2 \nu)^2},$$ $$\tan\lambda(E,\ell=1)\stackrel{E\to 0}{\longrightarrow} \frac{1+\nu(s-\nu)}{s-2\nu},$$ $$\tan\xi(E,\ell=1)\stackrel{E\to 0}{\longrightarrow} k^3 \overline{V} \frac{(1+\nu^2)(1-2s \nu+\nu^2)}{(s-2\nu)(1-3\nu^2)},$$ where we have defined the mean $p$-wave scattering volume $$\overline{V}=\frac{\pi}{9}\frac{(n-2)^{(n-8)/(n-2)}}{\Gamma^2\left(\frac{3}{n-2}\right)}R_n ^3.$$ We note that in the case of $n=4$, in the $p$-wave case one has to add the $\propto k^2$ term to the phase shift coming from the long range nature of the potential, as discussed in [@Sadeghpour2000]. This term gives the leading order contribution and modifies the threshold behavior of the elastic rate constant.

Elastic and reactive rate
-------------------------

Formulas derived in the previous section enable us to calculate the low energy limits of the reactive and elastic rate constants using - and some algebraic transformations. For the reactive rate constant we obtain $$\begin{aligned}
\label{Kl0}
\mathcal{K}^{\mathrm}{re}_{00} & \stackrel{E\to0}{\longrightarrow}  2 g \frac{h}{\mu} \bar{a} y \frac{1+(s-\nu)^2}{1+y^2(s-\nu)^2}, \\
\label{Kl1}
\mathcal{K}^{\mathrm}{re}_{1m} & \stackrel{E\to0}{\longrightarrow}  2 g \frac{h}{\mu } k^2 \overline{V} y \frac{1+\nu^2}{\nu^2} \frac{1+(s-\nu)^2}{y^2(s-\nu+\nu^{-1})^2+(s\nu^{-1}-2)^2},\end{aligned}$$ while for the elastic one $$\label{Ke0}
\mathcal{K}^{\mathrm}{el}_{00} \stackrel{E\to0}{\longrightarrow}  2 g \frac{h}{\mu} k\bar{a}^2 \frac{s^2+y^2\left(1+\nu^2-s\nu\right)^2}{1+y^2(s-\nu)^2},$$ $$\label{Ke1}
\begin{split}
\mathcal{K}^{\mathrm}{el}_{1m} \stackrel{E\to0}{\longrightarrow}  2 g \frac{h}{\mu} k^5 \overline{V}^2 \left(\frac{1+\nu^2}{1-3\nu^2}\right)^2 \times \\ \times \frac{(1-2s\nu+\nu^2)^2+y^2(s+\nu-s\nu^2+\nu^3)^2}{(s-2\nu)^2+y^2(1+(s-\nu)\nu)^2}.
\end{split}$$

We note that in the universal regime ($y=1$) all the above formulas reduce to the form which is independent of the $s$ parameter $$\mathcal{K}^{\mathrm}{re}_{00} \to  2 g \frac{h}{\mu} \bar{a}, \,\,\,\,\, \mathcal{K}^{\mathrm}{re}_{1m} \to  