Yukawa-Type Hyperfine Potential

• Yukawa-type hyperfine potential is a finite-range, spin-dependent interaction that models spin-spin couplings using an exponential distance decay.
• It replaces the traditional delta-function interaction with a mass-dependent range to regularize short-distance singularities and introduce medium-range effects.
• Precise parameter fits in quark models and muonic hydrogen systems demonstrate that this interaction kernel improves predictions of hadron spectra and binding energies.

A Yukawa-type hyperfine potential is a finite-range, spin-dependent two-body interaction characterized by an exponential ("Yukawa") spatial dependence and employed in quantum few-body systems to model spin-spin couplings, particularly in contexts where one-gluon-exchange or one-meson-exchange mechanisms are expected. It generalizes the traditional contact (delta-function) hyperfine interaction by incorporating a mass-dependent range, thereby regularizing short-distance singularities and introducing medium-range effects essential for describing hadron spectra and hadronic molecules. Yukawa-type hyperfine potentials are relevant both in quark models of multiquark states and in low-energy hadron-lepton systems.

1. Explicit Formulation and Theoretical Foundations

In quark models, the Yukawa-type hyperfine interaction is implemented within the nonrelativistic Hamiltonian as a color-spin term. In the case of the $T_{cc}$ tetraquark ($\bar u\bar d cc$), the Hamiltonian reads:

$$H = \sum_n \left(m_n + \frac{p_n^2}{2m_n}\right) - \frac{3}{4} \sum_{i<j} \frac{\vec\lambda^c_i}{2} \cdot \frac{\vec\lambda^c_j}{2} [V^{C}_{ij} + V^{CS}_{ij}],$$

with the hyperfine kernel:

$$V^{CS}_{ij}(r_{ij}) = \frac{\hbar^2 c^2 \kappa'_{ij}}{m_i m_j c^4} \frac{e^{-r_{ij}/r_{0ij}}}{r_{0ij} r_{ij}}\, (\vec\sigma_i \cdot \vec\sigma_j).$$

Here, $m_i$, $m_j$ are constituent quark masses, $\vec\sigma_i$ are Pauli spin operators, and $\vec\lambda_i^c$ are SU(3)-color Gell-Mann matrices. The coupling strength is:

$$\kappa'_{ij} = \kappa_0 [1 + \gamma \mu_{ij}], \quad \mu_{ij} = \frac{m_i m_j}{m_i + m_j},$$

and the range is parameterized as:

$r_{0ij} = \frac{1}{\alpha + \beta \mu_{ij}}.$

Fitted parameters are $\bar u\bar d cc$0 MeV, $\bar u\bar d cc$1 fm$\bar u\bar d cc$2, $\bar u\bar d cc$3 (MeV fm)$\bar u\bar d cc$4, and $\bar u\bar d cc$5 MeV$\bar u\bar d cc$6 (Noh et al., 2023).

Correspondingly, in hadron-lepton systems such as muonic hydrogen, the long-range Yukawa-type hyperfine arises from one-pion exchange:

$\bar u\bar d cc$7

with $\bar u\bar d cc$8 the pion mass. The exponential term yields the Yukawa potential, and the overall strength is computed from chiral Lagrangians and measured decay widths (Huong et al., 2015).

2. Motivation: Regularization and Physical Significance

The primary motivation for the Yukawa-type hyperfine form is to regularize the short-range singularity of the traditional one-gluon-exchange-induced contact (delta-function) interaction:

$\bar u\bar d cc$9

This point-like character is unphysical for extended quark/hadron wave functions, leading to the widespread use of smearing (e.g., Gaussian convolutions). The Yukawa kernel is finite for all $$H = \sum_n \left(m_n + \frac{p_n^2}{2m_n}\right) - \frac{3}{4} \sum_{i<j} \frac{\vec\lambda^c_i}{2} \cdot \frac{\vec\lambda^c_j}{2} [V^{C}_{ij} + V^{CS}_{ij}],$$0, removes the necessity for ad hoc smearing, and parametrically introduces a range $$H = \sum_n \left(m_n + \frac{p_n^2}{2m_n}\right) - \frac{3}{4} \sum_{i<j} \frac{\vec\lambda^c_i}{2} \cdot \frac{\vec\lambda^c_j}{2} [V^{C}_{ij} + V^{CS}_{ij}],$$1 linked to the reduced mass of the interacting quark pair or the mass of the exchanged meson.

