Hybrid vNRQCD/pNRQCD Lagrangian

• Hybrid vNRQCD/pNRQCD Lagrangian is an effective field theory combining velocity NRQCD and potential NRQCD to describe nonrelativistic heavy-quark systems with hybrid quarkonium and explicit soft radiation.
• It implements scale separation under hierarchies (m >> |p_Q| >> E_b) and bridges weak-coupling short-distance potentials with strong-coupling lattice QCD inputs for accurate spectral predictions.
• The formulation employs operator mixing and Hubbard–Stratonovich transformations to factorize quarkonium production and incorporate controlled spin-symmetry breaking corrections.

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 $H^i$ (Oncala et al., 2017). 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 $O_{\bf r}^a$ (Copeland et al., 28 May 2026). 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, $1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD}$ (Castellà, 2015).

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 (Castellà, 2015). For short-distance hybrid static energies, the relevant regime is weakly coupled pNRQCD,

$H^i$0

so that the octet potential is perturbative while the ultrasoft gluonic excitation remains nonperturbative (Castellà, 2015). In that setting, the hybrid is conceptually a nonrelativistic $H^i$1 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

$H^i$2

which is identified with a regime analogous to the strong-coupling regime of pNRQCD, where $H^i$3 is integrated out and the low-energy degrees of freedom are hadronic fields associated with static BO channels (Oncala et al., 2017). The low-lying $H^i$4 and $H^i$5 channels are then retained because the gap to the next static adiabatic surfaces is $H^i$6, numerically about $H^i$7 MeV from the lattice static spectrum (Oncala et al., 2017).

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 $H^i$8, soft gluons $H^i$9, and ultrasoft gluons $S_{\bf r}$0 (Copeland et al., 28 May 2026). Its purpose is not heavy-hybrid spectroscopy, but factorization of quarkonium production matrix elements and TMD soft transition functions while preserving explicit soft modes (Copeland et al., 28 May 2026, Copeland, 10 Jun 2026).

2. Degrees of freedom and operator organization

At short distance, the most direct pNRQCD operator-level statement of the hybrid degree of freedom is

$S_{\bf r}$1

where $S_{\bf r}$2 is the color-octet heavy-pair field and $S_{\bf r}$3 is a gluonic operator (Castellà, 2015). In the limit $S_{\bf r}$4, the gluonic spectrum reduces to gluelumps, and the BO channels $S_{\bf r}$5, $S_{\bf r}$6, and higher channels are interpreted as cylindrical-symmetry projections of gluelump multiplets. The channel classification follows $S_{\bf r}$7, with irreducible representations $S_{\bf r}$8, and weak-coupling pNRQCD predicts short-distance degeneracy patterns such as

$S_{\bf r}$9

For the lowest multiplet, $O_{\bf r}^a$0 and $O_{\bf r}^a$1 are both components of a $O_{\bf r}^a$2 gluelump multiplet (Castellà, 2015).

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 $O_{\bf r}^a$3, associated with $O_{\bf r}^a$4, and a hybrid field $O_{\bf r}^a$5, carrying a gluonic vector index $O_{\bf r}^a$6, used to package the nearly degenerate $O_{\bf r}^a$7 and $O_{\bf r}^a$8 static multiplet (Oncala et al., 2017). The reason for the vector index is explicit: at short distances both channels arise from different projections of the same operator $O_{\bf r}^a$9, with $m \gg |\mathbf p_Q| \gg E_b$0 transforming as $m \gg |\mathbf p_Q| \gg E_b$1 and the transverse projection as $m \gg |\mathbf p_Q| \gg E_b$2 (Oncala et al., 2017).

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

$m \gg |\mathbf p_Q| \gg E_b$3

which are interpreted as pNRQCD-like singlet and octet quarkonium fields carrying ultrasoft center-of-mass momentum and depending on the relative coordinate $m \gg |\mathbf p_Q| \gg E_b$4 (Copeland et al., 28 May 2026). In that formulation, the theory contains simultaneously explicit heavy quark and antiquark fields, soft and ultrasoft gluons, and composite singlet/octet bosonic fields (Copeland et al., 28 May 2026). 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 $m \gg |\mathbf p_Q| \gg E_b$5 explicitly (Copeland, 10 Jun 2026).

