These effective operators are typically the result of integrating out some degrees of freedom at a higher scale. The actually UV complete theory can contain only d=4 operators.zaman786 said:TL;DR Summary: how to add higher dimensional operator to SM
Hi, I Learned that we can add higher dimensional operator but they are non-renormalizable - but effect of higher dimensional operator is vanishes at low energy - my question is than how can we add higher dimensional operator at higher energy - like dimension 5 operator ( weinberg operator) which is non- renormalizable.
A higher-dimensional operator in the Standard Model (SM) refers to a term in the effective field theory Lagrangian that has mass dimension greater than four. These operators are suppressed by some high-energy scale, typically associated with new physics beyond the Standard Model, and become relevant at higher energies.
Higher-dimensional operators are needed in the Standard Model to account for effects of new physics that are not captured by the renormalizable (dimension-four) operators. They provide a way to systematically include the influence of unknown high-energy phenomena, potentially offering insights into phenomena such as neutrino masses, dark matter interactions, and baryon number violation.
To construct higher-dimensional operators in the Standard Model, you start by identifying the fields involved and their symmetries. You then write down all possible combinations of these fields and their derivatives that form Lorentz-invariant and gauge-invariant terms. These combinations are organized by their mass dimension, and terms with dimension greater than four are considered higher-dimensional operators. They are typically suppressed by powers of a high-energy scale, such as the Planck scale or the scale of new physics.
The energy scale, often denoted by Λ, plays a crucial role in higher-dimensional operators as it sets the suppression factor for these terms. The higher the dimension of the operator, the more it is suppressed by powers of 1/Λ. This means that the effects of higher-dimensional operators become significant only at energy scales close to Λ, making them relevant for high-energy processes or experiments probing beyond the Standard Model.
One common example of a higher-dimensional operator in the Standard Model is the dimension-five operator responsible for generating Majorana masses for neutrinos. This operator can be written as (LiH)(LjH)/Λ, where Li are the lepton doublets, H is the Higgs doublet, and Λ is the scale of new physics. This operator violates lepton number and, after electroweak symmetry breaking, gives rise to small neutrino masses, which are observed experimentally.