At short distances ($$H = \sum_n \left(m_n + \frac{p_n^2}{2m_n}\right) - \frac{3}{4} \sum_{i<j} \frac{\vec\lambda^c_i}{2} \cdot \frac{\vec\lambda^c_j}{2} [V^{C}_{ij} + V^{CS}_{ij}],$$2), the Yukawa form reduces to a contact-like $$H = \sum_n \left(m_n + \frac{p_n^2}{2m_n}\right) - \frac{3}{4} \sum_{i<j} \frac{\vec\lambda^c_i}{2} \cdot \frac{\vec\lambda^c_j}{2} [V^{C}_{ij} + V^{CS}_{ij}],$$3 behavior (modulo prefactors), while at finite $$H = \sum_n \left(m_n + \frac{p_n^2}{2m_n}\right) - \frac{3}{4} \sum_{i<j} \frac{\vec\lambda^c_i}{2} \cdot \frac{\vec\lambda^c_j}{2} [V^{C}_{ij} + V^{CS}_{ij}],$$4 it produces a smooth exponential tail. This structure makes the resulting hyperfine splittings sensitive to wave function components at both short and intermediate distances, rather than just at the origin as in the delta function case (Noh et al., 2023).

3. Parameter Determination and Simultaneous Spectrum Fits

The parameters of the Yukawa-type hyperfine potential are tightly constrained through simultaneous global fits to ground-state meson and baryon spectra. The full parameter set—including constituent quark masses, hyperfine and confining strengths, and range parameters—is optimized by minimizing the Pearson $$H = \sum_n \left(m_n + \frac{p_n^2}{2m_n}\right) - \frac{3}{4} \sum_{i<j} \frac{\vec\lambda^c_i}{2} \cdot \frac{\vec\lambda^c_j}{2} [V^{C}_{ij} + V^{CS}_{ij}],$$5 with respect to 33 hadron masses:

$$H = \sum_n \left(m_n + \frac{p_n^2}{2m_n}\right) - \frac{3}{4} \sum_{i<j} \frac{\vec\lambda^c_i}{2} \cdot \frac{\vec\lambda^c_j}{2} [V^{C}_{ij} + V^{CS}_{ij}],$$6

where $$H = \sum_n \left(m_n + \frac{p_n^2}{2m_n}\right) - \frac{3}{4} \sum_{i<j} \frac{\vec\lambda^c_i}{2} \cdot \frac{\vec\lambda^c_j}{2} [V^{C}_{ij} + V^{CS}_{ij}],$$7 are experimental uncertainties. The confining kernel is typically of Bhaduri type (linear plus Coulomb plus constant), and its parameters are included in the fit. Empirically, the calculated meson thresholds (e.g., $$H = \sum_n \left(m_n + \frac{p_n^2}{2m_n}\right) - \frac{3}{4} \sum_{i<j} \frac{\vec\lambda^c_i}{2} \cdot \frac{\vec\lambda^c_j}{2} [V^{C}_{ij} + V^{CS}_{ij}],$$8, $$H = \sum_n \left(m_n + \frac{p_n^2}{2m_n}\right) - \frac{3}{4} \sum_{i<j} \frac{\vec\lambda^c_i}{2} \cdot \frac{\vec\lambda^c_j}{2} [V^{C}_{ij} + V^{CS}_{ij}],$$9, $$V^{CS}_{ij}(r_{ij}) = \frac{\hbar^2 c^2 \kappa'_{ij}}{m_i m_j c^4} \frac{e^{-r_{ij}/r_{0ij}}}{r_{0ij} r_{ij}}\, (\vec\sigma_i \cdot \vec\sigma_j).$$0, $$V^{CS}_{ij}(r_{ij}) = \frac{\hbar^2 c^2 \kappa'_{ij}}{m_i m_j c^4} \frac{e^{-r_{ij}/r_{0ij}}}{r_{0ij} r_{ij}}\, (\vec\sigma_i \cdot \vec\sigma_j).$$1) agree with experiment within a few MeV, validating the chosen hyperfine structure (Noh et al., 2023).

4. Computational Implementation: Few-Body Solutions and Convergence

For multiquark systems, the spatial wave function is expanded in a basis of Jacobi-coordinate harmonic oscillators, with quantum numbers $$V^{CS}_{ij}(r_{ij}) = \frac{\hbar^2 c^2 \kappa'_{ij}}{m_i m_j c^4} \frac{e^{-r_{ij}/r_{0ij}}}{r_{0ij} r_{ij}}\, (\vec\sigma_i \cdot \vec\sigma_j).$$2 up to a cutoff $$V^{CS}_{ij}(r_{ij}) = \frac{\hbar^2 c^2 \kappa'_{ij}}{m_i m_j c^4} \frac{e^{-r_{ij}/r_{0ij}}}{r_{0ij} r_{ij}}\, (\vec\sigma_i \cdot \vec\sigma_j).$$3. The color, spin, and flavor parts are explicitly coupled to the required total quantum numbers, as for $$V^{CS}_{ij}(r_{ij}) = \frac{\hbar^2 c^2 \kappa'_{ij}}{m_i m_j c^4} \frac{e^{-r_{ij}/r_{0ij}}}{r_{0ij} r_{ij}}\, (\vec\sigma_i \cdot \vec\sigma_j).$$4 ($$V^{CS}_{ij}(r_{ij}) = \frac{\hbar^2 c^2 \kappa'_{ij}}{m_i m_j c^4} \frac{e^{-r_{ij}/r_{0ij}}}{r_{0ij} r_{ij}}\, (\vec\sigma_i \cdot \vec\sigma_j).$$5, $$V^{CS}_{ij}(r_{ij}) = \frac{\hbar^2 c^2 \kappa'_{ij}}{m_i m_j c^4} \frac{e^{-r_{ij}/r_{0ij}}}{r_{0ij} r_{ij}}\, (\vec\sigma_i \cdot \vec\sigma_j).$$6). The full few-body problem is solved via generalized eigenvalue diagonalization:

