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A covariant generalization of the one-dimensional cascade model for quark fragmentation functions is presented, so as to include the transverse momentum behaviour and the possibility to produce different particles at different vertices along the chain. In the scaling limit the exact solution is given, if the primordial function is of the type αzα−1·T(pT). For the more general case of factorizing primordial functions an analytic expression for the seagull effect is derived, which turns out to be independent of the functionT(pT).

The transverse momentum dependence of quark fragmentation functions from cascade models