Quark-Pair-Creation Model in QCD

• The quark-pair-creation model is a nonperturbative framework that generates q-q̄ pairs with vacuum quantum numbers (³P₀) for applications in hadron spectroscopy and strong decays.
• It facilitates the unquenching of hadronic states by coupling valence quarks to meson-meson continua, thereby quantifying self-energy shifts and decay widths.
• Enhanced formulations, including relativistic treatments and modified vertices with damping factors, improve convergence and precision in both heavy-quark and exclusive electroproduction analyses.

The quark-pair-creation model is a class of nonperturbative descriptions of $q\bar q$ production that is used in several distinct but related settings. In hadron spectroscopy and strong-decay phenomenology, it usually denotes the $^{3}P_0$ construction in which a light quark-antiquark pair is created with the vacuum quantum numbers $J^{PC}=0^{++}$, then recombines with the valence constituents of an initial hadron to produce open channels, higher-Fock components, and loop-induced self-energies (Tecocoatzi et al., 2015). In exclusive electroproduction, a single-pair version of the same idea is used to unfold flavor-dependent $q\bar q$-creation probabilities from measured cross-section ratios (Park et al., 2014). In strong-field QCD, the term also appears in the broader sense of Schwinger-like quark-pair production by color electric fields, with or without magnetic fields and longitudinal expansion (Tanji, 2010).

1. Canonical $^{3}P_0$ construction

In the standard hadronic formulation, the created pair carries the quantum numbers of the QCD vacuum: color singlet, flavor singlet, spin triplet, and relative $P$-wave, coupled to total $J=0$. This is the origin of the spectroscopic label $^{3}P_0$ and of the statement that the pair has $J^{PC}=0^{++}$ (Bijker et al., 2016). The nonrelativistic momentum-space transition operator is commonly written as

$$T^\dagger \;=\; -\,3\,\gamma\sum_m \langle 1m;1{-}m\mid 00\rangle \int d^3p_3\,d^3p_4\, \delta^{(3)}(\mathbf p_3+\mathbf p_4)\, \mathcal Y_1^m\!\Bigl(\tfrac{\mathbf p_3-\mathbf p_4}{2}\Bigr)\, \chi^{34}_{1,-m}\,\phi_0^{34}\,\omega_0^{34}\, b_3^\dagger(\mathbf p_3)\,d_4^\dagger(\mathbf p_4),$$

with $^{3}P_0$0 a dimensionless pair-creation strength, $^{3}P_0$1 the solid harmonic that enforces $^{3}P_0$2, $^{3}P_0$3 the spin-triplet wave function, and $^{3}P_0$4 the flavor- and color-singlet projectors (Chen et al., 2024).

Equivalent second-quantized forms emphasize the same ingredients. In one notation the operator is

$^{3}P_0$5

which makes explicit the zero total momentum of the created pair and the singlet structure in color and flavor space (Tecocoatzi et al., 2015). Some implementations multiply the vertex by a phenomenological form factor such as $^{3}P_0$6 or $^{3}P_0$7 to suppress high-momentum components (Tecocoatzi et al., 2015, Bijker et al., 2016).

The same vacuum-quantum-number logic can be expressed in coordinate space through the scalar bilinear

$^{3}P_0$8

or, in some charmonium applications,

$^{3}P_0$9

where the factor $J^{PC}=0^{++}$0 suppresses heavy-quark loop creation (Simonov et al., 2011, Kanwal et al., 2022). These equivalent representations are the basis for both decay calculations above threshold and virtual-loop dressings below threshold.

2. Unquenched quark model and continuum dressing

The main formal role of the quark-pair-creation model in spectroscopy is to unquench a valence hadron by coupling it to hadron-hadron continua. In the unquenched quark model, the physical state is expanded as

$J^{PC}=0^{++}$1

or, for baryons,

$J^{PC}=0^{++}$2

so that one-loop sea-quark effects appear as explicit baryon-meson or meson-meson components (Tecocoatzi et al., 2015, Bijker et al., 2017).

Projecting the Hamiltonian onto the dressed state yields the familiar self-energy formula

$J^{PC}=0^{++}$3

with $J^{PC}=0^{++}$4 above threshold (Tecocoatzi et al., 2015). In this sense the same operator governs both strong decays and virtual mass renormalization.

