Energy dependence of intrinsic charm production: Determining the best energy for observation

Background: A nonperturbative charm production contribution, known as intrinsic charm, was predicted in the early 1980s. Recent results have provided new evidence for its existence but further confirmation is needed.

Purpose: $J/\psi$ and $\overline{D}$ meson production are calculated with a combination of perturbative QCD and intrinsic charm to determine the best energy range to study intrinsic charm production.

Methods: $J/\psi$ and $\overline{D}$ meson production are calculated in perturbative QCD to next-to-leading order in the cross section. Cold nuclear matter effects, including nuclear modification of the parton densities and $p_T$ broadening by multiple scattering, are taken into account in the production of both; absorption by nucleons is also included for the $J/\psi$. Contributions from intrinsic charm are calculated assuming production from a $|uudc\overline{c}\rangle$ Fock state.

Results: The $J/\psi$ and $\overline{D}$ meson rapidity and $p_T$ distributions are calculated as a function of rapidity and transverse momentum $p_T$ over a wide range of center-of-mass energies with and without intrinsic charm in $p+p$ collisions. The nuclear modification factor, $R_{pA}$, is also calculated for $p+\mathrm{Pb}$ interactions at appropriate energies. Previous fixed-target data as a function of Feynman $x$, $x_F$, are also compared to calculations within the approach. Good agreement with the data is found when intrinsic charm is included.

Conclusions: The intrinsic charm signal may be largest at midrapidity for future low energy fixed target experiments such as the proposed $\mathrm{NA}60+$.

Figure 1

The value of ${\langle k_T^2\rangle }_p$ as a function of the center-of-mass energy, $\sqrt{s}$, in $p+p$ collisions obtained employing Eq. (5). The points show the energies at which ${\langle k_T^2\rangle }_p$ is calculated.

Figure 2

(a) The EPPS16 ratio for a lead nucleus, with uncertainties, is shown at the scale of the $J/\psi$ mass for gluons as a function of momentum fraction $x$. The central set is denoted by the solid curves while the dashed curves give the upper and lower limits of the uncertainty bands. (b) The $x_2$ range as a function of rapidity for six values of $\sqrt{s_{NN}}$ covering the range of energies studied with nuclear targets: 8.77 (solid red), 17.4 (dashed blue), 38.8 (dot-dashed black), 87.7 (dashed red), 200 (dot-dashed blue), and 5000 (solid black) GeV. The upper and lower dotted lines at $x_2=0.012$ and 0.2 represent the lower and upper limits of the antishadowing region for ${\mu }_F=3$ GeV.

Figure 3

The value of ${\sigma }_{\mathrm{abs}}$ as a function of the center-of-mass energy, $\sqrt{s_{NN}}$. The points show the energies used in this paper. The line is meant to guide the eye.

Figure 4

The probability distributions for $J/\psi$ [(a), (b)] and $\overline{D}$ [(c), (d)] production from a five-particle proton Fock state as a function of rapidity. In (a) and (c) the results are given for the default mass values, $m_c=1.27$ GeV and $m_q=0.3$ GeV (solid black), as well as for $m_c=1.5$ GeV and $m_q=0.3$ GeV (dot-dashed red) and for $m_c=1.27$ GeV and $m_q=0.02$ GeV (dashed blue). In (b) and (d) the results are shown for different values of the $k_T$ range for the light and charm quarks. The solid black curve employs the default values, $k_q^{\max }=0.2$ GeV and $k_c^{\max }=1.0$ GeV, while the red dot-dashed curve increases the default values by a factor of 2, $k_q^{\max }=0.4$ GeV and $k_c^{\max }=2$ GeV, and the blue dashed curves employs half the default values, $k_q^{\max }=0.1$ GeV and $k_c^{\max }=0.5$ GeV. All distributions are normalized to unity.

