---
title: Hybrid vNRQCD/pNRQCD Lagrangian
url: https://www.emergentmind.com/topics/hybrid-vnrqcd-pnrqcd-lagrangian
type: topic
---

# Hybrid vNRQCD/pNRQCD Lagrangian

The **hybrid vNRQCD/pNRQCD Lagrangian** denotes an effective-field-theory construction that combines elements of velocity NRQCD and potential NRQCD in order to describe nonrelativistic heavy-quark systems while retaining enough structure to handle either hybrid quarkonium degrees of freedom or production operators with explicit soft radiation. In the literature considered here, the phrase is used in two closely related senses. One is a **hybrid-hadron EFT** in which low-lying quarkonium and hybrid Born–Oppenheimer channels are encoded in hadronic fields such as a quarkonium singlet \(S\) and a hybrid vector field \(H^i\) [1702.03900]. The other is a **representation-changing EFT** obtained by applying a Hubbard–Stratonovich transformation to vNRQCD, so that explicit heavy quark and antiquark fields coexist with pNRQCD-like composite singlet and octet fields \(S_{\bf r}\) and \(O_{\bf r}^a\) [2605.30441]. This suggests that the term does not identify a single canonical Lagrangian, but rather a family of interpolating nonrelativistic EFT formulations constrained by the hierarchies \(m \gg |\mathbf p_Q| \gg E_b\) and, in the weak-coupling short-distance regime, \(1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD}\) [1510.04949].

## 1. Scale separation and EFT logic

The heavy-hybrid literature formulates the problem around the standard nonrelativistic hierarchy
\[
m \gg |\mathbf p_Q| \gg E_b ,
\]
with \(m\) the heavy-quark mass, \(|\mathbf p_Q|\sim 1/r\) the relative momentum scale, and \(E_b\) the binding-energy scale [1510.04949]. For short-distance hybrid static energies, the relevant regime is weakly coupled pNRQCD,
\[
1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD},
\]
so that the octet potential is perturbative while the ultrasoft gluonic excitation remains nonperturbative [1510.04949]. In that setting, the hybrid is conceptually a nonrelativistic \(Q\bar Q\) pair in a color-octet configuration coupled to an excited gluonic field.

A different but compatible hierarchy is used for the low-lying hybrid spectrum near the minima of the Born–Oppenheimer hybrid potentials. There the characteristic excitation energy is taken to satisfy
\[
E \sim \sqrt{\Lambda_{\rm QCD}^3/m_Q} \ll \Lambda_{\rm QCD} \ll m_Q ,
\]
which is identified with a regime analogous to the strong-coupling regime of pNRQCD, where \(\Lambda_{\rm QCD}\) is integrated out and the low-energy degrees of freedom are hadronic fields associated with static BO channels [1702.03900]. The low-lying \(\Pi_u\) and \(\Sigma_u^-\) channels are then retained because the gap to the next static adiabatic surfaces is \(O(\Lambda_{\rm QCD})\), numerically about \(400\) MeV from the lattice static spectrum [1702.03900].

The production-oriented hybrid vNRQCD/pNRQCD construction starts instead from the standard vNRQCD mode separation. The retained momentum regions are the usual hard, soft, potential, and ultrasoft regions, with potential heavy quarks satisfying \(p_c \sim (mv^2,\,m{\bf v})\), soft gluons \(A_q^\mu \sim (mv,\,m{\bf v})\), and ultrasoft gluons \(A^\mu_{us} \sim (mv^2,\,m{\bf v}^2)\) [2605.30441]. Its purpose is not heavy-hybrid spectroscopy, but factorization of quarkonium production matrix elements and TMD soft transition functions while preserving explicit soft modes [2605.30441; 2606.12175].

