# Lagrangian of Standard Model Deduction

• StephvsEinst
In summary, the Standard Model Lagrangian is based on postulated local symmetries and is tested against experimental observations. It consists of a spontaneously broken Yang-Mills theory with the gauge group SU(3)xSU(2)xU(1) and includes fermion fields transforming under different representations of the gauge groups. The terms in the Lagrangian can be derived by creating invariant objects under the symmetry transformations. The Higgs field plays a crucial role in the formation of these terms, such as the lepton Yukawa terms. The electroweak part of the Lagrangian can be found in the paper "The Standard Model Lagrangian: What Do We Know?".
StephvsEinst
Does anyone know where can I find the deduction of all terms of the updated SM lagrangian? Although I have already looked at some lagrangians and theories like local gauge invariance, Yang-Mills theory, feynman rules, spontaneous symmetry-breaking and others, I wanted to see the deduction and simplification of the updated SM lagrangian.

Thank you,
Jorge.

What do you mean by "deduction"? The Standard Model is postulated and tested against experiment. It was postulated on the basis of several experimental observations of the different interactions.

When I said deduction I was thinking of the mathematics behind the SM Lagrangian. Probably deduction wasn't the best word to use, instead I should have said "mathematical background".

It is essentially nothing else than a spontaneously broken Yang-Mills theory with the gauge group SU(3)xSU(2)xU(1) with a bunch of fermion fields transforming under different representations of the gauge groups.

StephvsEinst
Summed up looks easy xD
I will work on that, thank you for the reply.

I don't think you can deduce it out of somewhere... you take your local symmetries and create invariant objects under those symmetry transformations. The thing is that nothing (at least only by that principle) to write down many terms or insert different things. But if you write down some representations you'll have:
$(3 , 1,1)$ : a triplet under SU(3) alone, and SU(2) singlet . Here you can have right handed quarks, since they don't see the weak sector
$(3, 2, 1)$ : a triplet under SU(3), a doublet under SU(2). Here you can have left handed quarks , which can interact under both strong and weak interactions.
$(8, 1, 1)$ : the gauge bosons of SU(3)
$(1, 3,1 )$ : the gauge bosons of SU(2)
$(1,1,1)$ : Here you can have the right handed fermions.
$(1, 2, 1)$ :colorless SU(2) doublets... here you can have the Higgs or the left handed leptons.
The last 1 in the parenthesis could as well be changed...it's in fact the hypercharge of the particle (but because this can change from particle to particle I just wrote 1)
in a notations [SU(3) , SU(2), U(1)]... I don't know if I forgot anything...
Also for the SU(3) triplets you must have in mind that $3$ could as well be $\bar{3}$ (the complex repr). For SU(2) that's not the case, since $2 \equiv \bar{2}$.

One example to get an idea of a term, is to write down a mass term for the leptons... In that case you need something like : $m \bar{\psi}_L \bar{\psi}_R$ which means you need to mix the left and right handed leptons... going to the above scheme you have to make an invariant object with:
$(1,2,Y_1)$ and $(1,1,Y_2)$. The result of those two would be $(1,2,Y_1) \otimes (1,1,Y_2) = (1,2,Y_1+Y_2)$
In order to make an invariant object with those two, you need an extra SU(2) doublet (because a $2\otimes 2=3 \oplus 1$ contains the singlet) and with hypercharge $-Y_1 -Y_2$ - and that's the role for the Higgs... As a result you obtain the lepton Yuawa terms.
The result of those three combinations would have to be $(1,1,0)$

Last edited:
StephvsEinst
StephvsEinst

## 1. What is the Lagrangian of the Standard Model Deduction?

The Lagrangian of the Standard Model Deduction is a mathematical formula that describes the fundamental interactions and particles within the Standard Model of particle physics. It includes terms for the electromagnetic, strong, and weak forces, as well as the Higgs boson, quarks, leptons, and their interactions.

## 2. How is the Lagrangian of the Standard Model Deduction derived?

The Lagrangian of the Standard Model Deduction is derived using mathematical techniques such as group theory, gauge theory, and symmetry principles. It is based on experimental observations and predictions from theoretical models.

## 3. What is the significance of the Lagrangian of the Standard Model Deduction?

The Lagrangian of the Standard Model Deduction is significant because it provides a unified framework for understanding the fundamental building blocks of matter and their interactions. It has been extensively tested and is considered to be one of the most successful theories in physics.

## 4. Can the Lagrangian of the Standard Model Deduction be modified or improved?

Yes, the Lagrangian of the Standard Model Deduction is not a complete theory and there are still unanswered questions and discrepancies that require further research. Many physicists are working on extending or modifying the Standard Model to account for these limitations.

## 5. How does the Lagrangian of the Standard Model Deduction relate to other theories, such as general relativity?

The Lagrangian of the Standard Model Deduction and general relativity are two separate theories that describe different aspects of the universe. While the Standard Model describes the interactions between particles, general relativity describes the behavior of gravity. There have been attempts to unify these theories, but a complete and satisfactory solution has not yet been found.

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