Hadrons and Heavy Quarks

The useful starting point for hadron physics is an observable, not a model. A stable one-particle state is identified through its quantum numbers and pole; an unstable hadron through a pole of a coupled-channel amplitude; a current probe through invariant form factors; and a high-energy probe through a factorized light-ray correlator. Heavy-light and heavy-heavy systems then call for different effective expansions. This chapter keeps those statements separate while showing how each is connected to QCD.

Choose the observable before the expansion

The same hadron can appear in several mathematically different objects. Confusing them is a common source of overclaiming:

Physical question	QCD object	Quantity reported	Essential qualification
Which channel can create the state?	Gauge-invariant interpolating operator and two-point function	Spin, parity, flavor, charge conjugation where defined, and spectral support	An operator labels a channel; it need not isolate one state
Is an enhancement an unstable particle?	Analytically continued scattering amplitude	Pole position, sheet, residue, and channel couplings	A peak or a finite-volume level is not by itself a resonance
How does a local current resolve the hadron?	On-shell current matrix element	Lorentz-invariant form factors and their moments	The current, normalization, scheme, and kinematic region must be stated
How is longitudinal or transverse partonic structure encoded?	Renormalized bilocal operator with a Wilson line	PDF, GPD, or TMD at specified scales and scheme	These correlators have different kinematics and density interpretations
What simplifies when one quark is heavy?	HQET field with residual momentum	Heavy-quark symmetry relations and an expansion in $\Lambda_{\mathrm{QCD}}/m_Q$	The expansion describes one heavy source, not a slow heavy pair
What simplifies for a slow heavy pair?	NRQCD or potential-NRQCD operators	Velocity expansion, potentials, matrix elements, and threshold observables	The hierarchy $m\gg mv\gg mv^2$ must be checked rather than assumed

No single extraction method supplies all six kinds of information. Experimental amplitudes, Euclidean correlation functions, perturbative matching, and effective theories enter at different stages and carry different uncertainties.

Check your preparation

This is an unscored diagnostic. A “not yet” answer is simply a pointer to the shortest repair route.

Can you already…	Ready when you can…	Repair
read a spectral representation?	distinguish an operator overlap, a stable-state pole, and continuum support in a two-point function	review spectral decomposition and two-point functions
use $S$-matrix analyticity?	distinguish a physical-axis line shape from a pole reached on a specified sheet	review resonance poles, widths, and unstable particles
decompose a local-current matrix element?	build all independent Lorentz structures and impose current conservation	review form factors and local operator insertions
recognize factorization data?	separate a short-distance coefficient from a renormalized operator matrix element and its scale	review collinear factorization and operator PDFs
organize an effective expansion?	name the retained degrees of freedom, matching scale, and power-counting parameter	review heavy-particle EFT and HQET architecture and nonrelativistic potential EFT

The dependencies printed on each page are required for the derivation on that page. The order below is only suggested: begin at the first route whose readiness statement you can meet, and move sideways when your observable changes.

Chapter guide

1. Hadron Quantum Numbers and the QCD Spectrum constructs color-singlet channels, separates exact from approximate labels, and explains what a spectral level can and cannot mean.
2. Hadron Resonances and Coupled Channels defines an unstable hadron by amplitude poles and residues, with thresholds and Riemann sheets made explicit.
3. Hadron Form Factors and Current Structure decomposes local-current matrix elements and derives the standard spin-$\tfrac12$ electromagnetic form factors.
4. Partonic Structure, Spin, and Hadron Tomography distinguishes PDFs, GPDs, and TMDs by their operator definitions, kinematics, and valid interpretations.
5. Heavy-Quark Symmetry and HQET derives the static heavy-source limit, its spin-flavor symmetry, and the first $1/m_Q$ corrections.
6. Quarkonium and NRQCD treats a nonrelativistic heavy pair, from the scales $m$, $mv$, and $mv^2$ to NRQCD factorization and potential dynamics.

The first route supplies labels used by the next two. The fourth route branches from current structure into nonlocal operators. The last two routes are alternatives selected by the number and kinematics of heavy constituents; HQET is not a preliminary version of NRQCD.

Two heavy-quark limits that must not be merged

Feature	One heavy quark in a hadron	Heavy quark–antiquark pair
Momentum split	$p_Q=m_Qv+k$, with $k=O(\Lambda_{\mathrm{QCD}})$	relative momentum $p=O(mv)$ and energy $E=O(mv^2)$
Active low-energy fields	velocity-labelled heavy source plus light quarks and gluons	Pauli quark and antiquark fields; singlet/octet fields after a further matching step
Expansion	$\Lambda_{\mathrm{QCD}}/m_Q$ and perturbative matching	$v$, $\alpha_s$, and scale ratios among $m$, $mv$, $mv^2$, and $\Lambda_{\mathrm{QCD}}$
Leading symmetry or dynamics	heavy-quark spin and flavor symmetry	spin-independent potential dynamics, with spin effects suppressed in $v$ and $1/m$
Characteristic limitation	subleading currents, mass schemes, and finite-mass corrections	imperfect scale separation, potential renormalons, and process-dependent factorization proofs

This comparison is a decision rule, not a numerical claim. In borderline systems the relevant ratios must be estimated for the observable and precision under discussion.

Synthesis check

Use one concrete hadronic claim and answer these four questions without assigning yourself a score:

1. What is the observable? A successful answer writes the pole, matrix element, or bilocal correlator—not merely the name of a particle or method. If this is unclear, return to the observable map.
2. Which labels and kinematics are exact? A successful answer identifies conserved quantum numbers, momenta, channels, and any approximate symmetry separately. Repair this at the first route in the guide.
3. Which method connects QCD to the reported quantity? A successful answer states matching, analytic continuation, factorization, or correlation-function inversion and its assumptions. Repair this with the preparation diagnostic.
4. What could invalidate the inference? A successful answer names at least one threshold, scheme, scale-hierarchy, finite-volume, truncation, or model-dependence check relevant to the claim. Compare the two heavy-quark limits when heavy constituents are involved.

A complete synthesis keeps the observable definition, theoretical representation, extraction method, and uncertainty source in separate sentences before relating them.

Continue by purpose

• For perturbative production and evolution: continue to Perturbative QCD and Partons.
• For systematic matching and power counting: continue to Renormalization and Effective Field Theory.
• For finite-volume scattering information: continue to scattering amplitudes and resonance poles from finite-volume spectra.
• For current matrix elements from Euclidean correlators: continue to three-point functions, matrix elements, and disconnected contributions.
• For worked heavy-scale checks: reproduce the matching, running, and mass-scheme examples with the stated inputs.
• For the wider field-theory route: return to Gauge Theories and the Standard Model.

References

• Bodwin, Geoffrey T., Eric Braaten, and G. Peter Lepage. “Rigorous QCD Analysis of Inclusive Annihilation and Production of Heavy Quarkonium.” Physical Review D 51 (1995): 1125–1171; erratum 55 (1997): 5853. DOI · Open PDF
• Briceño, Raúl A., Jozef J. Dudek, and Ross D. Young. “Scattering Processes and Resonances from Lattice QCD.” Reviews of Modern Physics 90 (2018): 025001. DOI · Open PDF
• Diehl, Markus. “Generalized Parton Distributions.” Physics Reports 388 (2003): 41–277. DOI · Open PDF
• Neubert, Matthias. “Heavy-Quark Symmetry.” Physics Reports 245 (1994): 259–396. DOI · Open PDF
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014, chs. 26, 32, and 35. DOI