Color Gauge Invariant Diquark Theory

• The color gauge invariant diquark theory defines diquark fields with precise transformation properties, using covariant derivatives to ensure local SU(3) invariance.
• It constructs gauge-invariant Lagrangians that yield effective one-gluon exchange potentials, delineating nonzero and vanishing vertex contributions for multiquark interactions.
• Matching the derived potentials with phenomenological quark models provides robust parameters for exploring exotic hadronic states and multiquark dynamics.

A color gauge invariant theory of diquark interactions addresses the formulation, constraint structure, and physical consequences of diquark (two-quark) effective degrees of freedom in quantum chromodynamics (QCD) under exact local color SU(3) gauge invariance. Key elements include the identification of diquark transformation properties, construction of gauge-invariant Lagrangians (and resulting vertices), extraction of effective one-gluon-exchange (OGE) potentials, and the phenomenological determination of coupling constants. Lattice QCD, functional approaches, symmetry analyses, and effective field theory all converge to establish this theory as a robust basis for studies of multiquark and exotic hadronic states.

1. Diquark Fields, Gauge Transformations, and Covariant Derivatives

Diquark fields are constructed as color antitriplet combinations of two quark fields, with flavor/spin structures tailored to specific quantum numbers—light scalar ($S$), light axial-vector ($A_\mu$), heavy-light scalar ($S_Q$), and heavy-light axial-vector ($A_{Q\mu}$). Under a local SU(3)$_c$ gauge transformation $U(x)$, these fields transform as

$S \to U S U^T,\quad A_\mu \to U A_\mu U^T,$

where transposition results from the antisymmetric color contraction in the diquark. This guarantees that diquark fields transform homogeneously according to their color representation.

To maintain local color gauge invariance in the kinetic and interaction terms, covariant derivatives are defined to absorb the inhomogeneous transformation of the gauge field: $$D_\mu S = \partial_\mu S - i g G_\mu S - i g S G_\mu^T,$$ with $G_\mu$ the gluon field in the adjoint representation. For the axial-vector diquark, the structure generalizes to tensor indices,

$$D_\nu A_\mu = \partial_\nu A_\mu - i g G_\nu A_\mu - i g A_\mu G_\nu^T,$$

analogous rules apply for heavy-light diquark fields.

Gauge-invariant operators and Lagrangian densities are then constructed such that each diquark–(anti)diquark–gluon vertex arises from a term that transforms covariantly under color SU(3), e.g., for the $S$–$S$–$G$ vertex: $$\mathcal{L}'_S = \frac{1}{4} \langle D_\mu S D^\mu S^\dagger \rangle_{CF} - \frac{1}{4} m_S^2 \langle S S^\dagger \rangle_{CF},$$ where $\langle \cdot \rangle_{CF}$ denotes traces over color and flavor indices.

2. Gauge-Invariant Lagrangians and Vertex Construction

The theory constructs Lagrangians for all gauge-invariant diquark–diquark–gluon $(D\!D\!G)$ and mixed diquark–antidiquark–gluon vertices relevant to both light and heavy quark sectors:

• Scalar–Scalar–Gluon ($SSG$)
• Axial–Axial–Gluon ($AAG$)
• Scalar–Axial–Gluon ($SAG$)
• Heavy scalar ($S_Q$), heavy axial ($A_Q$) in all combinations.

The kinetic term uses the covariant derivatives. Interaction terms are built from commutators and traces appropriate for the quantum numbers and color structures. For example, the axial–axial–gluon interaction is

$$\mathcal{L}_{AAG} = i d_2 \langle A^\mu G_{\mu\nu}^T A^{\dagger\nu} \rangle_{CF},$$

where $d_2$ is a dimensionful coupling, and $G_{\mu\nu}$ is the gluon field strength tensor. Mixed vertices, such as heavy–light $S_QA_QG$ couplings, may involve antisymmetric tensors ($\epsilon^{\mu\nu\alpha\beta}$) and further flavor and color structure. The explicit construction ensures that under the full set of local color gauge transformations, all Lagrangian terms are invariant.

Importantly, due to the underlying (anti)symmetry of diquark and antidiquark fields in flavor and color, specific vertices identically vanish. For instance,

$\mathcal{L}_{SAG} = 0,$

since the trace structure for $SAG$ vanishes due to the mismatch between antisymmetric and symmetric combinations of color and flavor matrices. Consequently, transitions such as $S\bar{A}\to A\bar{S}$ are forbidden at the one-gluon-exchange level.

3. One-Gluon-Exchange Potentials and Their Structure

Starting from these gauge-invariant Lagrangians, the OGE effective potentials for diquark–antidiquark interactions have been derived using standard field-theoretic techniques (Breit equation and Fourier transforms). For example, the potential $V_{S\bar{S}\to S\bar{S}}(r)$ takes the form

$$V_{S\bar{S}}(r) = -\frac{3g^2\pi}{r} - 3g^2 m_S^2 \delta^3(\vec{r}),$$

with analogous expressions for $A\bar{A}\to A\bar{A}$ and for heavy diquark–antidiquark channels. Generically, the structure is

$$V(r) = \text{Coulomb} + \text{contact} + \text{tensor},$$

where the Coulomb term originates from the instantaneous static-gluon exchange, the contact term from nonrelativistic reductions of the vertex structures (including spin–spin interactions), and tensor terms arise due to the spin structure of the (axial-vector) diquark fields.

