There and Sharp Again:
The Circle Journey of Nucleons and Energy Deposition ^††thanks: Presented at the XXIXth International Conference on Ultra-relativistic Nucleus-Nucleus Collisions (Quark Matter 2022)

What is the appropriate nucleon size for high-energy scattering? We know little about it. In deep inelastic scattering (DIS) at low $x$, the dipole cross section at a given impact parameter is of the form $\frac{d\sigma_{q\bar{q}}}{d^{2}{\bf b}}\propto r^{2}\alpha_{s}xg(\mu^{2},x)T({\bf b})$, where $T({\bf b})$ weights the gluon density depending on the distance from the target’s center. The data-driven analysis by Caldwell and Kovalsky [2] based on the diffractive production of $J/\Psi$ at HERA concludes that $T({\bf b})$ can be modeled as a 2D Gaussian with width $0.35$ fm, corresponding to an rms radius in two dimensions, dubbed the two-gluon radius, of $\sqrt{2}w=0.50$ fm. In 2012, this $w$ has been used as input in the IP-Glasma model by Schenke, Tribedy, and Venugopalan [3], based on the color glass condensate (CGC) effective theory of high-energy QCD [4].

In a description of high-energy collisions, it is appropriate to consider as well the inner structure of the colliding nucleons. One introduces an effective parton structure by replacing Eq. (1) with

where now for each nucleon, $j$, one samples the centers, ${\bf x}_{q}$, of $n_{c}$ constituents from a Gaussian distribution of width $w$, and treats each sampled constituent as a Gaussian profile of width $w_{q}$. Some insights on the number of constituents and their widths can be obtained as well from diffractive production in DIS, by studying the incoherent J/$\Psi$ production, which probes fluctuations in the proton content. The data-driven analysis of Mäntysaari and Schenke [5] in the CGC framework infers three constituents of width $w_{q}=0.11$ fm. Note that the number of constituents is poorly constrained, and may be higher [6]. A nucleon size of about $0.4$ fm and three constituents of size $0.11$ fm are implemented in the latest version of IP-Glasma [7].

The second crucial ingredient for the geometry of the QGP is the energy deposition scheme. This refers to some function of $T_{A}({\bf x})$ and $T_{B}({\bf x})$ that turns them into an energy density in the transverse plane, corresponding to the initial condition for the subsequent QGP formation at proper time $\tau=0^{+}$. For high-energy nuclear scattering insights about the energy deposition come again from the CGC framework. A robust prediction of such a framework (which is nowadays textbook material [8]) is that the average energy density in the scattering of two CGCs at $\tau=0^{+}$ is of the form

The saturation scales, $Q^{2}_{A,B}$, involve the same function $T({\bf b})$ appearing in the dipole cross section, and as such are essentially proportional to $T_{A,B}$.¹¹1Note that in T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo the function $T_{A}$ includes only the nucleons that participate in the collisions, whereas all nucleons are considered in the nuclear profiles in IP-Glasma. The IP-Glasma model is based on such a scaling, as confirmed by comparisons with the IP-Jazma parametrization [9, 10], where in every collision event one takes $T^{00}\propto T_{A}T_{B}$.

Features of the CGC aside, one can confidently state that, up to quantitative corrections, experimental evidence in heavy-ion collisions points to an Ansatz of the kind $(T_{A}T_{B})^{m}$ for the initial energy density of the system. Note that the random normalizations, $\lambda_{j}$ in Eq. (1), and $\lambda_{q}$ in Eq. (2), are less important for the QGP geometry compared to $w$, $w_{q}$, or the energy deposition scheme, and will not be part of the present discussion.

The Duke group recognizes that this model provides geometric features for the initial profiles that are remarkably close to those calculated within IP-Glasma, in particular regarding the eccentricities, $\varepsilon_{n}$, which are the seeds of the anisotropic flow coefficients, $v_{n}$. The breakthrough of the T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo prescription is that it allows one to produce results of potentially similar quality as those obtained in IP-Glasma within a much lighter computational framework. The possibility of mass-producing millions of T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo initial conditions opens a new way to compare initial-state calculations to experimental data. In particular, one can precisely assess observables that are too costly for hydrodynamic simulations, such as higher-order fluctuations of $v_{n}$ [12], to scrutinize in great detail the role of the hydrodynamic response ($v_{n}\propto\varepsilon_{n}$). Thanks to T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo, the phenomenology of the soft sector of heavy-ion collisions enters a new era of precision.

We remark, then, that the latest version of the IP-Glasma model of initial condition also appears in 2020 [7]. As shown in Fig. 1, the predicted initial profile is sharp and lumpy, due to the fine nucleon structure. At the end of 2020, IP-Glasma and T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo models based on Bayesian analyses of nucleus-nucleus data are starkly inconsistent, both in terms of initial profiles and in terms of implemented bulk viscosities.

