Potential Nonrelativistic QCD (pNRQCD)

• pNRQCD is an effective field theory that separates energy scales in heavy-quark systems by integrating out hard (m) and soft (mv) modes.
• It formulates a Schrödinger-like Hamiltonian with nonlocal potentials, enabling precise ab initio predictions of heavy quarkonium spectra and radiative transitions.
• Lattice QCD and systematic power counting in v, αₛ, and 1/m are used to compute matching coefficients, underpinning its success in describing multiquark interactions.

Potential Nonrelativistic QCD (pNRQCD) is a systematically constructed effective field theory that bridges quantum chromodynamics (QCD) and nonrelativistic heavy-quark dynamics. It achieves a rigorous separation of energy scales in heavy quarkonium (and, by extension, multiquark systems of heavy quarks), allowing for ab initio calculations of @@@@2@@@@ spectra, transition rates, multiparticle forces, production cross sections, and matrix elements. pNRQCD is defined by integrating out hard ($m$) and soft ($mv$) modes, leaving as dynamical degrees of freedom heavy-quark singlet and octet wavefields and ultrasoft gluons or photons, with all interactions encoded as nonlocal potentials. The theory possesses a manifest power counting in $v$ (quark velocity), $\alpha_s$, and $1/m$, provides nonperturbative access to potential matching via lattice Wilson loops, and underpins a wide swath of heavy-quark physics, including radiative decays, hadroproduction, baryons, and multihadron forces. The following sections organize key technical features and results of pNRQCD as established in the arXiv research corpus.

1. Effective Field Theory Construction and Scale Hierarchy

Heavy quark systems exhibit a hierarchy of three separated energy scales:

• Hard scale ($m$): Heavy-quark mass, typically $1\!-\!5\,\textrm{GeV}$.
• Soft scale ($mv$): Relative momentum, a few $100\,\textrm{MeV}$.
• Ultrasoft scale ($mv^2$): Binding energy, $O(\Lambda_\textrm{QCD})$ or below.

The construction proceeds via sequential matching:

• QCD $\rightarrow$ NRQCD: Integrate out hard modes ($k\sim m$), obtaining NRQCD with two-component Pauli fields (quark $\psi$, antiquark $\chi$), soft and ultrasoft gluons, and a Lagrangian expanded in $1/m$ and $\alpha_s$.
• NRQCD $\rightarrow$ pNRQCD: Integrate out soft modes ($k\sim mv$) using a multipole expansion in the relative coordinate $r$. The residual fields are color-singlet ($S$) and color-octet ($O$) $Q\bar Q$ propagators plus ultrasoft gluons.
• Power Counting: In weak coupling, one counts $r\sim 1/(mv)$, $\nabla_r\sim mv$, $\nabla_R\sim mv^2$, $E,B\sim (mv^2)^2$, $$E^\textrm{em},B^\textrm{em}\sim k_\gamma^2\sim(mv^2)^2$$ (Pietrulewicz, 2012, Koma et al., 2012, Pietrulewicz, 2013, Segovia et al., 2017).

2. Hamiltonian Structure and Matching of Potentials

The pNRQCD Hamiltonian is a nonrelativistic Schrödinger-like operator with potentials systematically expanded as:

$$H_{pNRQCD} = 2m + \frac{p^2}{m} + V^{(0)}(r) + \frac{1}{m}V^{(1)}(r) + \frac{1}{m^2}[V_\textrm{SI}^{(2)}(r) + V_\textrm{SD}^{(2)}(r)] + \cdots$$

• Static Potential $V^{(0)}(r)$: Universal form $-\alpha/r + \sigma r + \textrm{const}$, where $\alpha$ is the short-distance Coulombic coupling and $\sigma$ the string tension (Koma et al., 2012, Chaturvedi et al., 2018).
• Relativistic Corrections:
  • $O(1/m)$ Spin-Independent: $$V^{(1)}(r) \sim -9\alpha^2/(8r^2) + \sigma^{(1)}\ln r + \textrm{const}$$.
  • $O(1/m^2)$ Momentum-Dependent (Darwin, etc.): Full implementation requires additional lattice input.
  • $O(1/m^2)$ Spin-Dependent: Dominant term is spin-orbit,

  $$V_{LS}^{(2)}(r)\,L\!\cdot\!S = [\,c_S/2r\,dV^{(0)}/dr + (c_F/r)(V_1'(r) + V_2'(r))\,]\,L\!\cdot\!S$$

  with explicit parameterizations for $V_1', V_2'$ (Koma et al., 2012).

