Heavy Quark Spin Symmetry in QCD

• Heavy Quark Spin Symmetry is an approximate QCD symmetry that decouples heavy quark spins from light quark dynamics when mQ is much larger than ΛQCD.
• HQSS constrains the hadron spectrum by grouping states into doublets and singlets and simplifies the treatment of multi-quark systems by reducing independent parameters.
• This symmetry underpins precision modeling of heavy baryon decays and aids in extracting CKM parameters by reducing complex form factors to a universal function.

Heavy Quark Spin Symmetry (HQSS) is an approximate symmetry of Quantum Chromodynamics (QCD) that arises in the limit where the mass of a heavy quark ($m_Q$) is much larger than the characteristic QCD scale ($\Lambda_{\mathrm{QCD}}$) and the masses of light quarks. The central consequence is that spin–dependent interactions involving the heavy quark are suppressed by $1/m_Q$, causing the spin orientation of the heavy quark to decouple from the dynamics of the light degrees of freedom. This decoupling has far-reaching implications across hadron spectroscopy, electroweak decays, the structure of effective field theories, and the extraction of Standard Model parameters from experimental data.

1. Theoretical Foundations and Decoupling Structure

The essential underlying observation of HQSS is that for $m_Q \gg \Lambda_{\mathrm{QCD}}$, chromomagnetic interactions—responsible for spin–spin forces—become negligible, and the light cloud (the light quarks and gluons) evolves independently of the heavy quark spin. In the $m_Q \to \infty$ limit, the dynamics of the “light degrees of freedom” are characterized by a definite total angular momentum

$S_{\text{light}} = J - S_Q,$

where $J$ is the total hadron spin and $S_Q$ is the heavy quark spin. For a given $S_{\text{light}}$, hadrons will appear in doublets—with total spins $J = S_{\text{light}} \pm 1/2$—or singlets if $S_{\text{light}} = 0$.

When multiple heavy quarks are present, as in doubly or triply heavy baryons, HQSS predicts an even higher degree of decoupling: the total spin of the heavy subsystem ($S_h$) becomes a well-defined quantum number, and the individual heavy quark spins decouple from both each other and from the light degrees of freedom.

This symmetry underpins both the spectroscopic pattern and dynamical processes involving heavy hadrons.

2. Constraints on Non-Relativistic Quark Model Wave Functions

Non-relativistic constituent quark models (NRCQM) describe hadrons as bound states of constituent quarks interacting via spin-dependent and confining potentials. In the absence of HQSS, multi-component wave functions must consider all allowed spin couplings. However, HQSS imposes rigorous restrictions:

• For $\Lambda_Q$-type baryons ($I=0$, $S_{\text{light}}=0$), the HQSS-allowed component of the wave function is

$$|\Lambda_Q; J = 1/2, M_J\rangle = \{ |00\rangle_I \otimes |00\rangle_{\text{light}} \} \Psi_{ll}^{\Lambda_Q}(r_1, r_2, r_{12}) \otimes |Q; M_J\rangle,$$

with the spatial function symmetric in $r_1 \leftrightarrow r_2$.

• Components with $S_{\text{light}}=1$, which would be allowed under broader flavor symmetries, are forbidden by HQSS—such terms are suppressed by $1/m_Q$.
• The spatial part commonly employs a Jastrow-type ansatz:

$$\Psi_{qq'}^{B_Q}(r_1, r_2, r_{12}) = F^{B_Q}(r_{12}) \phi_q^Q(r_1) \phi_{q'}^Q(r_2),$$

with $F^{B_Q}(r_{12})$ set to model light-quark correlations and $\phi_q^Q(r)$ for heavy–light binding.

As a result, HQSS drastically reduces the number of independent variational parameters and fundamentally simplifies three-body baryon problems.

3. Universal Form Factors and Electroweak Decay Structure

HQSS specifically constrains matrix elements for electroweak transitions—such as $b\to c$ semileptonic decays of heavy baryons—by relating seemingly independent Lorentz structures to a small set of universal functions. For transitions near zero recoil and in the heavy quark limit,

$$\langle B_{c'}, r', p' | \bar{c} \gamma^\mu (1-\gamma^5) b | B_b, r, p \rangle \propto \bar{u}^{B_{c'}}_{r'}(p') \left\{ \gamma^\mu [F_1(w) - \gamma^5 G_1(w)] + v^\mu [F_2(w) - \gamma^5 G_2(w)] + v'^\mu [F_3(w) - \gamma^5 G_3(w)] \right\} u^{B_b}_r(p),$$

with $w = v\cdot v' \simeq 1$ at zero recoil. In the strict heavy quark limit, all 12 independent form factors reduce to a single universal function.

