Meson production

Once the flavours $\mathrm{q}_{i-1}$ and $\overline{\mathrm{q}}_i$ have been selected, a choice is made between the possible multiplets. The relative composition of different multiplets is not given from first principles, but must depend on the details of the fragmentation process. To some approximation one would expect a negligible fraction of states with radial excitations or non-vanishing orbital angular momentum. Spin counting arguments would then suggest a 3:1 mixture between the lowest lying vector and pseudoscalar multiplets. Wave function overlap arguments lead to a relative enhancement of the lighter pseudoscalar states, which is more pronounced the larger the mass splitting is [And82a].

In the program, six meson multiplets are included. If the nonrelativistic classification scheme is used, i.e. mesons are assigned a valence quark spin $S$ and an internal orbital angular momentum $L$, with the physical spin $s$ denoted $J$, $\mathbf{J} = \mathbf{L} + \mathbf{S}$, then the multiplets are:

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$L = 0$, $S = 0$, $J = 0$: the ordinary pseudoscalar meson multiplet;

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$L = 0$, $S = 1$, $J = 1$: the ordinary vector meson multiplet;

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$L = 1$, $S = 0$, $J = 1$: an axial vector meson multiplet;

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$L = 1$, $S = 1$, $J = 0$: the scalar meson multiplet;

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$L = 1$, $S = 1$, $J = 1$: another axial vector meson multiplet; and

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$L = 1$, $S = 1$, $J = 2$: the tensor meson multiplet.

Each multiplet has the full five-flavour setup of $5 \times 5$ states included in the program. Some simplifications have been made; thus there is no mixing included between the two axial vector multiplets.

In the program, the spin $S$ is first chosen to be either 0 or 1. This is done according to parameterized relative probabilities, where the probability for spin 1 by default is taken to be 0.5 for a meson consisting only of $\u$ and $\d$ quark, 0.6 for one which contains $\mathrm{s}$ as well, and $0.75$ for quarks with $\c$ or heavier quark, in accordance with the deliberations above.

By default, it is assumed that $L = 0$, such that only pseudoscalar and vector mesons are produced. For inclusion of $L = 1$ production, four parameters can be used, one to give the probability that a $S = 0$ state also has $L = 1$, the other three for the probability that a $S = 1$ state has $L = 1$ and $J$ either 0, 1, or 2.

For the flavour-diagonal meson states $\u\overline{\mathrm{u}}$, $\d\overline{\mathrm{d}}$ and $\mathrm{s}\overline{\mathrm{s}}$, it is also necessary to include mixing into the physical mesons. This is done according to a parameterization, based on the mixing angles given in the Review of Particle Properties [PDG88]. In particular, the default choices correspond to

$$\eta = \frac{1}{2} (\u\overline{\mathrm{u}}+ \d\overline{\mathrm{d}}) - \frac{1}{\sqrt{2}} \mathrm{s}\overline{\mathrm{s}}~;$$

$$\eta' = \frac{1}{2} (\u\overline{\mathrm{u}}+ \d\overline{\mathrm{d}}) + \frac{1}{\sqrt{2}} \mathrm{s}\overline{\mathrm{s}}~;$$

$$\omega = \frac{1}{\sqrt{2}} (\u\overline{\mathrm{u}}+ \d\overline{\mathrm{d}})$$

$$\phi = \mathrm{s}\overline{\mathrm{s}}~.$$ (221)

In the $\pi^0 - \eta - \eta'$ system, no account is therefore taken of the difference in masses, an approximation which seems to lead to an overestimate of $\eta'$ rates [ALE92]. Therefore parameters have been introduced to allow an additional `brute force' suppression of $\eta$ and $\eta'$ states.