Figure 5

The probability distributions for $J/\psi$ [(a), (c)] and $\overline{D}$ [(b), (d)] production from a five-particle proton Fock state as a function of $p_T$. In (a) and (c) the results are given for the default mass values, $m_c=1.27$ GeV and $m_q=0.3$ GeV (solid black), as well as for $m_c=1.5$ GeV and $m_q=0.3$ GeV (dot-dashed red) and $m_c=1.27$ GeV and $m_q=0.02$ GeV (dashed blue), In (b) and (d) results are shown for different values of the $k_T$ range for the light and charm quarks. The solid black curve employs the default values, $k_q^{\max }=0.2$ GeV and $k_c^{\max }=1.0$ GeV, while the red dot-dashed curve increases the default values by a factor of 2, $k_q^{\max }=0.4$ GeV and $k_c^{\max }=2$ GeV, and the blue dashed curve employs half the default values, $k_q^{\max }=0.1$ GeV and $k_c^{\max }=0.5$ GeV. The dotted magenta curves show the $p_T$ distributions for a single charm quark from the same state. All distributions are normalized to unity.

Figure 6

The probability distributions as a function of rapidity for $J/\psi$ (a) and $\overline{D}$ (b) mesons from a five-particle proton Fock state. The black curves are for fixed-target energies with $p_{\mathrm{lab}}=40$ (solid), 80 (dashed), 120 (dot-dashed), 158 (dotted), 450 (dot-dot-dot-dashed), and 800 (dot-dot-dash-dashed) GeV. The blue curves correspond to SMOG energies: $\sqrt{s}=69$ (solid), 87.7 (dashed), and 110.4 (dot-dashed) GeV. The red curves denote RHIC energies of 200 (solid) and 500 (dashed) GeV. Finally, the magenta curves show the distributions for LHC energies of $\sqrt{s}=5$ (solid), 7 (dashed), and 13 (dot-dashed) TeV. All distributions are normalized to unity.

Figure 7

The probability distributions as a function of rapidity for $J/\psi$ [(a), (c), (e)] and $\overline{D}$ [(b), (d), (f)] mesons from a five-particle proton Fock state. The green curves in all plots show the $p_T$ distributions integrated over all rapidity. The calculations shown for other energies all include cuts in rapidity. The black curves in (a) and (b) are for fixed-target energies with $p_{\mathrm{lab}}=40$ (solid), 80 (dashed), 120 (dot-dashed), 158 (dotted), 450 (dot-dot-dot-dashed), and 800 (dot-dot-dash-dashed) GeV. The blue curves in (a) and (b) correspond to SMOG energies: $\sqrt{s}=69$ (solid), 87.7 (dashed), and 110.4 (dot-dashed) GeV. The red curves in (c) and (d) denote the RHIC energies of 200 (solid) and 500 (dashed) GeV. Finally, the magenta curves in (e) and (f) show the distributions for LHC energies of $\sqrt{s}=5$ (solid), 7 (dashed), and 13 (dot-dashed) TeV. All distributions are normalized to unity and integrated over all rapidity.

Figure 8

The combined rapidity distributions for $J/\psi$ [(a), (c)] and $\overline{D}$ [(b), (d)] mesons, including both the typical perturbative QCD contribution and intrinsic charm from a five-particle proton Fock state. Three curves are shown for each energy. From lowest to highest (when separable) they show no intrinsic charm (pQCD only), $P_{\mathrm{ic}5}^0=0.1$%, and $P_{\mathrm{ic}5}^0=1$%. In (a) and (b) the results are shown for fixed-target and SMOG energies. From lowest to highest the curves represent $p_{\mathrm{lab}}=40$ GeV (red solid), 80 GeV (blue dashed), 120 GeV (black dot-dashed), 158 GeV (red solid), 450 GeV (blue dashed), 800 GeV (black dot-dashed), $\sqrt{s}=69$ GeV (solid red), 87.7 GeV (blue dashed), and 110.4 GeV (black dot-dashed). The vertical line with the green arrows shows the assumed rapidity acceptance of $0<y<1$. In (c) and (d) the RHIC energies of $\sqrt{s}=200$ GeV (solid red) and 500 GeV (blue dashed) along with the LHC energies of $\sqrt{s}=5$ TeV (red solid), 7 TeV (blue dashed), and 13 TeV (black dot-dashed) are given. The vertical lines show the assumed rapidity acceptance of $1.1<y<2.2$ for RHIC (dashed, with green arrows) and $2.5<y<5$ for LHC (solid, with magenta arrows).