Let us comment, then, on the Bayesian analysis of Ref. [18] (Duke-18) which uses the same model as in Duke-19, albeit including experimental data from proton-nucleus collisions. Following the inclusion of the nucleon sub-structure in IP-Glasma [19], the Duke-18 calculation introduces sub-nucleonic constituents in agreement with Eq. (2) to make the T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo Ansatz viable for proton-nucleus collisions. The notable result is that the $p=0$ Ansatz remains the only viable model as well when $p$+Pb data is included. This is probably driven by the $A$-$A$ data, as it is not known at present whether analyzing $p$-$A$ alone would give the same result. Similar results are obtained in the first Bayesian analysis performed within the Trajectum framework (Trajectum-20) [20]. Both Duke-18 and Trajectum-20 infer rather large nucleon sizes, $w\approx 0.9$ fm, though with a constituent size $w_{q}\approx 0.5$ fm. Comparing, then, these profiles with the JETSCAPE and Duke-19 ones in Fig. 1, we can see the additional short-range structures coming from the nucleon constituents. Note that this additional structure does not imply larger values of $\zeta/s$, which remains very close to zero [20].

The first such observable has been pointed out by Giacalone, Schenke, and Shen [21]. It is the Pearson correlation between the charged-hadron mean transverse momentum, $\langle p_{t}\rangle$, and anisotropic flow, $v_{n}^{2}$, denoted by $\rho(\langle p_{t}\rangle,v_{n}^{2})$ [22]. As emphasized by the ALICE collaboration [23], this correlator presents a unique model dependence. For instance, $\rho(v_{3}^{2},\langle p_{t}\rangle)$ is negative at almost all centralities in the JETSCAPE results, while it is always positive with IP-Glasma initial conditions, in agreement with experimental measurements [24, 23, 25]. This largely comes from the huge difference in the implemented size of nucleons. Even the model outcome of the sophisticated Bayesian analysis performed in 2021 within the Trajectum framework (Trajectum-21) by Nijs and van der Schee [26], with the well-structured initial profile shown in Fig. 1, yields hydrodynamic results for $\rho(v_{n}^{2},\langle p_{t}\rangle)$ that are inconsistent with data [25]. The reason is partly that the nucleon size remains too large, of order 0.85 fm.

One way to fix this issue is to look at the total inelastic nucleus-nucleus cross section, $\sigma_{AA}$. In T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo and IP-Glasma, the nucleon size determines the probability of interaction between two ions at a given impact parameter. As the total cross section is constrained fairly well by Glauber fits of multiplicity distributions, it provides an experimental handle on $w$. The latest Trajectum study (Trajectum-22) [27] addresses this problem by essentially enforcing the Bayesian analysis to return the correct value of $\sigma_{AA}$. Not only the nucleon size shrinks to a value close to 0.5 fm, but the exploration of a doubly-generalized average with an additional parameter, $q$,

shows that, when the $\sigma_{AA}$ constraint is properly considered and the computed $\rho(v_{n}^{2},\langle p_{t}\rangle)$ correlators are qualitatively consistent with data, the favored initial-state model has $q\approx 4/3$ and $p\approx 0$, i.e., $dE/dy\propto(T_{B}T_{B})^{2/3}$ [28]. The circle is closed (see Fig. 1). We are back to the nucleon size and energy deposition of Duke-16 (albeit with four constituents per nucleon, and $w_{q}\approx 0.4$ fm). Moreover, the sharper profile does now impact visibly the extracted $\zeta/s(T)$, whose maximum increases by a large factor [27].

To improve the state-of-the-art, one obvious suggestion is that of including in future Bayesian analyses multi-particle correlation observables that offer a stronger sensitivity to the initial condition. The natural candidates are the $\rho(v_{n}^{2},\langle p_{t}\rangle)$ correlators, though much insight would would come as well from the relative fluctuation of $v_{3}$, $v_{3}\{4\}/v_{3}\{2\}$, or the skewness of the fluctuations of $\langle p_{t}\rangle$ [12, 31]. At the price of increasing dramatically the computational effort, these observable would indeed permit global analyses to yield much tighter constraints on initial-state properties.

Second, we have to understand in greater detail the fundamental origin of the strong dependence of observables on model features. One way to do this is to supplement event-by-event calculations with a model-independent description of the initial states in terms of correlation functions of fluctuating fields. Techniques to do so have been discussed in the past [29, 30]. Does $\rho(v_{n}^{2},[p_{t}])$ depend strongly on the nucleon size because the latter modifies the local average of the energy density field, or because it modifies the local variance? What is the main difference between implementing $w_{q}=0.4$ fm as in Trajectum-22 and $w_{q}=0.11$ fm as in IP-Glasma? Is it in the local variance of the density field, or in its correlation length?

One final suggestion about the inclusion in Bayesian analyses of features of the nucleus lead-208 itself. Introducing a nuclear skin thickness parameter would be especially interesting, as this feature affects $\sigma_{AA}$ and the sharpness of the average density profiles. The nucleon density in a nucleus, $\rho_{m}$, can be written as $\rho_{m}=\rho_{p}+\rho_{n}$, where $\rho_{n(p)}$ is the density of neutrons(protons). While $\rho_{p}$ is known from low-energy experiments, $\rho_{n}$, and in particular its skin, is not. One could attempt, thus, at inferring the neutron density of ²⁰⁸Pb via the Bayesian analysis of high-energy data. This would in turn provide an independent estimate of the neutron skin of this nucleus, a hot topic in low-energy nuclear physics due to its relevance for the understanding of the properties of neutron stars [32, 33].