Covariant Instantaneous Approximation

• Covariant Instantaneous Approximation is a technique that isolates instantaneous contributions from covariant integrals, enabling the analysis of constrained-field dynamics within a Lorentz covariant framework.
• It is applied in both light-front QED and Bethe–Salpeter theory to reduce four-dimensional equations to manageable three-dimensional forms while preserving covariance.
• The method extracts non-propagating interaction terms using asymptotic analysis or transverse-momentum kernels, aiding in precise predictions for heavy-quark bound states and decay processes.

Searching arXiv for the specified papers and topic. Covariant Instantaneous Approximation denotes a class of covariant reductions in which “instantaneous” contributions are isolated without abandoning manifestly four-dimensional starting points. In the arXiv literature represented by two distinct but related usages, it refers, first, to a light-front perturbative construction in which instantaneous interactions of constrained fields are recovered from covariant one-loop integrals by a controlled asymptotic analysis of the loop energy $k^-$ and light-front endpoint regions (Patel et al., 2010); and, second, to a Bethe–Salpeter reduction in which the interaction kernel depends only on momentum components transverse to the hadron four-velocity, so that the four-dimensional bound-state equation becomes a three-dimensional Salpeter-type integral equation while retaining a covariant decomposition (Weng et al., 2010). In both settings, the central objective is the same: to encode or extract non-propagating dynamics in a way compatible with Lorentz structure.

1. Terminological scope and core idea

The phrase “covariant instantaneous” is used in two technically different senses.

Setting	Meaning of the approximation	Main consequence
Light-front QED	Instantaneous light-front contributions are extracted from covariant loop integrals through $k^-$ integration and asymptotic/light-front endpoint analysis	Equivalence between covariant perturbation theory and light-front time-ordered perturbation theory at one loop
Bethe–Salpeter bound states	The kernel is taken to depend only on momentum components transverse to the total hadron momentum or velocity	Reduction of a four-dimensional BS equation to a three-dimensional integral equation

In the light-front formulation, the approximation is not an ansatz for the kernel but a procedure for identifying the parts of a covariant integral that correspond to constrained-field interactions, notably instantaneous photon exchange and instantaneous fermion exchange (Patel et al., 2010). In the Bethe–Salpeter formulation, by contrast, the approximation is imposed directly on the interaction kernel: only transverse momentum differences such as $p_t-p_t'$ or $q_t-q_t'$ are retained, while longitudinal components are integrated out (Weng et al., 2010).

Taken together, these usages suggest a broader interpretation: covariant instantaneous constructions do not equate “instantaneous” with a preferred frame. Rather, they reorganize dynamics so that non-propagating or retardation-suppressed structures appear through covariant projections or asymptotic limits.

2. Light-front QED realization

In light-front coordinates,

$$x^{\pm}=\frac{x^0\pm x^3}{\sqrt{2}},\qquad x_\perp=(x^1,x^2),$$

with momenta

$$k^{\pm}=\frac{k^0\pm k^3}{\sqrt{2}},\qquad k_\perp=(k^1,k^2),$$

the scalar product is

$a\cdot b=a^+b^-+a^-b^+-a_\perp\cdot b_\perp,$

and the metric is specified by $g^{+-}=g^{-+}=1$, $g^{ij}=-\delta^{ij}$, $g^{++}=g^{--}=0$ (Patel et al., 2010). The gauge choice is the light-cone gauge

$k^-$0

with $k^-$1 and $k^-$2.

In light-front Hamiltonian QED, the constrained components $k^-$3 and $k^-$4 are eliminated through their equations of motion. This generates effective nonlocal interactions in the Hamiltonian: instantaneous fermion exchange,

$k^-$5

and instantaneous photon exchange,

$k^-$6

where $k^-$7 is the independent “good” component of the fermion field and $k^-$8 is the transverse photon field (Patel et al., 2010).