For doubly heavy baryons, for instance in $$\Xi_{bc}^{(\prime,*)} \rightarrow \Xi_{cc}^{(*)} \ell \bar{\nu}$$, all decay matrix elements can be expressed in terms of a unique Isgur–Wise function. This simplification is critical for the extraction of CKM parameters such as $|V_{cb}|$, and for precise calculation of decay rates and angular asymmetries. Corrections to these relations appear at $\mathcal{O}(\Lambda_{\mathrm{QCD}}/m_Q)$.

4. Hyperfine Mixing Effects and Physical State Structure

HQSS significantly impacts the interpretation of physical bc baryons. In the “$S_h$-basis,” states are classified by the spin of the two heavy quarks. However, the actual bc baryon eigenstates undergo hyperfine mixing due to non-negligible spin–spin interactions between heavy and light quarks, proportional to $1/m_Q$:

$$\begin{aligned} \hat{\Xi}_{bc} &= (\sqrt{3}/2) \Xi'_{bc} + (1/2) \Xi_{bc}, \ \hat{\Xi}'_{bc} &= - (1/2) \Xi'_{bc} + (\sqrt{3}/2) \Xi_{bc}. \end{aligned}$$

This mixing modifies decay widths. HQSS predicts that the ratio

$$R_1^\text{(phys)} = \frac{\Gamma(\Xi_{bc}^{(2)} \to \Xi_{cc}^*)}{\Gamma(\Xi_{bc}^{(1)} \to \Xi_{cc}^*)} \approx \tan^2 \theta$$

(where $\theta$ is the mixing angle between the “ideal” and physical states), provides a near model-independent measure of hyperfine mixing. Analogous sensitivity manifests in flavor-conserving electromagnetic decays, though further complicated by phase-space effects from small mass splittings.

5. Extension to Triply Heavy Baryons and Confinement Dynamics

In triply heavy baryons ($ccc$, $bcc$, $bbc$, $bbb$), HQSS manifests as separate spin symmetries for each heavy flavor. The matrix elements for semileptonic decays such as $ccb\to ccc$ can be parametrized by a single function (Isgur–Wise), and the relative weight of all Dirac/Lorentz structures is determined uniquely. For instance:

$$\langle \Omega^*_{ccc}, v, k, r' | j^\mu(0) | \Xi^{(*)}_{ccb}, v, r \rangle = 2 \eta(w) \bar{u'}^\mu u + \ldots$$

with $\eta(1) = 1$ in the heavy quark limit.

The form of the confinement potential in these systems is also relevant. Lattice QCD motivates the use of a genuine three-body ("Y-shaped") flux-tube potential

$V_{\rm conf}^{(Y)} = \sigma L_{\rm min},$

whereas traditional models use a sum of two-body potentials ("$\Delta$-shaped"). Replacement with the Y-shaped potential leads to moderate mass shifts (20–50 MeV), potentially impacting precise spectroscopy and decay widths. This refinement is necessary for detailed phenomenological predictions and tests the interplay between HQSS constraints and nonperturbative confinement physics (Flynn et al., 2011).

6. Implications for Hadron Spectroscopy, Dynamics, and Experiment

HQSS imparts a predictive pattern to the heavy baryon spectrum—doublets (for $S_{\text{light}} \neq 0$) and singlets (for $S_{\text{light}} = 0$), nearly degenerate in the heavy quark limit. For $b\to c$ semileptonic decays, the HQSS-imposed relations among form factors drastically reduce the number of independent theoretical ingredients, facilitating the extraction of Standard Model parameters and enabling cross-validation with lattice QCD and experiment.

Measurement of decay width ratios sensitive to hyperfine mixing provides experimental access to mixing angles and heavy quark mass corrections. The overall approach unifies the treatment of single, doubly, and triply heavy baryon systems within a symmetry-constrained framework.

7. Summary Table: Core HQSS Constraints in Heavy Baryons

System	HQSS Constraint	Consequence / Observable
$\Lambda_Q$ ($S_{\text{light}}=0$)	Only singlet light-quark structure allowed	Single allowed wave function, see Eq.(1)
Doubly heavy baryons ($bc$, $cc$)	$S_h$ is a good quantum number in $m_Q\to\infty$	Predicts hierarchy of decay rates; test via $R_1^{\rm (phys)}$
Heavy baryon decays ($b\to c$)	All form factors reduce to a single universal function	Precision in $|V_{cb}|$ extractions
Triply heavy baryons	Matrix elements per transition class tied to one form factor	Model-independent decay width ratios
Mixings (hyperfine, $bc$ baryons)	Physical states are rotated admixtures; HQSS predicts dependence of decay widths on mixing angle	Direct handle on hyperfine-induced effects

HQSS thus imposes a network of stringent constraints, organizing spectroscopic and decay properties while enabling quantitative connections between theoretical models, lattice simulations, and experimental observables. Its predictive reach remains central in contemporary heavy-flavor hadron physics and in the program of precision Standard Model tests.