Three Vector B Meson States

• Three vector B meson states are orbitally excited bottom quark systems characterized by quantum numbers 0⁺, 1⁺, and 2⁺, often exhibiting mixed quark–antiquark and four-quark content.
• Their properties are analyzed using Heavy Quark Effective Theory and coupled-channel formalisms to reveal mass shifts, narrow decay widths, and molecular binding features.
• Experimental measurements from collider data validate theoretical predictions by accurately recording masses and decay patterns, thus deepening our understanding of heavy-flavor spectroscopy.

Three vector B meson states refer primarily to the set of orbitally excited B mesons with quantum numbers $J^P = 0^+$, $1^+$, and $2^+$, as well as their strange partners, whose masses, widths, and decay mechanisms form a cornerstone of heavy-quark spectroscopy. These states arise through a combination of conventional quark–antiquark configurations and, in some cases, significant four-quark admixtures, shaping their spectral properties and decay phenomenology. Experimental and theoretical investigations of these vector B meson states—especially $B_1(5721)$, $B_2^*(5747)$, $B_{s1}(5830)$, and $B_{s2}^*(5840)$—have elucidated their assignment, molecular nature, mass shifts, and implications for heavy-flavor QCD.

1. Spectroscopic Structure and Quantum Number Assignments

The vector B meson states under consideration are predominantly the $1P$ orbitally excited states formed by a bottom quark ($b$) and a light anti-quark ($\bar{q}$ or $\bar{s}$). Their quantum numbers in the quark model are:

State	$J^P$	Dominant Assignment	Observed Mass [MeV/$c^2$]
$B_1(5721)$	$1^+$	$1P_1'$ ($j_q=3/2$)	$\sim$5721
$B_2^*(5747)$	$2^+$	$^3P_2$	$\sim$5747
$B_{s1}(5830)$	$1^+$	$1P_1'$ ($j_q=3/2$)	$\sim$5830
$B_{s2}^*(5840)$	$2^+$	$^3P_2$	$\sim$5840

The nomenclature is rooted in Heavy Quark Effective Theory (HQET), which organizes states by the total angular momentum of the light degrees of freedom ($j_q$). The model predicts distinct doublets: narrow $j_q=3/2$ states ($B_1$, $B_2^*$) with D-wave decays and broad $j_q=1/2$ states ($B_0^*$, $B_1'$) with S-wave decays (0906.0808, Yu et al., 2019).

2. Theoretical Models: Quark–Antiquark and Four-Quark Components

A coupled-channel formalism incorporating mixing between conventional $P$-wave $(q\bar{q})$ and four-quark (tetraquark) components has been used to describe anomalous properties such as mass shifts and narrow widths:

$$|\psi\rangle = \alpha_i |q\bar{q}\rangle_i + \beta_j |qq\bar{q}\bar{q}\rangle_j$$

where the mixing term $V$ connects the Hilbert spaces corresponding to each component (0711.2359). Even moderate mixing ($\sim 30\%$ four-quark content) can lower the masses of positive-parity states (especially $B_{sJ}$) below strong decay thresholds. For vector states ($1^-$), the structure remains essentially conventional, dominated by $q\bar{q}$, and mass shifts are negligible.

For molecular (meson–meson bound state) scenarios, Bethe–Salpeter-equation-based approaches and local hidden gauge unitarization have generated the $J^P=0^+$, $1^+$, and $2^+$ states dynamically through $\rho B^*$ and $\omega B^*$ interactions (Soler et al., 2015). The $J=2$ member corresponds well to the experimental $B_2^*(5747)$, and the $J=1$ state to $B_1(5721)$. The absence of attractive dynamics in the $I=3/2$ channel precludes exotic "triquark" mesons.

3. Experimental Determination: Masses, Widths, and Decay Properties

Precision studies from collider experiments (CDF and D0 at the Tevatron) have measured the masses and widths of $B^{**}$ and $B_s^{**}$ states (0906.0808). The narrow states,

• $B_1^0$ and $B_2^{*0}$ (nonstrange),
• $B_{s1}^0$ and $B_{s2}^{*0}$ (strange),

are reconstructed via decays involving intermediate $B^*$ mesons. Owing to missing soft photons, mass distributions are shifted; measured values are:

State	Mass [MeV/$c^2$]	Width [MeV/$c^2$]
$B_1^0$	$5720.6 \pm 2.4$	$\sim$10
$B_2^{*0}$	$5746.8 \pm 2.4$	$\sim$10
$B_{s1}^0$	$5829.4 \pm 0.7$	$\sim$1
$B_{s2}^{*0}$	$5839.6 \pm 1.1$	$\sim$1

These widths confirm the expectation that D-wave decays are suppressed relative to S-wave, yielding narrow resonances.