The one-loop quantities analyzed in this framework are the fermion self-energy $k^-$9, the vertex correction $p_t-p_t'$0, and the vacuum polarization $p_t-p_t'$1. Their covariant expressions can be decomposed into light-front time-ordered perturbation theory contributions by performing the $p_t-p_t'$2 integral. Propagating intermediate states arise from pole residues, whereas non-propagating instantaneous pieces are associated with constrained-field physics (Patel et al., 2010).

A central technical distinction is between the two-term and three-term light-cone gauge photon propagators. The commonly used two-term propagator is

$p_t-p_t'$3

or, in the light-front component notation of Hamiltonian LFQED,

$p_t-p_t'$4

It satisfies $p_t-p_t'$5, but $p_t-p_t'$6 off shell, so it is singly transverse. The three-term, doubly transverse propagator contains an additional piece proportional to $p_t-p_t'$7; it satisfies both $p_t-p_t'$8 and $p_t-p_t'$9, and its extra term is crucial for generating instantaneous photon exchange directly from $q_t-q_t'$0 integration (Patel et al., 2010).

The light-cone pole is treated with the Mandelstam–Leibbrandt prescription,

$q_t-q_t'$1

which provides a consistent causal definition of the light-cone singularity and controls unphysical artifacts in covariant integrals (Patel et al., 2010).

3. Asymptotic extraction of instantaneous terms

A generic one-loop covariant integral can be written as

$q_t-q_t'$2

with denominators linear in $q_t-q_t'$3 after using

$q_t-q_t'$4

and the analogous expression for $q_t-q_t'$5 (Patel et al., 2010). Pole residues generate the standard propagating light-front time orderings. The nontrivial content of the covariant instantaneous approximation in this setting is that endpoint and arc-at-infinity contributions, particularly when $q_t-q_t'$6 and $q_t-q_t'$7 approaches light-front endpoints, reproduce the non-propagating contributions represented in LF Hamiltonian language by $q_t-q_t'$8 and $q_t-q_t'$9 (Patel et al., 2010).

For the fermion self-energy, the covariant starting point is

$$x^{\pm}=\frac{x^0\pm x^3}{\sqrt{2}},\qquad x_\perp=(x^1,x^2),$$0

Using the two-term propagator $$x^{\pm}=\frac{x^0\pm x^3}{\sqrt{2}},\qquad x_\perp=(x^1,x^2),$$1, one isolates the asymptotic region $$x^{\pm}=\frac{x^0\pm x^3}{\sqrt{2}},\qquad x_\perp=(x^1,x^2),$$2 with $$x^{\pm}=\frac{x^0\pm x^3}{\sqrt{2}},\qquad x_\perp=(x^1,x^2),$$3. In that limit, the numerator behaves as

$$x^{\pm}=\frac{x^0\pm x^3}{\sqrt{2}},\qquad x_\perp=(x^1,x^2),$$4

and the mass shift decomposes as

$$x^{\pm}=\frac{x^0\pm x^3}{\sqrt{2}},\qquad x_\perp=(x^1,x^2),$$5

After the regulated $$x^{\pm}=\frac{x^0\pm x^3}{\sqrt{2}},\qquad x_\perp=(x^1,x^2),$$6 integration, one finds that $$x^{\pm}=\frac{x^0\pm x^3}{\sqrt{2}},\qquad x_\perp=(x^1,x^2),$$7 reproduces precisely the instantaneous photon exchange contribution $$x^{\pm}=\frac{x^0\pm x^3}{\sqrt{2}},\qquad x_\perp=(x^1,x^2),$$8, while $$x^{\pm}=\frac{x^0\pm x^3}{\sqrt{2}},\qquad x_\perp=(x^1,x^2),$$9 reproduces the infrared-singular part of the propagating contribution $$k^{\pm}=\frac{k^0\pm k^3}{\sqrt{2}},\qquad k_\perp=(k^1,k^2),$$0 (Patel et al., 2010). The final one-loop decomposition is