Tensor terms, proportional to $g^2/r^3$ and suppressed by additional diquark-mass denominators (e.g., $1/(m_S m_A)$), are numerically negligible except perhaps for light diquark systems, but the analysis shows their effects are subdominant relative to the primary Coulomb and contact contributions. Mixed scalar–axial transitions are forbidden due to the vanishing of $\mathcal{L}_{SAG}$.

4. Coupling Constant Matching and Phenomenological Input

To calibrate the effective Lagrangian parameters, the derived OGE potentials are directly compared with those used in the Godfrey–Isgur (GI) quark model and related quark–diquark phenomenology. This comparison establishes the matching prescription for coupling constants: $g^2/(4\pi) = \alpha_s(r), \quad g \approx 2.7,$ consistent with a running coupling tailored to typical diquark or tetraquark system size. Spin–dependent couplings $d_1, d_2$ can be inferred by matching coefficients in the tensor and contact terms to those in the GI model, with values $d_1 \sim 3.1\,\mathrm{GeV}^{-1}$, $d_2 \sim 10.5$ (for the axial channel), and analogous relations for heavy–light systems. This matching ensures a phenomenologically grounded gauge-invariant EFT for diquark interactions.

5. Structural and Symmetry Constraints: Vanishing Vertices and the Role of Mass

The theory exposes deep connections between color–flavor symmetry and physical interaction channels:

• The vanishing of $SAG$ and hence $S\bar{A}\to A\bar{S}$ transitions is a direct consequence of the antisymmetry (scalar diquark) and symmetry (axial diquark) of their respective wavefunctions in flavor and color.
• Tensor forces, while in principle present, are found to be negligible numerically due to suppression by diquark mass factors ($1/(m_S m_A)$) and small numerical coefficients at the relevant $g$.
• In heavy–light diquark sectors, leading $1/m_Q$ corrections are systematically included in the form of additional suppressed vertices and are matched to heavy-hadron observables.

These symmetry and mass constraints not only dictate which effective interactions survive but also guide the truncation structure of diquark effective field theory for practical applications.

6. Implications for Multiquark and Exotic Hadron Physics

The color gauge invariant theory of diquark interactions derived here provides a systematic, model-independent foundation for constructing the EFT of multi-diquark and light–heavy hadron systems:

• It resolves the matching ambiguities in traditional constituent-quark models by enforcing local color gauge invariance at the level of effective interactions.
• The absence of certain transitions and the suppression of tensor terms tightly constrain allowed structures in diquark–based models of exotics such as tetraquarks and pentaquarks.
• Calibration to detailed potentials in phenomenological models (e.g., GI) offers immediate applicability to spectroscopy calculations and lattice QCD comparisons.
• The formalism directly underpins the color organization of effective degrees of freedom used in high-density QCD and color superconductivity scenarios via gauge-invariant operators and couplings.

This framework thus bridges non-perturbative QCD, effective field theory, and phenomenologically parameterized quark models for a wide range of hadronic physics.

7. Summary Table: Gauge-Invariant Diquark–Gluon Vertices

Vertex Type	Lagrangian Structure	Nonzero/Vanishing
$SSG$	$\mathcal{L}_{SSG}$: from covariant derivative in kinetic	Nonzero
$AAG$	$$\mathcal{L}_{AAG} = i d_2 \langle A^\mu G_{\mu\nu}^T A^{\dagger\nu} \rangle$$	Nonzero
$S_QS_QG/A_QA_QG$	Analogs, with heavy flavor indices	Nonzero
$S_QA_QG$	$$d_{1Q}\, \epsilon^{\mu\nu\alpha\beta} \langle D_\mu S_Q G_{\nu\alpha}^T A_{Q\beta}^\dagger\rangle$$	Nonzero
$SAG$	Trace structure vanishes due to color/flavor symmetry	Zero

The definitive vanishing of $\mathcal{L}_{SAG}$ is dictated by the underlying flavor and color symmetry: scalar (antisymmetric) × axial (symmetric) × gluon (adjoint, symmetric) cannot yield a nonzero invariant tensor under SU(3). Thus, the process $S\bar{A}\to A\bar{S}$ is forbidden at tree-level in this theory.

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In summary, the color gauge invariant theory of diquark interactions is formulated by specifying the correct transformation properties and covariant derivatives for diquark fields, building fully gauge-invariant Lagrangians, and extracting effective gluon-mediated potentials. Potentials contain dominant Coulomb and contact terms, tensor contributions are negligible, and some interactions vanish identically due to flavor–color symmetry. The full matching to phenomenological models ensures the viability and predictive power of this framework in modeling multiquark and exotic hadrons—providing a standard for effective field theory descriptions rooted in QCD gauge invariance (Wang et al., 22 Sep 2025).