and the wave functions are obtained from diagonalizing the effective light-front Hamiltonian operator (\[eqn:Heff\]): $$\label{eqn:eigenvalue_equation}
H_\text{eff} |\psi_h(P, j, m_j)\rangle = M^2_h |\psi_h(P, j, m_j)\rangle.$$ where $P=(P^-, P^+, \vec P_\perp)$ is the 4-momentum of the particle; $j$ and $m_j$ are the particle’s total angular momentum and the magnetic projection, respectively.

The Fock space representation of quarkonium reads: $$\begin{gathered}
\label{eqn:Fock_expansion}
 |\psi_h(P, j, m_j)\rangle = 
\sum_{s, \bar s}\int_0^1\frac{\dd x}{2x(1-x)} \int \frac{\dd^2 k_\perp}{(2\pi)^3}
\, \psi^{(m_j)}_{s\bar s/h}(\vec k_\perp, x) \\
\times \frac{1}{\sqrt{N_c}}\sum_{i=1}^{N_c} b^\dagger_{s{}i}\big(xP^+, \vec k_\perp+x\vec P_\perp\big) d^\dagger_{\bar s{}i}\big((1-x)P^+,
-\vec k_\perp+(1-x)\vec P_\perp\big) |0\rangle. \end{gathered}$$ The coefficients of the expansion, $\psi^{(m_j)}_{s\bar s/h}(\vec k_\perp, x)$ are the valence sector LFWFs with $s$ ($\bar s$) representing the spin of the quark (antiquark). The quark and anti-quark creation operators $b^\dagger$ and $d^\dagger$ satisfy the canonical anti-commutation relations, $$\label{eqn:CCR}
 \big\{ b_{si}(p^+, \vec p_\perp), b_{s'i'}^\dagger(p'^+, \vec p'_\perp) \big\} 
= \big\{ d_{si}(p^+, \vec p_\perp), d_{s'i'}^\dagger(p'^+, \vec p'_\perp) \big\} 
= 2p^+(2\pi)^3\delta^3(p-p')\delta_{ss'}\delta_{ii'},$$ where $\delta^3(p-p') \equiv \delta(p^+-p'^+)\delta^2(\vec p_\perp-\vec p'_\perp)$. We have kept only the $q\bar q$ sector while, in principle, the $q\bar qg$ sector can be included by, e.g., a perturbative treatment [@Leitner:2010nx]. The hadron state vector can be orthonormalized according to the one-particle state \[cf. Eq. (\[eqn:CCR\])\]: $$\label{eqn:normalizaiton_of_state_vector}
 \langle \psi_h(P, j, m_j) | \psi_{h'}(P', j', 