SU(3) for "color"?

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Raphanus - 26 Jul 2008 17:23 GMT

Why is SU(3) assumed as the exact symmetry for the color quantum
number rather than SO(3)?
I understand SU(3) is an inexact symmetry for "flavor" but is there
some"deep" reason that "color" has to be SU(3)? Some experimental
determination? Thanks.

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Tom Roberts - 27 Jul 2008 02:35 GMT

> Why is SU(3) assumed as the exact symmetry for the color quantum
> number rather than SO(3)?

The standard model is inherently a quantum theory, which means it uses
complex numbers in essential ways. So SU(3) is the relevant group, not
SO(3).

> I understand SU(3) is an inexact symmetry for "flavor" but is there
> some"deep" reason that "color" has to be SU(3)?

Yes. The color SU(3) is (a subgroup of) the gauge group of the theory.
There are lots of diagrams that intermix flavors, so no flavor symmetry
can ever be exact for observable particles.

    Well, for hadrons at least. In the original standard model
    the neutrinos were massless and did not mix, and this let
    the lepton flavor symmetries be exact. Now that neutrinos
    have nonzero masses and do mix, lepton flavor conservation
    is not exact. For example, there is a nonzero probability
    for a muon to convert directly into an electron, as long
    as other symmetries can be satisfied (e.g. conservation of
    momentum require the presence of a nucleus to "absorb" the
    excess 4-momentum).

In the standard model, only the Higgs breaks the color symmetry.

> Some experimental
> determination?

Well, yes. Experiments confirm the standard model quite well. In it the
gauge symmetry is broken only by the Higgs, but the flavor symmetries
are not at all exact; in the hadron sector this is due to electroweak
interactions, and in the lepton sector this is due to neutrino mixing.

This is my understanding. I am not an expert on the
inner details of the standard model.

Tom Roberts

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Raphanus - 27 Jul 2008 04:08 GMT

> > Why is SU(3) assumed as the exact symmetry for the color quantum
> > number rather than SO(3)?
[quoted text clipped - 34 lines]
>
> Tom Roberts

I thank you for your answer. I have a bit more insight. For
representations of groups we use matrices with complex elements.
SO(3) inherently has real elements whereas SU(3) can have complex
numbers. But is it not true - that since we can represent i =sqrt(-1)
by a 2x2 real matrix that SU(3) can be mapped isomorphically into
SO(6)? Also what if we allowed quaternions as matrix elements? I
know that Gellman's 8 matrices are the generators of SU(3).

Are the number of generators dependent on the numerical field used in
the elements of the generators? Would the generators of SU(3) still
be 8 if we allowed only real elements in our representation?

Also, if I understand your answer the group representation of "color"
as SU(3) rests upon the success of the standard model?

"Color" - if we take the Omega minus as evidence - is an experimental
fact. Maybe my confusion is contained in the following question:

"Is a symmetry group an inherent part of nature or is it an inherent
part of our description of nature?"

Maybe it's philosophy :-) I thank you again..

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Natural Science Forum / Physics / Particle Physics / July 2008

SU(3) for "color"?