Nakanishi Integral Representation (NIR)

• Nakanishi Integral Representation is a covariant framework that expresses multi-leg quantum-field amplitudes using smooth, real-valued weight functions integrated over auxiliary parameters.
• It converts complex singular 4D integral equations into tractable lower-dimensional Fredholm problems, enabling direct solutions of bound-state and scattering equations in Minkowski space.
• Practical applications include relativistic bound states, light-front wave functions, and hadron structure analyses, with its results in close agreement with Euclidean methods.

The Nakanishi Integral Representation (NIR) is a covariant and analytic framework for expressing multi-leg quantum-field-theoretic amplitudes—most notably the Bethe–Salpeter amplitude for relativistic bound and scattering states—in Minkowski space. It encodes all external-momentum dependence in a universal denominator structure, relegating the nontrivial nonperturbative dynamics to smooth, real-valued "Nakanishi weight functions" integrated over auxiliary parameters. This approach provides a powerful route to solving bound-state and gap equations directly in the physical (Minkowski) domain, circumventing the singularities that obstruct naive direct Minkowski-space solution of such equations. Canonical areas of application include relativistic two-body bound and scattering problems, covariant formulations of hadron structure, and the calculation of light-front wave functions, form factors, and distribution amplitudes.

1. Mathematical Formulation and Structural Properties

The fundamental result of NIR is that a wide class of $n$-point amplitudes $F(p_i)$ can be expressed in the form

$$F(p_i) = \int\![d\alpha]\, \frac{\Phi(\alpha)}{\left[ \sum_i \alpha_i f_i(p) \right]^N},$$

where $\Phi(\alpha)$ is the Nakanishi weight function, $f_i(p)$ are linear functions of the external momenta, $[d\alpha]$ comprises Feynman-parameter integrations (with the constraint $\sum_i \alpha_i = 1$), and $N$ is determined by the number of propagators in the relevant Feynman graph. For Bethe–Salpeter amplitudes of two scalar particles, the homogeneous case reduces to

$$\Phi(k,p) = \int_{-1}^{1} dz \int_{0}^{\infty} d\gamma \; \frac{g(\gamma,z)}{\left[\gamma + k^2 + z\,k \cdot p - \kappa^2 + i\epsilon\right]^n}$$

with $g(\gamma,z)$ the real-valued Nakanishi weight function, $\kappa^2$ parametrizing the bound- or scattering-state energy, and $n$ set by the number of denominators (e.g., $n=3$ for three propagators) (Karmanov et al., 2018, Carbonell et al., 2017). This structure persists for fermionic systems, where the Dirac decomposition yields several such NIRs, each with different powers and operator content (Jia, 2 Feb 2024, Salme' et al., 2017).

For scattering amplitudes and multi-point functions, an analogous representation applies, but with the support of $g(\gamma,z)$ adjusted to allow virtualities appropriate for the continuum (e.g., $\kappa^2\leq 0$ for scattering states) (Frederico et al., 2011). For dressed propagators (self-energies), the NIR is equivalent to a Källén–Lehmann spectral representation, with dispersion integrals over a positive variable $s$ and real weight functions $\rho(s)$ (Mezrag et al., 2020).

The key analytical property is the uniqueness of the weight function $\Phi(\alpha)$, provided all kinematic dependence has been factored, as guaranteed by Nakanishi's theorem. This property ensures that different formulations (e.g., direct Minkowski, light-front, or dispersion-based) yield the same physical content (Frederico et al., 2013).