$$V^{CS}_{ij}(r_{ij}) = \frac{\hbar^2 c^2 \kappa'_{ij}}{m_i m_j c^4} \frac{e^{-r_{ij}/r_{0ij}}}{r_{0ij} r_{ij}}\, (\vec\sigma_i \cdot \vec\sigma_j).$$7

where $$V^{CS}_{ij}(r_{ij}) = \frac{\hbar^2 c^2 \kappa'_{ij}}{m_i m_j c^4} \frac{e^{-r_{ij}/r_{0ij}}}{r_{0ij} r_{ij}}\, (\vec\sigma_i \cdot \vec\sigma_j).$$8 is the basis overlap matrix. Convergence is achieved by increasing $$V^{CS}_{ij}(r_{ij}) = \frac{\hbar^2 c^2 \kappa'_{ij}}{m_i m_j c^4} \frac{e^{-r_{ij}/r_{0ij}}}{r_{0ij} r_{ij}}\, (\vec\sigma_i \cdot \vec\sigma_j).$$9 until successive mass shifts are below $m_i$01 MeV. Typical values are $m_i$1 for mesons and $m_i$2 for tetraquarks, allowing for robust characterization of both short- and medium-range correlations (Noh et al., 2023).

5. Phenomenological Impact: Tetraquark $m_i$3 and Hadronic Molecules

With the Yukawa-type hyperfine, the mass of the $m_i$4 tetraquark is predicted as $m_i$5 MeV, yielding a binding energy:

$m_i$6

This matches recent LHCb observations of $m_i$7 MeV and $m_i$8 MeV. When the Yukawa interaction is replaced by a narrow Gaussian "smeared delta" hyperfine (as in earlier work), the system is unbound by approximately 13 MeV. Thus, the Yukawa kernel provides an additional $m_i$9 MeV of attraction and is uniquely responsible for binding $m_j$0 below threshold in fully converged models (Noh et al., 2023).

Analyzing hyperfine contributions further reveals that the extra attraction is localized predominantly within the light $m_j$1 pair, where the Yukawa tail enhances short-range interactions relative to Gaussian forms.

6. Comparison with Other Forms and Compactness Diagnostics

Only quark models implementing a Yukawa-type hyperfine kernel succeed in reproducing the experimentally observed binding of $m_j$2; those with Gaussian or strictly contact interactions do not yield a bound state. The radial profiles of $m_j$3 show that Gaussian forms dominate at intermediate ranges ($m_j$4–$m_j$5 fm), while Yukawa forms dominate at $m_j$6 fm, amplifying attraction where compact diquark correlations are expected.

Quantitative compactness is probed via the RMS radius ratio:

$m_j$7

finding $m_j$8 and $m_j$9, both less than 1. This signals a compact, non-molecular tetraquark structure, further supported by wave function analysis (Noh et al., 2023).

7. Yukawa-Type Hyperfine in Muonic Atoms

In muonic hydrogen, the Yukawa-type hyperfine potential arises from single-pion exchange and constitutes the formally longest-range hadronic correction to the hyperfine splitting of atomic energy levels. The effective potential,

$\vec\sigma_i$0

modifies the $\vec\sigma_i$1 hyperfine splitting by

$\vec\sigma_i$2

as determined using first-order perturbation theory and precise vertex amplitudes from chiral expansions and decay data. This correction is negligible at current empirical precision (about $\vec\sigma_i$3 of the leading Fermi splitting), but it completes the catalogue of strong-interaction-induced effects. Refined calculations may incorporate improved form factors, chiral corrections, or lattice determinations of relevant parameters (Huong et al., 2015).

Contribution	Magnitude	Relative Importance
Yukawa (1-π-exchange)	$\vec\sigma_i$4 μeV	$\vec\sigma_i$5 of main split
Leading Fermi	$\vec\sigma_i$622,860 μeV	Dominant
Zemach radius	$\vec\sigma_i$7 μeV	$\vec\sigma_i$8
Proton polarizability	$\vec\sigma_i$9 μeV	$\vec\lambda_i^c$0

The Yukawa-type hyperfine structure thus provides both a physically motivated interaction kernel for multiquark hadrons and a framework for computing subtle hadronic corrections to atomic spectra.

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References:

• "Observation of $\vec\lambda_i^c$1 and a quark model" (Noh et al., 2023)
• "Single pion contribution to the hyperfine splitting in muonic hydrogen" (Huong et al., 2015)