Quark-Antiquark Bound State in Three-Dimensional Approach

Abstract

In this paper we develop a three-dimensional approach for describing meson bound states based on the momentum-helicity basis. To this end, we formulate the relativistic form of Lippmann-Schwinger equation in the momentum-helicity basis which leads to two sets of integral equations. Then we have solved these integral equations by inserting a spin dependent quark-antiquark potential model numerically as eigenvalue equations. Finally, we obtain the mass spectrum of the light mesons and compare these results with the results which are obtained in the partial wave representation and experimental data.

Appendix A: Connection to the Partial Wave Representation

In this section we make a connection between the two-body wave function components in the momentum-helicity basis and the partial wave projected components of wave function. For this purpose, we insert the completeness relation of the partial wave basis in the momentum-helicity representation of the wave function and we obtain:

$$\begin{array}{@{}rcl@{}} \langle\textbf{p};\hat{\textbf{p}}S\Lambda|\Phi_{j}^{M_{j}}\rangle&=&\sum\limits_{lS^{\prime}M^{\prime}_{j}}\int_{0}^{\infty} dp^{\prime}p^{\prime 2}\langle\textbf{p};\hat{\textbf{p}}S\Lambda|p^{\prime}(l S^{\prime})j M^{\prime}_{j}\rangle\langle p^{\prime}(l S^{\prime})j M^{\prime}_{j}|\Phi_{j}^{M_{j}}\rangle. \end{array} $$

(3.8)

The scalar product of the the momentum-helicity and the partial wave basis states is given by [12]:

$$\begin{array}{@{}rcl@{}} \langle\textbf{p};\hat{\textbf{p}}S\Lambda|p^{\prime}\,(l S^{\prime})j M^{\prime}_{j}\rangle&\!=\!&\sqrt{\frac{2l + 1}{4\pi}}\,\frac{\delta(p\!-\!p^{\prime})}{pp^{\prime}}\,\delta_{SS^{\prime}}\,e^{iM^{\prime}_{j}\varphi} \,d^{j}_{M^{\prime}_{j}\Lambda}(\theta)\,C(lSj;0\Lambda\Lambda). \end{array} $$

(3.9)

Inserting (3.9) into (3.8) yields:

$$\begin{array}{@{}rcl@{}} \langle\textbf{p};\hat{\textbf{p}}S\Lambda|\Phi_{j}^{M_{j}}\rangle&=&\,e^{iM_{j}\varphi}\,d^{j}_{M_{j}\Lambda}(\theta) \sum\limits_{l}\sqrt{\frac{2l + 1}{4\pi}}\,C(lSj;0\Lambda\Lambda)\,\mathrm{\Psi}_{lSj}(p), \end{array} $$

(3.10)

where we use $\langle p\,(l S)j M^{\prime }_{j}|\Phi _{j}^{M_{j}}\rangle = \delta _{M^{\prime }_{j}M_{j}}\mathrm {\Psi }_{lSj}(p)$. Ψ _{l S j} (p) are the partial wave components of wave function. In comparison with (2.6) and (2.12), (3.10) can be rewritten as:

$$\begin{array}{@{}rcl@{}} &&\Phi^{\Lambda M_{j}}_{jS}(p,\theta)=d^{j}_{M_{j}\Lambda}(\theta)\sum\limits_{l}\sqrt{\frac{2l + 1}{4\pi}}\,C(lSj;0 \Lambda\Lambda)\, \mathrm{\Psi}_{lSj}(p), \end{array} $$

(3.11)

$$\begin{array}{@{}rcl@{}} &&\Phi^{\Lambda}_{jS}(p)=\sum\limits_{l}\sqrt{\frac{2l + 1}{4\pi}}\,C(lSj;0\Lambda\Lambda)\,\mathrm{\Psi}_{lSj}(p), \end{array} $$

(3.12)

respectively. These two equations connect the partial wave and the momentum-helicity representation of wave function to each other. The inverse of these equations can be obtained as bellow. We insert the completeness relation of the momentum-helicity basis in the partial wave representation of wave function and we obtain:

$$\begin{array}{@{}rcl@{}} \langle p\,(l S)j M_{j}|\Phi_{j}^{M_{j}}\rangle&=&\sum\limits_{S^{\prime}\Lambda}\int d\textbf{p}^{\prime}\langle p\,(l S)j M_{j}|\textbf{p}^{\prime};\hat{\textbf{p}}^{\prime}S^{\prime}\Lambda\rangle\langle\textbf{p}^{\prime}; \hat{\textbf{p}}^{\prime}S^{\prime}\Lambda|\Phi_{j}^{M_{j}}\rangle. \end{array} $$

(3.13)

By inserting (3.9) and integration over p ^′, we have

$$\begin{array}{@{}rcl@{}} \mathrm{\Psi}_{lSj}(p)&\!=\!&2\pi\sqrt{\frac{2l\!+\!1}{4\pi}}\sum\limits_{\Lambda}\,C(lSj;0\Lambda\Lambda) \int_{-1}^{1} d \cos\theta^{\prime}\,d^{j}_{M_{j}\Lambda}(\theta^{\prime})\Phi^{\Lambda M_{j}}_{jS}(p,\theta^{\prime}). \end{array} $$

(3.14)

From (2.6) and (2.12), we can explicitly show

$$\begin{array}{@{}rcl@{}} \Phi^{\Lambda M_{j}}_{jS}(p,\theta^{\prime})&=&d^{j}_{M_{j}\Lambda}(\theta^{\prime}) \Phi^{\Lambda}_{jS}(p). \end{array} $$

(3.15)

Thus, (3.14) can be written as

$$\begin{array}{@{}rcl@{}} \mathrm{\Psi}_{lSj}(p)&=&\frac{\sqrt{4\pi(2l + 1)}}{2j + 1}\sum\limits_{\Lambda}\,C(lSj;0\Lambda\Lambda) \Phi^{\Lambda}_{jS}(p), \end{array} $$

(3.16)

where we have used the orthogonality relation for the rotation d-matrices

$$\begin{array}{@{}rcl@{}} \int_{-1}^{1}d \cos\theta^{\prime}\,d^{j}_{M_{j}\Lambda}(\theta^{\prime})\,d^{j}_{M_{j}\Lambda}(\theta^{\prime})= \frac{2}{2j + 1}. \end{array} $$

(3.17)