Towards quarkonium formation time determination

We propose a parametrization of the nuclear absorption mechanism relying on the proper time spent by $c\overline{c}$ bound states travelling in nuclear matter. Our approach could lead to the extraction of charmonium formation time. It is based on a large amount of proton-nucleus data, from nucleon-nucleon center-of-mass energies $\sqrt{s_{NN}}=27$ GeV to $\sqrt{s_{NN}}=5.02$ TeV, collected in the past 30~years, and for which the main effect on charmonium production must be its absorption by the nuclear matter it crosses.

The production of charmonia, cc bound states, is the object of forceful researches in proton-proton, proton-nucleus and nucleus-nucleus collisions. Their production is intrinsically a two-scale problem, that of the heavy-quark pair production, manageable with perturbative methods, and that of its hadronization, non-perturbative and due to the color confinement QCD property. Today, nearly all the models of charmonium production rely on a factorisation between the heavy-quark pair production and its hadronisation, the evolving heavy-quark pair being in a Color-Singlet (CS) or a Color-Octet (CO) state [1]. In protonnucleus collisions, several initial and final state effects, also called Cold Nuclear Matter (CNM) effects, can modify charmonium yields. Charmonia can be suppressed due to nuclear absorption [2], suffer multiple scatterings or lose their energy by radiation [3], in their way out of the nucleus overlapping region. They can also be broken by comovers [4][5] [6] [7] or be affected by the modification of the parton flux inside nuclei as encoded in nuclear PDFs [8] [9]. The relative importance of the above-cited effects depends essentially on the collision energy, the transverse momentum and the rapidity of the probe, together with the nuclear size [10][11][12].
In this Letter, we propose to exploit the charmonium nuclear absorption (or break-up) effect to explore the cc hadronization mechanism. After its production, the small radius cc pair is expected to bind into a larger radius colour neutral state [13] [14]. The latter may interact with the nucleons of the target nucleus in which it was produced, eventually leading to its suppression. This mechanism, firstly described in [ The crossing time of a cc pair in the rest frame of a nuclear target can be expressed as t = L/v, where L corresponds to the length of nuclear matter traversed by the cc pair and v is the velocity of the cc pair in the target rest frame, related to the cc momentum by Here, m is the mass of the cc system and γ corresponds to its Lorentz factor.
Thus, the proper time τ spent by the cc pair in the target nucleus can be expressed as: where y = 0.5 × ln((E + p z )/(E − p z )) and p T = p 2 − p 2 z are the rapidity and the transverse momentum of the cc state in the target frame respectively, and m 2 T = m 2 + p 2 T with m the mass of the cc state. For simplicity, we use, as a good approximation, L = r(A 1/3 − 1), where r = 0.85 fm and A is the atomic mass number of the target. In the following, we study charmonium production as a function of τ for the datasets reported in Table   I, recorded with various targets at various energies. We assume that the small radius cc pair, before it forms a charmonium, does not interact with the target nucleons on its path.
Phenomenologically, in case of a sizeable charmonium formation time, and thanks to nuclear absorption, charmonium yields as a function of τ should exhibit a plateau, followed by a suppression. Table II provides kinematical information for the datasets used in this Letter.
Since the cc bound-state average transverse momentum < p T > is usually not reported, we follow the results given in [27] and take: Bjorken-x x 2 and Feyman-x x F are calculated following eq. 2, taking m = 3.097 GeV/c 2 [28], the mass of the J/ψ : Because charmonia may suffer several cold nuclear matter effects, the data used in this letter are chosen to cover kinematical regions where nuclear effects but nuclear absorption do not apply, or, at least, are expected to be small. The criteria are: • x F must be close to zero, far from the energy loss [3] and saturation regimes, • x 2 must belong to the region [10 −2 , 10 −1 ], close to the transition between nPDF shadowing and anti-shadowing regions [8,9], where those effects are expected to be small, • quarkonium interaction with comoving hadrons must be small, limiting the use of ψ ′ to the low energy (fixed-target) data samples [29].
Relevant data are also required to extend over reduced τ ranges corresponding to reduced rapidity ranges. In the following, data points are reported with uncertainties on the τ values corresponding to the rapidity ranges in which data were recorded.
