Parton Distribution Functions (PDFs)

• PDFs are mathematical functions defining the probability of finding a parton with a given momentum fraction inside a proton, key to the QCD factorization framework.
• They are determined through global fitting procedures that incorporate experimental data and DGLAP evolution to control scale dependence and uncertainty.
• Robust PDF uncertainties critically affect predictions for collider processes, impacting both precision Standard Model tests and new physics sensitivity.

Parton Distribution Functions (PDFs) encode the non-perturbative structure of the proton by specifying the probability density to find a parton (quark, anti-quark, or gluon) of flavor $i$ carrying a fraction $x$ of the proton’s light-cone momentum, when probed at a resolution scale $\mu^2$. PDFs are a universal ingredient for all theoretical predictions of hard processes at hadron colliders within the QCD factorization framework. Their determination, evolution, and uncertainties critically affect precision Standard Model calculations and the sensitivity of searches for new physics.

1. Collinear Factorization and PDF Definition

Inclusive cross sections for high-energy hadronic processes are computed using the factorization theorem: $$\sigma_{h_1 h_2 \to X}(Q) = \sum_{i,j} \int_0^1\! dx_1 dx_2\, f_i(x_1,\mu^2)\, f_j(x_2,\mu^2)\; \hat\sigma_{ij}(x_1, x_2, \alpha_s(\mu), \ldots)$$ where:

• $f_i(x, \mu^2)$ is the PDF for parton $i$ at scale $\mu$,
• $\hat\sigma_{ij}$ is the short-distance, perturbatively calculable partonic cross section,
• $\mu$ is the factorization scale, separating the long-distance physics (absorbed into the PDFs) from perturbative short-distance interactions.

The PDF $f_i(x, \mu^2)dx$ corresponds to the probability to find a parton with flavor $i$ and light-cone momentum fraction in $[x, x+dx]$, in a proton boosted to infinite momentum.

Residual dependence on the factorization scale $\mu$ in finite-order predictions provides a means to estimate theoretical uncertainties from missing higher orders.

2. DGLAP Evolution: QCD Scale Dependence

The scale dependence of PDFs is governed by the DGLAP evolution equations: $$\frac{\partial f_i(x, \mu^2)}{\partial \ln \mu^2} = \sum_j \int_x^1 \frac{dz}{z}\; P_{ij}(z, \alpha_s(\mu))\; f_j\left(\frac{x}{z}, \mu^2\right)$$ where $P_{ij}(z, \alpha_s)$ are the QCD splitting functions. These functions are known exactly up to NNLO; partial information exists at N$^3$LO, including Mellin moments, small-$x$ and large-$x$ expansions, and large-$n_f$ asymptotics. Modern ‘aN$^3$LO’ (approximate N$^3$LO) fits incorporate this information to extrapolate unknown splitting functions for enhanced theoretical rigor.

DGLAP equations enable the evolution of PDFs from an initial scale $Q_0$ (where a parameterization is chosen) to any higher scale $\mu$ relevant for collider processes.

3. PDF Determination: Global Fitting Procedure

Since PDFs are inherently non-perturbative, they are extracted through global fits to diverse experimental data. The procedure is as follows:

• Assume an initial parameterization $f_i(x, Q_0^2; \{a_k\})$ at scale $Q_0$, with $\mathcal{O}$(30–50) parameters in polynomial forms or $\mathcal{O}$(100) in neural-network ansatz (e.g., NNPDF).
• Use DGLAP evolution to obtain $f_i(x, \mu^2)$ at any scale.
• Assemble a dataset including DIS (from fixed-target experiments and HERA, $10^{-4} \lesssim x \lesssim 0.5$), Drell–Yan and $W/Z$ production (Tevatron, LHC; constraining sea quarks), jets/dijets (medium and large-$x$ gluon), and top-quark pair and high-$p_T$ jets (large-$x$ gluon).
• Each datum comes with experimental uncertainties encoded in a covariance matrix $C^{\text{exp}}$.
• Minimize the loss function: $$\chi^2(\{a_k\}) = [D - T(\{a_k\})]^T (C^{\rm exp} + C^{\rm th})^{-1} [D - T(\{a_k\})]$$ where $T(\{a_k\})$ denotes theory predictions for a set of PDF parameters, and $C^{\rm th}$ is the theory covariance matrix accounting for missing higher-order uncertainties (MHOUs), estimated by scale variations correlated across processes.

