Quark flavours and transverse momenta

In order to generate the quark-antiquark pairs $\mathrm{q}_i \overline{\mathrm{q}}_i$ which lead to string breakups, the Lund model invokes the idea of quantum mechanical tunnelling, as follows. If the $\mathrm{q}_i$ and $\overline{\mathrm{q}}_i$ have no (common) mass or transverse momentum, the pair can classically be created at one point and then be pulled apart by the field. If the quarks have mass and/or transverse momentum, however, the $\mathrm{q}_i$ and $\overline{\mathrm{q}}_i$ must classically be produced at a certain distance so that the field energy between them can be transformed into the sum of the two transverse masses $m_{\perp}$. Quantum mechanically, the quarks may be created in one point (so as to keep the concept of local flavour conservation) and then tunnel out to the classically allowed region. In terms of a common transverse mass $m_{\perp}$ of the $\mathrm{q}_i$ and the $\overline{\mathrm{q}}_i$, the tunnelling probability is given by

$$\exp \left( -\frac{\pi m_{\perp}^2}{\kappa} \right) = \exp... ...right) \exp \left( -\frac{\pi p_{\perp}^2}{\kappa} \right) ~.$$ (220)

The factorization of the transverse momentum and the mass terms leads to a flavour-independent Gaussian spectrum for the $p_x$ and $p_y$ components of $\mathrm{q}_i \overline{\mathrm{q}}_i$ pairs. Since the string is assumed to have no transverse excitations, this $p_{\perp}$ is locally compensated between the quark and the antiquark of the pair. The $p_{\perp}$ of a meson $\mathrm{q}_{i-1}\overline{\mathrm{q}}_i$ is given by the vector sum of the $p_{\perp}$:s of the $\mathrm{q}_{i-1}$ and $\overline{\mathrm{q}}_i$ constituents, which implies Gaussians in $p_x$ and $p_y$ with a width $\sqrt{2}$ that of the quarks themselves. The assumption of a Gaussian shape may be a good first approximation, but there remains the possibility of non-Gaussian tails, that can be important in some situations.

In a perturbative QCD framework, a hard scattering is associated with gluon radiation, and further contributions to what is naïvely called fragmentation $p_{\perp}$ comes from unresolved radiation. This is used as an explanation why the experimental $\left\langle p_{\perp}\right\rangle$ is somewhat higher than obtained with the formula above.

The formula also implies a suppression of heavy quark production $u : d : s : c \approx$ $1 : 1 : 0.3 : 10^{-11}$. Charm and heavier quarks are hence not expected to be produced in the soft fragmentation. Since the predicted flavour suppressions are in terms of quark masses, which are notoriously difficult to assign (should it be current algebra, or constituent, or maybe something in between?), the suppression of $\mathrm{s}\overline{\mathrm{s}}$ production is left as a free parameter in the program: $\u\overline{\mathrm{u}}$ : $\d\overline{\mathrm{d}}$ : $\mathrm{s}\overline{\mathrm{s}}$ = 1 : 1 : $\gamma_s$, where by default $\gamma_s = 0.3$. At least qualitatively, the experimental value agrees with theoretical prejudice. There is no production at all of heavier flavours in the fragmentation process, but only in the hard process or as part of the shower evolution.