2. Reduction of Four-Dimensional Integral Equations

Applying NIR to the Bethe–Salpeter equation (BSE) involves several systematic steps:

1. Ansatz: The BSE amplitude is rewritten in Nakanishi form, with the original amplitude expressed as an integral over $g(\gamma,z)$.
2. Substitution: The NIR is substituted into both sides of the BSE, and kernel insertions (e.g., ladder, cross-ladder) are mapped onto the NIR denominator (Carbonell et al., 2017, Karmanov, 2021).
3. Light-Front Projection: Singular integral equations in $4D$ Minkowski space are rendered tractable by integrating over the "minus" light-front component ($k^-$), yielding a manifestly finite $2$-dimensional integral equation for $g(\gamma,z)$ (Gigante et al., 2017, Salme' et al., 2017).
4. Stieltjes Transform Inversion: When the left-hand side has a generalized Stieltjes structure in $\gamma$, the inverse transform is used to cast the equation in canonical form:

$$g(\gamma,z) = \int_{-1}^{1} dz' \int_{0}^{\infty} d\gamma' \; N(\gamma,z;\gamma',z') g(\gamma',z')$$

with the kernel $N$ determined analytically from the original kernel via a contour integral in the complex $\gamma$-plane (Karmanov et al., 2018, Carbonell et al., 2017, Karmanov, 2021).

1. Basis Expansion and Solution: In numerical work, $g(\gamma,z)$ is expanded on a basis of orthogonal polynomials (e.g., Laguerre in $\gamma$, Gegenbauer in $z$), converting the problem into a finite-dimensional matrix eigenproblem (Gigante et al., 2017, Frederico et al., 2013).

This strategy generalizes to systems with internal spin, momentum-dependent kernels, and inhomogeneous equations (for scattering), leading to coupled integral equations for multi-component weight functions (Jia, 2 Feb 2024, Sauli, 2019).

3. Construction and Uniqueness of the Nakanishi Kernel

A critical step is explicit construction of $N(\gamma,z;\gamma',z')$, the kernel of the canonical NIR equation. For ladder kernel models, closed analytic real-valued expressions—free of branch ambiguities—are obtained by combining change of variables in Feynman parameters with real-contour evaluations (Karmanov, 2021): $$N(\gamma, z; \gamma', z') = \int_0^1 dv \, n(\gamma, z; \gamma', z'; v)$$ where $n$ involves only rational functions and elementary integrals. In generalizations to cross-ladder or more complex irreducible kernels, all integrations can be performed in strictly real variables.

The NIR framework accommodates all physically relevant singularities in the denominator, factorizing them into a single analytic structure and relegating dynamical complexity to the smooth weight $g$. The regularity and support of $g(\gamma,z)$ are fixed by the analytic structure of the underlying field theory and by physical boundary conditions (normalization, support at $z=\pm 1$), and are preserved under basis expansion (Karmanov et al., 2018, Carbonell et al., 2017, Karmanov, 2021, Moita et al., 2022).

For the fermionic case (e.g., pseudoscalar mesons or quark–photon vertices), the kernel becomes a matrix $K_{ij}(\gamma, z; \gamma', z')$ that couples the independent Dirac structures. An explicit analytic construction connects the NIR of the BSE to the NIR of the bound-state wave function via functionals $\mathbb{M},\mathbb{B}$, yielding the closed eigenvalue problem $\phi = \lambda K \phi$ with all analytic kernels known (Jia, 2 Feb 2024, Salme' et al., 2017, Sauli, 2019).

4. Applications: Bound, Scattering, and Vertex Functions

The NIR has been implemented in a wide range of contexts:

• Relativistic Bound States: Direct solution of the Minkowski-space BSE for scalar and fermionic two-body systems, including ladder and cross-ladder kernels, excited states, and systems in lower spacetime dimensions (2+1) (Gigante et al., 2017, Pimentel et al., 2017, Frederico et al., 2013).
• Light-Front Wave Functions: The same Nakanishi weight $g(\gamma,z)$ generates valence-sector light-front (LF) wave functions via one-dimensional integral projections. All LF observables (momentum distributions, form factors) are then determined by $g$ (Carbonell et al., 2017, Moita et al., 2022, Melo et al., 2019).
• Scattering States: Implementation of the inhomogeneous BSE for two-body scattering amplitudes and calculation of baseline quantities such as the scattering length in the zero-energy and massless-exchange limits (Frederico et al., 2011).
• Dressed Self-Energies (Gap Equations): For propagators, the NIR yields Källén–Lehmann representations with real spectral densities $\rho(s)$, leading to tractable coupled integral equations for dynamical mass and wave function renormalizations, benchmarked against perturbative results (Mezrag et al., 2020).
• Covariant Hadron Structure: Applications to pion Bethe–Salpeter amplitudes and GPDs employ the NIR both for covariant model-building and for extraction of partonic light-front amplitudes, with systematic enforcement of chiral and symmetry constraints (Chouika et al., 2017, Moita et al., 2022, Moita et al., 2022).
• Quark-Photon Vertex: Nonperturbative vertex functions and their kernels can be cast into NIR form, yielding a closed set of integro-differential equations for the relevant weight functions (Sauli, 2019).

The NIR delivers direct Minkowski-space access to real-time properties, distribution amplitudes, and observables that are inaccessible in Euclidean formulations.

5. Physical Interpretation and Computational Strategy

The Nakanishi weight function $g(\gamma, z)$ acts as a two-dimensional spectral density encoding the dynamical content of the amplitude; $\gamma$ parameterizes the off-shell invariant mass, while $z$ acts as a generalized cosine angle or light-front variable. The NIR translates singular four-dimensional integral equations, intractable in Minkowski space due to the complex analytic structure, into smooth two-dimensional (or higher for multi-leg) Fredholm equations in real variables.

Numerically, the canonical approach is:

• Expand $g(\gamma,z)$ in a suitable double basis (Laguerre–Gegenbauer);
• Project the integral equation onto this basis, converting it to a generalized eigenvalue problem;
• Scan the eigenvalue parameter (typically the squared coupling constant or binding energy) to locate the physical spectrum;
• Use the resulting $g$ to compute all covariant and light-front observables.

Perfect agreement—within high numerical precision—between NIR-based Minkowski-space solutions and direct Euclidean-space (Wick-rotated) computations validates the consistency of the NIR formalism, both in the scalar and the fermionic (multi-component) sectors (Gigante et al., 2017, Frederico et al., 2013).

6. Extensions, Limitations, and Outlook

NIR is not limited to ladder approximation: explicit analytic construction of $N$ for arbitrary irreducible kernels (cross-ladder, vertex corrections, multi-boson exchange) is feasible and has been demonstrated (Karmanov, 2021). For massless theories and in the Wick–Cutkosky model, the NIR can accommodate distributional solutions in $\gamma$, highlighting its flexibility (Pimentel et al., 2017).

The NIR extends to processes beyond bound-state problems: calculation of gauge-boson and fermion self-energies (gap equations), applications to QCD-like theories with dynamical chiral symmetry breaking, and studies of excitonic bound states or response functions in reduced-dimensional condensed matter systems have already been realized (Mezrag et al., 2020, Gigante et al., 2017). For the description of GPDs, TMDs, and DDs, the NIR provides a systematic link between covariant amplitudes and light-front observables while enabling controlled incorporation of symmetry constraints and chiral theorems (Chouika et al., 2017, Moita et al., 2022).

A plausible implication is that when coupled with realistic spectral input for propagators or kernels (e.g., from lattice QCD or Dyson–Schwinger approaches), the NIR will enable fully covariant, dynamical, and real-time computations of hadron structure and scattering processes, even in strongly coupled, confining systems.

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References: (Gigante et al., 2017, Carbonell et al., 2017, Karmanov, 2021, Frederico et al., 2013, Salme' et al., 2017, Karmanov et al., 2018, Jia, 2 Feb 2024, Moita et al., 2022, Melo et al., 2019, Moita et al., 2022, Chouika et al., 2017, Mezrag et al., 2020, Frederico et al., 2011, Sauli, 2019, Pimentel et al., 2017)