Bethe–Salpeter and Faddeev Bound-State Equations

A relativistic bound state appears as a pole of a connected higher-point function. Equating the pole residues in its integral equation produces a homogeneous Bethe–Salpeter equation for two constituents or a Faddeev equation for three. The kernel cannot be chosen independently of the constituent self-energy when a Ward identity is meant to hold. And a Euclidean eigenvalue crossing is a physical mass only when the required timelike momentum and singularity structure are under analytic control.

Required background. Schwinger–Dyson hierarchies supplies dressed propagators and vertices, while spectral decomposition of two-point functions supplies the pole interpretation. Helpful background. 2PI and nPI effective actions supplies functional kernel differentiation, while nuclear few-body EFT supplies complementary three-body organization.

Pole extraction and the homogeneous equation

Let $G_0(P)$ be the disconnected product of two full constituent propagators and $K(P)$ the two-particle-irreducible kernel. With signs absorbed into the definition of $K$, the scattering matrix obeys

$$T=K+KG_0T.$$

Near an isolated stable bound-state pole,

$$T(P) \underset{P^2\to-M^2}{\sim} \frac{\Gamma(P)\,\overline\Gamma(P)} {P^2+M^2} +T_{\mathrm{regular}}.$$

Equating residues gives

$$\Gamma=KG_0\Gamma.$$

For a fermion–antifermion state with relative momentum $k$,

$$\Gamma(k;P) =\int_q K(k,q;P)\, S(q_+)\Gamma(q;P)S(q_-),$$

where

$$q_+=q+\eta P, \qquad q_-=q-(1-\eta)P.$$

The momentum-partition parameter $\eta$ is arbitrary in an exact Poincaré-covariant treatment. Residual $\eta$ dependence diagnoses a regulator, basis, or truncation violation.

The original pole construction is developed in Bethe and Salpeter 1951, §§ 2–4.

Unit residue and canonical normalization

The homogeneous equation fixes the shape but not the scale of $\Gamma$. Unit residue of the four-point pole gives the canonical normalization

$$2P_\mu =\overline\Gamma\, \frac{\partial}{\partial P_\mu} \left[ G_0^{-1}(P)-K(P) \right] \Gamma \bigg|_{P^2=-M^2},$$

where products include momentum integration, indices, and constituent propagators. If $K$ depends on $P$, differentiating only the propagators is incomplete. Different normalizations of $T$, $\Gamma$, and the state move conventional factors, but the pole residue is invariant.

An inhomogeneous current vertex has the same pole. Its residue determines a decay constant or form-factor coupling. Thus a mass eigenvalue without canonical normalization cannot support a correctly normalized matrix element.

Kernel and self-energy must be related

Suppose the dressed fermion satisfies

$$S^{-1}=S_0^{-1}+\Sigma[S].$$

A symmetry-preserving two-body kernel is generated, with the corresponding index ordering and sign convention, by

$$K=-\frac{\delta\Sigma}{\delta S}.$$

This relation makes the variation of the self-energy under a symmetry transformation match the ladder of two-body interactions. For chiral symmetry, let $\Gamma_5$ denote the renormalized pseudoscalar vertex whose tree Dirac structure is $\gamma^5$, and suppress its flavor generator. In the standard Euclidean DSE convention, the axial Ward–Takahashi identity is

$$\begin{aligned} P_\mu\Gamma_{5\mu}(k;P) ={}& S^{-1}(k_+)i\gamma^5 +i\gamma^5S^{-1}(k_-) \\ &-2i m_R\Gamma_5(k;P). \end{aligned}$$

In the chiral limit, a self-energy with dynamical scalar dressing and a kernel related by $K=-\delta\Sigma/\delta S$ produce the pseudoscalar Goldstone pole consistently. Choosing a more elaborate propagator self-energy and an unrelated simple ladder kernel generally breaks this identity. Munczek derives the functional relation and Goldstone consequence Munczek 1995, §§ II–IV.

