start Sym2Int

<<Sym2Int`

Define model

In Sym2Int, a model is defined by the gauge group and field content.

gaugeGroup[SMEFT]^={SU3,SU2,U1};

(* Number of flavors *)
nF=3;

fld1={"u",{3,1,2/3},"R","C",nF};
fld2={"d",{3,1,-1/3},"R","C",nF};
fld3={"Q",{3,2,1/6},"L","C",nF};
fld4={"e",{1,1,-1},"R","C",nF};
fld5={"L",{1,2,-1/2},"L","C",nF};
fld6={"H",{1,2,1/2},"S","C",1};
fields[SMEFT]^={fld1,fld2,fld3,fld4,fld5,fld6};

Results I: Physical basis

Dimension - 6

The list of operators is obtained with the GenerateListOfCouplings routine, which takes as input the maximal operator dimension to be considered.

opsD6=GenerateListOfCouplings[SMEFT,MaxOrder->6];

Two comments to interpret the results:

1) Notation used in Sym2Int: F1 = G, F2 = W, F3 = B.
2) The number of operators corresponds to the number of real parameters when the operator is self-conjugated and to the number of complex parameters otherwise.

Dimension - 8

opsD8=GenerateListOfCouplings[SMEFT,MaxOrder->8];

The results were not printed because there are too many operators. However, the user can print specific operators with the PickParticularOperators function.

PickParticularOperators[SMEFT,opsD8,"L L H H"]
PickParticularOperators[SMEFT,opsD8,"H* H F2 F2"]

PickParticularOperators[SMEFT,opsD8,"D D F2 F2 H* H"]

PickParticularOperators[SMEFT,opsD8,"L* L Q* Q H* H"]

Results II: Green basis

Change default options

We can now change some Sym2Int options and obtain new results. For instance, we can set $Sym2IntUseEOM to False so that Sym2Int does not apply the equations of motion to remove redundant operators. This will allow us to determine the operators in the Green basis, of interest for off-shell computations.

(* Apply EOMs redundancies? [default=True] *)
$Sym2IntUseEOM=False;

(* Apply IBP redundancies? [default=True] *)
$Sym2IntUseIBP=True;

Obtain and print new results

opsD8Green=GenerateListOfCouplings[SMEFT,MaxOrder->8];

PickParticularOperators[SMEFT,opsD8,"D D D D D D H* H"]
PickParticularOperators[SMEFT,opsD8Green,"D D D D D D H* H"]

PickParticularOperators[SMEFT,opsD8,"D D F3 F3 H* H"]
PickParticularOperators[SMEFT,opsD8Green,"D D F3 F3 H* H"]