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5.4.2 Flux-Tube Model

 The flux-tube model was originally considered in order to account for the mass spectra of hadrons and their strong decay processes which are related to quark pair creations. At first, the creations of quark pairs were assumed arbitrarily with appropriate operators resulting in models such as the 3P0 or the 3S1 model. However, it was thought later that the quark pair creations were controlled by gluonic degrees of freedom. It is well known that the gluonic degrees of freedom in bound state problems are not so simple as to be described by perturbative calculations only. The non-perturbative feature of gluonic interactions is one of the motives for consideration of the simple flux-tube formalism. The description of gluonic flux-tube was firstly attempted by a string picture. In quark pair creation model, the created quark pair breaks a flux-tube with equal probability amplitude anywhere along the string and in any state of string oscillation. The amplitude to decay into a particular final state is assumed to depend on the overlap of original wave functions of quarks and string with the final two state wave functions separated by the pair creation. For ground state strings connecting quark and antiquark in mesons, the amplitude $\gamma(\xi\vec{r}, \vec{\omega})$ to break at point $\frac{\vec{r}} {2} + \vec{\omega}$ was first assumed to be[28]

\begin{displaymath}\gamma(\xi\vec{r}, \vec{\omega}) = \omega_0 e^{-\frac{b}{2} \omega^{2}_{min}}
\end{displaymath} (5.19)

where $\gamma_0$ and b are parameters and

\begin{displaymath}\frac{\vec{r}} {2} + \vec{\omega} = \vec{r}_{\bar{q}} + \xi\vec{r} + \vec{\omega}_{min}
\end{displaymath} (5.20)

with $\vec{r} = \vec{r}_q - \vec{r}_{\bar{q}}$ being the difference between the quark and the antiquark position vectors. $\vec{\omega}_{min}$ is the vector with the shortest distance between any point on the vector $\vec{r}$ and the pair creation point, and $\xi $ measures the ratio of the distances from the original antiquark and quark positions to the projection point of the created quark pair along the vector $\vec{r}$ . The shape of equi-$\gamma $ surface looks like a cigar which is appropriate to describe a flux-tube of constant width with end caps. For general shapes, we can introduce a $\xi $-dependent factor $f(\xi)$ into the exponent of $\gamma $ such that

\begin{displaymath}\gamma_{f} = \gamma_0 e^{-\frac{1}{2} f(\xi)b\omega^{2}_{min}} .
\end{displaymath} (5.21)

Although it has been found that physical results are nearly independent of $f(\xi)$ for $\frac{1}{3} < f < 3$, the arbitrariness of $f(\xi)$ raises the problem of theoretical basis for the derivation of flux-tube overlap function $\gamma $. In fact, the form in Eq.(4-2-3) was derived by using harmonic oscillator wave functions for discrete string components. The form was even changed into the spherical one[29]

\begin{displaymath}\gamma(\vec{r}, \vec{\omega}) = \gamma_0 e^{-\frac{b}{2}\omega^2}
\end{displaymath} (5.22)

which is convenient for calculating physical amplitudes expanded in the harmonic oscillator wave function basis. The changes in the form of $\gamma $ indicate the fact that no firm theoretical grounds exist for treating gluonic flux-tubes.
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Next: 5.4.3 Construction of Topological Up: 5.4 Non-perturbative Topics Previous: 5.4.1 Jet Overlapping
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