The relation does not determine every phenomenological detail: a consistent truncation can still omit transverse vertex structures or crossed kernels. It ensures that the specified identity is violated only at the declared higher order.

Euclidean eigenvalues versus physical poles

Numerically one often solves

$$\lambda_n(P^2)\Gamma_n =KG_0\Gamma_n$$

and locates $\lambda_n(P^2)=1$. In Euclidean space, a physical mass requires $P^2=-M^2$. The constituent momenta $q_\pm$ are then complex, so their propagator singularities can restrict the accessible mass domain.

Three situations must be distinguished:

1. the equation is solved directly at the required complex momenta and the contour avoids or accounts for singularities;
2. an analytic representation supplies a justified continuation to $P^2=-M^2$;
3. eigenvalues computed only for spacelike $P^2>0$ are extrapolated across an unknown analytic region.

Only the first two can establish a pole under their stated assumptions. A stable polynomial fit in the third case is not a unique analytic continuation. Thresholds and resonances require the appropriate sheet and cannot be identified by a real Euclidean eigenvalue alone.

Three-body Faddeev organization

For three constituents, decompose the amplitude into components in which one constituent is the spectator:

$$\Psi=\Psi_1+\Psi_2+\Psi_3.$$

With pair kernels $K_i$ acting in the complementary pair,

$$\Psi_i =K_iG_0 \sum_{j\ne i}\Psi_j,$$

when only pairwise irreducible kernels are retained. A genuine three-body irreducible kernel $K_{123}$ requires a separate, nonduplicated component or an explicitly chosen partition; inserting the same $K_{123}G_0\Psi$ term into every $\Psi_i$ would count it three times. Whether $K_{123}$ is retained, power-counting suppressed, or modeled must be stated. Pairwise contributions already generated through dressed propagators or vertices must not be inserted again.

The closure and validation map locates kernel construction and analytic access. The functional-method validation comparison records eigenpair residuals, normalization, branch, kernel variation, and constituent–kernel covariance.

Common pitfalls

Using a kernel unrelated to the self-energy. This can destroy the Ward identity and move a protected Goldstone state. Kernel sophistication does not compensate for inconsistency.

Differentiating only G₀ in the normalization. A momentum-dependent kernel contributes to the pole residue.

Calling λ = 1 at spacelike momentum a mass. The physical pole is timelike. Complex momenta, singularities, and the continuation model must be controlled.

Exercises

1. Insert the pole form of $T$ into $T=K+KG_0T$ and derive the homogeneous equation.

Solution

The regular $K$ term has no pole. The pole part of the right-hand side is

$$KG_0 \frac{\Gamma\overline\Gamma}{P^2+M^2}.$$

Matching its residue to the left-hand residue gives $\Gamma=KG_0\Gamma$. Matching the conjugate residue gives the left eigenvector equation for $\overline\Gamma$.

2. Why does varying an ansatz parameter in $\Sigma$ require varying $K$ coherently?

Solution

The kernel is a functional derivative of the self-energy when the Ward identity is to be preserved. A shared ansatz parameter therefore changes both the constituent propagator and the interaction kernel, with correlated effects on the eigenvalue. Varying only one side breaks the identity and underestimates covariance.

Continue

Coupled Propagator, Vertex, and Bound-State Systems organizes the shared inputs and covariance. Complex-Momentum, Spectral, and Real-Time Information treats analytic continuation and singularity access explicitly.

References

• Bethe, Hans A., and Edwin E. Salpeter. “A Relativistic Equation for Bound-State Problems.” Physical Review 84 (1951): 1232–1242. DOI.
• Munczek, H. J. “Dynamical Chiral Symmetry Breaking, Goldstone’s Theorem, and the Consistency of the Schwinger–Dyson and Bethe–Salpeter Equations.” Physical Review D 52 (1995): 4736–4740. DOI.