Matchete example for LHC Precision program Benasque

Install Matchete

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

Load Matchete

<<Matchete`

There is a thorough documentation in the package folder

Definition of the Standard Model

Definition of the gauge groups

Gauge fields and couplings and fundamental and adjoint representations are automatically generated (other representations can also be defined)

DefineGaugeGroup[SU3c, SU@3, gs, G, FundAlphabet -> {"a","b","c","d","e","f"},AdjAlphabet ->{"A","B","C","D","E","F"}];		DefineGaugeGroup[SU2L, SU@2, gL, W, FundAlphabet -> {"i","j","k","l","m","n"}, AdjAlphabet -> {"I","J","K","L","M","N"}];
DefineGaugeGroup[U1Y,   U1, gY, B];

Definition of flavor index

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

Definition of fields

DefineField[q,Fermion,Indices->{SU3c[fund],SU2L[fund],Flavor},Charges->{U1Y[1/6]},Chiral-> LeftHanded,Mass->0];
DefineField[u,Fermion,Indices->{SU3c[fund],Flavor},Charges->{U1Y[2/3]},Chiral-> RightHanded,Mass->0];
DefineField[d,Fermion,Indices->{SU3c[fund],Flavor},Charges->{U1Y[-1/3]},Chiral-> RightHanded,Mass->0];
DefineField[l,Fermion,Indices->{SU2L[fund],Flavor},Charges->{U1Y[-1/2]},Chiral-> LeftHanded,Mass->0];
DefineField[e,Fermion,Indices->{Flavor},Charges->{U1Y[-1]},Chiral-> RightHanded,Mass->0];
(* Mass added as an interaction because it is tachyonic *)
DefineField[H,Scalar,Indices->{SU2L[fund]}, Charges->{U1Y[1/2]},Mass->0];

Definition of couplings

DefineCoupling[Yu,Indices->{Flavor,Flavor}];
DefineCoupling[Yd,Indices->{Flavor,Flavor}];
DefineCoupling[Ye,Indices->{Flavor,Flavor}];
DefineCoupling[μ,SelfConjugate->True,EFTOrder->1];
DefineCoupling[λ,SelfConjugate->True,EFTOrder->0];

Definition of couplings

YukawaLag =  Yu[p,r] Bar@q[α,i,p]**u[α,r] CG[eps[SU2L],{i,j}]Bar@H[j] + Yd[p,r] Bar@q[α,i,p]**d[α,r]H[i] + Ye[p,r] Bar@l[i,p]**e[r]H[i];
HiggsPotential = -μ[]^2 Bar@H[i]H[i] + λ[]/2 Bar@H[i]H[i]Bar@H[j]H[j];
LSM=FreeLag[] - PlusHc[YukawaLag] - HiggsPotential //RelabelIndices;

Printing the Lagrangian

HiggsPotential//NiceForm

HiggsPotential

LSM//NiceForm

Scalar-singlet extension

ResetAll[]

Definition of the model

Loading the SM

The Higgs potential parameters are relabeled to mH and λH.

LSM=LoadModel["SM"];
LSM//NiceForm

New field and couplings

Define new field

DefineField[S,Scalar,SelfConjugate->True,Mass->{Heavy,M}]

Define new couplings

DefineCoupling[μS,SelfConjugate->True]
DefineCoupling[κ,SelfConjugate->True]
DefineCoupling[λSH,SelfConjugate->True]
DefineCoupling[λS,SelfConjugate->True]

Lagrangian

Write interactions

Lint=(-(1/(3!))μS[]((S[])^3)-(1/(4!))λS[]((S[])^4)-(1/2)λSH[]Bar@H[i]H[i]S[]S[]-κ[]Bar@H[i]H[i]S[])//RelabelIndices;
Lint//NiceForm

Define Lagrangian extension

Lext=FreeLag[S]+Lint;

Define full UV Lagrangian

LUV=LSM+Lext;
LUV//NiceForm

CheckLagrangian@LUV

CheckLagrangian[LUV,DetailedOutput->True]

Matching to the effective Lagrangian

Tree-level

Matching

LEFT0=Match[LUV,LoopOrder->0,EFTOrder->6];
LEFT0-LSM//CollectOperators//NiceForm

Removing redundant operators off-shell

Apply IBP and other identities to reduce the output to a Green’s basis.

LEFT0OffShell=LEFT0//GreensSimplify;
LEFT0OffShell-LSM //CollectOperators//NiceForm

Removing redundant operators on-shell

Simplify to the physical basis using field redefinitions.

LEFT0OnShell=LEFT0//EOMSimplify;
LEFT0OnShell -LSM //CollectOperators//HcSimplify//NiceForm

One-loop

Matching

LEFT=EchoTiming@Match[LUV,LoopOrder->1,EFTOrder->6]/.(ϵ^(-1))->0;
LEFT-LSM//CollectOperators//HcSimplify//NiceForm

Removing redundant operators off-shell

Apply IBP and other identities to reduce the output to a Green’s basis.

LEFTOffShell=LEFT//GreensSimplify;
LEFTOffShell-LSM//CollectOperators//HcSimplify//NiceForm

Removing redundant operators on-shell

Simplify to the physical basis using field redefinitions.

LEFTOnShell=EchoTiming@EOMSimplify[LEFT];
LEFTOnShell-LSM //CollectOperators//HcSimplify//NiceForm

Print the effective coupling that was introduced

PrintEffectiveCouplings[LEFTOnShell]

Selecting specific operator classes

SelectOperatorClass[LEFTOnShell,{Bar@H,H,Bar@l,l},1]//GreensSimplify//NiceForm