Non-Relativistic QCD Hamiltonian

• Non-relativistic QCD Hamiltonian is a framework that integrates out high-energy QCD modes to focus on low-energy dynamics of heavy quarks, like bottomonium and charmonium.
• It employs a systematic operator expansion including terms up to O(p^6) and one-loop radiative corrections to improve dispersion matching and mass renormalisation.
• Key lattice techniques such as Fat³ smearing and stability parameter tuning reduce discretisation errors, enabling precise quarkonium spectroscopy and decay prediction.

Non-relativistic QCD (NRQCD) Hamiltonians provide a systematic framework for describing low-energy dynamics of heavy quarks, notably bottomonium and charmonium states, within both continuum and lattice approaches. Central aspects include operator expansions in powers of momentum and velocity, accurate matching of lattice and continuum dispersion relations, and operator improvement through radiative corrections and lattice stabilisation techniques. These structures underpin quantitative predictions in heavy quark physics at the percent-level and are foundational for calculations of quarkonium spectra, transition rates, and non-perturbative QCD phenomena.

1. Hamiltonian Structure and Operator Expansion

The NRQCD Hamiltonian is constructed by integrating out hard QCD modes above the heavy quark mass scale, leading to a non-relativistic theory with quark fields that evolve under an effective Hamiltonian: $$H = H_0 + \delta H_{\text{kin}} + \delta H_{\text{rad}}$$ where $H_0$ contains the leading kinetic term, $\delta H_{\text{kin}}$ incorporates higher-order corrections in the operator expansion up to $\mathcal{O}(p^6)$, and $\delta H_{\text{rad}}$ includes radiative (loop-level) shifts (Davies et al., 2018).

Leading and Subleading Operators

• Leading kinetic: $aH_0 = -\Delta^{(2)} / (2 a m_b)$, where $\Delta^{(2)}$ is the lattice Laplacian with link-pair cancellations to suppress tadpoles.
• $\mathcal{O}(v^4)$ corrections: Operators such as $(\Delta^{(2)})^2$, $i[\nabla \cdot \tilde{E} - \tilde{E} \cdot \nabla]$, $$\sigma \cdot (\nabla \times \tilde{E} - \tilde{E} \times \nabla)$$, $\sigma \cdot \tilde{B}$, and $\Delta^{(4)}$ are included with matching coefficients $c_i$.
• $\mathcal{O}(p^6)$ corrections: Operators like $(\Delta^{(2)})^3$, $\Delta^{(6)}$, and $(\Delta^{(2)} \Delta^{(4)})$ provide further discretisation error reduction.

2. Radiative Improvement and Dispersion-Relation Matching

One-loop radiative corrections renormalise the heavy quark mass and introduce zero-point energy shifts:

• Mass renormalisation $m_b \to Z_m m_b$ affects the dispersion relation.
• Zero-point energy shift $W_0 = \Sigma_0^{(0)}$ enters additively.
• Matching to continuum NRQCD is realized via the on-shell energy expansion: $$\omega_{0}(p) = \frac{p^2}{2m_b} - \frac{(p^2)^2}{8m_b^3} + \frac{(p^2)^3}{16m_b^5}$$ and at one-loop,

$$\omega(p) = \omega_0(p) - \alpha_s \left[W_0 + W_1 \frac{(p^2)^2}{8 m_b^2} + W_2 p^4\right] + \mathcal{O}(\alpha_s^2)$$

yielding kinetic matching coefficients $c_1^{(1)} = -W_1/ (1/m_b + 1/2n)$ and $c_5^{(1)} = 24 m_b W_2$, so all $\mathcal{O}(\alpha_s p^4)$ mismatches between lattice and continuum are eliminated (Davies et al., 2018).

3. Lattice Stabilisation and Operator Improvement Techniques

Stability Parameter $n$ and Fat$^3$ Smearing

• The evolution operator $e^{-aH_0/2n}$ requires $p^2 < 4n a m_b$ for stability, with $n=4$ the typical choice.
• Fat$^3$ smearing replaces every thin gauge link by a locally averaged link, projected back to $U(3)$, nearly eliminating unphysical tadpole contributions. This permits setting $u_0 \to 1$ and reduces the magnitude of radiative corrections, making expansion truncations reliable (Davies et al., 2018).

Tadpole Improvement and Matching Scale

Radiative coefficients are either tadpole-improved via mean-field rescaling ($u_0 = \langle \mathrm{Tr}\,U \rangle / 3$ in Landau gauge) or, equivalently, Fat$^3$ smearing can be used (where all mean-field counterterms vanish). The $\alpha_s$ coupling is evaluated at a process-dependent BLM scale $q^*$, with $q^*\,a \sim 2.0$–$3.0$ for $c_1$, $c_5$ and $q^*\,a \sim 1$–$1.5$ for $Z_m$, $W_0$ (Davies et al., 2018). This ensures perturbative control and correct matching of kinetic terms.