The inclusion of $C^{\rm th}$ in the $\chi^2$ function marks a methodological advance in recent fits, allowing systematic theory uncertainties to be propagated alongside experimental ones.

4. Theoretical Uncertainties and Approximate N$^3$LO PDFs

Approximate N$^3$LO PDF fits (e.g., MSHTaN3LO, NNPDF4.0 aN$^3$LO) introduce new strategies for handling incomplete splitting functions—either via nuisance parameters in the prior or by supplementing $C^{\rm th}$ with extra variations. These fits observe robust perturbative convergence and only modest shifts in PDF central values when upgrading from NNLO to aN$^3$LO, except for the large-$x$ gluon where corrections can reach a few percent.

Uncertainty estimates for key LHC processes at NNLO and NNLO+MHOU are summarized as follows:

Process	PDF uncertainty (NNLO)	PDF + MHOU (NNLO)
Higgs (ggF, 13 TeV)	2.0 %	2.3 %
$W/Z$ total	1.2 %	1.4 %
High-mass Drell–Yan ($m_{\ell\ell} \gtrsim 1$ TeV)	8–12 %	9–14 %
Jet $p_T \gtrsim 1$ TeV	10–15 %	12–18 %

PDF uncertainties remain a leading source of error, particularly for Higgs cross sections and high-mass new-physics searches. At $x \gtrsim 0.5$ uncertainties increase rapidly due to scarce DIS data and limited jet coverage.

5. PDF Uncertainties and New Physics Sensitivity

Large-$x$ PDFs are critical for new physics searches probing high-energy tails, where Standard Model Effective Field Theory (SMEFT) operators can yield deviations. Care must be taken that global fits do not absorb genuine BSM signals into the PDFs, especially when fitting LHC tail data.

Neglecting the interplay between large-$x$ PDF uncertainty and Wilson-coefficient extraction weakens SMEFT bounds, particularly for operators like $\hat{W}$ and $\hat{Y}$, as according to SMEFT studies, PDF shifts in antiquark densities can mimic energy dependence of contact terms.

Only with complementary low-energy constraints—e.g., new lepton–nucleon DIS data from an EIC, LHeC, or forward neutrino collider—can PDFs and new physics effects be robustly disentangled, underpinning the necessity for coordinated high-/low-energy experimental programs.

6. Outlook: HL-LHC, Future Facilities, and PDF Precision

The HL-LHC will deliver greatly increased statistics for vector-boson, top, and jet measurements, shrinking experimental PDF uncertainties and making theory systematics—MHOUs, aN$^3$LO splitting functions, electroweak corrections—ever more prominent. Continued progress hinges on:

• Systematic development of approximate N$^3$LO fits based on all available theoretical input,
• Full integration of theory covariance matrices in global analysis,
• Joint exploitation of collider and low-/intermediate-energy DIS data, including nuclear effects.

Reduction of overall theoretical uncertainty will enable sharpened sensitivity to subtle SM deviations—e.g., in Higgs couplings, $W$ mass determinations, and indirect searches for new interactions. The synergy between HL-LHC and next-generation lepton–hadron colliders is expected to drive PDF uncertainties to the percent-level domain.

7. Implications for QCD Phenomenology and Experimental Analysis

Improved PDFs, together with state-of-the-art theoretical frameworks, will enable precision QCD phenomenology, permitting measurements and SM tests at unprecedented accuracy. As PDF errors are systematically reduced, new challenges arise in the estimation and propagation of theory uncertainties, necessitating the ongoing development of advanced statistical and theoretical tools.

PDF uncertainties constitute a dominant component of error budgets for benchmark processes (Higgs, $W/Z$, top), backgrounds for high-mass new physics searches, and the calibration of luminosity via standard candles. Their reliable quantification is essential for maximizing the discovery potential and precision of current and future collider experiments.