Figure 9

The combined $p_T$ distributions for $J/\psi$ [(a), (c)] and $\overline{D}$ [(b), (d)] mesons, including both the typical perturbative QCD contribution and intrinsic charm from a five-particle proton Fock state. The calculated $p_T$ distributions are integrated over all rapidity in (a) and (b) but limited to $0<y<1$ in (c) and (d). Three curves are shown for each energy. From lowest to highest (when separable) they show no intrinsic charm (pQCD only), $P_{\mathrm{ic}5}^0=0.1$%, and $P_{\mathrm{ic}5}^0=1$%. The results are shown for fixed-target and SMOG energies. From lowest to highest the curves represent $p_{\mathrm{lab}}=40$ GeV (red solid), 80 GeV (blue dashed), 120 GeV (black dot-dashed), 158 GeV (red solid), 450 GeV (blue dashed), 800 GeV (black dot-dashed), $\sqrt{s}=69$ GeV (solid red), 87.7 GeV (blue dashed), and 110.4 GeV (black dot-dashed).

Figure 10

The combined $p_T$ distributions for $J/\psi$ [(a), (c)] and $\overline{D}$ [(b), (d)] mesons, including both the typical perturbative QCD contribution and intrinsic charm from a five-particle proton Fock state. Three curves are shown for each energy. From lowest to highest (when separable) they show no intrinsic charm (pQCD only), $P_{\mathrm{ic}5}^0=0.1$%, and $P_{\mathrm{ic}5}^0=1$%. In (a) and (b) the RHIC energies of $\sqrt{s}=200$ GeV (solid red) and 500 GeV (blue dashed) are shown in the rapidity range $1.1<y<2.2$. In (c) and (d) the distributions for LHC energies of $\sqrt{s}=5$ TeV (red solid), 7 TeV (blue dashed), and 13 TeV (black dot-dashed) are given for $2.5<y<5$.

Figure 11

The exponent $\alpha$ as a function of $x_F$ for $J/\psi$ production at fixed-target energies: NA60 at $p_{\mathrm{lab}}=158$ GeV [6] (blue), NA3 at $p_{\mathrm{lab}}=200$ GeV [1] (magenta), NA60 at $p_{\mathrm{lab}}=400$ GeV [6] (cyan), NA50 at $p_{\mathrm{lab}}=450$ GeV [5] (green), E866 at $p_{\mathrm{lab}}=800$ GeV [7] (black), and HERA-B at $p_{\mathrm{lab}}=920$ GeV [9] (red). The points are the experimental data while the curves of the same color are calculations made to match the energy of the measurement. Calculations without intrinsic charm are shown in (a) while calculations with $P_{\mathrm{ic}5}^0=0.1$%, 0.3%, and 1% are shown in (b)–(d) respectively.

Figure 12

The $p+p$ and $p+\mathrm{Pb}$ (per-nucleon) $J/\psi$ distributions at $p_{\mathrm{lab}}=40$ and 800 GeV and $\sqrt{s}=200$ GeV as a function of rapidity. The red curves show the results for $p+p$ collisions while the blue curves show the $p+\mathrm{Pb}$ distributions. Three curves are shown: no intrinsic charm (pQCD only, solid), $P_{\mathrm{ic}5}^0=0.1$% (dashed), and $P_{\mathrm{ic}5}^0=1$% (dot-dashed). No $J/\psi$ absorption by nucleons is considered in the $p+\mathrm{Pb}$ calculation.

Figure 13

The nuclear suppression factor $R_{p\mathrm{Pb}}$ as a function of rapidity for $J/\psi$ [(a), (c), (e)] and $\overline{D}$ [(b), (d), (f)] mesons, including both the typical perturbative QCD contribution and intrinsic charm from a five-particle proton Fock state. The intrinsic charm contribution is varied in the panels from no intrinsic charm (pQCD only) [(a) and (b)], $P_{\mathrm{ic}5}^0=0.1$% [(c) and (d)], and $P_{\mathrm{ic}5}^0=1$% [(e) and (f)]. The red curves include EPPS16 modifications of the parton densities only while the blue curves include nuclear absorption of the $J/\psi$. (There is no absorption of the $\overline{D}$ mesons in cold nuclear matter.) The line types denote different energies: $p_{\mathrm{lab}}=40$ GeV (solid), 158 GeV (dashes), 800 GeV (dot-dashed), $\sqrt{s}=87.7$ GeV (dotted), 200 GeV (dot-dot-dot-dashed), and 5 TeV (dot-dot-dash-dashed).