## 2. Degrees of freedom and operator organization

At short distance, the most direct pNRQCD operator-level statement of the hybrid degree of freedom is
\[
H(\mathbf R,\mathbf r,t)=H^a(\mathbf R,t)\mathcal O^{a\dagger}(\mathbf R,\mathbf r,t),
\]
where \(\mathcal O^a\) is the color-octet heavy-pair field and \(H^a\) is a gluonic operator [1510.04949]. In the limit \(r\to 0\), the gluonic spectrum reduces to gluelumps, and the BO channels \(\Sigma_u^-\), \(\Pi_u\), and higher channels are interpreted as cylindrical-symmetry projections of gluelump multiplets. The channel classification follows \(D_{\infty h}\), with irreducible representations \(\Lambda_\eta^\sigma\), and weak-coupling pNRQCD predicts short-distance degeneracy patterns such as
\[
\Pi_u-\Sigma_u^- ,\qquad \Pi_g-\Sigma_g^{+\prime},\qquad \Delta_g-\Sigma_g^- -\Pi_g',\qquad \Delta_u-\Pi_u'-\Sigma_u^+ .
\]
For the lowest multiplet, \(\Pi_u\) and \(\Sigma_u^-\) are both components of a \(1^{+-}\) gluelump multiplet [1510.04949].

This short-distance identification motivates the field content of the strong-coupling hybrid EFT. The low-energy degrees of freedom are a quarkonium singlet field \(S({\bf R},{\bf r},t)\), associated with \(\Sigma_g^+\), and a hybrid field \(H^i({\bf R},{\bf r},t)\), carrying a gluonic vector index \(i\), used to package the nearly degenerate \(\Pi_u\) and \(\Sigma_u^-\) static multiplet [1702.03900]. The reason for the vector index is explicit: at short distances both channels arise from different projections of the same operator \({\rm tr}({\bf B}({\bf 0},t)\,O({\bf 0},{\bf r},t))\), with \(\hat{\bf r}\cdot{\bf B}\) transforming as \(\Sigma_u^-\) and the transverse projection as \(\Pi_u\) [1702.03900].

The production-factorization literature uses a different field organization. Starting from vNRQCD, a Hubbard–Stratonovich transformation introduces auxiliary or composite singlet and octet bosonic fields
\[
S_{\bf r}(x), \qquad O^a_{\bf r}(x),
\]
which are interpreted as pNRQCD-like singlet and octet quarkonium fields carrying ultrasoft center-of-mass momentum and depending on the relative coordinate \({\bf r}\) [2605.30441]. In that formulation, the theory contains simultaneously explicit heavy quark and antiquark fields, soft and ultrasoft gluons, and composite singlet/octet bosonic fields [2605.30441]. The later production analysis states the same conceptual point in a more operational form: the Hubbard–Stratonovich transformation replaces the vNRQCD potential operator “with color-singlet and color-octet composite fields coupled to quark-antiquark pairs,” and the displayed production matrix elements use the singlet field \(S_0\) explicitly [2606.12175].

## 3. Explicit Lagrangian structures

For the low-lying hybrid spectrum, the explicit strong-coupling pNRQCD-type hybrid Lagrangian is
\[
{\cal L}={\rm tr} \left( {H^i}^\dagger\left( \delta_{ij}i\partial_0-{h_H}_{ij}\right)H_j \right),
\]
with
\[
{h_{H}}_{ij}=\left(-\frac{\nabla^2}{m_Q}+V_{\Sigma_u^-}(r)\right)\delta_{ij}
+\left(\delta_{ij}-\hat r_i\hat r_j\right)\left[V_{\Pi_u}(r)-V_{\Sigma_u^-}(r)\right].
\]
The radial component \(H_\parallel \sim \hat r_iH^i\) propagates with \(V_{\Sigma_u^-}\), while the transverse components propagate with \(V_{\Pi_u}\). At leading order this Hamiltonian is spin independent and preserves heavy-quark spin symmetry [1702.03900]. The corresponding quarkonium Hamiltonian is
\[
h_s=-\frac{\boldsymbol{\nabla}^2}{m_Q}+V_{\Sigma_g^+}(r).
\]