We propose the following parametrization of the nuclear absorption [10,11] based on the proper time spent in the nucleus by the quarkonia (or cc precursor): where σ cc pA is the charmonium production cross section in pA collisions, A the atomic mass number of the target nucleus, σ cc pp the charmonium production cross section in pp collisions, σ abs the charmonium absorption (or break-up) cross section, ρ 0 = 0.170 fm −3 the nuclear density and βγcτ (with β = v/c and c the speed of light) is the length of nuclear matter traversed by the cc pair. Assuming that, before charmonium state formation time τ 0 , the small radius cc pair does not interact with the target nucleons on its path, we propose, first, based on eq. 3, to introduce τ 0 in the step function: where all cc bound states start suffering nuclear absorption when reaching τ 0 . Figure 1 shows the J/ψ and ψ ′ production cross sections measured by the NA51 [19] and NA50 [16] experiments at √ s N N = 29.1 GeV, as functions of τ as defined in eq. 1. In both cases, a structure appears, made of a plateau, followed by a suppression. Taking σ cc pp , σ abs and τ 0 as free parameters, χ 2 -minimization fits based on eq. 4 give τ J/ψ 0 = 0.08 ± 0.04 fm/c and τ ψ ′ 0 = 0.10 ±0.04 fm/c, for the J/ψ and the ψ ′ respectively. For completeness, the values of σ cc pp and σ abs are reported in Table III. More realistically, considering that cc hadronization follows the standard decay law dN cc = −λN cc dt, with λ = 1/γτ 0 and t = γτ , the charmonium survival probability follows the sigmoid function: experimental results do not permit to obtain reliable minimizations without constraining the parameters. We therefore take the results obtained with the step function (eq. 4) as fixed input parameters for the sigmoid function (eq. 5).
We now consider experimental results recorded in different experimental conditions, as reported in Table I. Because quarkonium cross section depends on center-of-mass energy, cross section ratios, such as the nuclear modification factor R AB and the ψ ′ over J/ψ cross sections ratio σ ψ ′ /σ J/ψ are appropriate quantities to compare data from various experiments at various energies. Figure 2 shows the J/ψ nuclear modification factor R AB as a function of τ for the SPS, RHIC and LHC experimental data, listed in Tables I and II, where the uncertainties on the NA51 pp J/ψ cross section have been propagated to the ratio.
As in Figure 1, a plateau is observed for small values of τ . Moreover, although recorded at very different energies, NA50 and PHENIX data, in the region τ > 0.1 fm/c, follow a similar trend, consistent with a suppression scenario depending on geometrical effect such as nuclear absorption. The results of a fit based on eq. 4 are reported in Table III with τ J/ψ 0 = 0.10 ± 0.02 fm/c, in agreement with the value obtained for Figure 1. Beside, since experimental data have been recorded in different kinematical regimes, the βγ factor depends on the data sample, preventing reporting the fit results on the plot. We instead report on the (data − f it)/data ratio where, as for Figure 1, the results obtained with eq. 4 are used as fixed input parameters for the sigmoid function. The corresponding χ-square per degree of freedom, χ 2 4 /ndf = 0.65 and χ 2 5 /ndf = 0.90 for the step (eq. 4) and sigmoid (eq. 5) functions respectively, indicate good agreement between data and fit. Figure 3 shows the ψ ′ /J/ψ cross section ratio as a function of τ for several data collected at various energies by the CERN NA51 [19] and NA50 [16] experiments, and the Fermilab E288 [20], E771 [21] and E789 [22] experiments. PHENIX and LHC data are not considered here since, because of the large center-of-mass energy, ψ ′ production is expected to be significantly [1] J. P. Lansberg, Phys. Rep. 889, 1 (2020).
[2] C. Gerschel  between data and eq. 5 sigmoid function (close diamonds), all parameters being determined with fits based on eq. 4. The Grey band shows the effects of τ 0 uncertainties on the sigmoid function.  and y are the rapidity of the cc state in the center-of-mass, laboratory and target frames respectively, and < P T > its average transverse momentum as discussed in the text; x 2 and x F are the Bjorken-x and Feynman-x respectively as defined in eq. 2, considering smaller and larger y values.