3. Explicit Lagrangian structures

For the low-lying hybrid spectrum, the explicit strong-coupling pNRQCD-type hybrid Lagrangian is

$m \gg |\mathbf p_Q| \gg E_b$6

with

$m \gg |\mathbf p_Q| \gg E_b$7

The radial component $m \gg |\mathbf p_Q| \gg E_b$8 propagates with $m \gg |\mathbf p_Q| \gg E_b$9, while the transverse components propagate with $1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD}$0. At leading order this Hamiltonian is spin independent and preserves heavy-quark spin symmetry (Oncala et al., 2017). The corresponding quarkonium Hamiltonian is

$1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD}$1

The combined quarkonium-plus-hybrid EFT is then

$1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD}$2

with

$1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD}$3

This is the central explicit low-energy hybrid pNRQCD Lagrangian in the literature surveyed here (Oncala et al., 2017). A closely related conference formulation presents the same structure as

$1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD}$4

supplemented by a leading hybrid hyperfine operator,

$1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD}$5

all embedded in an NRQCD/pNRQCD framework constrained by lattice QCD, weak-coupling pNRQCD, and the QCD effective string theory (Soto, 2017).

The production-oriented hybrid vNRQCD/pNRQCD Lagrangian is structurally different. It begins from the vNRQCD Lagrangian with heavy fields $1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD}$6, $1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD}$7, ultrasoft gauge fields, explicit soft-gluon seagull couplings, and an instantaneous potential $1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD}$8 (Copeland et al., 28 May 2026). A Hubbard–Stratonovich term is then added to linearize the singlet and octet potential channels using the fields $1/r \sim |\mathbf p_Q| \gg \Lambda_{\rm QCD}$9 and $m \gg |\mathbf p_Q| \gg E_b ,$0, with Coulomb kernels

$m \gg |\mathbf p_Q| \gg E_b ,$1

The resulting hybrid Lagrangian contains heavy fermions, soft and ultrasoft gluons, and singlet/octet composites at the same time (Copeland et al., 28 May 2026). After integrating out the heavy quarks, one recovers the leading pNRQCD composite-field Lagrangian,

$m \gg |\mathbf p_Q| \gg E_b ,$2

This is the paper’s explicit interpolation between vNRQCD and pNRQCD (Copeland et al., 28 May 2026).

4. Static potentials, BO channels, and nonadiabatic structure

The heavy-hybrid EFT is anchored by the static-energy formula

$m \gg |\mathbf p_Q| \gg E_b ,$3

up to next-to-leading order in the multipole expansion (Castellà, 2015). Here $m \gg |\mathbf p_Q| \gg E_b ,$4 is the perturbative octet potential, $m \gg |\mathbf p_Q| \gg E_b ,$5 is the gluelump mass extracted from an adjoint correlator, and $m \gg |\mathbf p_Q| \gg E_b ,$6 is the first nontrivial multipole correction allowed by rotational invariance. In a BO-channel language, this becomes a diagonal potential matrix with entries

$m \gg |\mathbf p_Q| \gg E_b ,$7

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 (Castellà, 2015).

In the explicit strong-coupling hybrid EFT, the static BO potentials are inserted directly into the Hamiltonian. The quarkonium potential is approximated by

$m \gg |\mathbf p_Q| \gg E_b ,$8

with $m \gg |\mathbf p_Q| \gg E_b ,$9 and $m$0 (Oncala et al., 2017). For the $m$1 channel,

$m$2

with $m$3, $m$4, numerically $m$5 and $m$6 (Oncala et al., 2017). For $m$7, a rational-plus-linear ansatz is used, with short-distance degeneracy and long-distance effective-string constraints imposed explicitly (Oncala et al., 2017).

