A quantum critical point (QCP) is currently being conjectured for the BaFe$_2$(As$_{1-x}$P$_x$)$_2$ system at the critical value $x_{rm c} approx$ 0.3. In the proximity of a QCP, all thermodynamic and transport properties are expected to scale with a single characteristic energy, given by the quantum fluctuations. Such an universal behavior has not, however, been found in the superconducting upper critical field $H_{rm c2}$. Here we report $H_{rm c2}$-data for epitaxial thin films extracted from the electrical resistance measured in very high magnetic fields up to 67 Tesla. Using a multi-band analysis we find that $H_{rm c2}$ is sensitive to the QCP, implying a significant charge carrier effective mass enhancement at the doping-induced QCP that is essentially band-dependent. Our results point to two qualitatively different groups of electrons in BaFe$_2$(As$_{1-x}$P$_x$)$_2$. The first one (possibly associated to hot spots or whole Fermi sheets) has a strong mass enhancement at the QCP, and the second one is insensitive to the QCP. The observed duality could also be present in many other quantum critical systems.
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