Three-Point QCD Sum Rules Overview

• Three-point QCD sum rules are nonperturbative tools that compute hadronic form factors and coupling constants from three-hadron vertices.
• They employ operator product expansion, QCD vacuum condensates, and double Borel transforms to match theoretical and phenomenological representations.
• Numerical analysis with Borel stability and continuum subtraction yields robust couplings essential for hadronic interaction models.

Three-point QCD sum rules are nonperturbative field-theoretical tools for computing hadronic form factors and strong coupling constants associated with three-hadron vertices, utilizing operator product expansion (OPE), QCD vacuum condensates, and Borel-transformed dispersion relations. In this framework, three-point correlation functions are constructed from appropriate interpolating currents for the hadrons of interest. The theoretical (QCD) side is expanded in terms of local operators up to a chosen dimension, and the phenomenological (hadronic) side models the physical resonances and continuum. Matching both sides after Borel transformation and continuum subtraction yields a sum rule for the desired form factor as a function of the squared momentum transfer. Numerical analysis and parameter fitting enable the extraction of quantitative couplings relevant for hadron interactions in nuclear and hadronic matter.

1. Formulation of the Three-Point Correlation Function

A three-point QCD sum rule for a vertex $A\to B+C$ is based on the correlator

$$\Pi(x, y) = i^2 \int d^4x\,d^4y\; e^{-ip\cdot x} e^{ip'\cdot y} \langle 0 | T\{ J_B(y) J_C(0) \bar{J}_A(x) \} | 0 \rangle,$$

where $J_{A,B,C}$ are local interpolating currents matching the quantum numbers of the external hadrons. For instance, in the paper of the $\Sigma_b N B$ vertex, the currents are

$$\begin{aligned} J_{\Sigma_b}(x) &= \epsilon_{ijk}\,[u^{i T}(x) C \gamma_\mu d^j(x)] \gamma_5 \gamma^\mu b^k(x), \ J_N(y) &= \epsilon_{ijk}\,[u^{i T}(y) C \gamma_\mu u^j(y)] \gamma_5 \gamma^\mu d^k(y), \ J_B(0) &= \bar u(0) \gamma_5 b(0), \end{aligned}$$

with $p$ for $\Sigma_b$, $p'$ for $N$, $q$ for $B$, and $p = p' + q$ (Yu et al., 2016).

2. Phenomenological Side and Hadronic Saturation

On the hadronic side, the correlator is saturated by complete sets of hadronic states. The ground-state pole contribution gives a double (or triple) pole structure in external momenta. For each current, the overlap with physical states is parameterized: $$\begin{aligned} \langle 0 | J_N | N(p') \rangle &= \lambda_N u_N(p'), \ \langle 0 | J_B | B(q) \rangle &= i \frac{m_B^2 f_B}{m_u + m_b}, \ \langle \Sigma_b(p) | \bar{J}_{\Sigma_b} | 0 \rangle &= \lambda_{\Sigma_b} \bar{u}_{\Sigma_b}(p), \end{aligned}$$ and the vertex matrix element is parameterized by the strong form factor: $$\langle N(p') B(q) | \Sigma_b(p) \rangle = G_{\Sigma_b N B}(Q^2) \, \bar{u}_N(p') i\gamma_5 u_{\Sigma_b}(p).$$ This leads to the hadronic representation: $$\Pi^{\rm HAD}(p^2, p'^2, q^2) \approx \frac{\lambda_N \lambda_{\Sigma_b} i m_B^2 f_B}{m_u + m_b} \frac{G_{\Sigma_b N B}(Q^2)}{(p^2 - m_{\Sigma_b}^2)(p'^2 - m_N^2)(q^2 - m_B^2)} + \dots$$ after appropriate Dirac projections.

3. QCD Representation: Operator Product Expansion

On the OPE side, all fields are Wick-contracted, and full quark propagators are used:

• For heavy quarks:

$$S_Q(x) = \int \frac{d^4k}{(2\pi)^4} e^{-ik\cdot x} \left[ \frac{\not{k} + m_Q}{k^2 - m_Q^2} - \frac{g_s G_{\alpha\beta} \sigma^{\alpha\beta} (\not{k} + m_Q)}{(k^2 - m_Q^2)^2} + \dots \right]$$

• For light quarks:

$$S_q(x) = i\not{x}/(2\pi^2 x^4) - m_q/(4\pi^2 x^2) - \langle \bar{q} q \rangle/12 (1 - i m_q \not{x}/4) - x^2 m_0^2 \langle \bar{q} q \rangle/192 (1 - i m_q \not{x}/6) + \dots$$

After Fourier transformation and Feynman parametrization, one arrives at double dispersion representations of the form: $$\Pi_i^{\rm OPE}(p^2, p'^2, Q^2) = \int ds\, ds' \frac{ \rho_i^{\rm pert}(s,s',Q^2) + \sum_{d} \rho_i^{(d)}(s,s',Q^2) }{(s - p^2)(s' - p'^2)}$$ with $d$ running over condensate contributions up to dimension 5 ($\langle \bar{q}q \rangle$, $\langle G^2\rangle$, $\langle \bar{q}g_s \sigma G q \rangle$). Explicit spectral densities are provided for the relevant Dirac structures [(Yu et al., 2016), see Eqs. (15)-(19)].

