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.
Q&A (4869)
