Coupled Propagator, Vertex, and Bound-State Systems

A propagator, vertex, and bound-state calculation is one coupled inference problem. The same renormalized inputs and vertex structures can enter a self-energy, a Ward or Slavnov–Taylor identity, a two-body kernel, and the final spectrum. Solving each equation with a separately convenient ansatz can produce individually smooth curves that are mutually inconsistent. The dependency order and shared covariance must therefore be part of the result.

Required background. Closure, symmetry, and branch selection supplies ansatz and branch records, while Bethe–Salpeter and Faddeev equations supplies the pole kernel. Helpful background. Functional-RG applications supplies an alternative flow-based closure and variation sequence.

Unknowns and dependencies

Collect all unknown functions into

$$\mathcal U =\left( D_{\mathrm{gh}}, D_{\mathrm{g}}, S, \Gamma_{\mathrm{ghg}}, \Gamma_{\mathrm{qg}}, K_{\mathrm{BS}}, \Gamma_{\mathrm{BS}} \right).$$

A coupled system has the schematic form

$$\mathcal R_i[\mathcal U;\alpha]=0,$$

where $\alpha$ contains renormalized masses and couplings, gauge and regulator choices, ansatz parameters, and boundary data. Dependencies are not necessarily acyclic: the quark–gluon vertex affects the quark propagator, which enters the vertex equation again.

A useful solution order is:

1. fix regulator, renormalization scheme, momentum routing, and tensor bases;
2. solve a symmetry-closed gauge or matter propagator–vertex subsystem;
3. construct $K_{\mathrm{BS}}=-\delta\Sigma/\delta S$ or its declared approximation from the same retained structures, following the symmetry-preserving relation of Munczek 1995, §§ II–IV;
4. solve the bound-state eigenproblem and canonical normalization;
5. feed any required vertex or polarization response back and repeat to a joint residual tolerance;
6. vary the closure coherently and propagate the resulting changes to every derived observable.

If backreaction is neglected, that is an approximation edge in this dependency graph, not an implementation detail.

Shared renormalization conditions

At a spacelike subtraction point represented by Euclidean momentum $p_E^2=\mu^2$, a fermion example may impose

$$\left.S_R^{-1}(p_E)\right|_{p_E^2=\mu^2} =i\gamma_E^\mu p_{E\mu}+m_R,$$

while gauge and ghost dressings are normalized separately. Vertex renormalization constants are linked to wavefunction and coupling renormalization by Ward or Slavnov–Taylor identities. Using propagators renormalized in one scheme with a vertex model fitted in another requires an explicit conversion.

Cutoff removal must be checked for the whole coupled system. A counterterm that removes a propagator divergence can alter the functional derivative that defines a bound-state kernel. Refitting a derived kernel after every cutoff change can hide rather than remove this inconsistency.

Joint residual and branch tests

Define dimensionless component residuals

$$r_i =\frac{\lVert\mathcal R_i[\mathcal U]\rVert_i} {\epsilon_i+\lVert\mathcal U_i\rVert_i},$$

with documented norms, momentum weights, and floors $\epsilon_i$. A joint stopping condition is

$$\max_i r_i<\varepsilon_{\mathrm{solve}}.$$

Monitoring only the last-updated equation can miss a limit cycle in another subsystem. Useful checks include:

• forward and reverse sweeps through all coupled equations;
• direct substitution after interpolation between grids;
• Jacobian or linear-response tests near a converged solution;
• parameter homotopy from a controlled weak-coupling or massive branch;
• multiple initial seeds and algorithms.

If two branches satisfy the same residual threshold, numerical convergence has not selected the physical one. Symmetry, positivity where applicable, action or thermodynamic information, and external data remain necessary.

