# Asymptotics of the Wigner $6j$-Symbol

So, in doing some numerical computations in QFT, I've run into the following Wigner $$6j$$-Symbol:

$$\left\{ \begin{array}{ccc} x & J_1 & J_2 \\ \frac{N}{2} & \frac{N}{2} & \frac{N}{2} \\ \end{array} \right\}.$$

In the regime where $$x \ll J_1,J_2,N$$ and $$J_1 \approx J_2 \approx N$$, and $$N$$ is large. I would like to know if there is an asymptotic formula for such a symbol, or if one can be derived. Using symmetries we can get

$$\left\{ \begin{array}{ccc} x & \frac{1}{2} \left(J_1+J_2\right) & \frac{1}{2} \left(J_1+J_2\right) \\ \frac{N}{2} & \frac{1}{2} \left(N+J_1-J_2\right) & \frac{1}{2} \left(N-J_1+J_2\right) \\ \end{array} \right\}.$$

Perhaps this could help, I'm really not sure.

The source for this is the book of Varshalovich et al, Quantum Theory of angular momentum. In section 9.8 one can find the following: $$\left\{\begin{array}{ccc} a&b&c\\ d+R&e+R& f+R\end{array}\right\} \approx \frac{(-1)^{a+b+d+e}}{\sqrt{2R(2c+1)}}C^{c\gamma}_{a\alpha;b\beta}$$ where $$C^{c\gamma}_{a\alpha;b\beta}$$ is a Clebsch Gordan coefficient, and where $$\alpha=f-e, \beta=d-f, \gamma=d-e$$. This expression is valid in the limit where $$R\gg 1$$.

(I have never personally used this but with a few simple test using Mathematica gives a pretty good estimate. For instance, with $$(a,b,c,d,e,f)=(3,3,2,2,4,4)$$ and $$R=75$$, the $$6j\approx -0.01763$$ while the approximate expression gives $$-0.01781$$.)

For the asymptotic behavior of the Wigner 6j-Symbol when all the coefficients but one grow, you can use the Edmonds formula. In your case it reads as:

$$\left\lbrace\begin{matrix} x & J_1 & J_2 \\ \frac{N}{2} & \frac{N}{2} & \frac{N}{2}\end{matrix}\right\rbrace \approx \dfrac{(-1)^{J_2+N+x}}{\sqrt{(2J_2+1)(N+1)}}d^x_{J1-J2, 0}(\phi),$$

where $$d^x_{J1-J2, 0}(\phi)$$ is the small Wigner d-matrix and

$$\cos(\phi)=\frac{1}{2}\sqrt{\dfrac{J_1(J_1+1)}{N/2(N/2+1)}}.$$

Here it is a good reference https://aip.scitation.org/doi/10.1063/1.532474 .

• Welcome !Could you elaborate the key points of this publication here? This would make your answer within this thread a lot more useful. Commented Oct 23, 2020 at 8:57