The combined quarkonium-plus-hybrid EFT is then
\[
\begin{aligned}
\mathcal{L}_{H+S}=&\ {\rm tr}\left( S^\dagger [i\partial_0-h_s]S\right)
+ {\rm tr}\left( H^{i\dagger}[i\delta_{ij}\partial_0-{h_H}_{ij}]H^j\right) \\
&+ {\rm tr}\left(S^\dagger  V_S^{ij} \left\{ \sigma^i \, , H^j\right\}+\,{\rm H.c.}\right) ,
\end{aligned}
\]
with
\[
V_S^{ij}=\delta^{ij}V_S^\Pi(r)+\hat r^i\hat r^j\left(V_S^\Sigma(r)-V_S^\Pi(r)\right).
\]
This is the central explicit low-energy hybrid pNRQCD Lagrangian in the literature surveyed here [1702.03900]. A closely related conference formulation presents the same structure as
\[
L = {\rm tr}\left(H^{i\dagger}\left(\delta_{ij}i\partial_0-h_{H\,ij}\right)H^j\right),
\qquad
L_{\rm mixing} = {\rm tr}\left[S^\dagger V_S^{ij}\{\sigma^i,H^j\}+{\rm h.c.}\right],
\]
supplemented by a leading hybrid hyperfine operator,
\[
i\epsilon^{ijk} V_S(r)\, {\rm tr}\left(H^{i\dagger}[\sigma^k,H^j]\right),
\]
all embedded in an NRQCD/pNRQCD framework constrained by lattice QCD, weak-coupling pNRQCD, and the QCD effective string theory [1709.08038].

The production-oriented hybrid vNRQCD/pNRQCD Lagrangian is structurally different. It begins from the vNRQCD Lagrangian with heavy fields \(\psi_{\bf p}\), \(\chi_{\bf p}\), ultrasoft gauge fields, explicit soft-gluon seagull couplings, and an instantaneous potential \(V({\bf p},{\bf q})\) [2605.30441]. A Hubbard–Stratonovich term is then added to linearize the singlet and octet potential channels using the fields \(S_{\bf r}\) and \(O_{\bf r}^a\), with Coulomb kernels
\[
V^{(1)}(r) = - C_F \frac{\alpha_s}{r}, \qquad
V^{(8)}(r) = \left(\frac{C_A}{2}-C_F\right)\frac{\alpha_s}{r}.
\]
The resulting hybrid Lagrangian contains heavy fermions, soft and ultrasoft gluons, and singlet/octet composites at the same time [2605.30441]. After integrating out the heavy quarks, one recovers the leading pNRQCD composite-field Lagrangian,
\[
{\cal L}^{(0)} = \int d^3{\bf r}\,  S^\dagger_{\bf r} \left(i\partial_0 + \frac{\nabla^2_{\bf r}}{m}-V^{(1)}(r)\right)S_{\bf r}
+ \int d^3{\bf r}\,  O_{\bf r}^{a \dagger} \left(i\partial_0 + \frac{\nabla_{\bf r}^2}{m}-V^{(8)}(r)\right)O_{\bf r}^a .
\]
This is the paper’s explicit interpolation between vNRQCD and pNRQCD [2605.30441].

## 4. Static potentials, BO channels, and nonadiabatic structure

The heavy-hybrid EFT is anchored by the static-energy formula
\[
E_H=2m+V_H ,
\qquad
V_H(r)=V_o(r)+\Lambda_H+b_H r^2
\]
up to next-to-leading order in the multipole expansion [1510.04949]. Here \(V_o(r)\) is the perturbative octet potential, \(\Lambda_H\) is the gluelump mass extracted from an adjoint correlator, and \(b_H r^2\) is the first nontrivial multipole correction allowed by rotational invariance. In a BO-channel language, this becomes a diagonal potential matrix with entries
\[
V_{\Lambda_\eta^\sigma}(r)=V_o(r)+\Lambda_H+b_{\Lambda_\eta^\sigma}r^2+\cdots .
\]
The actual spectroscopy calculation is then a mixed construction: the short-distance form is constrained by weak-coupling pNRQCD, while longer-distance potentials are taken from lattice-QCD static energies and fitted or interpolated phenomenologically [1510.04949].

