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Generations and Chan-Paton factors

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Al.R...@gmail.com

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Jul 7, 2007, 4:42:00 AM7/7/07
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Consider QCD. Consider the QCD scale, which really are two scales in
the theory -the chyral scale and the dimensional transmutation of QCD
coupling- and two scales in practice: the pion and the proton. It is a
mistery why the charged leptons (and their related cousins, some 2nd
and 3rd generation quarks) like to live in these scales too. And we
are not going to solve it now. There is no law of the physics asking
them to live there precisely, just keep this detail in your mind.

Consider the QCD string. We can attach quarks to the extremes of the
string. This was called the Chan-Paton, or Paton-Chan factor, in the
old times, and nowadays is something related to D-Branes, the singular
manifolds where the string terminates. It is not by random than string
physicsts like to walk to Benasque Hospital, where the road finishes.
Kind of D-brane feeling.

But quarks have mass, and then in order to attach them at the ends of
the string we have a clear distintion: those which have masses a lot
greater than the QCD string scale, and those which have a mass similar
or less than the QCD string scale.

I thought about this back in 2005. I do not like the result. When I
was younger, I liked to think that generations were Nature's answer to
the question of quantization ambiguity, by providing three different
scales to calculate differentials. Today I like to think that
generations have some deep algebraic or geometric reason, related to
Bott periodicity and to the Leech lattice, or to the GUT groups, or to
NCG, or all together. Also I know that we need generations to have CP
violation, but it could be more a consequence than a cause.

So here you have the postulate: the number of different terminated
strings, which marks different Regge trajectories starting with spin 0
bosons, must coincide with the number of different standard model
fermions having the same charge. Of course spin counts: a positron has
TWO states. It can even be an useful postulate. For instance if you
formulate a theory with some technihiggs whose coupling to the string
is of the same order than its coupling to fermions, then the
perturbative corrections to it can be arranged to cancel as in
supersymmetry.

Of course you have a small version and a big version of the postulate:
you can ask it only for quarks, or also extend it to leptons. Only the
extension can be used for the cancellation mentioned above, but even
the quark-only version already decrees than the number of generations
must be equal or greater than 3, and that if it is n=3 then there is
only a very massive quark, in the UP species.

So, equations:

Let n be the number of generations, r the number of light quarks of
the "down" species, and s the number of light quarks of the "up"
species. Both quarks and antiquarks can be attached to the ends of the
string. Perhaps we are surpassing Chan and Paton slightly here, but we
do not want limit ourselves to mesons. Or, we tell that when one
string divides into two, the charges in the new line do not need to
keep the new strings uncoloured.

Obviously
0 <= r <=n
0 <= s <=n

for charge +1/3 (string u-d)

2 n = r*s

for charge -2/3 (string d-d)

2 n = r*(r+1)/2

and obviously r,s and n are integers, so the above equation alone has
solutions n=3, 14, 33, ...

Lets solve the pair to build the detailed table. Using 2 s=(r+1) and
2n=r*s

r s n
0 x x
1 1 x
2 x x
3 2 3
5 3 x
7 4 14
9 5 x
11 6 33
13 7 x
15 8 60
.. ... ...
4k+3 2k+2 4k^2+7k+3

So the smallest solution is n=3 and implies r=3, s=2, that means 5
light flavours and one "up"-like quark beyond the QCD scale. This is
the structure of the standard model. Thus, the SM gauge groups and
representations plus our postulate above imply the number of
generations and single out the special role of the top quark.

Now consider leptons. For charge +1 we have the equation 2n= r*s, same
as above, so no new information is added except that the postulate
keeps working.

For charge 0, hmm, we need a small trick: to argue that, as it happens
with the flavour octet, one degree of freedom is lost. In that way we
have the equation
4 n= r^2+s^2-1
It is tentative, because we do not know a lot about the neutrino
sector. But if taken as true, it fixes n=3.

All the setup can be a coincidence, but it is interesting to keep an
eye on it. Even if the real origin of generations comes from other
source, as we all expect.

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