Hump-backed distribution without jet reconstruction in direct-$\gamma$-hadron correlations

Borghini and Wiedemann proposed using the hump-backed or $\xi=\ln(1/z)$ distribution of jet fragments, which is a signature of QCD coherence for small values of particle momentum fraction, $z=p/E_{\rm jet}$, to explore the medium-modification of jets in heavy ion collisions. The use of the $\xi$ variable would emphasize the increase in the emission of fragments at small $z$ due to the medium induced depletion of the number of fragments at large $z$. It was presumed that full jet reconstruction would be required. However, one of the original measurements of the $\xi$ distribution in $e^+ e^-$ collisions on the $Z^0$ resonance at LEP was made using the inclusive distribution of $\pi^0$, which could be plotted in either the $z$ or the $\xi$ variable since the energy of the jets for di-jet events was known. A similar state of affairs exists for direct-$\gamma$-hadron correlations in p-p and A+A collisions since, modulo any $k_T$ effect, the jet recoiling from a direct-$\gamma$ has equal and opposite transverse momentum to the precisely measured $\gamma$. Thus, the $x_E$ or $z_T$ distribution of the away-side hadrons from a direct-$\gamma$ represents the away-jet fragmentation function, as suggested by Wang, Huang and Sarcevic, so that $dN/d\xi=z\,dN/dz$ can be derived. Examples from RHIC measurements will be given and compared to previous results.