Applications to sea-quark observables in the proton are a standard testbed. One UQM calculation gives the Gottfried integral $J^{PC}=0^{++}$5, a strange magnetic moment $J^{PC}=0^{++}$6, and a strange radius $J^{PC}=0^{++}$7, all from the same continuum-dressed wave function (Tecocoatzi et al., 2015). In baryon spectroscopy, the same mechanism enhances $J^{PC}=0^{++}$8 from $J^{PC}=0^{++}$9 keV to $q\bar q$0 keV, to be compared with the quoted experimental value $q\bar q$1 keV, and reduces the naive constituent-quark-model neutron $q\bar q$2-decay axial coupling from $q\bar q$3 to about $q\bar q$4, or $q\bar q$5 once further $q\bar q$6 loops are included (Bijker et al., 2017). A pion-cloud UQM implementation for baryons gives $q\bar q$7 and lowers $q\bar q$8 from 1 to about $q\bar q$9 (Bijker et al., 2016).

Heavy quarkonia provide another benchmark. In one UQM treatment the net shifts are of order $^{3}P_0$0 MeV in charmonium and $^{3}P_0$1 MeV in bottomonium, while relative shifts can be tens of MeV near open-flavor thresholds (Tecocoatzi et al., 2015). These results established the $^{3}P_0$2 vertex as a standard continuum-coupling kernel, but they also exposed a recurring problem: without additional suppression, loop effects can become too large for precision spectroscopy.

3. Relativistic formulations and first-principles support

Several lines of work seek to ground or extend the phenomenological $^{3}P_0$3 picture. A nonperturbative derivation from QCD string breaking starts from the QCD partition function with a heavy $^{3}P_0$4 Wilson loop and a light-quark loop, leading to an effective bilinear light-quark kernel

$^{3}P_0$5

In the small-correlation-length limit this kernel becomes local and purely scalar,

$^{3}P_0$6

with $^{3}P_0$7 (Simonov et al., 2011). The dominant scalar term is therefore linear in the distances to the static sources, flavor blind, and, according to the same analysis, in good agreement with the $^{3}P_0$8 model. Its momentum-space average can be parameterized by $^{3}P_0$9, and fits to $P$0, $P$1, and $P$2 yield mutually consistent values in that range (Simonov et al., 2011).

A complementary route starts from Landau-gauge QCD Green’s functions with insertions of a constant chromoelectric field. Because the background selects the $P$3-axis, the natural low-energy channels are $P$4, $P$5, and $P$6 rather than a fully spherical $P$7 basis. In that framework the produced pair is dominated at sub-GeV momenta by the $P$8 channel, while ultrarelativistic fermions are rather ejected with $P$9 quantum numbers (Alkofer et al., 2023). This suggests that the conventional $J=0$0 operator is an effective low-momentum reduction of a more detailed QCD amplitude, not merely an ad hoc rule.

Relativistic versions of the quark-pair-creation model replace nonrelativistic mock-meson states by boosted states with Wigner rotations and exact Lorentz kinematics. The relativistic transition operator derived from

$J=0$1

contains the same $J=0$2, $J=0$3, color-singlet, flavor-singlet structure as the nonrelativistic one, but with explicit factors $J=0$4 and Wigner-rotation matrices in the helicity amplitude (Zhou et al., 2020, Gao et al., 23 Apr 2026). A 2026 comparison finds that the relativistic and nonrelativistic QPC models yield decay-width predictions of comparable overall quality, so the nonrelativistic model remains adequate for most practical applications; however, the relativistic model shows a stronger suppression of off-shell amplitudes in the high-energy region, by up to an order of magnitude at $J=0$5 GeV, which improves the convergence of unquenched self-energies (Gao et al., 23 Apr 2026).

The relativistic Friedrichs-Lee embedding makes this continuum structure analytically explicit. There the discrete state couples to a continuum through a QPC form factor $J=0$6, and the analytic structure is controlled by

$J=0$7

For the lowest $J=0$8, $J=0$9 channel, coupling a bare $^{3}P_0$0 state to $^{3}P_0$1 produces two second-sheet poles, $^{3}P_0$2 MeV and $^{3}P_0$3 MeV, associated in that analysis with the $^{3}P_0$4 and probably the $^{3}P_0$5 (Zhou et al., 2020).

4. Modified vertices, ultraviolet suppression, and heavy-quark spectroscopy

A major technical issue in unquenched spectroscopy is that the original $^{3}P_0$6 operator can generate excessively large negative self-energies. To address this, modified $^{3}P_0$7 vertices introduce damping in both the relative momentum of the created pair and the spatial separation between the creation point and the parent hadron. A representative choice is

$^{3}P_0$8

where $^{3}P_0$9, $J^{PC}=0^{++}$0, and $J^{PC}=0^{++}$1 is a spatial range (Chen et al., 2024). In the cited implementation the fitted values are $J^{PC}=0^{++}$2 for $J^{PC}=0^{++}$3, $J^{PC}=0^{++}$4 for $J^{PC}=0^{++}$5, $J^{PC}=0^{++}$6, and $J^{PC}=0^{++}$7 (Chen et al., 2024).