$$k^{\pm}=\frac{k^0\pm k^3}{\sqrt{2}},\qquad k_\perp=(k^1,k^2),$$1

with

$$k^{\pm}=\frac{k^0\pm k^3}{\sqrt{2}},\qquad k_\perp=(k^1,k^2),$$2

$$k^{\pm}=\frac{k^0\pm k^3}{\sqrt{2}},\qquad k_\perp=(k^1,k^2),$$3

Here $$k^{\pm}=\frac{k^0\pm k^3}{\sqrt{2}},\qquad k_\perp=(k^1,k^2),$$4 is instantaneous fermion exchange and $$k^{\pm}=\frac{k^0\pm k^3}{\sqrt{2}},\qquad k_\perp=(k^1,k^2),$$5 is instantaneous photon exchange (Patel et al., 2010).

The same logic applies to the vertex correction. Propagating time orderings arise from residues at the poles of the triangle integral, while the instantaneous photon exchange piece is recovered from the large-$$k^{\pm}=\frac{k^0\pm k^3}{\sqrt{2}},\qquad k_\perp=(k^1,k^2),$$6 asymptotics when the internal photon plus-momentum approaches an external plus-momentum, $$k^{\pm}=\frac{k^0\pm k^3}{\sqrt{2}},\qquad k_\perp=(k^1,k^2),$$7 or $$k^{\pm}=\frac{k^0\pm k^3}{\sqrt{2}},\qquad k_\perp=(k^1,k^2),$$8. The resulting structure is of the form

$$k^{\pm}=\frac{k^0\pm k^3}{\sqrt{2}},\qquad k_\perp=(k^1,k^2),$$9

which matches the $a\cdot b=a^+b^-+a^-b^+-a_\perp\cdot b_\perp,$0 insertion in LFTOPT (Patel et al., 2010).

For vacuum polarization,

$a\cdot b=a^+b^-+a^-b^+-a_\perp\cdot b_\perp,$1

only propagating fermion-pair intermediate states occur at one loop in LFTOPT. Even so, the asymptotic method captures the endpoint-sensitive part of the propagating diagram,

$a\cdot b=a^+b^-+a^-b^+-a_\perp\cdot b_\perp,$2

from the limits $a\cdot b=a^+b^-+a^-b^+-a_\perp\cdot b_\perp,$3, $a\cdot b=a^+b^-+a^-b^+-a_\perp\cdot b_\perp,$4, and $a\cdot b=a^+b^-+a^-b^+-a_\perp\cdot b_\perp,$5 (Patel et al., 2010).

The principal conceptual point is that the third term of the doubly transverse propagator makes instantaneous photon exchange explicit at the level of the covariant integrand, whereas the two-term propagator permits the same physics to be recovered from asymptotic regions of the loop integral. This directly addresses the misconception that instantaneous photon exchange in light-front QED can be generated only by explicitly inserting the three-term propagator (Patel et al., 2010).

4. Bethe–Salpeter formulation for heavy diquarks and doubly heavy baryons

In the Bethe–Salpeter literature, the covariant instantaneous approximation is formulated through longitudinal and transverse projections defined relative to the bound-state four-velocity. For a heavy diquark,

$a\cdot b=a^+b^-+a^-b^+-a_\perp\cdot b_\perp,$6

and for a quark–diquark baryon,

$a\cdot b=a^+b^-+a^-b^+-a_\perp\cdot b_\perp,$7

with $a\cdot b=a^+b^-+a^-b^+-a_\perp\cdot b_\perp,$8 (Weng et al., 2010). “Instantaneous” means that the interaction kernel depends only on transverse momentum differences, such as $a\cdot b=a^+b^-+a^-b^+-a_\perp\cdot b_\perp,$9 or $g^{+-}=g^{-+}=1$0, not on longitudinal variables. “Covariant” means that these decompositions are defined with the hadron velocity rather than a fixed spatial direction (Weng et al., 2010).

