\le l \le L_{\mathrm{max}}.$$ As such, the $N_{\max}$-truncation provides a natural pair of UV and IR cutoffs: $\Lambda_\textsc{uv} \simeq b\sqrt{N_{\max}}$, $\lambda_\textsc{ir} \simeq b/\sqrt{N_{\max}}$, where $b=\kappa$ is the oscillator basis energy scale parameter. $L_{\max}$ represents the resolution of the basis in the longitudinal direction. Namely, the basis cannot resolve physics at: $\Delta x\lesssim L_{\max}^{-1}$ [@Wiecki:2014ola]. The complete basis is reached by taking $N_{\max}\to\infty, L_{\max} \to \infty$.

The eigenvalues of the parity and charge conjugation operators can be extracted from the basis representation of the LFWFs as [@Li:2015zda], $$\begin{aligned}
 (-i)^{2j}\mathsf P =\,& \langle\psi_{-m_j}|\hat m_P | \psi_{m_j}\rangle = \sum_{n,m,l,s,\bar s} (-1)^m \psi_{-m_j}^*(n, -m, l, -s, -\bar s)
\psi_{m_j}(n, m, l, s, \bar s). \label{eqn:LFP}\\
 \mathsf C =\,& \langle\psi_{m_j}|\hat C | \psi_{m_j}\rangle = \sum_{n,m,l,s,\bar s} (-1)^{m+l+1} \psi_{m_j}^*(n, m, l, \bar s, s)
\psi_{m_j}(n, m, l, s, \bar s). \label{eqn:C}\end{aligned}$$

Generalizing Light-Front Holography
-----------------------------------

Before proceeding to the full diagonalization, it is worth looking at the results without the one-gluon exchange, where the solutions are analytical. The mass eigenvalues are: $$M^2_{n,m,l} = (m_q+m_{\bar q})^2 + 2\kappa^2(2n+|m|+l+1) + \frac{\kappa^4}{(m_q+m_{\bar q})^2}l(l+1).$$ *Here $l$ is the longitudinal quantum number, not the orbital angular momentum*. The corresponding wave functions are: $$\psi_{nml}(\vec k_\perp, x) = \phi_{nm}(\vec k_\perp/\sqrt{x(1-x)}) \chi_l(x).$$

![ Comparison 