Wilson coefficients $c_F, c_S$ are fixed by perturbative matching at leading order, with corrections computed via field-theoretic matching or lattice correlators. Poincaré invariance constrains the potential forms, enforcing the Gromes relation and its generalizations (Berwein et al., 2018).

3. Nonperturbative Determination and Lattice Input

The key insight of pNRQCD is that potentials $V^{(0)}(r), V^{(1)}(r),$ and $V_{SD}^{(2)}(r)$ are not model parameters but are directly computable from gauge-invariant correlators:

• Static Potential: Extracted from Polyakov-loop or Wilson-line correlators in the limit $T\to\infty$.
• $O(1/m)$ Correction: Computed via integral of two color-electric field insertions as $$-\int_0^\infty dt\, t\, \langle E_i^a(t) E_i^a(0)\rangle_W$$.
• Spin-Orbit Correction: From color-magnetic and color-electric insertions, $$\epsilon_{ijk}\int dt\, t\, \langle B^j(t) E^k(0)\rangle_W$$ (Koma et al., 2012).

Lattice QCD provides direct nonperturbative access to these correlators, enabling parameter-free predictions for spectra and level spacings.

4. Spectroscopy: Schrödinger Equation and Relativistic Corrections

The quantum mechanical pNRQCD Hamiltonian is solved for charmonium ($m_c \sim 1.27$ GeV) and bottomonium ($m_b \sim 4.20$ GeV) using the radial Schrödinger equation:

$$\Bigg[ -\frac{1}{m}\left( \frac{d^2}{dr^2} + \frac{2}{r}\frac{d}{dr} -\frac{L(L+1)}{r^2}\right) + V^{(0)}(r) \Bigg]u_{nL}(r) = E_{nL}^{(0)} u_{nL}(r)$$

Corrections are added perturbatively:

• $O(1/m)$ Shift: $$\Delta E^{(1)} = \langle \psi^{(0)}_{nL} | V^{(1)} | \psi^{(0)}_{nL} \rangle / m$$
• $O(1/m^2)$ Spin-Orbit: $$\Delta E^{(2)}_{SD} = \langle \psi^{(0)}_{nL} | V_{SD}^{(2)} | \psi^{(0)}_{nL} \rangle / m^2$$ (Koma et al., 2012)

Empirically, inclusion of $O(1/m)$ corrections is essential: without them, the static spectrum is systematically too high and compressed; with them, the level spacings and overall pattern match experimental data to $\lesssim 30$ MeV below open-flavor thresholds. Spin-orbit terms reproduce bulk $^3P_J$ splitting but require tensor terms for fine-structure ordering.

5. E1 Electric Dipole Transitions and Relativistic Corrections

The pNRQCD formalism enables systematic, model-independent calculation of E1 (electric dipole) rates up to $O(v^2)$:

$$\Gamma_{n^3P_J \to n'^3S_1 \gamma} = \frac{4}{9} \alpha_{\rm em} e_Q^2 k_\gamma^3 [I_3(nP\to n'S)]^2 \left\{ 1 + R - \frac{k_\gamma^2}{60} \frac{I_5}{I_3} - \frac{k_\gamma}{6m} + \frac{(c_F^{em}-1)}{2m}\left[ \frac{J(J+1)}{2} - 2 \right] \right\}$$

where $$I_N(nP\to n'S) = \int_0^\infty dr\, r^N R_{n'S}(r) R_{nP}(r)$$ and $R$ encodes wave-function corrections from $1/m$ and $1/m^2$ terms (Pietrulewicz, 2013, Brambilla et al., 2012, Pietrulewicz, 2012, Segovia et al., 2017).

