by the transverse mass $\sqrt{\mu^2+p_\perp^2}$. This gives $$\frac{d\mathcal{P}}{d^2 p_\perp}\propto \kappa \exp\left(-\,
  \frac{\pi\,}{\kappa}(\mu^2+p_\perp^2)\right), 
\label{eq:schwinger2}$$ which integrated over $p_\perp$ gives the result in [eq. (\[eq:schwinger\])]{}, with the replacement $e\mathcal{E} \rightarrow \kappa$.

The result in [eq. (\[eq:schwinger2\])]{} can be used to estimate the relative production of quarks with different flavour, and the distribution in $p_\perp$. As it is not possible to theoretically determine the effective quark masses to be used in [eq. (\[eq:schwinger2\])]{}, it is in practice necessary to tune the $s/u$ ratio to experimental data. Fits to LEP data give $s/u \approx 0.2$ [@Hamacher:1995df; @Sjostrand:2014zea], which is not inconsistent with [eq. (\[eq:schwinger2\])]{} for reasonable quark masses. This mechanism also implies that charm and heavier quarks cannot be produced in the soft hadronization process; they can only be produced in an initial perturbative phase. For the transverse momenta it gives a Gaussian $p_\perp$-distribution with $\sqrt{\langle
  p_\perp^2\rangle}\approx 0.25\,\,\mathrm{GeV}$. Phenomenological fits to data will, however, also include effects from soft gluons below a necessary $p_\perp$ cut in the perturbative parton shower. Tunes to data therefore give a somewhat wider distribution, with a width $\sigma_{p_\perp}\approx 0.32$ GeV.

Space-time picture and longitudinal momentum distribution {#sec:spacetime}
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The strange quark fraction and the $p_\perp$-distribution are governed by the 