Inclusive Quarkonium Photoproduction

• Inclusive quarkonium photoproduction is the production of heavy quarkonium states via quasi-real photon interactions in ultra-peripheral collisions, serving as a probe of small-x gluon densities.
• The process relies on NRQCD factorization, incorporating both color-singlet and color-octet mechanisms to explain observed cross sections and kinematic distributions.
• Experimental strategies employ rapidity gaps, centrality selection, and forward neutral vetoes to isolate the inclusive photoproduction signal from overwhelming hadronic backgrounds.

Inclusive quarkonium photoproduction refers to the process in which a quasi-real photon, typically emitted in ultra-peripheral collisions (UPCs) of protons and/or nuclei, interacts with a hadronic target to produce a heavy quarkonium state (such as $J/\psi$, $\psi(2S)$, $\Upsilon(nS)$, $\eta_c$, $\eta_b$), together with additional hadronic activity, excluding purely exclusive or elastic topologies. The study of this process provides a direct probe of gluon distributions at very small Bjorken-$x$ and offers stringent tests of non-relativistic QCD (NRQCD) factorization, color-octet and singlet mechanisms, and diffraction dynamics in QCD.

1. Theoretical Framework and Factorization

The inclusive quarkonium photoproduction cross section in hadronic collisions is described by NRQCD factorization, in which the cross section is written as a sum over heavy-quarkonium Fock states weighted by short-distance coefficients and nonperturbative long-distance matrix elements (LDMEs) (Wu et al., 2020, Goncalves et al., 2018, Zhan et al., 2020). The essential structure reads: $\sigma(\gamma + p \to Q + X) = \sum_n \sigma_{\hat}( \gamma + i \to Q\bar{Q}[n] + X ) \langle O_n^Q \rangle$ where $n$ labels the intermediate $c\bar{c}$ (or $b\bar{b}$) state ($^3S_1^{[1]}$, $^1S_0^{[8]}$, $^3S_1^{[8]}$, $^3P_J^{[8]}$), $i$ runs over the partons in the target (mostly gluons, but also quarks for quark-induced subprocesses), and $\langle O_n^Q \rangle$ are process-independent LDMEs.

The underlying hard photoproduction process at leading order is typically $\gamma + g \to Q\bar{Q}[n] + g$, but also includes $\gamma + q \to Q\bar{Q}[n] + q$ and various resolved contributions, especially at high energies (Wu et al., 2020). For diffractive production, the resolved Pomeron model is used, in which the proton emits a Pomeron carrying its own partonic structure (Goncalves et al., 2017).

Color-singlet (CS) and color-octet (CO) mechanisms are both included. The CO channels are particularly important for matching data on $J/\psi$ and back-to-back polarization observables and are required for $\eta_c$ and $\eta_b$ photoproduction, where the CS channel is absent at leading order (Goncalves et al., 2018).

2. Photon Fluxes and Target Structure

Photon emission in UPCs is modeled via the equivalent-photon (Weizsäcker–Williams) approximation, where the flux is proportional to the square of the electromagnetic charge of the emitter and depends on Lorentz boost and impact parameter.

For protons: $$\frac{dN_\gamma^p}{d\omega} = \frac{\alpha_{\rm em}}{2\pi\omega} \left[ 1+\left(1-\frac{2\omega}{\sqrt{s}} \right)^2 \right] \left(\ln \Omega - \frac{11}{6} + \frac{3}{\Omega} - \frac{3}{2\Omega^2} + \frac{1}{3\Omega^3} \right)$$ with $\Omega=1+0.71\,\text{GeV}^2/Q_{\min}^2$, $Q_{\min}^2\approx(\omega/\gamma_L)^2$ (Goncalves et al., 2017).

For nuclei (charge $Z$), the flux is enhanced $\propto Z^2$ and takes the form: $$\frac{dN_\gamma^A}{d\omega} = \frac{2Z^2\alpha_{\rm em}}{\pi \omega} \left[ \bar{\eta} K_0(\bar{\eta}) K_1(\bar{\eta}) - \frac{\bar{\eta}^2}{2} (K_1^2(\bar{\eta}) - K_0^2(\bar{\eta}))\right]$$ where $\bar{\eta} = \omega (R_1 + R_2) / \gamma_L$ (Goncalves et al., 2017, Lynch, 21 Jan 2025).

Diffractive processes rely on the resolved Pomeron flux, parameterized using H1 fits to HERA data, including the Pomeron's partonic PDFs and trajectory parameters (Goncalves et al., 2017, Wu et al., 2020).

3. Kinematic Observables and Cross Sections

The inclusive cross section can be written as a convolution over the photon flux and the photon-nucleon cross section: $$\sigma_{h_1 h_2 \to h_1 \otimes Q + X \otimes h_2} = \int d\omega\, n_{h_1}(\omega)\, \sigma_{\gamma h_2 \to Q+X}(W_{\gamma h_2}) + (h_1 \leftrightarrow h_2)$$ with $W_{\gamma h_2}^2 = 2\omega \sqrt{s_{NN}}$ (Goncalves et al., 2018, Lynch, 21 Jan 2025). The rapidity of the quarkonium is $y = \ln(2\omega/M_Q)$, invertible to $\omega=(M_Q/2) e^y$.