Relativistic effects are dominated by subleading potentials, especially $V^{(1)}$, underscoring the need for precise nonperturbative determination. For bottomonium, $O(v^2)$ corrections suppress leading-order rates by $10$–$30\%$; in charmonium, the suppression can reach $20$–$60\%$.

6. Inclusive Production: Factorization, LDMEs, and Gluonic Correlators

pNRQCD provides first-principles expressions for NRQCD long-distance matrix elements (LDMEs) in terms of wave-functions and universal gluonic correlators:

• Color-Singlet S-Wave: $$\langle O^H(^3S_1^{[1]}) \rangle = 2N_c \frac{3}{4\pi}|R_H(0)|^2$$
• Color-Octet: All LDMEs expressible as $|R_H(0)|^2$ times a gluonic correlator, e.g.,

$$\langle O^H(^3S_1^{[8]}) \rangle = \frac{1}{2N_c m^2} \frac{3 |R_H(0)|^2}{4\pi} \mathcal{E}_{10;10}$$

and similarly for $^1S_0^{[8]}, ^3P_J^{[8]}$ (Brambilla et al., 2022, Brambilla et al., 2022, Brambilla et al., 2021).

The correlators are flavor- and excitation-independent, dramatically reducing nonperturbative unknowns. Phenomenological fits to LHC data yield universality relations and robust predictions for cross sections and polarizations.

7. Applications to Multiquark Systems, Baryons, and Exotic States

Extensions to triply-heavy baryons, tetraquarks, and composite dark sectors employ the same hierarchical potential structure:

• Three-Body Potentials: NNLO includes an intrinsic three-body term, generally small ($30$–$40$ MeV shift in baryon masses) (Llanes-Estrada et al., 2013, Assi et al., 2023).
• Quantum Monte Carlo Diagonalization: High-precision mass predictions achieved with two-parameter variational or QMC schemes (Assi et al., 2023).
• Multihadron Interactions: LO and NLO van der Waals forces between color-singlet hadrons vanish, only entering at NNLO via dipole-dipole $1/R^3$—too weak to bind unless additional dynamics is present (Assi et al., 13 Aug 2025).

8. Symmetries, Poincaré Algebra, and Constraints on Potentials

Enforcing Poincaré invariance in pNRQCD imposes strict constraints on potential structures:

• Boost Generators: Nonlinear realizations fix all forms up to $O(1/M^2)$ (Berwein et al., 2018).
• Relations Among Potentials: Spin–orbit and tensor interactions become functionals of the static potential and its derivative; Gromes and BBV relations are derived directly from Lorentz symmetry.
• Reduction of Nonperturbative Inputs: Only one static profile and its derivatives are required for all $1/M^2$ spin-dependent interactions, unifying weak and strong coupling treatments.

9. Limitations and Outlook

The pNRQCD framework rests on the hierarchy $m \gg mv \gg mv^2 \gg \Lambda_\textrm{QCD}$, hence its scope is restricted to heavy-flavor sectors and fails for systems where soft or ultrasoft scales are not perturbative. Light-quark effects, chiral transitions, and non-QCD nonperturbative forces require extensions beyond strict pNRQCD (Koma et al., 2012, Assi et al., 13 Aug 2025). Continued progress hinges on lattice determination of matching coefficients, computation of higher-body potentials, inclusion of ultrasoft corrections, and systematic treatment of coupled channels or decay processes.

---

In summary, pNRQCD constitutes the authoritative, model-independent framework for heavy quarkonium and multiquark spectroscopy, radiative transitions, and production phenomenology, with first-principles nonperturbative inputs supplanting model dependence throughout its applications (Koma et al., 2012, Pietrulewicz, 2013, Brambilla et al., 2012, Pineda, 2013, Brambilla et al., 2021, Brambilla et al., 2022, Brambilla et al., 2022).