Figure 14

The $p+p$ and $p+\mathrm{Pb}$ (per-nucleon) $J/\psi$ distributions at $p_{\mathrm{lab}}=40$ and 800 GeV and $\sqrt{s}=200$ GeV as a function of $p_T$ at forward (a) and backward (b) rapidity. The red curves show the results for $p+p$ collisions while the blue and black curves show the $p+\mathrm{Pb}$ distributions without and with an enhanced intrinsic $k_T$ kick respectively. Three curves are shown in each case: no intrinsic charm (pQCD only, solid), $P_{\mathrm{ic}5}^0=0.1$% (dashed), and $P_{\mathrm{ic}5}^0=1$% (dot-dashed). No $J/\psi$ absorption by nucleons is considered in the $p+\mathrm{Pb}$ calculation.

Figure 15

The nuclear suppression factor $R_{p\mathrm{Pb}}$ as a function of transverse momentum at forward rapidity for $J/\psi$ [(a), (c), (e)] and $\overline{D}$ [(b), (d), (f)] mesons, including both the typical perturbative QCD contribution and intrinsic charm from a five-particle proton Fock state. The intrinsic charm contribution is varied in the panels from no intrinsic charm (pQCD only) [(a) and (b)], $P_{\mathrm{ic}5}^0=0.1$% [(c) and (d)], and $P_{\mathrm{ic}5}^0=1$% [(e) and (f)]. The red curves include the EPPS16 modifications of the parton densities only while the blue curves also include nuclear absorption of the $J/\psi$. The magenta curves include the EPPS16 modifications as well as $k_T$ broadening while the cyan curves include EPPS16, nuclear absorption for the $J/\psi$, and $k_T$ broadening. (There is no absorption of the $\overline{D}$ mesons in cold nuclear matter.) The line types denote different energies: $p_{\mathrm{lab}}=40$ GeV (solid), 158 GeV (dashes), 800 GeV (dot-dashed), $\sqrt{s_{NN}}=87.7$ GeV (dotted), 200 GeV (dot-dot-dot-dashed), and 5 TeV (dot-dot-dash-dashed). Note that the rapidity range is $0<y<1$ for all energies except the two highest where the rapidity range is $1.1<y<2.2$ for 200 GeV and $2.5<y<5$ for 5 TeV.

Figure 16

The nuclear suppression factor $R_{p\mathrm{Pb}}$ as a function of transverse momentum at backward rapidity for $J/\psi$ [(a), (c), (e)] and $\overline{D}$ [(b), (d), (f)] mesons, including both the typical perturbative QCD contribution and intrinsic charm from a five-particle proton Fock state. The intrinsic charm contribution is varied in the panels from no intrinsic charm (pQCD only) [(a) and (b)], $P_{\mathrm{ic}5}^0=0.1$% [(c) and (d)], and $P_{\mathrm{ic}5}^0=1$% [(e) and (f)]. The red curves include the EPPS16 modifications of the parton densities only while the blue curves include nuclear absorption of the $J/\psi$. The magenta curves include the EPPS16 modification as well as $k_T$ broadening while the cyan curves include EPPS16, nuclear absorption for the $J/\psi$, and $k_T$ broadening. (There is no absorption of the $\overline{D}$ mesons in cold nuclear matter.) The line types denote different energies: $p_{\mathrm{lab}}=40$ GeV (solid), 158 GeV (dashes), 800 GeV (dot-dashed), $\sqrt{s_{NN}}=87.7$ GeV (dotted), 200 GeV (dot-dot-dot-dashed), and 5 TeV (dot-dot-dash-dashed). Note that the rapidity range is $-1<y<0$ for all energies except the two highest, where the rapidity ranges are $-2.2<y<-1.1$ for 200 GeV and $-5<y<-2.5$ for 5 TeV.