For the heavy diquark, the four-dimensional BS equation is

$g^{+-}=g^{-+}=1$1

with $g^{+-}=g^{-+}=1$2 and $g^{+-}=g^{-+}=1$3 (Weng et al., 2010). Under CIA, the kernel is written as

$g^{+-}=g^{-+}=1$4

$g^{+-}=g^{-+}=1$5

The first term is one-gluon exchange; the second is scalar confinement with an explicit subtraction term that removes the infrared singularity at equal transverse momenta (Weng et al., 2010).

After integrating over the longitudinal component $g^{+-}=g^{-+}=1$6, the scalar BS function $g^{+-}=g^{-+}=1$7 satisfies a three-dimensional equation of Salpeter type. This reduction is the operational content of the approximation: four-dimensional covariant dynamics are traded for a three-dimensional integral equation in transverse momentum space (Weng et al., 2010).

The same strategy is used for doubly heavy baryons treated as heavy-diquark–light-quark systems in the heavy quark limit. The BS equation is

$g^{+-}=g^{-+}=1$8

with $g^{+-}=g^{-+}=1$9, $g^{ij}=-\delta^{ij}$0, and

$g^{ij}=-\delta^{ij}$1

where the effective diquark–gluon vertex is

$g^{ij}=-\delta^{ij}$2

Under CIA,

$g^{ij}=-\delta^{ij}$3

$g^{ij}=-\delta^{ij}$4

Integrating over $g^{ij}=-\delta^{ij}$5 yields coupled three-dimensional equations for $g^{ij}=-\delta^{ij}$6 in the scalar-diquark case and $g^{ij}=-\delta^{ij}$7 in the axial-vector case (Weng et al., 2010).

The amplitude basis is correspondingly reduced. For the heavy diquark,

$g^{ij}=-\delta^{ij}$8

for the scalar diquark and

$g^{ij}=-\delta^{ij}$9

for the axial-vector diquark. For the doubly heavy baryon,

$g^{++}=g^{--}=0$0

in the scalar-diquark sector and

$g^{++}=g^{--}=0$1

in the axial-vector sector (Weng et al., 2010). Normalization conditions are derived after CIA and longitudinal integration for both diquarks and baryons, preserving the covariant structure of the state normalization (Weng et al., 2010).

An important physical point is that the approximation does not neglect internal heavy-diquark transverse motion. The internal momentum in the heavy–heavy subsystem scales like $g^{++}=g^{--}=0$2, so $g^{++}=g^{--}=0$3 must be retained even at leading order in $g^{++}=g^{--}=0$4 (Weng et al., 2010).

5. Kernels, numerical implementation, and phenomenology

The Bethe–Salpeter implementation uses constituent masses

$g^{++}=g^{--}=0$5

for the heavy quarks, together with $g^{++}=g^{--}=0$6 and $g^{++}=g^{--}=0$7 in the diquark kernel. In the baryon calculation, the light-quark masses are

$g^{++}=g^{--}=0$8

with $g^{++}=g^{--}=0$9, and the confinement parameter $k^-$00 is varied in the range $k^-$01–$k^-$02 (Weng et al., 2010). A small regulator $k^-$03 is introduced, and the subtraction terms in $k^-$04 and $k^-$05 remove singularities at equal transverse momenta. Numerically, the three-dimensional momentum integrals are discretized into $k^-$06 segments, converting the integral equations into matrix eigenvalue problems for $k^-$07 and $k^-$08 (Weng et al., 2010).

The heavy diquark masses are found to be independent of diquark spin at leading order; only the heavy-quark flavors matter. For the doubly heavy baryons, the predicted mass ranges are as follows (Weng et al., 2010).

