# What is the full Lagrangian of SU(3)xSU(2)xU(1) model?

1. Jan 4, 2015

### nguyendung

What is the full Lagrangian of Standard Model?How can we build a Lagrangian that satisfies both the symmetry SU(3) and the symmetry SU(2) at the same time?

2. Jan 4, 2015

3. Jan 4, 2015

### nguyendung

What do X,Y stand for?

4. Jan 4, 2015

### ShayanJ

I'm not sure but I think they're referring to X and Y bosons. But hey...they're part of a GUT! and as far as I know, no GUT is part of SM yet!
So either I'm wrong in the first sentence, or I'm wrong in the last sentence, or I'm wrong in thinking that picture shows the SM's Lagrangian.

EDIT: This is X,Y free!

5. Jan 4, 2015

### vanhees71

Hm, I don't think that it makes much sense to stare at the Standard Model Lagrangian in full glory after Higgsing it. Even in unitary gauge, which is useful at the tree level, you won't understand much what's behind it. I recommend to study, how it's constructed from the fundamental symmetries underslying the standard model, starting from Quantum Electrodynamics QED (an Abelian U(1) gauge theory), then Quantum Chromodynamics (an un-Higgsed non-Abelian gauge theory with SU(3) as the gauge group) QCD and finally Quantum Flavor Dynamics QFD (a non-Abelian higgsed gauge theory with SU(2)$\times$U(1) as gauge group). The latter is the most complicated, because you have to deal with the Higgs mechanism and also with mixing (in the original version only for the quarks, nowadays also for the neutrinos).

A very good book on this subject is

Otto Nachtmann, Elementary Particle Physics, Springer 1990

Unfortunately neutrino oscillations are not treated in this book, because this subject came under closer consideration only a bit later after neutrino oscillations were discovered, which solved the "solar neutrino puzzle", but anyway this book explains very well the foundation of the Standard Model with the minimum prerequesites, which are a usual quantum-mechanics 1 lecture. Quantum field theory is explained as far as needed to understand the construction of the Standard Model Lagrangian, and also the experimental foundations are well treated.

6. Jan 4, 2015

Staff Emeritus
First, the only good reason to look at the whole Lagrangian is to put it on a T-shirt. More insight can be gained by looking at the pieces.

Second, it pains me to say that Shyan doesn't really know what he's talking about with the X and Y. These have nothing to do with GUTs. These are so-called "Popov Ghosts" and are an unnecessary complication at this level.

Third, the way you can have both SU(2) and SU(3) symmetries (your original question) is by having every field having it's place in these SU(2) and SU(3) multiplets. For example, an electron is in an SU(2) doublet but an SU(3) singlet.

7. Jan 4, 2015

### ShayanJ

I know what you mean by that pain but I can agree about having that pain only when you're talking about someone who actually claims to know QFT and SM well. I never claimed such a thing. I just gave the OP a picture of the SM's Lagrangian and when he asked about X and Y, I said I'm not sure! So I see no reason for you to feel that pain!

8. Jan 4, 2015

### dextercioby

The Standard Model Lagrangian density has to include the BRST ghost fields which are inherent to the correct gauge-fixing of the components: E-W + QCD. I mean just consider the QCD model: would you write the Lagrangian density without the ghosts? What purpose would that serve you? You can't put it in the path integral (generating functional for the non-connected Green functions), you can't use for the S-matrix, so it's useless.

9. Jan 4, 2015

### nguyendung

This idea occur when I was thinking about whether there is a limit at which SUSY leading to normal QFT.Whether SUSY Lagrangian is a generalized of the Lagrangian of QFT.Because we can build SUGRA but we still can not build a QFT of gravity.(I am reading Weinberg Vol 3 to page 209.)

10. Jan 4, 2015

### ChrisVer

You just let fields that exist in some representations of those gauge groups and you make singlet combinations out of them, keeping in mind what each field stands for (for example fermions will have Dirac Lagrangians)....
The gauge fixing terms appear just by letting gauge transfo....

11. Jan 9, 2015

### nguyendung

I have heard that almost SUGRA theories is not renormalizable(I am reading SUSY but not still SUGRA),then why do we still investigate SUGRA theories?

12. Jan 9, 2015

### samalkhaiat

From your first post, it is very clear that you have no knowledge about ordinary QFT let alone supersymmetric QFT. In order to benefit from PF, I suggest that you ask questions that match your level of understanding. Otherwise, you will be wasting your and other people time.

13. Jan 9, 2015

### ChrisVer

In general neither the SM has to contain renormalizable terms.... you can as well write order 5 (eg See-Saw mechanism) or order 6 (eg proton decay suppression) or [...] operators ,suppressed by the theory's scale/cutoff (for example the Planck Mass scale). That's because the SM itself is an effective field theory [the existence of Landau's pole in QED for example, is "showing" something like that]... In practise it's the super-renormalizable terms that bring problems into the theory (such as the Higg's mass term or if you add a cosmological constant), since they require some fine-tuning ,and things like SUSY or SUGRA or other theories come in the game.
As for SUGRA, why would you expect for it to be RN, since gravity itself is not RN?
In general an Effective Field Theory doesn't have to contain renormalizable terms.... and SUGRA might be such a theory (I'm not sure about this comment : as for example coming from a low-energy limit of Superstring theory)....

But in general it would be good to listen to samalkhaiat comment [specializing on you as a person/individual]... In general, however, I tend to disagree because there are other people who can read the answer , understand it, find it helpful etc [good for the rest of the community]...

Last edited: Jan 9, 2015
14. Jan 9, 2015

### nguyendung

Please tell me how can we use a QFT that is nonrenormalizable in pactice such as Effective Field Theory.How can we make the results finite with that theory?

15. Jan 10, 2015

### nguyendung

Effective Field Theory implies some "cut off" so then being free of infinities?

16. Jan 10, 2015

### nguyendung

When we calculate the loops we must integrate over momenta in a range that the effective theory having meaning?

17. Jan 10, 2015

### ChrisVer

Yes and Yes...
The infinities in general appear near a pole, and the pole is the natural cut-off of your theory, and at the pole the couplings become infinite (the theory can't be treated perturbatively). Of course the "perturbative methods" can be broken way below the cutoff (for the coupling constant to become non-perturbative you don't have to expect it to reach infinity, but some large number, so the theory fails even before the cut-off).
The SM cutoff is when new theories appear, and it's a good theory up to those theories. For QED for example, the Landau Pole appears way above our expectation of "new physics" (such as Quantum Gravity), so from the UV spectrum the SM is quite "protected". To the IR spectrum the QCD for example stops working, and you are using another effective theory (such as the meson-interactions EFT).
The integrations are done from some IR cutoff (could be zero if the theory is free from IR-divergencies) up to the UV cutoff (could be infinity in a QFT that is not effective). Your theory will give results that will depend on the cutoff...
The same occurs for the Higgs Mass term, which divergies quadratically (like $\delta m_H \sim \Lambda_{cutoff}^2$). And many others.
In general it's good, before looking at SUSY or SUGRA, to have a look in QFTs. Of course you can find subjects in both these theories, that you can understand without the need of QFTs (in most grad-level courses, the effective theories are dealt with in a second semester's course- such as QFT 2, since most of the times you are introduced in path integrals in the ends of the 1st or the beginnings of the 2nd semesters' courses.), but at the same time you will have to accept some things a-priori without being able to understand them (leave aside proving them), such as the Renormalizability or the Running couplings and so on.