$\psi \left(2S\right)$ versus $J/\psi$ suppression in proton-nucleus collisions from factorization violating soft color exchanges

We argue that the large suppression of the $\psi \left(2S\right)$ inclusive cross section relative to the $J/\psi$ inclusive cross section in proton-nucleus (p+A) collisions can be attributed to factorization breaking effects in the formation of quarkonium. These factorization breaking effects arise from soft color exchanges between charm-anticharm pairs undergoing hadronization and comoving partons that are long lived on time scales of quarkonium formation. We compute the short distance pair production of heavy quarks in the color glass condensate (CGC) effective field theory and employ an improved color evaporation model (ICEM) to describe their hadronization into quarkonium at large distances. The combined CGC+ICEM model provides a quantitative description of $J/\psi$ and $\psi \left(2S\right)$ data in proton-proton (p+p) collisions from both RHIC and the LHC. Factorization breaking effects in hadronization, due to additional parton comovers in the nucleus, are introduced heuristically by imposing a cutoff $\mathrm{\Lambda }$, representing the average momentum kick from soft color exchanges, in the ICEM. Such soft exchanges have no perceptible effect on $J/\psi$ suppression in p+A collisions. In contrast, the interplay of the physics of these soft exchanges at large distances, with the physics of semihard rescattering at short distances, causes a significant additional suppression of $\psi \left(2S\right)$ yields relative to that of the $J/\psi$. A good fit of all RHIC and LHC $J/\psi$ and $\psi \left(2S\right)$ data, for transverse momenta $P_{\perp }\le 5$ GeV in p+p and p+A collisions, is obtained for $\mathrm{\Lambda }\sim 10$ MeV.

Figure 1

Schematic diagram of Onium production in p+A collisions. The red blob represents parton hard scattering at short distances. Vertical gluons represent multiple gluon scattering (with typical net momentum exchange of order $Q_s$, the saturation scale) off the target nucleus, expressed through a lightlike Wilson line. Soft gluon exchange between produced $c\overline{c}$ pair and comover spectators at larger distances are shown as orange vertical gluons. The three time scales $(t_t,t_c,t_f)$ discussed in the text are also illustrated in the figure.

Figure 2

An illustration of $c\overline{c}$ pair production and hadronization in p+A collisions. The left side of the final-state cut (represented by a dashed vertical line) is the scattering amplitude while the right side is its complex conjugate. The blobs represent the charmonium final state. Gluon exchanges of the $c\overline{c}$ with the target, carrying momenta of order $Q_s$, are shown in black. Soft color exchanges with momentum resolution $\mathrm{\Lambda }$ are shown in red. See text for further discussion.

Figure 3

$M$ distribution of the $c\overline{c}$ pair production cross section for different rapidities and initial saturation scales in the target nucleus. The figure (a) [(b)] is the result at a RHIC (LHC) energy with $m=1.3$ GeV. The boundaries of the $M$ integral for ($m_{J/\psi }, m_{\psi \left(2S\right)}, 2m_D$) are specified.

Figure 4

Differential cross section as a function of $P_{\perp }$ for $J/\psi$ and $\psi \left(2S\right)$ production in p+p collisions at RHIC and the LHC in the CGC+ICEM framework. The various lines correspond to the results for the different values of $\sqrt{s}$ or $y$ range. The uncertainty bands reflect the quark mass dependence: $m=(1.3–1.4)$ GeV, though the width of the bands is narrow. Data are taken from Refs. [47, 48, 49, 50, 51, 52].

Figure 5

Rapidity distributions of $J/\psi$ and $\psi \left(2S\right)$ in p+p collisions at RHIC and LHC. The various lines correspond to the results for the different values of $\sqrt{s}$. The uncertainty bands reflect the quark mass dependence. Data are taken from Refs. [47, 48, 49, 50, 51, 52].

Figure 6

Ratios of the differential cross section for $J/\psi$ production in p+p collisions at $\sqrt{s}=13$ TeV relative to that for $\psi \left(2S\right)$ production. The uncertainty bands reflect the quark mass dependence. Data from Refs. [52].

Figure 7

$P_{\perp }$ distribution of forward $J/\psi$ and $\psi \left(2S\right)$ production in p+A collisions at the LHC for varying values of $\mathrm{\Lambda }$. The quark mass is fixed as $m=1.3\mathrm{GeV}$. The initial saturation scale for the target nucleus is chosen to be $Q_{s0,A}^2=2Q_{s0,\mathrm{p}}^2$. Data are taken from Refs. [29, 53].

Figure 8

(a) Nuclear modification factors of $J/\psi$ and $\psi \left(2S\right)$ vs $P_{\perp }$. Data are taken from Refs. [29, 53]. (b) $y$ dependence of nuclear modification factors of $J/\psi$ and $\psi \left(2S\right)$. Data are taken from Refs. [29, 30]. The dark shaded uncertainty band corresponds to a fixed value of $\mathrm{\Lambda }=15$ MeV but varying $Q_{s0,A}^2=(1.5–2.0)Q_{s0,\mathrm{p}}^2$. The light shaded uncertainty bands display the further uncertainties in varying $\mathrm{\Lambda }$ in the range $\mathrm{\Lambda }=10–20$ MeV on top of the variations in the choice of $Q_{s0,A}^2$.

Figure 9

Comparison to $J/\psi$ data and predictions for $\psi \left(2S\right)$ data at the LHC for $\sqrt{s}=8.16$ TeV/nucleon. Notations are the same as Fig. 8. $\mathrm{\Lambda }$ is fixed to be the same at both $\sqrt{s}=5.02\mathrm{TeV}$ and $\sqrt{s}=8.16\mathrm{TeV}$. ALICE preliminary data are taken from Ref. [54]. LHCb data are taken from Ref. [55].

Figure 10

Comparison to $J/\psi$ data and predictions for $\psi \left(2S\right)$ data at RHIC for $\sqrt{s}=0.2$ TeV/nucleon. Notations are the same as Fig. 8 but here $\mathrm{\Lambda }=7.5$ MeV is the fixed valued for the dark shaded band. The light bands reflect the variation in $\mathrm{\Lambda }=5–10\mathrm{MeV}$. Data on $J/\psi$ production in d+Au collisions are taken from Refs. [56, 57]. STAR preliminary data are from Ref. [58].

Figure 11

(a) $P_{\perp }$ dependence and (b) rapidity dependence of ratios of $R_{\mathrm{pA}}$ of $J/\psi$ relative to that of $\psi \left(2S\right)$ at the LHC obtained by varying the shown values of $\mathrm{\Lambda }$. Uncertainty bands reflect the dependence of the model on the initial saturation scale. Data are taken from Refs. [29, 30].

Figure 12

Ratios of $R_{\mathrm{pA}}$ for $J/\psi$ relative to the $R_{\mathrm{pA}}$ for $\psi \left(2S\right)$ at RHIC. Notations are the same as in Fig. 11. PHENIX data are taken from Refs. [28, 32]. STAR preliminary data are available in Ref. [58].