Quarkonium and Nonrelativistic QCD

Quarkonium contains a heavy quark and antiquark moving nonrelativistically relative to one another. Its controlled description separates the hard mass $m$, relative momentum $mv$, and binding-energy scale $mv^2$: matching QCD at $m$ gives NRQCD, while a further matching step at $mv$ gives potential NRQCD when that separation is justified. Spectroscopy, annihilation, threshold production, and inclusive production use related operator expansions but do not share one universal potential-only description.

Required background. Nonrelativistic potential EFT supplies the two-body expansion; factorization and operator structure supplies multiscale matching. Helpful background. Heavy-quark symmetry and HQET sharpens the one-heavy-source contrast; collinear factorization and operator PDFs supplies initial-state factorization for hadronic production.

A heavy pair creates three dynamical scales

In the quarkonium rest frame, nonrelativistic kinematics assigns

$$|\mathbf p|\sim mv, \qquad E-2m\sim mv^2, \qquad v\ll1.$$

The ideal hierarchy is

$$m\gg mv\gg mv^2,$$

but the placement of $\Lambda_{\mathrm{QCD}}$ relative to $mv$ and $mv^2$ determines which ingredients are perturbative. If $mv\gg\Lambda_{\mathrm{QCD}}$, the soft matching can be weakly coupled; if $mv$ is comparable to the confinement scale, potential terms require nonperturbative input. Calling a constituent “heavy” establishes $m\gg\Lambda_{\mathrm{QCD}}$, not automatically the full pair hierarchy.

Integrating out fluctuations of virtuality $m^2$ leaves two independent Pauli fields: $\psi$ annihilates the heavy quark and $\chi$ creates the heavy antiquark. A representative bilinear sector is

$$\begin{aligned} \mathcal L_{\mathrm{NRQCD}}={}& \psi^\dagger\left( iD_0+\frac{\mathbf D^2}{2m} +c_F(\mu)\frac{g_s\boldsymbol\sigma\cdot\mathbf B}{2m} +\cdots\right)\psi\\ &+\text{the charge-conjugate antiquark terms} +\mathcal L_{4f}+\mathcal L_{\mathrm{light}}. \end{aligned}$$

The ellipsis includes relativistic kinetic, Darwin, and spin-orbit terms. The coefficients such as $c_F$ are fixed by matching on-shell QCD amplitudes at the hard scale and then running; setting every coefficient to one is only tree-level matching. Four-fermion operators in $\mathcal L_{4f}$ encode short-distance annihilation and production channels. The energy-scale and velocity organization is given systematically in Bodwin, Braaten, and Lepage 1995, §§ II.A–II.C.

The heavy pair has both color-singlet and color-octet configurations,

$$3\otimes\bar3=1\oplus8.$$

With $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$, normalized color tensors are $\delta_{ij}/\sqrt{N_c}$ and $\sqrt2,T^a_{ij}$. Their orthogonality and the completeness identity

$$\delta_{il}\delta_{kj} =\frac1{N_c}\delta_{ij}\delta_{kl} +2T^a_{ij}T^a_{kl}$$

provide a useful group-factor check. A physical quarkonium state is a color singlet, but color-octet pair operators can contribute inside a gauge-invariant matrix element through accompanying gluonic degrees of freedom.

Matching currents and inclusive reactions

Near threshold, a relativistic vector current matches onto nonrelativistic pair operators. For the spin-triplet spatial component,

$$\bar Q\gamma^iQ =c_v(\mu)\,\psi^\dagger\sigma^i\chi +\frac{d_v(\mu)}{6m^2} \psi^\dagger\sigma^i \left(-\frac{i}{2}\overleftrightarrow{\mathbf D}\right)^2\chi +\cdots.$$

Both the QCD current and the leading Pauli bilinear have mass dimension three, so $c_v$ is dimensionless; the derivative operator needs its explicit $1/m^2$. Hard corrections live in $c_v,d_v$, while repeated potential exchange and long-distance bound-state dynamics live in NRQCD matrix elements or Green functions. Counting the same momentum region in both pieces would double count it.