State	Mass range
$k^-$09	$k^-$10
$k^-$11	$k^-$12
$k^-$13	$k^-$14
$k^-$15	$k^-$16
$k^-$17	$k^-$18
$k^-$19	$k^-$20

The $k^-$21 mass is stated to agree with experiment, $k^-$22 MeV. The masses and BS wave-function amplitudes are found to be independent of the spins of the heavy diquarks and the doubly heavy baryons at leading order, consistent with heavy diquark spin symmetry (Weng et al., 2010).

The formalism is also used to construct effective diquark–gluon vertices and form factors. For the scalar diquark,

$k^-$23

while for the axial-vector diquark,

$k^-$24

Numerically, $k^-$25, $k^-$26, and $k^-$27 is neglected at small and intermediate $k^-$28. The form factor satisfies $k^-$29 and $k^-$30 (Weng et al., 2010).

For non-leptonic decays emitting a pseudoscalar meson, the analysis uses factorization, the pseudoscalar decay-constant relation

$k^-$31

and the two-body width formula

$k^-$32

Representative predictions include, for $k^-$33 emission in units of $k^-$34 GeV,

$k^-$35

$k^-$36

$k^-$37

$k^-$38

Further representative values are given for $k^-$39, $k^-$40, and $k^-$41 emission, including

$k^-$42

in units of $k^-$43 GeV,

$k^-$44

in units of $k^-$45 GeV, and

$k^-$46

in units of $k^-$47 GeV (Weng et al., 2010).

6. Interpretation, applicability, and limitations

The two main uses of the covariant instantaneous approximation share a common methodological claim: one can isolate instantaneous physics without giving up a covariant starting point. In light-front QED, the physical reason given is that the third term of the doubly transverse propagator carries the non-propagating longitudinal structure associated with the elimination of $k^-$48, whereas the asymptotic-limit method with the two-term propagator extracts the same structure from the large-$k^-$49, endpoint behavior of the covariant integrand (Patel et al., 2010). In Bethe–Salpeter theory, the analogous claim is that a kernel depending only on transverse momentum differences can still be formulated covariantly because the longitudinal/transverse split is defined by the hadron velocity $k^-$50 rather than a fixed frame (Weng et al., 2010).

These constructions also delimit what is and is not included. In the light-front QED case, the explicit demonstration is at one loop for self-energy, vacuum polarization, and, by extension, the vertex. At higher loops, endpoint and arc contributions proliferate, and overlapping singularities require careful treatment. The method relies on light-cone gauge with the ML prescription; with the three-term propagator instantaneous photon exchange appears directly through $k^-$51 integration, whereas with the two-term propagator it is obtained asymptotically (Patel et al., 2010). The endpoint divergences discussed there are described as spurious light-front divergences distinct from physical soft or collinear infrared singularities, which still require standard regulators such as a small photon mass $k^-$52 or dimensional regularization (Patel et al., 2010).

In the Bethe–Salpeter case, the approximation is justified as suitable for weakly bound heavy systems, but it remains an instantaneous-kernel approximation. Dynamics beyond instantaneous kernels or significant retardation effects are not captured. The calculation is performed at leading order in the $k^-$53 expansion, so spin-dependent splittings between scalar and axial diquarks and between baryon spin-$k^-$54 and spin-$k^-$55 states are deferred to $k^-$56. The use of free, constituent-like propagators for light quarks and effective diquarks is explicitly identified as an approximation; a more rigorous treatment based on Dyson–Schwinger equations and the axial-vector Ward–Takahashi identity is acknowledged as technically challenging and left for future improvement (Weng et al., 2010).

Within these limits, the two formulations establish complementary versions of the same general principle. In perturbative light-front QED, covariant instantaneous approximation means extracting the constrained-field content of amplitudes from asymptotic regions of covariant loop integrals. In Bethe–Salpeter bound-state theory, it means imposing a covariant instantaneous kernel that yields tractable three-dimensional equations. Both uses make instantaneous dynamics calculable while preserving an explicitly covariant organizational framework (Patel et al., 2010, Weng et al., 2010).