Three Quarks Confined by an Area-Dependent Potential in Two Dimensions

Abstract

We study the low-lying parts of the spectrum of three-quark states with definite permutation symmetry bound by an area-dependent three-quark potential. Such potentials generally confine three quarks in non-collinear configurations, but classically allow for free (unbound) collinear motion. We use our previous work to evaluate the low-lying parts of the spectrum in a non-adiabatic approximation. We show that the eigen-energies are positive and discrete, i.e., that the system is quantum-mechanically confined in spite of the classically allowed free collinear motion.

Appendix

Equation (1) can be re-written as a function of (the absolute value of) only one O(3) (hyper-)spherical harmonic in the shape (hyper-)space: the $|{Y}_{10}(\alpha , \phi )|$:

$$\begin{aligned} \frac{2}{R^2} |{\varvec{\rho }} \times {\varvec{\lambda }}|= & {} |\cos \alpha | = \sqrt{\frac{4 \pi }{3}} |{Y}_{10}(\alpha , \phi )| . \ \end{aligned}$$

(6)

Now, the absolute value of $|{Y}_{10}(\alpha , \phi )|$ can be expressed as $\sqrt{{Y}_{10}^{*}(\alpha , \phi ){Y}_{10}(\alpha , \phi )}$ and the O(3) Clebsch–Gordan expansion can be applied to ${Y}_{10}^{*}(\alpha , \phi ){Y}_{10}(\alpha , \phi )$, which contains only the (obviously even) values of $L=0,2$, as in Eq. (A12) of Ref. [3].

$$\begin{aligned} \frac{2}{R^2} |{\varvec{\rho }} \times {\varvec{\lambda }}|= & {} \sqrt{\frac{1}{3}} \sqrt{1 + \frac{2}{\sqrt{5}} \frac{Y_{20}(\alpha , \phi )}{{Y}_{00} (\alpha , \phi )}}. \ \end{aligned}$$

(7)

The square root can be expanded in a Taylor-like series, the first two terms of which coincide with the expansion in Legendre polynomials, or O(3) spherical harmonics, and for $L=0$, even in O(4) hyper-spherical harmonics

$$\begin{aligned} \frac{2}{R^2} |{\varvec{\rho }} \times {\varvec{\lambda }}|= & {} \sqrt{\frac{1}{3}} \left( 1 + \frac{1}{\sqrt{5}} \frac{Y_{20}(\alpha , \phi )}{{Y}_{00} (\alpha , \phi )} + \cdots \right) . \ \end{aligned}$$

(8)

Manifestly the Legendre polynomial expansion, Eq. (8) is limited to even-order $J=0,2,4, \ldots$ terms only,

$$\begin{aligned} V_\mathrm{HY}(R, \alpha , \phi )= & {} \frac{k}{2} \left( v_{a} ({\varvec{\rho }}^2 + {\varvec{\lambda }}^2) + v_{b} |{\varvec{\rho }} \times {\varvec{\lambda }}| \right) . \end{aligned}$$

(9)

$$\begin{aligned}= & {} \frac{k}{2} R^2 \left( v_{a} + v_{b} \frac{1}{2}\sqrt{\frac{1}{3}} \left( 1 + \frac{1}{\sqrt{5}} \frac{Y_{20}(\alpha , \phi )}{{Y}_{00} (\alpha , \phi )} + \cdots \right) \right) \nonumber \\= & {} \frac{k}{2} R^2 \frac{v_0^{HY}}{\sqrt{4 \pi }} \left( 1 + \frac{v^{HY}_2}{v^{HY}_0} \sqrt{{4 \pi }} {Y}_{20}(\alpha , \phi ) + \cdots \right) . \ \end{aligned}$$

(10)

Note, however, that $v_{b}/v_{a} \ne v^{HY}_2/v^{HY}_0$. In particular the additive constant in the expansion Eq. (8) is important, as it ensures the (overall) positivity of this potential and leads to the change of “effective couplings”

$$v_{00}^{HY} = \sqrt{4\pi } \left( v_{a} + v_{b} \frac{1}{2}\sqrt{\frac{1}{3}} \right) ,$$

and

$$v_{2}^{HY} = v_{b} \frac{1}{2}\sqrt{\frac{4 \pi }{15}} .$$

These two equations in turn lead to

$$\frac{v_{20}^{HY}}{v_{00}^{HY}} = \frac{v_{b} \frac{1}{2}\sqrt{\frac{4 \pi }{15}}}{\sqrt{4\pi } \left( v_{a} + v_{b} \frac{1}{2}\sqrt{\frac{1}{3}} \right) } = \frac{v_{b}}{{2}\sqrt{15} \left( v_{a} + v_{b} \frac{1}{2}\sqrt{\frac{1}{3}} \right) },$$

and in particular in the $v_a \rightarrow 0$ limit, this ratio for the HY potential equals that of the pure area potential:

$${\lim }_{v_a \rightarrow 0} \left( \frac{v_{20}^{HY}}{v_{00}^{HY}}\right) = \frac{1}{\sqrt{5}}.$$