Consider a model with an exact ##SU(N_{TC})## techni-color symmetry and a ##SU(N_{TF})_L\otimes SU(N_{TF})_R## global techni-flavour symmetry which is spontaneously broken to the diagonal sub-group ##SU(N_{TF})## by condensates producing techni-pions (TC\pi) and techni-baryons(TCb).(adsbygoogle = window.adsbygoogle || []).push({});

What i'm trying to understand is how the various TCb transform under ##SU(N_{TF})## and ##SU(2)_{spin}##.

Because the wave-function is totally antisymmetric in techni-color we expect that TCb must be fully symmetric in spin and techni-flavour. Following the line of thought of Georgi's chapter 15 I have embedded ##SU(N_{TF})## and ##SU(2)_{spin}## in a ##SU(2N_{TF})##. The TCb transform under this ##SU(2N_{TF})## like a ##N_{TC}## completely symmetric combination, or in Young tableaux notation like a tableaux with ##N_{TC}## horizontal box. Now Georgi says that to understand what representation of ##SU(N_{TF})## and ##SU(2)_{spin}## are contained in the fully symmetric representation of ##SU(2N_{TF})## we must take the tensor product of all tableaux with ##N_{TC}## boxes of ##SU(N_{TF})## and ##SU(2)_{spin}## and see if they contain the fully symmetric representation. The results (see for example Chivukula, R.S. & Walker, T.P., 1989.Technicolor cosmology, Boston, MA: Boston Univ.) should be the following:

-For ##N_{TC}=3##

-For ##N_{TC}=4##

The problem is that, except for the ##N=3## case, I'm not able to obtain the fully symmetric representation by taking the tensor product of this tableaux.

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# Technibaryon transformation

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