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Is there a concise-but-thorough statement of the Standard Model?

+ 7 like - 0 dislike
134 views

I’m a grad student in high-energy physics. I’m familiar enough with the Standard Model, but I’ve always wondered whether there existed a canonical statement of, effectively, “what we talk about when we talk about the Standard Model”. Obviously the SM didn’t spring fully-formed from the pen of one author, but since its inception surely someone has compiled our current understanding into one document?

(The closest things I’ve found to this are the Wikipedia article and the explanatory chapters of the PDG.)

This post imported from StackExchange Physics at 2015-04-18 02:26 (UTC), posted by SE-user bdesham
asked Oct 29, 2011 in Theoretical Physics by bdesham (35 points) [ no revision ]
You see the periodic table of particles shown in the Wikipedia link a lot in talks, and the presenter often alludes to the $SU(3) \times SU(2) \times U(1)$ group structure and leaves it at that.

This post imported from StackExchange Physics at 2015-04-18 02:26 (UTC), posted by SE-user dmckee
i.imgur.com/pu7WM.png :)

This post imported from StackExchange Physics at 2015-04-18 02:26 (UTC), posted by SE-user Qmechanic
The most concise I've seen is An Introduction to the Standard Model of Particle Physics by Cottingham and Greenwood. It's short book, the meat of it is about 150 pages.

This post imported from StackExchange Physics at 2015-04-18 02:26 (UTC), posted by SE-user mtrencseni
What's missing from the Wikipedia article? It gives the entire field content and all interactions in 2-component form. It was written precisely to answer this question.

This post imported from StackExchange Physics at 2015-04-18 02:26 (UTC), posted by SE-user Ron Maimon
hep-ph/0405097 writes down the Lagrangian in 5 lines on the first page. hep-ph/9810316 is a review. Veltman's Diagrammatica has all the Feynman rules.

This post imported from StackExchange Physics at 2015-04-18 02:26 (UTC), posted by SE-user Mitchell Porter
@RonMaimon Quantisation is missing, or at least the tree level perturbative expansion. The wikipedia article does not go into it. That is good to be concise, and then adequate to the answer here, but it means that some implicit interactions, as WW-gamma or three gammas, are not shown explicitly; you must go to the Diagrammar, as Mitchell points out.

This post imported from StackExchange Physics at 2015-04-18 02:26 (UTC), posted by SE-user arivero

1 Answer

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Perhaps the most concise statement of the standard model is in terms of the Georgi-Glashow SU(5) GUT. I know that you were not asking for a theory beyond the standard model, only the standard model itself. But the Georgi Glashow description provides a natural understanding of all the features of the standard model, particularly the strange hypercharge assignments.

Recall that SU(5) is defined by all 5 by 5 unitary matrices. The gauge group of the standard model consists of an arbirary SU(2) matrix W in the top 2 by 2 corner block, an arbitrary SU(3) matrix G in the lower 3 by 3 corner block, and e^{iY}, where Y=diag(-1/2,-1/2,1/3,1/3,1/3). These generate a natural SU(3)xSU(2)xU(1) subgroup of SU(5).

The matter in the standard model consists of a left handed fermion T in the antisymmetric tensor SU(5) representation $T_{\mu\nu} = - T_{\nu\mu}$, where $\mu$ and $\nu$ are SU(5) indices, and a right handed vector in the defining representation $V^\mu$.

The vector V splits into the first two components and the last 3 components. The first two components are a doublet L under SU(2), and the last 3 are a triplet D under SU(3) (both defining representations). From the form of Y, the U(1) charge of the doublet L is -1/2, and of the triplet is 1/3.

The antisymmetric tensor consists of an upper 2 by 2 block, which is an antisymmetric SU(2) tensor, a singlet (SU(2) antisymmetric 2 tensors are just scalars), a middle 2 by 3 block, which is an SU(2) doublet SU(3) triplet, and a lower 3 by 3 block, which is an SU(3) triplet (written as an antisymmetric tensor).

To find the Y charge of the pieces of a 2-tensor, you perform a Y U(1) transformation. The rows and columns independently get multiplied by the phase factors, and so you add up the appropriate phases for each position. The top 2 by 2 block has Y charge 1, the bottom 3 by 3 block has Y charge -2/3, and the 2 by 3 blocks on top and bottom have Y charge 1/6.

So you get a list of objects:

  • SU(2) doublet SU(3) triplet of charge 1/6
  • SU(2) singlet of charge 1
  • SU(3) triplet of charge 2/3
  • SU(2) double of charge 1/2
  • SU(3) triplet of charge -1/3

Together with an SU(2) doublet Higgs with charge 1/2 (which can be thought of as the top two components of a Higgs defining SU(5) represetation), these are the fields of the standard model.

The Georgi Glashow model allows for a statement of the standard model: it consists of a defining and antisymmetric 2-tensor of SU(5), coupled to the subset of the gauge bosons of SU(5) corresponding to the embedding of SU(2),SU(3), and U(1) (with different gauge couplings for each gauge subgroup), together with the most general renormalizable interaction between these and a scalar Higgs with the same SU(2) and charge numbers as the top-two components of a defining representation.

I think that the standard model only became "the standard model" in 1974 when Georgi and Glashow published their GUT.

answered Nov 8, 2011 by Ron Maimon (7,535 points) [ revision history ]
edited Apr 18, 2015 by Ron Maimon
This is the earliest occurrence (1978) of the phrase "standard model" that I have found: prd.aps.org/abstract/PRD/v17/i1/p275_1

This post imported from StackExchange Physics at 2015-04-18 02:26 (UTC), posted by SE-user Mitchell Porter
@Porter That's very interesting.

This post imported from StackExchange Physics at 2015-04-18 02:26 (UTC), posted by SE-user QuantumDot

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