= \frac{1}{2}\Big(\sqrt{\frac{x'}{x}}\vec k_\perp-\sqrt{\frac{x}{x'}}\vec k'_\perp\Big)^2 +
\frac{1}{2}\Big(\sqrt{\frac{1-x'}{1-x}}\vec k_\perp-\sqrt{\frac{1-x}{1-x'}}\vec k'_\perp \Big)^2 
+ \frac{1}{2}(x-x')^2\Big( \frac{m_q^2}{xx'}+\frac{m_{\bar q}^2}{(1-x)(1-x')}\Big) + \mu_g^2.$$

Longitudinal Confinement
------------------------

In Eq. (\[eqn:Heff\]), the term $\kappa^4 \vec\zeta^2_\perp \equiv \kappa^4 x(1-x)\vec r^2_\perp$ is the “soft-wall” confinement from light-front holography, which is introduced in the massless case. For heavy quarkonium, the quark masses and the longitudinal dynamics cannot be ignored[^2] and we introduce a longitudinal confining interaction to complete the transverse holographic confinement. The form of the longitudinal confinement is designed to produce a power-law behavior for the distribution amplitudes $\phi(x)\sim x^a(1-x)^b$ at the endpoints (cf. [@Gutsche:2012ez; @Gutsche:2013zia; @Chabysheva:2012fe]).

We fix the strength of the longitudinal confinement by matching to the transverse holographic confinement in the non-relativistic limit. Therefore, rotational symmetry is retained in the heavy-quark limit. Another advantage of this choice for the longitudinal confinement is that it produces, without the one-gluon exchange, analytic solutions. Therefore, it affords computational convenience within the basis function method (see Sect. \[sec:3.1\]). In the massless limit, our wave function (without the one-gluon exchange) reduces to the soft-wall wave function of Brodsky and de Téramond[^3] [@Brodsky:2014yha]. It has been suggested that in the massless limit one can choose the longitudinal confining strength to be independent of the quark 