QCD (1708.00770v1)

Abstract: In these lecture notes I describe the theory of QCD and its application, through perturbation theory, at particle colliders.

• The paper offers a comprehensive guide to applying perturbative QCD at collider experiments by integrating theory with practical computational techniques.
• It presents detailed methodologies including parton distribution functions, renormalization group analysis, and resummation strategies critical for precision predictions.
• Key insights cover automated NLO/NNLO computations and Monte Carlo event generators, ensuring reliable implementation for high-energy physics research.

The document is a comprehensive set of lecture notes titled "QCD" by E. Laenen (1708.00770), designed as an introduction and an advanced guide to Quantum Chromodynamics (QCD) and its application—particularly through perturbation theory—in the context of high-energy particle colliders like the LHC. The notes balance theoretical fundamentals, formal methods, and practical computational strategies for real-world QCD applications at the energy frontier.

Here’s a structured and detailed summary emphasizing practical implementation and applications:

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1. Introduction and Motivation

• QCD is not merely a background to seek new particles at colliders, but a rich and fundamental aspect of the Standard Model.
• At high-energy colliders, e.g., the LHC, proton-proton collisions are inherently partonic (quark, antiquark, gluon) and produce diverse hadronic final states. Understanding QCD dynamics is essential for reliable background estimation and precision measurements.

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2. Fundamentals: Partons and Hadrons

Parton Model and QCD Lagrangian

• Six quark flavors, each in three color states (red, green, blue) interacting via SU(3) non-abelian gauge theory.
• The QCD Lagrangian encapsulates the dynamics of quarks (with spins, flavors, and colors) and gluons (eight color-charged gauge bosons).
• Hadronic states (protons, neutrons, mesons) are color-singlet combinations.

Symmetry and Spectroscopy

• QCD respects color SU(3) local symmetry and, in the massless limit, exhibits approximate global flavor and chiral symmetries important for understanding hadron mass spectra and interactions.
• Phenomena like confinement (no free quarks) and color screening in hadronization are explained.

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3. Parton Model & Deep-Inelastic Scattering (DIS)

• Describes the experimental evidence for parton (quark and gluon) constituents inside hadrons, measurable via deep-inelastic scattering and the Drell-Yan process.
• Parton Distribution Functions (PDFs): Quantify the probability of finding a parton of type $i$ carrying momentum fraction $x$ inside the proton.
  • PDFs are universal and essential inputs for collider predictions.
  • Practical extraction involves fits to a range of experimental data sets, using parametrizations (e.g., neural networks — NNPDF, or analytical forms — MSTW, CTEQ) evolved with the DGLAP equations.
• Examples of how inclusive cross-sections for processes like $e^+e^- \to$ hadrons and Drell-Yan lepton-pair production are connected to PDFs.

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4. Renormalization, Running, and Regularization in QCD

• QCD, like other quantum field theories, requires regularization of infinities (done via lattice or, for perturbative calculations, usually dimensional regularization).
• Renormalization group equations dictate how the coupling $\alpha_s$ (strong coupling constant) varies with energy scale (asymptotic freedom).
• The $\beta$-function governs "running"; for QCD with $N_F$ flavors, it predicts weakening of coupling at high energies—a unique non-abelian feature allowing perturbative calculations.

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5. Higher-Order QCD: Practical Aspects

Parton Distribution Functions (PDFs) — Implementation

• PDFs are extracted through global fits to experimental data (DIS, Drell-Yan, jets, vector boson production).
• PDF uncertainties are critical in precision predictions; often, multiple PDF sets are compared to estimate systematic uncertainties.
• Modern fits may use artificial neural networks for unbiased functional forms (NNPDF), or targeted parametric forms (MSTW, CTEQ).

Collider Observables

• Calculations at next-to-leading (NLO) and next-to-next-to-leading order (NNLO) are now standard for LHC processes.
• All calculations hinge on factorization theorems, separating perturbative ("short distance") and non-perturbative (PDFs/hadronization) physics.
• Infrared and Collinear Safety: Essential concept for meaningful observables; e.g., jet algorithms and event shapes like thrust are constructed to avoid divergences.

Monte Carlo Event Generators

• Practical QCD analyses often require matching of fixed-order calculations to parton showers and hadronization models for event simulation—enabling direct comparison with experimental data.

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6. Modern Methods in QCD Calculations

Spinor Helicity & Recursion

• Use of spinor helicity formalism dramatically simplifies computation of multi-leg scattering amplitudes, replacing conventional trace techniques with efficient algebra involving spinor products.
• Parke-Taylor (MHV) amplitudes and recursion relations (Berends-Giele, BCFW) enable analytic or semi-automatic computation of complex processes involving many external gluons or quarks.

NLO (and Beyond) Automation ("NLO Revolution")

• Automation and standardization of NLO calculations using unitarity methods, integrand reduction (OPP), and on-shell techniques.
• Tools/frameworks: BLACKHAT, CutTools, Rocket, MCFM, MadGraph5_aMC@NLO, POWHEG BOX, etc.
• These enable the computation of higher-order corrections for multi-particle processes crucial for LHC analyses.

NNLO Progress

• Essential for percent-level predictions at LHC Run 2 and beyond; requires advanced handling of multi-loop integrals and infrared subtraction.
• Tools like FORM facilitate algebraic manipulation at this level.

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7. All-Order Resummation Techniques

Resummation (Sudakov Logs, Threshold Effects, SCET)

• In some kinematic regions (phase space edges, small jet masses, threshold production), fixed-order expansions fail due to large logarithms.
• Resummation methods (e.g., threshold and transverse momentum resummation), sometimes formulated via Soft-Collinear Effective Theory (SCET), systematically sum these logarithms to all orders in $\alpha_s$.
• Enables precise predictions for observables like Higgs/transverse momentum, $t\bar{t}$ production at threshold, jet vetoes, etc.

Exponentiation and Webs

• Structures like "webs" (collections of eikonal diagrams) govern exponentiation of soft divergences in QCD and underlie the resummation formalism.

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8. Implementation Guidance

• Cross sections are computed by convolving partonic cross sections (obtained at appropriate order using modern techniques) with PDFs extracted via global fits.
• For precise collider predictions:
  • Use NLO/NNLO calculations for both matrix elements and PDF sets.
  • Ensure that kinematic cuts and observables are IR safe.
  • Incorporate matching/merging to parton showers for fully exclusive event predictions.
  • Estimate theoretical uncertainties via scale variations, PDF uncertainties, and comparison of resummation schemes.

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9. Summary Tables and Useful Formulae

The notes include comprehensive appendices listing:

• Conventions for indices, cross section and decay rate formulae, spinor and Dirac algebra, group theory, SM parameters.
• Feynman rules and integration techniques, essential for practical calculations.

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Takeaways for Practical QCD in Collider Physics

• Understand factorization: Separate perturbative from non-perturbative, use universal PDFs.
• Choose proper calculation tools: For NLO/NNLO, use automated frameworks and validate with multiple PDF sets and Monte Carlo generators.
• Resummation is essential where large logs appear, use SCET or threshold resummation techniques as appropriate.
• Analysis reproducibility: Adhere to common conventions, and use community PDF sets and codes to ensure compatibility and reliability.

These notes, while technical, are focused on empowering researchers to implement, compute, and deploy QCD predictions at high-energy colliders with practical guidance for both the conceptual framework and modern computational techniques.