Next we turn to a unified theory that is not so `grand': its gauge group is not a simple Lie group, as it was for the and theories. This theory is called the Pati-Salam model, after its inventors ; it has gauge group , which is merely semisimple.
We might imagine the theory as an answer to this question:
Why are the hypercharges in the Standard Model what they are?The answer it provides is something like this:
Because is the actual gauge group of the world, acting on the representation .But there are other intriguing patterns in the Standard Model that does not explain--and these lead us in different directions.
First, there is a strange similarity between quarks and leptons. Each generation of fermions in the Standard Model has two quarks and two leptons. For example, in the first generation we have the quarks and , and the leptons and . The quarks come in three `colors': this is a picturesque way of saying that they transform in the fundamental representation of on . The leptons, on the other hand, are `white': they transform in the trivial representation of on .
Second, there is a strange difference between left- and right-handed fermions. The left-handed ones participate in the weak force governed by , while the right-handed ones do not. Mathematically speaking, the left-handed ones live in a nontrivial representation of , while the right-handed ones live in a trivial one. The nontrivial one is , while the trivial one is :
Following Pati and Salam, let us try to sculpt a theory that makes
these ideas precise. In the last two sections, we saw some of the
ingredients we need to make a grand unified theory: we need to extend
the symmetry group
to a larger group using an inclusion
We now use the same methods to chip away at our current challenge. We asked if leptons correspond to a fourth color. We already know that every quark comes in three colors, , , and , which form a basis for the vector space . This is the fundamental representation of , the color symmetry group of the Standard Model. If leptons correspond to a fourth color, say `white', then we should use the colors , , and , as a basis for the vector space . This is the fundamental representation of , so let us take that group to describe color symmetries in our new GUT.
Now has an obvious inclusion into , using block
We can do even better if we start with the splitting
So, if we choose a splitting
we should again look at the subgroup that preserves this splitting.
This works very much as it did for . We want a map
If we let
via this map, the `quark part'
transforms as though it has
hypercharge : that is, it gets multiplied by a factor of .
Meanwhile, the `lepton part' transforms as though it has
hypercharge , getting multiplied by a factor of .
So, as a representation of
, we have
The right-handed leptons do not work this way. That is a problem we need to address. But this brings us to our second question, which was about the strange difference between left- and right-handed particles.
Remember that in the Standard Model, the left-handed particles live
in the fundamental rep of on
, while the right-handed
ones live in the trivial rep on
. Physicists write this
by grouping left-handed particles into `doublets', while leaving the
right-handed particles as `singlets':
If we restrict these representations to the `left-handed' subgroup, we obtain:
Now let us try to combine these ideas into a theory with symmetry group . We have seen that letting act on is a good way to unify our treatment of color for all the left-handed fermions. Similarly, the dual representation on is good for their antiparticles. So, we will tackle color by letting act on the direct sum . This space is 8-dimensional. We have also seen that letting act on is a good way to unify our treatment of isospin for left- and right-handed fermions. This space is 4-dimensional.
, and the Standard Model representation is
32-dimensional, let us take the tensor product
How can we map
are several possibilities. Our work so far suggests this option:
Let us see what this gives. The Pati-Salam representation
is a direct sum
of four irreducibles:
Given our chosen map from
we can work out which representations of the
these four spaces give.
For example, consider . We have already seen that under our
If we go ahead and do the other four cases, we see that everything works except for the hypercharges of the right-handed particles--and their antiparticles. Here we just show results for the particles:
|The Pati-Salam Model -- First Try|
|Particle||Hypercharge: predicted||Hypercharge: actual|
The problem is that the right-handed particles are getting the
same hypercharges as their left-handed brethren.
To fix this problem, we need a more clever map from
. This map must behave
differently on the factor of
, so the hypercharges
come out differently. And it must take advantage of the
right-handed copy of , which acts nontrivially only on the
right-handed particles. For example, we can try this map:
|The Pati-Salam Model -- Second Try|
|Particle||Hypercharge: predicted||Hypercharge: actual|
Miraculously, all the hypercharges match if we choose .
So, let us use this map:
Similarly, we can create a kind of `Pati-Salam code' to specify an
isomorphism of Hilbert spaces
The color comes from of course, which we already decreed to be spanned by , , and . For antiparticles, we also require anticolors, which we take to do be the dual basis , , and , spanning .
It is now easy, with our knowledge of how the Pati-Salam model is to work, to
construct this code. Naturally, the left-handed quark doublet corresponds to
the left-isospin up and down states, which come in all three colors
This table defines an isomorphism of Hilbert spaces
As with and , we can summarize all the results of this section in a commutative square:
The Pati-Salam representation and especially the homomorphism look less natural than the representation of on and the homomorphism . But appearances can be deceiving: in the next section we shall see a more elegant way to describe them.