Inclusive Dijet Cross-Sections

• Inclusive dijet cross-sections are defined as the differential rates of events with at least two high-pT jets, characterized by invariant mass and rapidity separation.
• NLO and NNLO pQCD calculations, together with modern PDFs and non-perturbative/electroweak corrections, form the backbone of accurate theoretical predictions.
• Experimental analyses employ advanced jet algorithms, precise calibrations, and unfolding techniques to minimize uncertainties and validate QCD dynamics.

Inclusive dijet cross-sections quantify the production rates of two-jet systems in high-energy hadronic, leptonic, or nuclear collisions without imposing restrictions on extra radiation (i.e., inclusive in all additional jet emission). These observables are central to testing perturbative quantum chromodynamics (pQCD), constraining parton distribution functions (PDFs), exploring QCD at high energies, and probing new physics. Inclusive dijet cross-sections are defined and measured in diverse hard-scattering environments: proton–proton (pp), proton–antiproton (p$\bar{\text{p}}$), electron–proton (ep) in both deep inelastic scattering (DIS) and photoproduction regimes, and ultraperipheral heavy-ion collisions (UPCs) via photon-induced processes.

1. Defining Inclusive Dijet Cross-Sections

The inclusive dijet cross section, $\sigma_\text{dijet}$, is the differential or double-differential rate for events containing at least two jets above given $p_T$ and rapidity thresholds, as a function of variables such as the dijet invariant mass $m_{jj}$ and rapidity separations. The general experimental definition is

$\frac{d^2\sigma}{dm_{jj} \, dy^*}$

where $m_{jj} = \sqrt{(p_1 + p_2)^2}$ is the invariant mass of the two highest-$p_T$ jets, and $y^* = \frac{1}{2}|y_1 - y_2|$ is half their rapidity separation. Alternative projections use the boost of the dijet system, $y_{\text{boost}} = \frac{1}{2}(y_1+y_2)$, or the largest absolute rapidity, $|y|_{max}$, depending on the analysis focus (Collaboration, 22 Dec 2025, 1711.02692, Collaboration, 2011, Collaboration, 2011, Collaboration et al., 2010).

At leading order, inclusive dijet production in hadron–hadron collisions is described by the convolution of PDFs and the $2\to2$ QCD partonic subprocess matrix elements

$$d^2\sigma = \sum_{i,j} \int dx_i \, dx_j \, f_i(x_i,\mu_F) f_j(x_j,\mu_F) \, d\hat{\sigma}_{ij\to 2\,\text{jets}}(\mu_R, \mu_F)$$

with appropriate phase-space cuts.

2. Theoretical Frameworks and Corrections

Precise predictions require next-to-leading order (NLO), and for the latest LHC results, full colour next-to-next-to-leading order (NNLO) pQCD calculations, matched to modern PDF sets and corrected for non-perturbative (NP) and electroweak (EW) effects (Collaboration, 22 Dec 2025, 1711.02692).

Key Elements of Theory Prediction

• PDFs: Modern global fits (ATLASPDF21, CT18NNLO, NNPDF4.0, MSHT20) matched to the factorization scheme used in NLO/NNLO computations (Collaboration, 22 Dec 2025).
• Scale choice: Most analyses adopt $\mu_R = \mu_F = m_{jj}$ for central values, with 7-point scale variations to assess theoretical uncertainties (Collaboration, 22 Dec 2025, 1711.02692).
• NP corrections: Derived from the ratio of particle-level to parton-level cross sections using MC event generators (Pythia8, Herwig7), accounting for hadronization, multiple parton interactions (MPI), and underlying event (UE) effects. Corrections are large at low $m_{jj}$ ($+$5–10%), approaching unity at high mass (Collaboration, 22 Dec 2025, Collaboration, 2011).
• EW corrections: At high $m_{jj}$, full $$\mathcal{O}(\alpha^2) + \mathcal{O}(\alpha_s^2\alpha)$$ effects can yield corrections up to $+10\%$ in central rapidity slices (Collaboration, 22 Dec 2025).

Across all environments, scale variations, PDF uncertainties, and NP corrections define the dominant theoretical error envelope, with NNLO predictions reducing scale uncertainties to the few-percent level for inclusive-jet observables, but residual normalization discrepancies persist at the 10–20% level in the dijet channel (Collaboration, 22 Dec 2025).

