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

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.)

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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. –  dmckee Oct 29 '11 at 19:29
–  Qmechanic Oct 30 '11 at 20:29
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. –  mtrencseni Nov 3 '11 at 8:51
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. –  Ron Maimon Nov 5 '11 at 5:57
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. –  Mitchell Porter Nov 5 '11 at 6:54

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.

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This is the earliest occurrence (1978) of the phrase "standard model" that I have found: prd.aps.org/abstract/PRD/v17/i1/p275_1 –  Mitchell Porter Jul 9 '12 at 8:30
@Porter That's very interesting. –  QuantumDot Aug 19 '12 at 5:15