Saturation momentum scale extracted from semi-inclusive transverse spectra in high-energy pp collisions

Geometric scaling is well confirmed for transverse momentum distributions observed in proton-proton collisions at LHC energies. We introduced multiplicity dependence on a saturation momentum of the geometrical scaling, assuming the scaling holds for semi-inclusive distributions as well as for inclusive distributions. The saturation momentum is usually given by Bjorken's $x$ variable, but redefinition of the scaling variable can make the saturation momentum a function of collision energy $W$. We treat the energy as a free parameter (denoted $W^*$ to distinguish it from $W$) and associate the energy-dependent saturation momentum $Q_{\rm sat}(W^*)$ with particle number density. By using $Q_{\rm sat}(W^*)$ for a scaling variable $\tau$, we show semi-inclusive distributions can be geometrically scaled. i.e., all semi-inclusive spectra observed at $W$=0.90, 2.76 and 7.00 TeV overlap one universal function. The particle density dependences of mean transverse momentum $\langle p_{\rm T} \rangle$ for LHC energies scales in terms of $Q_{\rm sat}(W^*)$. Furthermore, our model explains a scaling property of event-by-event $p_{\rm T}$ fluctuation measure $\sqrt{C_m}/\langle p_{\rm T}\rangle$ at LHC energies for pp collisions, where $C_m$ is two-particle transverse momentum correlator. Our analysis of the $p_{\rm T}$ fluctuation makes possible to evaluate a non-perturbative coefficient of the gluon correlation function.