3. Experimental Methodologies and Unfolding Techniques

All modern inclusive dijet measurements deploy infrared- and collinear-safe jet algorithms (anti-$k_T$, $k_T$; $R=0.4$–$0.7$), with jets reconstructed at the particle level within precise $p_T$ and $y$ acceptances (Collaboration, 22 Dec 2025, 1711.02692, Collaboration, 2011, Collaboration, 2010). Key experimental strategies include

• Data selection: Dijet events are defined with leading/subleading jet $p_T$ and rapidity requirements to ensure full trigger efficiency, minimize detector effects, and maintain perturbative stability.
• Jet calibration: Sequential MC-based JES corrections, in-situ calibrations (multijet balance, $\gamma+$jet), and pile-up mitigation yield $\sim$1–10% JES uncertainty, the dominant experimental error (Collaboration, 22 Dec 2025, 1711.02692).
• Unfolding: Response matrices constructed from MC simulations (Pythia, Herwig) enable correction for resolution and acceptance (ITERATIVE DYNAMICALLY STABILIZED, IDS, or TUnfold methods) (Collaboration, 22 Dec 2025, Collaboration, 2021, Collaboration, 2011). Closure and data-driven ttests constrain model bias and stability.
• Systematics: JES (2–22% across phase space), JER ($<$2%), unfolding bias ($<$1%), pile-up, luminosity (sub-percent), and dedicated detector effects are all propagated to final uncertainties.

Differential cross sections are quoted with full covariance matrices, and typical statistical uncertainties are sub-dominant except at highest $m_{jj}$. Tabulated results uniformly show the cross sections falling steeply over 6–9 orders of magnitude as $m_{jj}$ is increased from $\sim 100$ GeV to $\sim 10$ TeV (Collaboration, 22 Dec 2025, 1711.02692, Collaboration, 2011, Collaboration et al., 2010).

4. Physics Implications and QCD Dynamics

Inclusive dijet cross-section measurements directly test QCD dynamics at large momentum transfer and extreme parton kinematics, providing strong constraints on PDFs and $\alpha_s$.

• PDF sensitivity: Access to quark and gluon densities at high $x$ via $m_{jj}$ dependence ($x \sim m_{jj}/\sqrt{s} \cdot \exp(\pm y^*)$), essential for SM and BSM searches (Collaboration, 22 Dec 2025, Malaescu, 2012, Collaboration, 2011, Collaboration et al., 2010).
• QCD verification: Agreement with NLO and NNLO predictions within combined theory+experimental uncertainties validates perturbative QCD up to $Q^2$ of order $10^6$ GeV$^2$ and $m_{jj}\approx 10$ TeV (Collaboration, 22 Dec 2025).
• Nuclear and photon PDFs: In heavy-ion UPCs, inclusive dijet photoproduction is sensitive to nuclear PDF modifications (shadowing/antishadowing) at the 10–20% level, with comparable theoretical uncertainties; measurement constrains gluon densities in nuclei (Guzey et al., 2018).
• DIS and photoproduction: Inclusive dijet cross sections in DIS at HERA constrain the gluon PDF in the proton and the photon structure, with NLO QCD describing data to $\pm$5% (Kuprash, 2011, Collaboration, 2010).

Ancillary cross-section ratios, such as the Mueller–Navelet/inclusive ratio as a function of rapidity separation, isolate BFKL logarithms and QCD radiation patterns at high $\Delta y$ (Collaboration, 2021).

5. Main Experimental Results and Comparison with Predictions

A representative summary of recent extensive measurements demonstrates the key features:

Experiment	$\sqrt{s}$	$m_{jj}$ Range	$y^*$ Range	Precision/Uncertainty	Theory/Experiment Agreement
ATLAS (2025) (Collaboration, 22 Dec 2025)	13 TeV	0.24–10 TeV	$<3.0$	$<$5% (central), up to 20% (fwd)	NNLO overestimates data by 15–20%, shape well-described; improvement at low $y^*$
ATLAS (2017) (1711.02692)	13 TeV	0.3–9 TeV	$<3.0$	6–30%	NLO within uncertainties; mild excess in theory at forward/high $m_{jj}$
CMS (2011) (Collaboration, 2011)	7 TeV	0.2–3.5 TeV	$<2.5$	10–60% (JES dom.)	NLO+NP accurate over 8 orders of magnitude
DØ (2010) (Collaboration et al., 2010)	1.96 TeV	0.15–1.3 TeV	$<2.4$	6–45%	NLO+NP describes spectrum, large PDF uncertainties at high mass
HERA (ZEUS/H1) (Collaboration, 2010, Kuprash, 2011)	318 GeV	$M_{jj}$ up to 120 GeV	–	4–10%	NLO QCD within 5–10%; constrains gluon and photon densities

