Matchete Tutorial

EFT Foundations and Tools 2023 — MITP

Matching the (Subscript[S, 1]) leptoquark model to the SMEFT

Installing Matchete

Only required if you have not installed Matchete before.

(*Import["https://gitlab.com/matchete/matchete/-/raw/master/install.m"]*)

Matchete requires Mathematica 12.0.0 or newer!

Load Matchete

<<Matchete`

If you want to update

(*CheckForUpdate[]*)

Overview of all Matchete functions

?Matchete`*

Define the Standard Model

The SM is predefined in Matchete and can be loaded by:

ℒSM=LoadModel["SM"];

This defines all gauge symmetries, fields, couplings, and the Lagrangian. We can print the latter by:

ℒSM//NiceForm

Explicit SM definitions [can be skipped if LoadModel[“SM”] was used]

For convenience you can find below the definitions included in the predefined SM model file.
This sections does not need to be evaluated if you have run LoadModel[“SM”].

Define symmetries

QCD

(*
DefineGaugeGroup[SU3c,SU[3],gs,G,
	FundAlphabet->{"a","b","c","d","e","f"},
	AdjAlphabet->{"A","B","C","D","E","F"}
]
*)

This automatically defines a gauge coupling gs[] and the field-strength tensor G[μ,ν].
Furthermore it assigns characters for the printing of gauge indices.

Weak force

(*
DefineGaugeGroup[SU2L,SU[2],gL,W,
	FundAlphabet->{"i","j","k","l","m","n"},
	AdjAlphabet->{"I","J","K","L","M","N"}
]
*)

Hypercharge

(*
DefineGaugeGroup[U1Y,U1,gY,B]
*)

Define flavor indices

(*
DefineFlavorIndex[Flavor,3,
	IndexAlphabet->{"p","r","s","t","u","v"}
]
*)

Define field content

Fermions

Quark doubled

(*
DefineField[q,Fermion,
	Indices->{SU3c[fund],SU2L[fund],Flavor},
	Charges->{U1Y[1/6]},
	Chiral->LeftHanded,
	Mass->0
]
*)

Up-type quark singlet

(*
DefineField[u,Fermion,
	Indices->{SU3c[fund],Flavor},
	Charges->{U1Y[2/3]},
	Chiral->RightHanded,
	Mass->0
]
*)

Down-type quark singlet

(*
DefineField[d,Fermion,
	Indices->{SU3c[fund],Flavor},
	Charges->{U1Y[-1/3]},
	Chiral->RightHanded,
	Mass->0
]
*)

Lepton doublet

(*
DefineField[l,Fermion,
	Indices->{SU2L[fund],Flavor},
	Charges->{U1Y[-1/2]},
	Chiral->LeftHanded,
	Mass->0
]
*)

Charged lepton singlet

(*
DefineField[e,Fermion,
	Indices->{Flavor},
	Charges->{U1Y[-1]},
	Chiral->RightHanded,
	Mass->0
]
*)

Higgs field

(*
DefineField[H,Scalar,
	Indices->{SU2L[fund]},
	Charges->{U1Y[1/2]},
	Mass->0
]
*)

The Higgs field is defined as massless, since its mass parameter is tachyonic. We will define it as a coupling below.

Define couplings

Yukawa couplings

(*
DefineCoupling[Yu,Indices->{Flavor,Flavor}]
*)

(*
DefineCoupling[Yd,Indices->{Flavor,Flavor}]
*)

(*
DefineCoupling[Ye,Indices->{Flavor,Flavor}]
*)

Higgs mass and coupling

Higgs mass

(*
DefineCoupling[μ,SelfConjugate->True,EFTOrder->1] (* μ has light mass dimension one *)
*)

Higgs quartic coupling

(*
DefineCoupling[λ,SelfConjugate->True,EFTOrder->0] (* λ has light mass dimension zero *)
*)

Defining the SM Lagrangian

Yukawa interaction Lagrangian

(*
ℒYukawa=Ye[p,r] Bar[l[i,p]]**e[r]H[i]+
	Yd[p,r]Bar[q[a,i,p]]**d[a,r]H[i]+
	Yu[p,r]Bar[q[a,i,p]]**u[a,r]CG[eps[SU2L],{i,j}]Bar[H[j]]//PlusHc;
%//HcSimplify//NiceForm
*)

Scalar potential

(*
ℒH=-((μ[])^2)Bar[H[i]]H[i]+((λ[])/2)Bar[H[i]]H[i]Bar[H[j]]H[j];
%//NiceForm
*)

Full SM Lagrangian

(*
ℒSM=FreeLag[q,d,u,l,e,H,B,W,G]-ℒYukawa-ℒH//RelabelIndices;
%//HcSimplify//NiceForm
*)

Definition of the (Subscript[S, 1]) leptoquark model

Task: implement the (Subscript[S, 1]) leptoquark model.

Define the (Subscript[S, 1]) field

?DefineField

Define (Subscript[S, 1]) couplings

?DefineCoupling

Define the (Subscript[S, 1]) interaction Lagrangian

Hints: 
- The charge conjugate is specified by: CConj[...].
- The ϵ  tensor is defined as a Clebsch-Gordan coefficients by: CG[eps[SU2L],{i,j}].

Define entire (Subscript[S, 1]) Lagrangian

?FreeLag

Define full BSM Lagrangian

Tree-level

Fierz identities not yet included due to evanescent operators.
No other simplifications required in this case.