In the explicit strong-coupling hybrid EFT, the static BO potentials are inserted directly into the Hamiltonian. The quarkonium potential is approximated by
\[
V_{\Sigma_g^+}(r)\approx -\frac{k_g}{r}+\sigma_g r+E_g^{Q\bar Q},
\]
with \(k_g=0.489\) and \(\sigma_g=0.187\ {\rm GeV}^2\) [1702.03900]. For the \(\Sigma_u^-\) channel,
\[
V_{\Sigma_u^-}(r)=\frac{\sigma_s}{r}+\kappa_s r +E_s^{Q\bar Q},
\]
with \(\sigma_s=\sigma_g/8\), \(\kappa_s=\kappa_g\), numerically \(\sigma_s=0.061\) and \(\kappa_s=0.187\ {\rm GeV}^2\) [1702.03900]. For \(\Pi_u\), a rational-plus-linear ansatz is used, with short-distance degeneracy and long-distance effective-string constraints imposed explicitly [1702.03900].

The key dynamical refinement beyond a naive BO treatment is the appearance of nonadiabatic \(\Lambda\)-doubling couplings. The heavy-pair angular kinetic operator contains the terms
\[
\mathbf{L}_{Q\overline{Q}}^2
=\mathbf{L}^2-2K_n^2+\mathbf{K}^2-L_-K_+-L_+K_- ,
\]
whose last two pieces raise or lower \(\Lambda\) and generate channel mixing [1510.04949]. In the small-\(r\) gluelump limit, this yields coupled radial equations of the form
\[
\left(-\frac{\partial_r^2}{2\mu}+\frac{C_{lk}^{\Lambda\,0}}{2\mu r^2}+E_{n\Lambda_\eta^\sigma}(r)\right)\psi_N^\Lambda
-\frac{C_{lk}^{\Lambda\,-}}{2\mu r^2}\psi_N^{\Lambda-1}
-\frac{C_{lk}^{\Lambda\,+}}{2\mu r^2}\psi_N^{\Lambda+1}
=E_N\psi_N^\Lambda .
\]
In matrix notation,
\[
H_{\rm hyb}= -\frac{1}{2\mu}\partial_r^2 +\frac{1}{2\mu r^2}\,\mathcal C_{lk} +\mathcal E_n(r),
\]
with \(\mathcal E_n(r)\) diagonal in the \(\Lambda\) basis and \(\mathcal C_{lk}\) containing diagonal centrifugal terms and off-diagonal \(\Lambda\to\Lambda\pm1\) couplings [1510.04949]. This is the clearest Hamiltonian-level template for reconstructing a second-quantized hybrid pNRQCD or vNRQCD/pNRQCD Lagrangian.

## 5. Mixing with quarkonium and spin-symmetry violation

The leading hybrid–quarkonium mixing term,
\[
{\rm tr}\left(S^\dagger  V_S^{ij} \left\{ \sigma^i \, , H^j\right\}+\,{\rm H.c.}\right),
\]
is an \(O(1/m_Q)\) operator and is explicitly spin dependent [1702.03900; 1709.08038]. Because it contains \(\sigma^i\), it breaks heavy-quark spin symmetry. In the spin decomposition
\[
S=\frac{1}{\sqrt{2}}\left(S_0+\sigma^kS_1^k\right), \qquad
H^j=\frac{1}{\sqrt{2}}\left(H_0^j+\sigma^iH_1^{ji}\right),
\]
the mixing becomes
\[
2V_S^{ij}\left({S_1^i}^\dagger H_0^j+{S_0}^\dagger H_1^{ji} \right),
\]
so spin-0 hybrids mix with spin-1 quarkonia, and spin-1 hybrids mix with spin-0 quarkonia [1702.03900]. This is the mechanism used to generate large spin-symmetry-violating phenomenology in the hybrid spectrum.

