Using $SU(3)$ Clebsch-Gordan coefficients to find the singlet state

Question

I tried to use this online CG calculator to find the singlet state in the $SU(3)$ relation: $$ \textbf{3}~\otimes~\bar{\textbf{3}} =\textbf{1}~\oplus~\textbf{8} $$ Taking the basis of $\bf 3$ and $\bar{\bf 3}$ as $|r\rangle,|g\rangle,|b\rangle$ and $|\bar r\rangle,|\bar g\rangle,|\bar b\rangle$, respectively, this calculator gives the following CG coefficients: $1/\sqrt 3, -1/\sqrt 3, 1/\sqrt 3$. I know however that the singlet is supposed to be: $$ |0\rangle = \frac{|\bar rr\rangle + |\bar gg\rangle + |\bar bb\rangle}{\sqrt 3} $$ i.e. without a minus sign. Where did I go wrong?

Answer 1

This depends very much on the phases of the states. Basically, it could be that $\vert \text{your } \bar d\rangle$ is $- \vert \text{their } \bar d\rangle$, as you suggest.

The difficulty is there is non standard phases for states so it's quite difficult to use someone else's table without first doing some checks.

IIRC this code uses for basis states the Gelfand-Tseitlin basis (which is pretty much the standard basis for $\mathfrak{su}(n)$ calculations), and the matrix elements for any irreps are actually given in a paper by the same authors. Soooo... what you need to do is construct the $(1,0)$ and the $(0,1)$ matrices (or $\boldsymbol{3}$ and $\boldsymbol{\bar 3}$ if you prefer this sadistic notation of HEP), compare the signs of the paper with your own sign convention, and infer the relative phases between their $\vert \bar r\rangle, \vert \bar g\rangle$ and $\vert \bar d\rangle$ with your states. (Constructing the $(1,0)$ isn't strictly necessary but if you can't get your $(1,0)$ matrix elements to agree with theirs you're in trouble.)

The lesson here is that, for anything beyond $\mathfrak{su}(2)$, phases have to be checked very very carefully and it's not so easy to just use existing codes as tools just "off-the-shelf".

Aside: if you need another source, there this recent C++ code:

https://gitlab.com/tdytrych/su3lib

which will give you reduced CGs (have to be multiplied by an $\mathfrak{su}(2)$ CG). The code is an upgrade of an older code and is based on the gold standard for $\mathfrak{su}(3)$:

Draayer JP, Akiyama Y. Wigner and Racah coefficients for SU 3. Journal of Mathematical Physics. 1973 Dec;14(12):1904-12.

It must be run locally though, it is also based on the GT basis, and there's no $\mathfrak{su}(n)$ version that I'm aware. The online code you're using has a different way of disentangling outer multiplicities (i.e. repeated representations); the symmetries of the Draayer code are described in the JMP paper.