These operators are embedded in a coupled-channel Schrödinger problem,

$J^{PC}=0^{++}$8

solved in a Gaussian expansion method basis. The resulting generalized eigenvalue problem mixes a bare $J^{PC}=0^{++}$9 sector with explicit meson-meson components through $$T^\dagger \;=\; -\,3\,\gamma\sum_m \langle 1m;1{-}m\mid 00\rangle \int d^3p_3\,d^3p_4\, \delta^{(3)}(\mathbf p_3+\mathbf p_4)\, \mathcal Y_1^m\!\Bigl(\tfrac{\mathbf p_3-\mathbf p_4}{2}\Bigr)\, \chi^{34}_{1,-m}\,\phi_0^{34}\,\omega_0^{34}\, b_3^\dagger(\mathbf p_3)\,d_4^\dagger(\mathbf p_4),$$0 (Chen et al., 2024). The same framework is used in dedicated charmonium calculations with $$T^\dagger \;=\; -\,3\,\gamma\sum_m \langle 1m;1{-}m\mid 00\rangle \int d^3p_3\,d^3p_4\, \delta^{(3)}(\mathbf p_3+\mathbf p_4)\, \mathcal Y_1^m\!\Bigl(\tfrac{\mathbf p_3-\mathbf p_4}{2}\Bigr)\, \chi^{34}_{1,-m}\,\phi_0^{34}\,\omega_0^{34}\, b_3^\dagger(\mathbf p_3)\,d_4^\dagger(\mathbf p_4),$$1, $$T^\dagger \;=\; -\,3\,\gamma\sum_m \langle 1m;1{-}m\mid 00\rangle \int d^3p_3\,d^3p_4\, \delta^{(3)}(\mathbf p_3+\mathbf p_4)\, \mathcal Y_1^m\!\Bigl(\tfrac{\mathbf p_3-\mathbf p_4}{2}\Bigr)\, \chi^{34}_{1,-m}\,\phi_0^{34}\,\omega_0^{34}\, b_3^\dagger(\mathbf p_3)\,d_4^\dagger(\mathbf p_4),$$2, $$T^\dagger \;=\; -\,3\,\gamma\sum_m \langle 1m;1{-}m\mid 00\rangle \int d^3p_3\,d^3p_4\, \delta^{(3)}(\mathbf p_3+\mathbf p_4)\, \mathcal Y_1^m\!\Bigl(\tfrac{\mathbf p_3-\mathbf p_4}{2}\Bigr)\, \chi^{34}_{1,-m}\,\phi_0^{34}\,\omega_0^{34}\, b_3^\dagger(\mathbf p_3)\,d_4^\dagger(\mathbf p_4),$$3, and $$T^\dagger \;=\; -\,3\,\gamma\sum_m \langle 1m;1{-}m\mid 00\rangle \int d^3p_3\,d^3p_4\, \delta^{(3)}(\mathbf p_3+\mathbf p_4)\, \mathcal Y_1^m\!\Bigl(\tfrac{\mathbf p_3-\mathbf p_4}{2}\Bigr)\, \chi^{34}_{1,-m}\,\phi_0^{34}\,\omega_0^{34}\, b_3^\dagger(\mathbf p_3)\,d_4^\dagger(\mathbf p_4),$$4 intermediate channels (Chen et al., 2023).