The existence and tensor structure of the mixing are derived from NRQCD at \(O(1/m_Q)\). The relevant NRQCD Lagrangian contains the chromomagnetic Pauli term
\[
\mathcal{L}_{NRQCD}= \psi^\dagger\left[ iD_0+\frac{\bf D}^2}{2m_Q}+gc_F \frac{\boldsymbol{\sigma}\cdot{\bf B}}{2m_Q}\right]\psi
+\chi^\dagger\left[ iD_0-\frac{\bf D}^2}{2m_Q}-gc_F \frac{\boldsymbol{\sigma}\cdot{\bf B}}{2m_Q}\right] \chi ,
\]
and the weak-coupling pNRQCD matching identifies the crucial operator
\[
{\mathcal{L}'}_{pNRQCD}= \frac{gc_F}{2m_Q} {\rm Tr} \left( {\rm O}^\dagger({\bf r}, {\bf R}, t)\, {\bf B}({\bf R}, t)\,\{ \boldsymbol{\sigma} , {\rm S}({\bf r}, {\bf R}, t)\} \right) + {\rm H.c.}
\]
[1702.03900]. Since that interaction is \(r\)-independent at leading order, the short-distance constraint is
\[
V_S^\Sigma(r)=V_S^\Pi(r)= \pm\frac{c_F \lambda^2}{m_Q},
\]
with \(\lambda\sim \Lambda_{\rm QCD}\) [1702.03900]. At long distances, effective string theory implies
\[
V_S^\Sigma(r)\propto \frac{1}{m_Q r^3}, \qquad
V_S^\Pi(r)\propto \frac{1}{m_Q r^2},
\]
with coefficients involving \(g\Lambda'\) and \(g\Lambda'''\), estimated in the analysis as \(g\Lambda'\sim -59\,{\rm MeV}\) and \(g\Lambda'''\sim \pm 230\,{\rm MeV}\) [1702.03900].

A common misconception is that the hybrid EFT can be truncated to a single BO channel without qualitative loss. The coupled-channel analyses argue against that. In the lowest multiplet, \(\Sigma_u^-\) and \(\Pi_u\) mixing through \(\Lambda\)-doubling splits opposite-parity partners that would otherwise be degenerate [1510.04949], and quarkonium–hybrid mixing can become effectively leading when hybrid and quarkonium levels are close, despite being formally \(1/m_Q\) suppressed [1709.08038].

## 6. Hubbard–Stratonovich formulation, decoupling, and production factorization

The production version of the hybrid vNRQCD/pNRQCD Lagrangian is introduced to prove factorization of quarkonium production matrix elements. Its central device is a Hubbard–Stratonovich transformation, described as an exact mathematical operation that replaces four-fermion operators in a Lagrangian with auxiliary bosonic fields coupled to fermions [2606.12175]. Applied to the vNRQCD potential sector, it trades the nonlocal potential operator for singlet and octet composite fields while keeping the explicit soft and ultrasoft sectors [2605.30441; 2606.12175].

After this transformation, the theory may be viewed in two equivalent limits. Integrating out \(S_{\bf r}\) and \(O_{\bf r}^a\) returns the original vNRQCD formulation; integrating out the heavy quarks reproduces the leading pNRQCD Lagrangian [2605.30441]. The hybrid character of the formulation lies precisely in this coexistence of explicit heavy fields, soft modes, ultrasoft modes, and pNRQCD-like composites.

Ultrasoft decoupling is implemented at leading order by a BPS-type field redefinition,
\[
\psi_{\bf p}(x) \to U_v(x)\,\psi'_{\bf p}(x),\qquad
U_v(x) = {\cal P}\exp\bigg(-ig \int_{-\infty}^0 ds\, A_0 (x+s)\bigg),
\]
which removes the leading ultrasoft \(A^0\) coupling from the lowest-order Lagrangian and pushes it into subleading interactions and currents [2605.30441]. The dressed ultrasoft fields are then
\[
{\bfcal B}^k_{us} = \frac12 U_v^\dagger \epsilon^{ijk} G^{ij}_{us} U_v, \qquad
{\bfcal E}_{us}^i = -\frac1g U_v^\dagger(\boldsymbol{\nabla}^i - g {\bf A}_{us}^i)U_v .
\]

Soft decoupling is subtler. The analysis states explicitly that nonrelativistic propagators do not eikonalize, so the soft interaction cannot be removed by an exact Wilson-line decoupling; the relevant soft operator is not unitary [2605.30441]. Instead, a production-specific result is established: when the quarkonium production operator creates the \(Q\bar Q\) pair locally, \(r=0\), the soft couplings generated during the Hubbard–Stratonovich transformation vanish in the production matrix elements, and a factorization between soft and ultrasoft sectors is permitted at leading order in the velocity power counting [2605.30441; 2606.12175].

