Single sum of squares of Clebsch–Gordan coefficients

Question

Let $C^{j_3 m_3}_{j_1 m_1 j_2 m_2}$ be the standard Clebsch–Gordan coefficients of $\operatorname{SU}(2)$. They obey the orthogonality relation $$ \sum_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (m_3 - m_1)} \right)^2 = 1.$$ My question is about what can be said if I remove the sum over $j_3$. Does there exist a bound for $$ \max_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 (m_3 - m_1)} \right)^2 $$ in terms of $m_1$, $j_1$ and $j_2$? Of course it is $\leq 1$, but I am interested in whether this expression decays in $j_1$ and $j_2$, or some combination thereof. Positive or negative statements, or a reference, would be very useful.

Answer 1

Consider the sum of Clebsch–Gordan coefficients,$^\ast$ $$J= \sum_{m_3=-j_3}^{j_3} \left(C^{j_3,m_3}_{j_1, m_1; j_2, (m_3 - m_1)} \right)^2$$ with $2j_1,2j_2,2j_3\in\mathbb{N}$ and $2m_1\in\mathbb{Z}$. For an nonvanishing sum we also need $j_3\in\{|j_1-j_2|,|j_1-j_2|+1,\ldots j_1+j_2-1, j_1+j_2\}$ and $m_1\in\{-j_1,-j_1+1,\ldots,j_1-1, j_1\}$.
With these restrictions the sum is independent of $m_1$, so I may set $m_1=j_1$. The sum also increases monotonically with increasing $j_3$, reaching its maximal value $J_{\rm max}$ for $j_3=j_1+j_2$, hence $$J_{\rm max}=\sum_{m=-j_2}^{j_2} \left(C^{j_1+j_2,j_1+m}_{j_1, j_1; j_2, m} \right)^2=\sum_{m=-j_2}^{j_2}\frac{\Gamma (2 j_2+1) \Gamma (2 j_1+j_2+m+1)}{(1+2j_2)\Gamma (2 j_1+2 j_2+1) \Gamma (j_2+m+1)}$$ $$\qquad=\frac{1+2j_1+2j_2}{(1+2j_1)(1+2j_2)}.$$

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_{$^\ast$ Mathematica normalizes these coefficients such that $\sum_{j_3}J=1+2j_2$. Here I use the normalization of the OP, where $\sum_{j_3}J=1$.}