4. Double Borel Transformation and Continuum Subtraction

To suppress higher states and enhance ground-state contributions, a double Borel transform is performed: $$\mathcal{B}_{p^2 \to M_1^2} \mathcal{B}_{p'^2 \to M_2^2} \left[ \frac{1}{(s - p^2)} \right] = e^{-s/M_1^2}$$ Applying this to both sides yields for the OPE side: $$\Pi_i^{\rm OPE}(M_1^2, M_2^2, Q^2) = \int_{s_{\rm min}}^{s_0} ds \int_{u_{\rm min}}^{u_0} ds' \, e^{-s/M_1^2 - s'/M_2^2} [ \rho_i^{\rm pert} + \rho_i^{\rm non-pert} ]$$ Here, $s_0$, $u_0$ are chosen as effective continuum thresholds (typically $(m_{\text{hadron}}+0.5\,\text{GeV})^2$) to effect quark–hadron duality.

5. Matching, Extraction of the Form Factor, and Coupling

The sum rule for the form factor follows by equating the Borelized hadronic and OPE representations for a fixed Lorentz structure: $$\lambda_N \lambda_{\Sigma_b} \frac{i m_B^2 f_B}{m_u + m_b} G_{\Sigma_b N B}(Q^2) e^{-m_{\Sigma_b}^2 / M_1^2} e^{-m_N^2 / M_2^2} = \text{RHS}_\text{OPE}(M_1^2, M_2^2, Q^2)$$ Solving for $G_{\Sigma_b N B}(Q^2)$ gives the sum rule.

Numerically, the form factor is computed in the space-like region $Q^2 \gtrsim 1$ GeV$^2$ over a discretized grid. These points are fitted to an analytical ansatz: $$G(Q^2) = C_1 \exp\left(-\frac{Q^2}{C_2 + C_3 e^{-Q^2/C_4}}\right)$$ with parameters $C_i$ extracted from the fit [(Yu et al., 2016), Eq. (21) and Table 1]. The physical on-shell coupling is defined at $Q^2 = -m_B^2$, with final values: $$G_{\Sigma_b N B} = 0.43 \pm 0.01~\text{GeV}^{-1}, \qquad G_{\Sigma_c N D} = 3.76 \pm 0.05~\text{GeV}^{-1}.$$

6. Numerical Analysis, Borel Stability, and Theoretical Inputs

The stability and reliability of the extraction are ensured as follows:

• Borel windows: For $\Sigma_b N B$, $M_1^2 \in [7,14]$ GeV$^2$, $M_2^2 \in [3,7]$ GeV$^2$; for $\Sigma_c N D$, $M_1^2 \in [3,7]$ GeV$^2$, $M_2^2 \in [2,6]$ GeV$^2$.
• Input condensates and parameters:

$$\langle \bar{q}q \rangle = - (0.24~\text{GeV})^3,~~ \langle \alpha_s G^2/\pi \rangle = 0.022~\text{GeV}^4,~~ m_0^2 = 0.8~\text{GeV}^2,$$

$$f_B = 248~\text{MeV},~~ f_D = 205.8~\text{MeV},~~ \lambda_N=1.1 \times 10^{-3}~\text{GeV}^6,~~ \lambda_{\Sigma_b}=0.062~\text{GeV}^3,~~ \lambda_{\Sigma_c}=0.045~\text{GeV}^3.$$

• Pole dominance and OPE convergence are confirmed within these windows (Figs. 1–4 in (Yu et al., 2016)), indicating a pronounced Borel plateau and small dependence of $G$ on Borel parameters.

7. Physical Interpretation and Applications

The couplings $G_{\Sigma_b N B}$ and $G_{\Sigma_c N D}$ are essential inputs for hadronic models of $B$ and $D$ meson–nucleon dynamics, with implications for nuclear matter effects, heavy-ion collisions, and experiments such as PANDA. The control over systematic uncertainties stems from careful choice of Borel windows, input parameters, continuum thresholds, and robust fitting. Comparison with existing models and sum rule calculations shows the results are compatible within nominal uncertainties (Yu et al., 2016).

The methodology is general and applies to a broad class of hadronic vertices, with the overall workflow summarized as:

• Formulate the three-point correlator using suitable interpolating currents.
• Compute the OPE to the desired dimension (here, up to dimension-5 condensates).
• Model the phenomenological side and isolate the ground-state contribution.
• Execute a double Borel transform and subtract continuum duality contributions.
• Match QCD and hadronic representations for the targeted Lorentz structure.
• Fit the computed form factor and extrapolate to the relevant kinematic point for the coupling.
• Quantify uncertainties from all sources.

This approach is the standard for extracting nonperturbative hadron–hadron couplings where experimental data or lattice QCD calculations are challenging or unavailable.