The resulting matrix elements factorize into a composite-field matrix element, identified with the wave function at the origin, and state-independent vacuum correlators of chromoelectric or chromomagnetic gluon fields. For example,
\[
\langle O^V(^1S_0^{[8]})\rangle
= \frac{1}{3 N_cM^2}\, \langle {\cal B}\rangle\, \frac{3|R_V(0)|^2}{2\pi},
\]
\[
\langle O^V(^3S_1^{[8]})\rangle
= \frac{1}{N_cM^2}\, \langle {\cal EE}\rangle\, \frac{3|R_V(0)|^2}{2\pi},
\]
and
\[
\langle O^V(^3P_J^{[8]})\rangle
= \frac{1}{36N_c}(2J+1)\, \frac{3|R_V(0)|^2}{2\pi}\, \langle {\cal E}\rangle
\]
[2605.30441; 2606.12175]. The same logic extends to TMD soft transition functions, where explicit soft Wilson lines \({\cal S}_v\) and \({\cal S}_n\) remain and all state dependence is again isolated into \(|R_V(0)|^2\) [2605.30441; 2606.12175].

## 7. Higher-order bilinear sector, interpretation, and limitations

Any hybrid vNRQCD/pNRQCD construction still requires a heavy-quark bilinear sector with correct chromoelectric, chromomagnetic, and higher-derivative couplings. A relevant input is the on-shell-matched NRQCD bilinear Lagrangian through \(1/m^5\),
\[
\mathcal L = h^\dagger \Big(iD_0 + \frac{\mathbf D^2}{2m} + \cdots\Big) h
\]
augmented by the operators with coefficients \(c_F\), \(c_D\), \(c_S\), \(c_{W1}\), \(c_{W2}\), \(c_{p'p}\), \(c_M\), the \(X_i\), and the \(Y_i\) [2004.10545]. The symbolic coefficients are given in terms of the Dirac and Pauli form factors \(F_1\), \(F_2\), and their derivatives, for example
\[
c_F = F_1+F_2,\qquad
c_D = F_1+2F_2+8F_1',\qquad
c_S = F_1+2F_2,\qquad
c_M=\frac{F_2}{2}+4F_1' .
\]
These coefficients do not by themselves define a hybrid vNRQCD/pNRQCD Lagrangian, but they supply precisely the single-heavy-quark Wilson coefficients that feed the bilinear sector, one-gluon matching, and higher-order spin-dependent interactions [2004.10545].

Several limitations recur across the literature. The BO/pNRQCD heavy-hybrid formulations are not purely weak-coupling pNRQCD treatments of the whole spectrum; they are mixed constructions anchored by short-distance EFT and extended with lattice static energies [1510.04949]. The explicit low-energy hybrid Lagrangian of the strong-coupling approach includes only the \(1/m_Q\) correction relevant to quarkonium–hybrid mixing, while additional \(1/m_Q\) corrections to \(h_s\), \(h_H\), and further hybrid spin-dependent terms are omitted in the phenomenology [1702.03900]. The production-factorization papers do not provide a single complete standalone hybrid Lagrangian with every interaction written out; rather, they use the hybrid framework operationally and establish decoupling and factorization only at leading order in the velocity expansion, with soft decoupling relying on the production-specific \(r=0\) limit [2606.12175; 2605.30441]. Finally, the high-order bilinear coefficients are basis dependent because they are obtained by on-shell matching and field redefinitions, so they must be translated carefully before use in another EFT basis [2004.10545].

Taken together, these constructions define the modern meaning of the hybrid vNRQCD/pNRQCD Lagrangian: a nonrelativistic EFT architecture in which composite singlet/octet or hybrid BO fields are combined with selected vNRQCD mode content, lattice-constrained static potentials, and controlled \(1/m_Q\) spin-breaking operators, so that one can treat either hybrid spectroscopy and quarkonium–hybrid mixing or production-matrix-element factorization within a common effective-field-theory language [1510.04949; 1702.03900; 2605.30441].

Source: https://www.emergentmind.com/topics/hybrid-vnrqcd-pnrqcd-lagrangian