**2.5 Discussion: entanglement entropy in proton-proton collisions**

The material presented in Section 2 provide evidence for an unconventional mechanism of apparent thermalization in high energy *pp* collisions. The data shows that the effective thermal temperature *Tth* is non-universal and that it is proportional to the hard scale temperature parameter of the collision *T*, that is, to the momentum transfer, with *T* ≃4*:*2 *Tth*. Strikingly, this conclusion seems to apply even to the Higgs boson production, suggesting that even in this very hard process the QCD radiation may be affected by thermalization. Moreover, we have found that the thermal component of the spectrum is entirely absent in diffractive production (even though many hadrons are still produced in this case) – this again points to the non-universal, process-dependent, nature of thermalization.

The theory and the analyses of the data discussed in Section 2 appear to be consistent with the proposition that thermalization in these high energy collisions is induced by quantum entanglement. That the effective temperature determined from the data is proportional to the momentum transfer *Q* in the collision that provides the UV cutoff for the quantum modes, as expected. Notably, inclusive charged hadron and Higgs boson transverse momentum distributions, in which the typical momentum transfers are vastly different are in agreement in this analysis. It is seen that the thermal component is present in both cases, event though the values of the effective temperature differ by over an order of magnitude.<sup>1</sup>

In diffractive events studied in Section 2, it is clearly seen that where studies of the coherent response of the entire proton in this scattering, there is no associated entanglement entropy [15], and that therefore there should be no thermal component to the transverse momentum distribution. The data confirms this prediction in diffractive Drell-Yan production analyzed in this section, as well as by the diffractive deep-inelastic scattering data shown in [20].

The findings presented here appear to support the proposition that a deep connection between quantum entanglement and thermalization in high-energy hadron collisions, and that this proposed link should be further investigated. Possibilities include the following as non-exhaustive examples. Combining measurements of the structure functions with the study of hadronic final states, especially in the target fragmentation region in deep inelastic scattering at the future Electron Ion Collider. Studies of the thermal component and the corresponding effective temperature in hard processes characterized by different momentum transfers in proton-proton, proton-nucleus and nucleus-nucleus collisions at RHIC and the LHC. Already, analysis of Pb–Pb HI collision data also points to a picture of thermalization as a result of quantum entanglement at high energies [9]. An investigation of the dependence of the apparent thermalization on rapidity – as depicted in **Figure 2**, suggesting that the thermal component and the corresponding effective temperature in hard processes characterized by different momentum transfer would be interesting. It suggests that thermalization is achieved faster if a measurement is performed in a smaller rapidity interval.