Perform the tree-level matching your model:

?Match

Manual Fierzing [predefined]

Currently Fierz identities still have to be implemented manually. Below you can find the Fierz identities required for the current example.

FierzIDs={
(* eu *)
Bar[Field[e,Fermion,{Index[s_,Flavor]},{}]]**DiracProduct[GammaCC,Proj[-1]]**Transp[Bar[Field[u,Fermion,{Index[a_,SU3c[fund]],Index[t_,Flavor]},{}]]] Transp[Field[e,Fermion,{Index[p_,Flavor]},{}]]**DiracProduct[GammaCC,Proj[1]]**Field[u,Fermion,{Index[a_,SU3c[fund]],Index[r_,Flavor]},{}]:>(1/2)Bar[e[s]]**γ[ν]**e[p]Bar[u[a,t]]**γ[ν]**u[a,r]
,
(* lq *)
Bar[Field[l,Fermion,{Index[i_,SU2L[fund]],Index[s_,Flavor]},{}]]**DiracProduct[GammaCC,Proj[1]]**Transp[Bar[Field[q,Fermion,{Index[a_,SU3c[fund]],Index[j_,SU2L[fund]],Index[t_,Flavor]},{}]]] Transp[Field[l,Fermion,{Index[i_,SU2L[fund]],Index[p_,Flavor]},{}]]**DiracProduct[GammaCC,Proj[-1]]**Field[q,Fermion,{Index[a_,SU3c[fund]],Index[j_,SU2L[fund]],Index[r_,Flavor]},{}]:>(1/2)Bar[l[i,s]]**γ[ν]**l[i,p]Bar[q[a,j,t]]**γ[ν]**q[a,j,r]
,
(* lq' *)
Bar[Field[l,Fermion,{Index[i_,SU2L[fund]],Index[s_,Flavor]},{}]]**DiracProduct[GammaCC,Proj[1]]**Transp[Bar[Field[q,Fermion,{Index[a_,SU3c[fund]],Index[j_,SU2L[fund]],Index[t_,Flavor]},{}]]] Transp[Field[l,Fermion,{Index[j_,SU2L[fund]],Index[p_,Flavor]},{}]]**DiracProduct[GammaCC,Proj[-1]]**Field[q,Fermion,{Index[a_,SU3c[fund]],Index[i_,SU2L[fund]],Index[r_,Flavor]},{}]:>(1/4)Bar[l[i,s]]**γ[ν]**l[i,p]Bar[q[a,j,t]]**γ[ν]**q[a,j,r]+CG[gen[SU2L[fund]],{A,i,j}]CG[gen[SU2L[fund]],{A,ii,jj}]Bar[l[i,s]]**γ[ν]**l[j,p]Bar[q[a,ii,t]]**γ[ν]**q[a,jj,r]
,
(* lequ *)
Bar[Field[l,Fermion,{Index[j_,SU2L[fund]],Index[s_,Flavor]},{}]]**DiracProduct[GammaCC,Proj[1]]**Transp[Bar[Field[q,Fermion,{Index[a_,SU3c[fund]],Index[i_,SU2L[fund]],Index[t_,Flavor]},{}]]] Transp[Field[e,Fermion,{Index[p_,Flavor]},{}]]**DiracProduct[GammaCC,Proj[1]]**Field[u,Fermion,{Index[a_,SU3c[fund]],Index[r_,Flavor]},{}]:>-(1/2)Bar[l[j,s]]**e[p]Bar[q[a,i,t]]**u[a,r]+(1/8)Bar[l[j,s]]**σ[ν1,ν2]**e[p]Bar[q[a,i,t]]**σ[ν1,ν2]**u[a,r]
,
Bar[Field[e,Fermion,{Index[r_,Flavor]},{}]]**DiracProduct[GammaCC,Proj[-1]]**Transp[Bar[Field[u,Fermion,{Index[a_,SU3c[fund]],Index[p_,Flavor]},{}]]] Transp[Field[l,Fermion,{Index[j_,SU2L[fund]],Index[t_,Flavor]},{}]]**DiracProduct[GammaCC,Proj[-1]]**Field[q,Fermion,{Index[a_,SU3c[fund]],Index[i_,SU2L[fund]],Index[s_,Flavor]},{}]:>-(1/2)Bar[e[r]]**l[j,t]Bar[u[a,p]]**q[a,i,s]+(1/8)Bar[e[r]]**σ[ν1,ν2]**l[j,t]Bar[u[a,p]]**σ[ν1,ν2]**q[a,i,s]
};

Tree-level EFT Lagrangian in Warsaw basis

Derive the tree-level EFT Lagrangian in the Warsaw basis by applying the Fierz identities.

Full one-loop matching

Matching

Now perform the full one-loop matching for your model:

?Match

Simplifications

Perform the off-shell simplifications on the resulting EFT Lagrangian.

?GreensSimplify

Apply the field redefinitions to the resulting EFT Lagrangian. (This also includes off-shell simplifications, thus GreensSimplify does not have to be called in advance in principle.)

?EOMSimplify

Extract leptonic dipole operators

Extract the leptonic dipole operators (Subscript[Q, eW]) and (Subscript[Q, eB]) from the fully simplified one-loop EFT Lagrangian.

?SelectOperatorClass