The numerical effect of the damping is substantial. In one calculation the shift of $$T^\dagger \;=\; -\,3\,\gamma\sum_m \langle 1m;1{-}m\mid 00\rangle \int d^3p_3\,d^3p_4\, \delta^{(3)}(\mathbf p_3+\mathbf p_4)\, \mathcal Y_1^m\!\Bigl(\tfrac{\mathbf p_3-\mathbf p_4}{2}\Bigr)\, \chi^{34}_{1,-m}\,\phi_0^{34}\,\omega_0^{34}\, b_3^\dagger(\mathbf p_3)\,d_4^\dagger(\mathbf p_4),$$5 becomes $$T^\dagger \;=\; -\,3\,\gamma\sum_m \langle 1m;1{-}m\mid 00\rangle \int d^3p_3\,d^3p_4\, \delta^{(3)}(\mathbf p_3+\mathbf p_4)\, \mathcal Y_1^m\!\Bigl(\tfrac{\mathbf p_3-\mathbf p_4}{2}\Bigr)\, \chi^{34}_{1,-m}\,\phi_0^{34}\,\omega_0^{34}\, b_3^\dagger(\mathbf p_3)\,d_4^\dagger(\mathbf p_4),$$6 MeV instead of $$T^\dagger \;=\; -\,3\,\gamma\sum_m \langle 1m;1{-}m\mid 00\rangle \int d^3p_3\,d^3p_4\, \delta^{(3)}(\mathbf p_3+\mathbf p_4)\, \mathcal Y_1^m\!\Bigl(\tfrac{\mathbf p_3-\mathbf p_4}{2}\Bigr)\, \chi^{34}_{1,-m}\,\phi_0^{34}\,\omega_0^{34}\, b_3^\dagger(\mathbf p_3)\,d_4^\dagger(\mathbf p_4),$$7 MeV with the unmodified $$T^\dagger \;=\; -\,3\,\gamma\sum_m \langle 1m;1{-}m\mid 00\rangle \int d^3p_3\,d^3p_4\, \delta^{(3)}(\mathbf p_3+\mathbf p_4)\, \mathcal Y_1^m\!\Bigl(\tfrac{\mathbf p_3-\mathbf p_4}{2}\Bigr)\, \chi^{34}_{1,-m}\,\phi_0^{34}\,\omega_0^{34}\, b_3^\dagger(\mathbf p_3)\,d_4^\dagger(\mathbf p_4),$$8, while $$T^\dagger \;=\; -\,3\,\gamma\sum_m \langle 1m;1{-}m\mid 00\rangle \int d^3p_3\,d^3p_4\, \delta^{(3)}(\mathbf p_3+\mathbf p_4)\, \mathcal Y_1^m\!\Bigl(\tfrac{\mathbf p_3-\mathbf p_4}{2}\Bigr)\, \chi^{34}_{1,-m}\,\phi_0^{34}\,\omega_0^{34}\, b_3^\dagger(\mathbf p_3)\,d_4^\dagger(\mathbf p_4),$$9 shifts by $^{3}P_0$00 MeV instead of $^{3}P_0$01 MeV (Chen et al., 2024). A related charmonium study reports that the improved operator reduces mass shifts by $^{3}P_0$02 on average, with $^{3}P_0$03 changing from $^{3}P_0$04 MeV to $^{3}P_0$05 MeV (Chen et al., 2023). These results are the basis for the frequent criticism that the plain $^{3}P_0$06 operator overestimates unquenching unless supplemented by QCD-motivated suppression.

The $^{3}P_0$07 is the best-known application. In a modified-operator UQM study, the bare $^{3}P_0$08 at $^{3}P_0$09 MeV acquires a shift of about $^{3}P_0$10 MeV, and after a slight retuning of charm-sector parameters the final mass becomes $^{3}P_0$11 MeV, with a dominant charmonium component of about $^{3}P_0$12 and meson-meson components of about $^{3}P_0$13 (Tan et al., 2019). Another implementation quotes $^{3}P_0$14 MeV, $^{3}P_0$15 MeV, and $^{3}P_0$16 MeV, with $^{3}P_0$17, $^{3}P_0$18, and $^{3}P_0$19 (Chen et al., 2024). By contrast, a self-consistent refit with a standard $^{3}P_0$20 vertex and intermediate states summed up to $^{3}P_0$21 excitations finds that most charmonium levels shift only modestly after refitting, but the $^{3}P_0$22 multiplet moves upward by order $^{3}P_0$23 MeV and the erstwhile $^{3}P_0$24 $^{3}P_0$25 state is pushed to $^{3}P_0$26 GeV, well above $^{3}P_0$27 threshold, thereby favoring a molecular interpretation of $^{3}P_0$28 (Kanwal et al., 2022). The coexistence of these outcomes shows that the inferred composition of near-threshold states is highly sensitive to the detailed choice of pair-creation kernel and renormalization strategy.

5. Exclusive electroproduction and strangeness suppression

A particularly transparent use of a quark-pair-creation model occurs in exclusive two-body electroproduction, where only a single $^{3}P_0$29 pair is assumed to be created. The underlying tunneling picture is the Schwinger-like flux-tube formula

$^{3}P_0$30

with a phenomenological strangeness-suppression factor

$^{3}P_0$31

In high-energy fragmentation models such as LUND and PYTHIA, $^{3}P_0$32 is the canonical value (Park et al., 2014).

