Multiplicity-dependent saturation momentum in $p$-Pb collisions at 5.02 TeV

Semi-inclusive transverse momentum spectra observed in proton-proton and proton-lead nuclear collisions at LHC energies obey a geometric scaling with a scaling variable using multiplicity-dependent saturation momentum. The saturation momentum extracted from the experimental data is proportional to the 1/6 power of the hadron multiplicity in the final state. On the other hand, the system's transverse size is proportional to the 1/3 power of the multiplicity, and the saturation momentum and the transverse size of the system are strongly correlated with the hadron multiplicity in the final state. Since the saturation momentum is proportional to the average transverse momentum of hadrons, one predicts average transverse momentum is also proportional to the 1/6 power of the multiplicity, which is consistent with experimental results at the LHC energy. We found that a nuclear modification factor $R_{\rm pPb}$ calculated by the multiplicity-dependent saturation momentum decreases at $p_{\rm T} \lesssim$ 1GeV/$c$ and that our model can partially explain the $R_{\rm pPb}$'s behavior thought to be caused by nuclear shadowing. On the other hand, Cronin enhancement experimentally observed at $2 \lesssim p_{\rm T} \lesssim $ 6 GeV/$c$ is not reproduced. However, the experimental result, including the Cronin effect, can be reproduced well by introducing $p_{\rm T}$ dependence as a 4$\sim$5\% correction to the multiplicity-dependent saturation momentum.We also discuss a relation between the geometric scaling in the semi-inclusive distributions and the string percolation model.