Derivation of standard model lagrangian

[resurgance2001]

What's the simplest, most direct way to derive the lagrangian of the SM?

I saw earlier today:

L[S M] = L[Dirac] + L[mass] + L[Gauge] + L[Gauge/psi]
That seems like a good starting point. I like it because it says the SM Lagrangian is simply the sum of four lagrangians. The next step seems to be that one needs to say what each of the four parts are.

I am trying to get at this by using the absolute minimum amount tough mathematical concepts/language, at least to start with.

 

[Orodruin]

What exactly do you mean by "derive the SM lagrangian"? Like any physical model, it is an attempt to describe Nature in a certain mathematical framework. As such, it is not something you can derive from first principles, but something you must postulate and test experimentally for verification.

You might just as well ask how to derive the fact that Newton's gravitational potential follows Poisson's equation.

 

[resurgance2001]

Ok - we'll that's helpful. But then even if it is something that is postulated, are there not some (simple) reasons for postulating it. I once read that Gell Mann had seen patterns amongst the fundmental particles that were known during his day, and that was what led him to make the contribution(s) he made to sorting it all out.

I mean can it for example be compared to the periodic table, which can be partly 'derived' or deduced (inducted?) by looking at patterns amongs the various elements? You will guess perhaps from the level of these questions that I am only an absolute (or nearly) beginner at studying this, so please excuse any ignorance. Thanks

 

[Orodruin]

The reason for postulating it in the first place was that it was (and still is) a good description of what we observe in Nature. Since then, several predictions have been confirmed to high accuracy, which solidifies the model as a good description.

If you are asking why we use U(1)xSU(2)xSU(3) as the gauge group, the answer is simply that this is what best describes what we observe and, for example, U(1)xU(1)xU(1) would be a bad description.

 

[resurgance2001]

But surely there must have been a reason why someone(s) thought that particular group would work? Or did tegu randomly try out different groups until they found one that fitted the data?

How does the group relate to the Lagrangian? Once you decide on the group, does that determine the lagrangian or is it the other way round? Cheers

 

[JorisL]

But surely there must have been a reason why someone(s) thought that particular group would work? Or did tegu randomly try out different groups until they found one that fitted the data?

How does the group relate to the Lagrangian? Once you decide on the group, does that determine the lagrangian or is it the other way round? Cheers

Generally I think of it as follows. (Warning, this is a simplified picture which for me has had its merits)

We know what our world looks like. And which symmetries are present. (in the particle landscape)
We want to Lagrangian (or perhaps more general the action) to reflect these symmetries.
When you know the symmetries a term has, we can use this to construct a suitable Lagrangian and then check our candidate Lagrangian.

If you google "construction of standard model lagrangian" will surely lead to more detailed explanations/calculations.

 

[vanhees71]

I think the best treatment of how the standard model is constructed from the observations in high-energy particle physics over the last 5-6 decades is given in

Nachtmann, O.: Elementary Particle Physics - Concepts and Phenomenology, Springer-Verlag, 1990

It's only missing the fascinating new insights on neutrino oscillations, but to understand how the standard model (being finalized in 1971 by 't Hooft's PhD thesis on the renormalizability of "un-Higgsed" and "Higgsed" non-abelian gauge theories or, in the strong-interaction sector, the discovery of asymptotic freedom of QCD by Wilczek&Gross and Politzer) came into being, it's a great source.

 

[RUTA]

The actual form of the SM Lagrangian density is a bit more involved than your simplified source would indicate http://nuclear.ucdavis.edu/~tgutierr/files/sml.pdf

 

[vanhees71]

This is fun, but totally useless. Of course, in Nachtmann at the end you get this Lagrangian too, but in an understandable form!

 

[resurgance2001]

I have now read that it is the group that is postulate and that the Lagrangian is chosen to fit the symmetry of the group - e.g. U(1) x SU(2) x SU(3)

My dumb question for today is how is the group and Lagrangian connected. If the form of the Lagrangian is deduced or obtained from the group, how is this done? :) cheers