4. Fully Improved Lattice NRQCD Hamiltonian

Integrating all components, the improved Hamiltonian used in simulations is: $$\begin{aligned} aH =& -(1+\alpha_s Z_m^{(1)})\,\frac{\Delta^{(2)}}{2\, a m_b} -(1+\alpha_s c_1^{(1)})\,\frac{(\Delta^{(2)})^2}{8 (a m_b)^3} +i(1+\alpha_s c_2^{(1)})\,\frac{\nabla \cdot \tilde{E} - \tilde{E} \cdot \nabla}{8 (a m_b)^2} \ & -(1+\alpha_s c_3^{(1)})\,\frac{\sigma \cdot (\nabla \times \tilde{E} - \tilde{E} \times \nabla)}{8 (a m_b)^2} -(1+\alpha_s c_4^{(1)})\,\frac{\sigma \cdot \tilde{B}}{2 a m_b} +(1+\alpha_s c_5^{(1)})\,\frac{\Delta^{(4)}}{24 a m_b} \ & -(1+\alpha_s c_6^{(1)})\,\frac{(\Delta^{(2)})^2}{16 n (a m_b)^2} -(1+\alpha_s c_{(p^2)^3}^{(1)})\, \frac{[1-(a m_b)^2/(6n^2)] (\Delta^{(2)})^3}{16 (a m_b)^5} \ & -(1+\alpha_s c_{p^6}^{(1)})\,\frac{\Delta^{(6)}}{180 a m_b} +(1+\alpha_s c_{p^2 p^4}^{(1)})\, \frac{\Delta^{(2)} \Delta^{(4)}}{48 (a m_b)^3} +\alpha_s W_0^{(1)}\, \mathbf{1} + \cdots \end{aligned}$$ where each coefficient $c_i$ is expanded as $1 + \alpha_s(q^*) c_i^{(1)}$, with all operators discretised on the lattice. Annotations indicate mapping to continuum terms and stabilization mechanisms (Davies et al., 2018).

5. Implications for Spectroscopy and Precision Calculations

By systematically including radiative corrections and higher-order momentum operators, discrete errors are reduced to $\mathcal{O}((pa)^6)$, rotational symmetry is restored (as confirmed through momentum tests), and the precision of quarkonium mass predictions is improved to the 1–2% regime (Davies et al., 2018). Mass renormalisation and zero-point energy shifts allow conversion between lattice results and continuum pole masses, crucial for physical observables.

Lattice ensembles incorporating $u$, $d$, $s$, and $c$ vacuum polarisation are used to non-perturbatively determine states such as $\Upsilon(1S)$, $\eta_b(1S)$, and to tune the $b$-quark mass. Improvements in action further stabilise time-evolution algorithms and correct for high-momentum artefacts.

6. Relation to Continuum and Effective Field Theory Approaches

The NRQCD Hamiltonian is tightly connected with pNRQCD and continuum effective descriptions. In pNRQCD, the Hamiltonian incorporates matched potentials (static, $1/m$, $1/m^2$, spin-dependent) with coefficients determined up to two-loop order (Mishima et al., 2024, Segovia et al., 2017), using either two-step or single-step matching procedures, IBP reduction to master integrals, and differential equations in the velocity expansion parameter. These ensure that all heavy-quarkonium binding energies and transitions can be consistently computed up to N$^4$LO in the weak-coupling regime. The approach also provides a basis for further extensions, such as BCS quasiparticle formation in non-perturbative QCD Hamiltonians (Yepez-Martinez et al., 2021), supporting dynamical generation of constituent quark masses and correlations in low-energy hadron spectra.

7. Summary Table: Key Features of Improved Lattice NRQCD Hamiltonian

Feature	Description	Impact
Operator Expansion	Includes terms up to $\mathcal{O}(p^6)$	$\mathcal{O}((pa)^6)$ error control
Radiative Corrections	One-loop matching for mass and kinetic terms	1–2% level mass accuracy
Fat$^3$ Smearing	Removes unphysical tadpoles, sets $u_0 \to 1$	Perturbative convergence, stable fits
Stability Parameter $n$	Ensures $p^2 < 4n a m_b$ for stable evolution	No transfer-matrix sign flips
Matching Scale $q^*$	BLM scale-setting for $\alpha_s$ in $V$-scheme	Reliable coefficient expansion

The non-relativistic QCD Hamiltonian, in both lattice and continuum forms, is now a rigorously improved tool for heavy quark physics. Its comprehensive operator content, precise radiative matching, and stabilization techniques enable systematic reduction of discretisation and perturbative errors, supporting high-precision spectroscopy, decay, and transition predictions for quarkonium systems.