or $\beta$ from now on.

In the presence of the one-gluon exchange term, we use these analytic functions as a basis to expand the LFWFs in, $$\label{eqn:basis_representation}
 \psi_{ss'/h}(\vec k_\perp, x) = \sum_{n, m, l} \psi_h(n, m, l, s, s') \, \phi_{nm}(\vec k_\perp/\sqrt{x(1-x)}) \chi_l(x).
$$ Here the coefficients $ \psi_h(n, m, l, s, s')$ are obtained from diagonalization. The basis is constructed to conserve the magnetic projection of the total angular momentum: $ m_j = m + s + s'$.

Performing a 2D Fourier transformation gives the LFWFs in coordinate space. The Fourier transformation of a HO function is a HO function with a relative phase, which simplifies the expression greatly. $$\widetilde \psi_{ss'/h}(\vec r_\perp, x) = \sqrt{x(1-x)} \sum_{n, m, l} \psi_h(n, m, l, s, s') \, \widetilde \phi_{nm}(\sqrt{x(1-x)} \vec
r_\perp) \chi_l(x).$$ Here $\widetilde \phi_{nm}$ is the 2D HO in coordinate space: $$\widetilde \phi_{nm}(\vec \rho_\perp; b^{-1}) = b \sqrt{\frac{n!}{\pi(n+|m|)!}} (b\rho_\perp)^{|m|} \exp\big(-b^2\rho^2_\perp/2\big)
L_n^{|m|}(b^2\rho_\perp^2) \exp\big[ \imag m\theta_\rho + \imag \pi (n+|m|/2) \big].$$

In practical calculations, the basis is truncated and wave functions are obtained in the basis expansion. Following Refs. [@Vary:2009gt; @Wiecki:2014ola; @Li:2015zda], we truncate the transverse and the longitudinal bases separately by their energies: $$2n + |m| + 1 \le N_\mathrm{max}, \quad 0 