What is the Equation on My CERN T-Shirt?

[Coin]

So I was lucky enough to visit CERN earlier this year, and they were selling t-shirts with a long equation on them. And I, like an idiot tourist, somehow jumped to the conclusion this was the standard model Lagrangian and got the t-shirt.

Well, it's still a pretty cool t-shirt, but then I went and looked up the SM lagrangian and it seemed to come in two flavors, one four times this long and one about 50 times this long. Oops...

I lack a particle physics background, so-- I think I know what some of these symbols might mean, but honestly, not most of them! Any help appreciated, what is this equation I am wearing?

 

[Simon_Tyler]

If you assume that there's lots of suppressed flavour and gauge indices, then it's the Lagrangian for a general renormalizable quantum field theory. Since there are no explicit fermion mass terms, it's probably meant to be shorthand for the Standard Model (which assumes all fermion masses come from the Higgs mechanism).

The 1st line is the kinetic and self-interaction terms for a Yang-Mills (gauge) field.
The 2nd line is the kinetic term for some fermions and their minimal coupling to the Yang-Mills field.
The 3rd line is some Yukawa couplings - interactions between the fermions and scalars.
The 1st term in the 4th line is the kinetic term for the scalar fields and its minimal coupling to the Yang-Mills field.
The 2nd term in the 4th line is an arbitrary scalar potential (scalar mass term and self-couplings). This term probably is meant to have a vacuum solution that breaks some of the gauge symmetry and thus gives mass to the fermions via the Yukawa interaction.

PS. I'm jealous - I have not yet got to visit CERN!

 

[Coin]

Thanks much, that helps! And yeah, I was very grateful to my sister, who was the reason I was in Switzerland in the first place :)

I think your description may help me decode some of this if I look on my own, a few specific things I'm confused about though (sorry if these are dumb questions)--

My first naive guess looking at that first term is that F is force, M is mass, V is velocity, it looks like it's describing something like momentum?! But... what on Earth is this notation with MV being a subscript and then a superscript? Is this some sort of tensor thing?

What is the repeated "+ h.c." and why write it twice? (Whoops, maybe I should have smoothed out the shirt before taking the picture...)

So is the V(Φ) / "scalar potential" meant to be the Higgs mechanism or some placeholder for it?

I am glad that my guess, that this was meant to represent the standard model, turned out to be correct... my plan when I bought this shirt was that I'm going to wait for the LHC to find a flaw in the standard model... then sew on a patch "fixing" the equation :D

 

[The_Duck]

I think the shirt is meant specifically to represent the Higgs mechanism.

For the first line: this term describes the "nonabelian gauge field." This is like a generalized version of the electromagnetic field; in the standard model it is the common origin of the W and Z bosons and the photon (Gluons come from a separate nonabelian gauge field that is not involved in the Higgs mechanism). In this term the sub- and superscripts are not m and v but the Greek letters mu and nu, which are commonly used as indices in relativistic physics. It is "some sort of tensor thing:" F is called the "field strength tensor." It is analagous to the http://en.wikipedia.org/wiki/Electromagnetic_tensor" .

h.c. stands for "hermitian conjugate" and means "the hermitian conjugate of the previous term." You can make a real number out of a complex number by adding its complex conjugate. The Lagrangian must be a hermitian operator, which is like a real number. You can make a hermitian operator from a general operator by adding its hermitian conjugate, so often pairs of hermitian conjugate terms will appear in a Lagrangian and are abbreviated in this way.

V(Φ) is the potential energy of the Higgs field, Φ. It's shape is a http://en.wikipedia.org/wiki/Mexican_hat_potential#Mathematical_example:_the_Mexican_hat_potential", which induces "spontaneous symmetry breaking" which gives mass to fermions through the term on the third line and to the W and Z bosons through the first term on the last line.

 

[vanhees71]

I think that's the famous T-shirt with the standard-model Lagrangian written by John Ellis. It's of course the Lagrangian, before the spontaneous symmetry breaking is made explicit, and that's much shorter than the one written in terms of "physical fields".

 

[maxverywell]

^Lagrangian density.

 

[element4]

It looks more like a U(1) gauge theory with Dirac fermions and a Higgs field (scalar field + Yukawa + ...).

If it is the standard model, then its written in a very implicit notation. It is conventional to suppress indices on the Dirac fermions, but the trace in the SU(3)xSU(2)xU(1) space for the Yang-Mills term is always written in usual notational conventions (as far as I know).

 

[robert2734]

Doesn't a Yang Mill field have three indices?

 

[Simon_Tyler]

@robert2734 It depends what you mean by the "Yang-Mills" fields... The potential has one Lorentz index and one adjoint rep group/colour index. The latter can be written as a pair of fundamental rep group indices. It can also have "flavour" indices if the group is not simple (eg the standard model group U(1)xSU(2)xSU(3)).
the Yang-Mills fields strength has the same colour and flavour indices, but it has 2 Lorentz indices.

@element4
As I said in my first post, there could be (and probably are) "lots of suppressed flavour and gauge indices". What you make explicit really depends on the context and who's writing. Often I'll write tr(F.F) to mean the trace over Lorentz indices, colour indices and flavour indices. And I'd use Tr(F.F) to include the trace over the continuous position index - ie the integral over space-time.