Coherent kernel variation

Suppose a vertex parameter $\alpha_v$ changes the fermion self-energy:

$$\Sigma=\Sigma[S,\Gamma_{\mathrm{qg}}(\alpha_v)].$$

Then a symmetry-related kernel changes as

$$\frac{\mathrm dK}{\mathrm d\alpha_v} =-\frac{\mathrm d}{\mathrm d\alpha_v} \frac{\delta\Sigma}{\delta S}.$$

Varying the propagator solution while freezing $K$ breaks this shared dependence. Conversely, varying only the kernel double counts an independent uncertainty that is actually correlated with the constituent dressing. Symmetry-preserving truncation schemes make this relation explicit Bender, Roberts, and von Smekal 1996, §§ 2–4.

Propagating covariance

Let observables $y_a$ depend on inputs $\alpha_i$ through the re-solved coupled system. Linear propagation uses the total response

$$J_{ai} =\frac{\mathrm dy_a}{\mathrm d\alpha_i}, \qquad C_y =JC_\alpha J^{\mathsf T}.$$

Implicit differentiation of $\mathcal R[\mathcal U(\alpha),\alpha]=0$ gives

$$\frac{\mathrm d\mathcal U}{\mathrm d\alpha_i} =- \left( \frac{\partial\mathcal R}{\partial\mathcal U} \right)^{-1} \frac{\partial\mathcal R}{\partial\alpha_i}.$$

Finite differences must re-solve the whole system and follow the same branch. One-at-a-time envelope variation is useful for sensitivity screening but does not reconstruct covariance when parameters were fitted together or one structure enters several equations.

Truncation changes are often discrete rather than statistical. Their spread should be reported separately from $C_y$ unless a probabilistic model for the closure family is justified.

A coupled gauge-sector and meson example

In a symmetry-preserving pseudoscalar calculation:

• the gauge and ghost sector supply the running interaction;
• the quark equation produces $S$ with scalar and vector dressings;
• the quark–gluon vertex ansatz enters $\Sigma$;
• differentiating the same $\Sigma$ constructs the meson kernel;
• the axial Ward identity tests their consistency;
• the Bethe–Salpeter eigenvalue and canonical normalization produce the mass and decay amplitude.

A vertex change that shifts the quark mass function and the meson kernel in opposite directions can leave the meson mass deceptively stable. That cancellation is correlated sensitivity, not evidence that each input is accurate. Reporting the mass alone hides it.

The functional-equation closure and validation map displays the full inference sequence. The functional-method validation comparison requires joint solver checks, coherent truncation variation, independent benchmarks, and covariance.

Common pitfalls

Mixing kernels from different closures. A propagator and bound-state kernel can each look reasonable while violating their shared Ward identity.

Freezing upstream uncertainties. A derived mass inherits propagator, vertex, renormalization, continuation, and solver uncertainties with covariance.

Treating cancellation as validation. Stability of one output can result from compensating errors. Held-out observables and component responses are needed.

Exercises

1. Derive the implicit-response equation for $\mathrm d\mathcal U/\mathrm d\alpha_i$.

Solution

Differentiate $\mathcal R(\mathcal U(\alpha),\alpha)=0$:

$$\frac{\partial\mathcal R}{\partial\mathcal U} \frac{\mathrm d\mathcal U}{\mathrm d\alpha_i} +\frac{\partial\mathcal R}{\partial\alpha_i} =0.$$

If the Jacobian is invertible on the selected branch, multiply by its inverse to obtain the displayed formula. A nearly singular Jacobian signals large sensitivity or a branch turning point.

2. Two fitted vertex parameters have correlation coefficient $-0.9$. Why is adding their separate mass shifts in quadrature unreliable?

Solution

The covariance contribution contains $2J_1J_2C_{12}$. With correlation $-0.9$, this cross term can nearly cancel or enhance the diagonal terms depending on the response signs. Quadrature assumes zero covariance and therefore misstates the propagated parameter uncertainty.

Continue

Complex-Momentum, Spectral, and Real-Time Information adds continuation uncertainty to the dependency system. Functional-Method Validation and Error Control turns the component responses into a bounded claim.

References

• Bender, A., C. D. Roberts, and L. von Smekal. “Goldstone Theorem and Diquark Confinement Beyond Rainbow–Ladder Approximation.” Physics Letters B 380 (1996): 7–12. 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.