These results establish the robustness of pQCD and provide stringent constraints on PDFs and theoretical modeling. No significant deviation from the Standard Model is reported over the entire measured phase spaces (Collaboration, 22 Dec 2025, 1711.02692, Malaescu, 2012, Collaboration, 2011).

6. Inclusive Dijet Cross-Sections in Nuclear and Lepton-Induced Collisions

In ultraperipheral Pb–Pb collisions, the cross section for inclusive dijet photoproduction

$$\sigma(\mathrm{Pb\,Pb}\to\mathrm{Pb} + 2\,\mathrm{jets} + X)$$

factors into a convolution over the equivalent-photon flux, nuclear and photon PDFs, and the hard partonic cross section. The NLO spectrum is sensitive to nuclear PDF modifications at the level of 10–20% in $x_A\sim10^{-3}$–$10^{-1}$, sufficient for constraints on unmeasured gluon shadowing (Guzey et al., 2018). Theoretical uncertainties are dominated by gluon shadowing ambiguities, with NLO residual scale uncertainty notably smaller. Ratios of UPC cross sections cancel systematic errors and further enhance nPDF sensitivity.

At HERA, inclusive dijet cross sections in both NC DIS ($Q^2>125$ GeV$^2$) and photoproduction ($Q^2\sim 0$) have been precisely measured. The NLO theoretical description includes full scale and PDF variations, photon PDF ambiguities, and hadronization corrections. In photoproduction, observed sensitivity to the photon PDF is manifest in the $x_\gamma^{obs}$ distribution, with unresolved-photon processes dominant at $x_\gamma^{obs}<0.75$ and resulting in substantial (10–15%) theoretical ambiguities (Kuprash, 2011).

In $e+A$ collisions, the Color Glass Condensate (CGC) formalism permits NEikonal (NEik) corrections to the strict eikonal limit, incorporating finite target width, transverse field insertions, and dynamical target effects. The NEik contributions can be $\mathcal{O}(10\%)$ for EIC kinematics, parameterized by decorated Wilson-line correlators (Tymowska et al., 2023).

7. Implications for QCD Phenomenology and Future Directions

Inclusive dijet cross-section data across hadron–hadron, lepton–hadron, and nuclear environments play a critical role in

• PDF global fits: Directly influencing gluon and valence PDFs at high $x$, illuminating the $x$ and $Q^2$ dependencies inaccessible in inclusive DIS or low-$p_T$ Drell–Yan data (Collaboration, 22 Dec 2025, Malaescu, 2012, Collaboration et al., 2010).
• Precision SM parameters: Extending $\alpha_s$ running tests to multi-TeV scales; constraining higher-order QCD and EW corrections formally and phenomenologically (Collaboration, 22 Dec 2025, 1711.02692).
• Nuclear modifications: Providing input for nPDF fits, probing shadowing, antishadowing, and EMC effects, particularly via UPCs (Guzey et al., 2018).
• Discriminating QCD evolution: Ratios of inclusive and Mueller–Navelet cross sections as a function of rapidity separation $\Delta y$ probe the interplay between DGLAP and BFKL evolution; deviations may hint at the onset of high-energy logarithm resummation (Collaboration, 2021).
• Prospects: Incorporation of unfolded and fully normalized measurements with covariance matrices into global fits; extension to three-jet, heavy-flavor tagged final states; development of full NNLO and small-$x$ resummation for precision comparison (Guzey et al., 2018, Collaboration, 22 Dec 2025).

A plausible implication is that future measurements extended in energy, luminosity, final-state complexity, and with reduced systematic/theoretical uncertainty, will further sharpen constraints on QCD evolution, the PDFs of protons, nuclei, and photons, and the search for physics beyond the Standard Model.