For an inclusive annihilation decay, the factorized NRQCD form is

$$\Gamma(H\to X) =\sum_n\frac{2\operatorname{Im}f_n(\mu)}{m^{d_n-4}} \langle H|O_n(\mu)|H\rangle.$$

The coefficients $f_n$ describe hard annihilation; the operators $O_n$ project the pair onto definite spin, orbital, and color configurations. Velocity scaling orders their matrix elements. Inclusive production is organized analogously,

$$d\sigma(A+B\to H+X) =\sum_n d\hat\sigma(A+B\to Q\bar Q[n]+X;\mu) \langle0|O_n^H(\mu)|0\rangle.$$

Initial-state PDFs and their factorization scale are additional inputs for hadronic beams. Color-octet terms are required by the operator expansion and renormalization, but their matrix elements are not literal classical probabilities. The production formula also has a more process- and kinematics-sensitive factorization status than the basic matching of the NRQCD Lagrangian; its domain and power corrections must be stated Bodwin, Braaten, and Lepage 1995, § VI.

From NRQCD to potential NRQCD

When the soft scale $mv$ can be integrated out, potential NRQCD uses pair fields depending on center coordinate $\mathbf R$ and separation $\mathbf r$. Its leading structure is

$$\begin{aligned} \mathcal L_{\mathrm{pNRQCD}}=\int d^3r\,\bigl\{& S^\dagger(i\partial_0-h_s)S +O^\dagger(iD_0-h_o)O\\ &+V_A\,[O^\dagger\,\mathbf r\cdot g_s\mathbf E\,S+\text{h.c.}] +\cdots\bigr\}, \end{aligned}$$

where

$$h_s=\frac{\mathbf p^2}{m}+V_s(r;\mu)+\text{spin and relativistic corrections}.$$

For equal masses the reduced mass is $m/2$, hence the relative kinetic term is $\mathbf p^2/m$. The singlet and octet potentials are matching coefficients, not fundamental instantaneous forces at all scales. Ultrasoft gluons remain dynamical and couple through the multipole operator $\mathbf r\cdot\mathbf E$. The construction, including weakly and strongly coupled regimes, is reviewed in Pineda 2012, §§ 2–4 and Brambilla et al. 2005, §§ III–IV.

At weak coupling, the leading color-singlet potential is

$$V_s(r)=-\frac{C_F\alpha_s}{r}, \qquad C_F=\frac{N_c^2-1}{2N_c}.$$

The Coulomb solution immediately gives

$$p_B=\frac{mC_F\alpha_s}{2}, \qquad p_n=\frac{p_B}{n}, \qquad E_n=-\frac{m(C_F\alpha_s)^2}{4n^2}.$$

This derivation uses the reduced mass $m/2$ and measures $E_n$ relative to the pair threshold $2m$. It displays $p\sim mv$ and $E\sim mv^2$ explicitly. Spin-dependent splittings enter through chromomagnetic and higher-order potentials, parametrically beyond this leading Coulomb result.

A reproducible hierarchy check

A reproducible calculation uses the exact benchmark

$$C_F=\frac43, \qquad \alpha_s=\frac3{10}, \qquad m=10.$$

It gives

$$p_B=2, \qquad v=\frac{p_B}{m}=\frac15, \qquad E_1=-\frac25=-mv^2.$$

These identities check the reduced mass, color factor, and scale counting. They are a weak-coupling toy benchmark, not a precision prediction for any physical quarkonium state.

Choose the representation from the hierarchy

Situation	Appropriate representation	Inputs and matching	Principal limitation
$v$ not demonstrably small	relativistic QCD amplitude or correlation function	relativistic current/operator renormalization	no controlled velocity expansion
$m\gg mv,mv^2,\Lambda_{\mathrm{QCD}}$	NRQCD	hard coefficients at $m$; velocity-ordered bilinear and four-fermion matrix elements	truncation in $v$ and $1/m$; operator mixing
$m\gg mv\gg mv^2$ with separable soft scale	potential NRQCD	potentials matched at $mv$; ultrasoft fields at $mv^2$	weak- versus strong-coupling potential input; multipole expansion
Threshold production	matched current times nonrelativistic Green function	hard, soft, potential, and ultrasoft resummation; mass scheme	width effects, nonresonant background, double counting
Inclusive production at high momentum	short-distance production coefficients times NRQCD matrix elements	initial-state factorization, channel basis, evolution	process-dependent proof, large logarithms, polarization and power corrections

Different observables can place the same state in different rows. The representation is selected by active momentum regions and desired accuracy, not by the particle name alone.