The key dynamical refinement beyond a naive BO treatment is the appearance of nonadiabatic $m$8-doubling couplings. The heavy-pair angular kinetic operator contains the terms

$m$9

whose last two pieces raise or lower $|\mathbf p_Q|\sim 1/r$0 and generate channel mixing (Castellà, 2015). In the small-$|\mathbf p_Q|\sim 1/r$1 gluelump limit, this yields coupled radial equations of the form

$|\mathbf p_Q|\sim 1/r$2

In matrix notation,

$|\mathbf p_Q|\sim 1/r$3

with $|\mathbf p_Q|\sim 1/r$4 diagonal in the $|\mathbf p_Q|\sim 1/r$5 basis and $|\mathbf p_Q|\sim 1/r$6 containing diagonal centrifugal terms and off-diagonal $|\mathbf p_Q|\sim 1/r$7 couplings (Castellà, 2015). 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,

$|\mathbf p_Q|\sim 1/r$8

is an $|\mathbf p_Q|\sim 1/r$9 operator and is explicitly spin dependent (Oncala et al., 2017, Soto, 2017). Because it contains $E_b$0, it breaks heavy-quark spin symmetry. In the spin decomposition

$E_b$1

the mixing becomes

$E_b$2

so spin-0 hybrids mix with spin-1 quarkonia, and spin-1 hybrids mix with spin-0 quarkonia (Oncala et al., 2017). 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 $E_b$3. The relevant NRQCD Lagrangian contains the chromomagnetic Pauli term

$E_b$4

and the weak-coupling pNRQCD matching identifies the crucial operator

$E_b$5

(Oncala et al., 2017). Since that interaction is $E_b$6-independent at leading order, the short-distance constraint is

$E_b$7

with $E_b$8 (Oncala et al., 2017). At long distances, effective string theory implies

$E_b$9

with coefficients involving $H^i$00 and $H^i$01, estimated in the analysis as $H^i$02 and $H^i$03 (Oncala et al., 2017).

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, $H^i$04 and $H^i$05 mixing through $H^i$06-doubling splits opposite-parity partners that would otherwise be degenerate (Castellà, 2015), and quarkonium–hybrid mixing can become effectively leading when hybrid and quarkonium levels are close, despite being formally $H^i$07 suppressed (Soto, 2017).

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 (Copeland, 10 Jun 2026). 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 (Copeland et al., 28 May 2026, Copeland, 10 Jun 2026).

After this transformation, the theory may be viewed in two equivalent limits. Integrating out $H^i$08 and $H^i$09 returns the original vNRQCD formulation; integrating out the heavy quarks reproduces the leading pNRQCD Lagrangian (Copeland et al., 28 May 2026). 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,

$H^i$10

which removes the leading ultrasoft $H^i$11 coupling from the lowest-order Lagrangian and pushes it into subleading interactions and currents (Copeland et al., 28 May 2026). The dressed ultrasoft fields are then

$H^i$12

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 (Copeland et al., 28 May 2026). Instead, a production-specific result is established: when the quarkonium production operator creates the $H^i$13 pair locally, $H^i$14, 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 (Copeland et al., 28 May 2026, Copeland, 10 Jun 2026).

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,

$H^i$15

$H^i$16

and

$H^i$17

(Copeland et al., 28 May 2026, Copeland, 10 Jun 2026). The same logic extends to TMD soft transition functions, where explicit soft Wilson lines $H^i$18 and $H^i$19 remain and all state dependence is again isolated into $H^i$20 (Copeland et al., 28 May 2026, Copeland, 10 Jun 2026).

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 $H^i$21,

$H^i$22

augmented by the operators with coefficients $H^i$23, $H^i$24, $H^i$25, $H^i$26, $H^i$27, $H^i$28, $H^i$29, the $H^i$30, and the $H^i$31 (Huang et al., 2020). The symbolic coefficients are given in terms of the Dirac and Pauli form factors $H^i$32, $H^i$33, and their derivatives, for example

$H^i$34

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 (Huang et al., 2020).

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 (Castellà, 2015). The explicit low-energy hybrid Lagrangian of the strong-coupling approach includes only the $H^i$35 correction relevant to quarkonium–hybrid mixing, while additional $H^i$36 corrections to $H^i$37, $H^i$38, and further hybrid spin-dependent terms are omitted in the phenomenology (Oncala et al., 2017). 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 $H^i$39 limit (Copeland, 10 Jun 2026, Copeland et al., 28 May 2026). 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 (Huang et al., 2020).

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 $H^i$40 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 (Castellà, 2015, Oncala et al., 2017, Copeland et al., 28 May 2026).