The CLAS analysis of

$^{3}P_0$33

uses a “single-pair-creation, lowest-mass-hadron” ansatz. The virtual photon couples to a valence $^{3}P_0$34 or $^{3}P_0$35 quark in the charge-squared ratio $^{3}P_0$36; one $^{3}P_0$37 pair is then produced with probabilities $^{3}P_0$38, $^{3}P_0$39, or $^{3}P_0$40; and the struck quark plus the new antiquark form the lightest allowed pseudoscalar meson while the remnant diquark plus the new quark form the associated lowest-mass baryon (Park et al., 2014). This yields

$^{3}P_0$41

with the factor $^{3}P_0$42 in the $^{3}P_0$43 channel coming from the $^{3}P_0$44 $^{3}P_0$45–$^{3}P_0$46 content of $^{3}P_0$47 (Park et al., 2014).

From the reported $^{3}P_0$48-integrated ratios,

$^{3}P_0$49

the extraction gives

$^{3}P_0$50

and therefore

$^{3}P_0$51

Depending on which ratio is used and on whether one forces $^{3}P_0$52, the quoted ranges are $^{3}P_0$53 and $^{3}P_0$54, in agreement with the canonical high-energy value (Park et al., 2014).

The significance of this result is methodological as well as phenomenological. Because only one pair is created, no cascade-decay modeling is needed, and the mapping from final hadrons to pair-creation probabilities is unusually transparent. The same analysis also states the main limitations: no $^{3}P_0$55-channel exchange, no vector-meson admixtures, no multi-step final-state interactions, and a rigid “lightest-hadron” recombination rule (Park et al., 2014).

6. Strong color fields, Schwinger production, and glasma dynamics

In another major usage, quark-pair-creation models describe vacuum decay in strong color-electric backgrounds rather than hadronic recombination. For homogeneous color fields the vacuum persistency probability is written as

$^{3}P_0$56

and the QCD analogue of Schwinger’s formula in parallel electric and magnetic fields can be expressed as a sum over Landau levels (Hidaka et al., 2011). The transverse mass-squared is

$^{3}P_0$57

The lowest Landau level, $^{3}P_0$58, $^{3}P_0$59, gives $^{3}P_0$60, so for $^{3}P_0$61 it dominates the rate; in the massless limit with $^{3}P_0$62, $^{3}P_0$63, whereas for pure electric field $^{3}P_0$64 one recovers the finite Schwinger series (Hidaka et al., 2011).

A dynamical back-reaction treatment of a uniform SU(3) color electric field Abelianizes the background into effective color charges $^{3}P_0$65 and describes pair creation through time-dependent Bogoliubov coefficients $^{3}P_0$66 and anomalous distributions $^{3}P_0$67 (Tanji, 2010). The induced current

$^{3}P_0$68

drives plasma oscillations of the electric field. In an approximate treatment that neglects the polarization current,

$^{3}P_0$69

and the first zero of the field occurs at $^{3}P_0$70 (Tanji, 2010). The same framework shows Pauli blocking, quantum interference in momentum space, pressure anisotropy reduction, enhancement by a parallel magnetic field, and chiral-charge generation through

$^{3}P_0$71

(Tanji, 2010).

For the expanding glasma, a simplified massless QED model in Milne coordinates $^{3}P_0$72 yields the coupled equations

$^{3}P_0$73

with solutions

$^{3}P_0$74

Unlike the nonexpanding case, the oscillation amplitude decays as $^{3}P_0$75 because of the explicit $^{3}P_0$76 damping term (Iwazaki, 2011).

A different strong-field program uses a flavor-dependent contact interaction within Schwinger-Dyson equations. There the dressed quark mass $^{3}P_0$77 decreases with the field, and for $^{3}P_0$78, $^{3}P_0$79 the pseudo-critical field is $^{3}P_0$80. The pair-production rate,

$^{3}P_0$81

or, at leading order,

$^{3}P_0$82

grows rapidly for $^{3}P_0$83. Increasing $^{3}P_0$84 lowers $^{3}P_0$85, increasing $^{3}P_0$86 raises it, and for $^{3}P_0$87 the transition becomes first order at $^{3}P_0$88 (Ahmad et al., 2023).

Taken together, these strong-field studies concern vacuum decay rather than hadronic decays, but they show that “quark-pair creation” has a broader nonperturbative meaning in QCD: flux-tube breaking in hadrons, flavor-resolved pair creation in exclusive reactions, and Schwinger production in intense classical color fields are technically different realizations of the same underlying problem of creating $^{3}P_0$89 pairs from nonperturbative QCD backgrounds.