Method and uncertainty handoff

Quantity	Long-distance object	Short-distance input	Required validation and uncertainty sources
Energy level or threshold line shape	potential Green function or relativistic correlator	mass scheme, potential coefficients, current matching	scale variation, ultrasoft terms, finite width, nonperturbative potential, continuum limit
Inclusive annihilation width	NRQCD four-fermion matrix elements	imaginary matching coefficients	operator mixing, velocity truncation, relativistic corrections, matrix-element provenance
Inclusive production cross section	production matrix elements $\langle O_n^H\rangle$	partonic coefficients and, for hadrons, PDFs	factorization domain, correlated fit uncertainty, feed-down definition, scale dependence
Spin splitting	chromomagnetic and spin-dependent potentials	$c_F$ and higher matching coefficients	power counting, discretization or potential model, coupled thresholds

The pole mass and perturbative static potential each carry a leading infrared renormalon; it cancels in consistent physical combinations such as the static energy. Using a short-distance heavy-quark mass and a correspondingly subtracted potential makes that cancellation explicit. A precision result must state the mass and potential schemes rather than attach an independent uncertainty to each ambiguous quantity Brambilla et al. 2005, §§ IV.A–IV.B.

Independent checks and failure modes

• Scale hierarchy: estimate $v$, $mv$, and $mv^2$ and compare each with $\Lambda_{\mathrm{QCD}}$. If adjacent scales do not separate, the corresponding matching expansion is not controlled.
• Kinetic normalization: for equal masses, use reduced mass $m/2$, so the relative Hamiltonian begins with $\mathbf p^2/m$. A factor-of-two error spoils both $p_B$ and $E_n$.
• Color algebra: singlet and octet projectors must be orthogonal and complete. Potential color factors and four-fermion matching channels must use the same generator normalization.
• Dimensions: Pauli bilinears have dimension three; derivative currents and dimension-$d_n$ four-fermion operators require the shown powers of $m$.
• Velocity counting: kinetic and leading potential energies are both $O(mv^2)$. A term counted smaller but iterated at leading order signals inconsistent mode or power assignment.
• Matching independence: a physical result must be independent of factorization scales to the computed order. Residual scale dependence estimates missing orders but is not itself a proof of convergence.
• Bound-state versus resonance physics: open-flavor thresholds can invalidate a single-channel potential picture. Coupled-channel poles then require the resonance analysis.
• Production ceiling: NRQCD production factorization and universality must be qualified by process, momentum region, and perturbative/power accuracy; fitting many channels does not establish the theorem.

Common pitfalls

Using HQET for both members of the pair. Removing each rest mass while integrating out the antiquark erases the potential region responsible for binding. NRQCD keeps both Pauli fields active.

Calling the static potential an observable. It is a scheme- and scale-dependent matching coefficient. Only a complete energy or amplitude, with the mass and ultrasoft pieces combined consistently, is physical.

Dropping color-octet operators because the hadron is a singlet. The full gauge-invariant state is a singlet, but short-distance pair configurations and their accompanying gluons include octet channels required by matching and renormalization.

Applying Coulomb formulas outside weak coupling. The analytic benchmark tests normalization and hierarchy. It does not replace nonperturbative input when $mv$ is near $\Lambda_{\mathrm{QCD}}$ or coupled thresholds are important.

Informal self-check

Derive the Coulombic ground-state momentum and energy for an equal-mass heavy pair and verify the calculation benchmark.

Answer

The reduced mass is $\mu=m/2$. For a potential $-C_F\alpha_s/r$, the Bohr momentum is $p_B=\mu C_F\alpha_s=mC_F\alpha_s/2$, and $E_n=-\mu(C_F\alpha_s)^2/(2n^2)=-m(C_F\alpha_s)^2/(4n^2)$. With $C_F=4/3$, $\alpha_s=3/10$, and $m=10$, one obtains $p_B=2$, $v=p_B/m=1/5$, and $E_1=-2/5=-mv^2$.

Handoffs

• Send heavy-pair scale estimates, color/spin channel, matching scheme, and desired observable to the appropriate NRQCD or potential-EFT calculation.
• Send open-channel thresholds and scattering quantum numbers to the coupled-channel resonance route when a potential-only state description fails.
• Send short-distance hadronic production coefficients and initial-state definitions to the collinear factorization route.
• For thermal screening, dissociation, and regeneration, continue to quarkonium in-medium dynamics and regeneration with the vacuum matching and scale conventions fixed.

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 · Erratum DOI · Open PDF
• Brambilla, Nora, Antonio Pineda, Joan Soto, and Antonio Vairo. “Effective-Field Theories for Heavy Quarkonium.” Reviews of Modern Physics 77 (2005): 1423–1496. DOI · Open PDF
• Pineda, Antonio. “Review of Heavy Quarkonium at Weak Coupling.” Progress in Particle and Nuclear Physics 67 (2012): 735–785. DOI · Open PDF