Predictions for $pp$ collisions at $\sqrt{s}=13$ TeV using the resolved Pomeron model yield:

• $J/\psi$: $d\sigma/dy|_{y=0} \approx 0.5$ nb; $\sigma_{\rm tot}=3.4$ nb
• $\psi(2S)$: $d\sigma/dy|_{y=0} \approx 0.2$ nb; $\sigma_{\rm tot}=1.47$ nb
• $\Upsilon$: $d\sigma/dy|_{y=0} \approx 1$ pb; $\sigma_{\rm tot}=11.1$ pb

These distributions are symmetric in rapidity. The transverse-momentum distributions fall with power-laws, $d\sigma/dp_T \propto 1/p_T^n$ ($n \approx 4$–$6$), which allows inclusive production to dominate exclusive at large $p_T$ (Goncalves et al., 2017).

For $\eta_c$, only CO channels contribute, with predictions at mid-rapidity in $pp$ (13 TeV): $d\sigma/dY|_{Y=0} \simeq 0.27$ nb; $\sigma_{\rm tot} = 3.49$ nb (Goncalves et al., 2018).

4. Higher-Order QCD Effects and Quark-Initiated Channels

NLO corrections in collinear factorization for $\gamma + g \to Q + X$ are known to be unstable at high center-of-mass energy ($\sqrt{s_{\gamma p}}$), yielding negative cross sections and excessive factorization-scale dependence. Resummation of high-energy logarithms via high-energy factorization (HEF, DLA) restores physical positivity and reduces uncertainties (Lansberg et al., 2023). HEF resums terms $\propto \alpha_s^n \ln^{n-1}(\hat{s}/M^2)$, yielding agreement with HERA and LHC UPC data.

Inclusion of quark-initiated subprocesses ($\gamma q$, $qg$, $qq$) in the NRQCD framework is essential for reconciling theoretical predictions with HERA data, particularly for $J/\psi$ and $\Upsilon$ (Wu et al., 2020). At the LHC, quark-involved channels can contribute up to 8% of $d\sigma/dy$ and 6% of $d\sigma/dp_T$, with relative fractions tabulated below:

Collision	$d\sigma/dy|_{y=0}$ [nb]	Quark Fraction
$pp$	$3.5$	$8\%$
$p$Pb	$28$	$7\%$
PbPb	$150$	$6\%$

(Wu et al., 2020)

5. Experimental Strategies and Background Suppression

The primary experimental challenge is to extract the inclusive photoproduction signal, which is typically orders of magnitude smaller than hadronic quarkonium production. Isolation leverages several key selections (Lansberg et al., 27 Nov 2025, Lynch, 21 Jan 2025, Lansberg et al., 2024):

• Centrality Selection: Selecting the most peripheral events (80–100%) via Zero Degree Calorimeter (ZDC), which eliminates 94% of hadronic backgrounds but retains nearly all the photoproduction signal.
• Rapidity Gap: Requiring an absence of additional charged tracks/calorimeter activity in the direction of the photon emitter (typically the Pb-going side in $p$Pb), e.g., a minimum gap $\Delta\eta_\gamma \gtrsim 5$ suppresses hadronic background to below $|\text{B}/\text{S}| \sim 10^{-3}$.
• Forward Neutral Veto: ZDC veto on neutron emission on the photon emitter side, exploiting the low probability of photonuclear breakup in photoproduction compared to frequent neutron emission in hadronic collisions.
• Pileup Mitigation: Low-pileup running, vertex association, and timing detectors are required for $pp$; $p$Pb and PbPb are naturally low-pileup.

Detection efficiencies after all cuts in $p$Pb are estimated at the few percent level; predicted yields are $\mathcal{O}(10^3$–$10^4$) inclusive $J/\psi$ and $\Upsilon$ events per rapidity unit in Run 3+4 luminosities.

6. Phenomenological Implications and Comparison with Exclusive Photoproduction

Inclusive diffractive photoproduction cross sections are consistently found to be an order of magnitude or more smaller than exclusive photoproduction in the same kinematic region. For example, at $\sqrt{s}=13$ TeV, $\sigma_{\rm incl}/\sigma_{\rm excl} \lesssim 0.05$–$0.1$ for $J/\psi$, $\psi(2S)$, and $\Upsilon$ (Goncalves et al., 2017).

Exclusive production is characterized by exponentially damped $p_T$ distributions ($d\sigma/dp_T \propto e^{-B_Vp_T^2}$), while inclusive photoproduction follows a much slower power-law ($d\sigma/dp_T \propto 1/p_T^n$). At high $p_T$ ($\gtrsim$ several GeV), the inclusive channel overtakes the exclusive one.

Inclusive photoproduction provides sensitivity to gluon PDFs at small $x$ ($x \sim 10^{-5}$ in high-$W$ LHC UPCs) and discriminates NRQCD color-octet mechanisms, particularly through $p_T$ and $y$ spectra and polarization observables (Lansberg et al., 2024, Lansberg et al., 27 Nov 2025, Zhan et al., 2020).

7. Outlook and Future Directions

Measurement of inclusive quarkonium photoproduction at the LHC, particularly in $p$Pb and PbPb UPCs, will extend $W_{\gamma p}$ coverage to $\sim 1.3$ TeV, a regime inaccessible at HERA or EIC (Lansberg et al., 2024). These data will

• constrain small-$x$ gluon densities,
• test NRQCD through $p_T$ and rapidity spectra,
• allow extraction of individual color-octet LDMEs in heavy quarkonia,
• and provide benchmarks for perturbative and non-perturbative QCD dynamics.

Proposed analysis techniques—including rapidity gap and neutron veto strategies, as well as Jacquet-Blondel kinematic reconstruction—are directly applicable with current LHC detectors (Lansberg et al., 2024, Lynch, 21 Jan 2025). Ongoing theoretical developments in high-energy resummation and resolved photon/pomeron modeling further enhance the reliability and interpretation of future experimental results (Lansberg et al., 2023).