Strong decay assignments follow the $^3P_0$ model, which predicts partial width ratios and selection rules: for example, the ratio $$\Gamma(B_2^*(5747) \to B\pi) / \Gamma(B_2^*(5747) \to B^*\pi) \approx 1.18$$ matches the measured $1.12$; observing forbidden decay channels can distinguish $2^{3}S_1$ from $2^{1}S_0$ assignments for higher $B_J$ states (Yu et al., 2019).

Electromagnetic decays are sensitive probes: mixing suppresses radiative widths, yielding ratios as large as $$\Gamma(1^+ \to 1^+\gamma)/\Gamma(1^+ \to 0^+\gamma)\sim 100$$ in mixed configurations, versus $\sim 1$ in the pure $q\bar{q}$ case (0711.2359).

4. Dynamical Generation and Molecular Interpretations

Bethe–Salpeter analyses in the hidden gauge approach (Soler et al., 2015) have shown that vector B meson states can emerge as molecular bound states,

• $J=0^+$, $1^+$, and $2^+$ in $I=1/2$ channels, with masses $M\sim 5812{-}5817$ MeV ($J=0^+,1^+$), $M\sim 5745$ MeV ($J=2^+$). Decays of the $J=2^+$ state are consistent with experimental $B_2^*(5747)$ width once box diagram contributions to the $\pi B$ decay channel are included; $J=1^+$ state width is essentially zero unless additional decay modes are present. The absence of molecular states in $I=3/2$ reflects the repulsive interaction in this sector.

Three-body extension studies (Deng et al., 6 Nov 2024, Garcilazo et al., 2018) utilize Faddeev equations and six-body Gaussian expansion methods to show that a compact tri-meson configuration,

$$\left[[\bar{B} \bar{B}^*]^1_0\,\bar{B}^*\right]^0_{1/2}$$

is bound $\sim$10 MeV below its constituent threshold, predominantly stabilized by $\sigma$-meson exchange, and exhibits strong spatial correlation analogous to pn pairs in nuclear physics.

5. Weak, Electromagnetic, and Strong Decays: Analytical Frameworks and Observables

For weak decays into vector or radially excited states, the improved Bethe–Salpeter method (Zhou et al., 2020) provides overlap integrals of Salpeter wave functions, elucidating the nodal structures that suppress transition amplitudes:

$I \sim \int d^3q\, \psi_B(q) \psi^*_{2S}(q)$

with cancellation across nodes leading to branching ratios $\sim 10^{-5}{-}10^{-4}$, two orders of magnitude below decays into ground-state vectors.

In quasi-two-body decays with intermediate vector resonances, the factorization-assisted topological amplitude (FAT) approach (Zhou et al., 2023, Ou-Yang et al., 17 Jun 2025) incorporates nonperturbative parameters and Breit–Wigner propagators:

$$L^{BW}(s) = \frac{1}{s - m_V^2 + i\, m_V\, \Gamma_V(s)}$$

This formalism yields branching fractions compatible with perturbative QCD and QCD factorization, with direct CP asymmetries predicted via the interference of amplitudes carrying different strong and weak phases.

Final-state interactions that dominate color-suppressed processes, notably $B \to \psi K$ with vector charmonium states (Yuan et al., 15 Apr 2025), highlight long-distance hadronic loop contributions; for $\psi(4160)$, large deviations from naive factorization indicate possible exotic content.

6. Multiquark and Three-Meson Bound States: Universality and Exceptions

In the context of three-body universality, effective field theory calculations (Lin et al., 2017) have ruled out universal three-meson B/B* bound states (Efimov effect) based on spin-isospin structure, even for meson molecules such as $Z_b(10610)$ and $Z'_b(10650)$, though the scattering lengths in some channels are extremely large ($\sim$99 fm), close to but insufficient for trimer formation.

The existence of a three B-meson bound state ($T_{bbb}$) (Garcilazo et al., 2018) requires a deeply-bound $T_{bb}$ tetraquark, with $T_{bbb}$ lying about 90 MeV below any possible three-meson threshold, possibly providing a benchmark for lattice QCD and a target for future experimental searches.

7. Vector B Mesons in Nuclear Medium

Studies of $B$ meson–nucleus bound states in the quark meson coupling model (Mondal et al., 29 Jul 2024) show B mesons (especially B$^-$) can form deeply bound states ($\sim$30–33 MeV binding) at the nuclear center, stabilized by scalar ($\sigma$, $\delta$) and vector ($\omega$, $\rho$) mean fields, and for the charged case, the Coulomb interaction. Such states are significantly more bound than analogous $K$ or $D$ mesons and may provide sensitive probes for in-medium modification of hadron properties in upcoming facilities (PANDA, J-PARC-E29, JLab).

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Collectively, the paper of three vector B meson states reveals a complex interplay of heavy-quark symmetry, potential quark–antiquark and multiquark configurations, molecular binding mechanisms, and rich experimental signatures in spectroscopy and decay observables. The convergence of theoretical models, precise experimental measurements, and advances in many-body quantum techniques continues to advance the characterization of these states and their role in the broader framework of hadron physics.