CO D EX GUIDE

CoDEx Parafernalia

CoDEx is a package which can be used to calculate Wilson Coefficients (both tree level and one loop) for a given BSM theory in which heavy degrees of freedom with spin 0, 1/2 or 1. The Wilson Coefficients will be evaluated at the scale of the mass of the heavy particle.

Functions available to the User Functions available to the User

There are some key functions, which let the user to define a list of BSM particles (along with their properties), build a Lagrangian, calculate the effective Wilson Coefficients

Help Function

CoDExHelp —

This function opens the CoDEx guide, with all help files listed

re — re[expression] gives only the real part of expression.

im — im[expression] gives only the imaginary part of expression.

abs — Provides the norm of a vector. abs[A]= Sqrt[A.dag[A]]

hermitianConjugate — hermitianConjugate[a]=ComplexExpand[Conjugate[a]]

dag — Stands for the Dagger operation on some expression. That expression must be a list with more than one element. Otherwise, use hermitianConjugate.

commutator — commutator[a,b]=a.b - b.a (a and b should be matrices)

antiCommutator — anticommutator[a,b]=a.b + b.a (a and b should be matrices)

pau — Slightly modified version of PauliMatrix. pau[4]=*PauliMatrix[4]. Otherwise pau[i] = PauliMatrix[i].

tau — tau[a] = (1/2) * pau[a]

del — del is a variation of a partial derivative operator. In del[a,expression], a is the Lorentz index, and runs from 0 to 3.

defineHeavyFields

treeOutput ▪ initializeLoop ▪ loopOutput ▪ codexOutput

formPick ▪ texTable ▪ RGFlow

Standard Model (SM) gauge couplings Standard Model (SM) gauge couplings

The SM gauge couplings :

gY , gW and gS are , SU(2)_L and SU(3)_C gauge couplings respectively.

Standard Model (SM) Fields Standard Model (SM) Fields

While writing the Beyond Standard Model (BSM) Lagrangian, the following SM fields have to be used.

In this CoDEx version, we consider only one generation Standard Model fields.

Fermion sector:

The Leptons:

left handed :

Left handed electrons →

, where 'i' runs over Lorentz indices.

Left handed neutrinos →

, where 'i' runs over Lorentz indices.

So, Lepton Doublet, lep[1] →

right handed :

Lorentz Conjugate:

left handed :

right handed :

eLb1[i] , nuLb1[i] and eRb1[i] are the Lorentz conjugates (

Ψ ^† γ ⁰

) of eL1[i] , nuL1[i] and eR1[i] respectively.

The Quarks:

left handed :

Left handed up type quarks →

, where 'i' runs over Lorentz indices.

Left handed down type quarks→

, where 'i' runs over Lorentz indices.

So, Quark Doublet, qdub[1,1] →

right handed :

Lorentz Conjugates:

left handed :

right handed :

uLb1[1,i] , dLb1[1,i], uRb1[1,i] and dRb1[1,i] are the Lorentz conjugates (

Ψ ^† γ ⁰

) of uL1[1,i], dL1[1,i], uR1[1,i] and dR1[1,i] respectively.

Bosonic sector:

The SM bosonic sector consists of the SM Higgs doublet, three SU(2)_L gauge bosons and eight SU(3)_C gauge bosons (gluons).

H — SM Higgs doublet

A — SM Isospin gauge bosons. Acts like a vector with three components. Each component behaves like a spin 1 field.

bYvec — SM abelian gauge boson. Acts like a spin 0 field.

gluons — SM color gauge bosons. Acts like a vector with eight components. Each component behaves like a spin 1 field.

Field Strength

bYst — U(1) field strength

W — SU(2)_L field strength

glust — SU(3)_C field strength

Following are examples of these in component form:

Operator Bases Operator Bases

CoDEx can find Wilson coefficients according two bases of SMEFT operators, namely "Warsaw" and "SILH". They are listed below. Their details can be obtained in this page. Below are the operators in a grid format.