Covariant Reggeization in High-Energy Physics

• Covariant Reggeization is a method that implements Regge behavior within a covariant tensor and effective-action framework.
• It replaces fixed-spin exchanges with moving Regge poles, ensuring the preservation of current, gauge, conformal, or general covariance.
• This approach underpins high-energy processes in hadronic diffraction, pion photoproduction, QCD effective theories, conformal field theories, and gravity.

Covariant Reggeization designates a class of constructions in which Regge behavior is implemented without abandoning the covariant tensor, amplitude, or effective-action structure of the underlying theory. In the cited literature, the term appears in hadronic diffraction, gauge-invariant charged pion photoproduction, Lipatov’s high-energy effective field theory, conformal field theory, AdS/CFT, and gravity. Across these settings, the recurring operation is the replacement of fixed-spin exchange by a moving Regge pole $J=\alpha(t)$, together with a covariant organization of tensor structures, partial waves, or Reggeon fields, while preserving current conservation, gauge invariance, conformal covariance, or general covariance as appropriate (Ryutin, 21 Jul 2025, Montana et al., 2024, Nefedov et al., 2016, 0910.2746, 0710.5480, Lipatov, 2011).

1. Covariant tensor basis and the Regge-pole replacement

In the most explicit general formulation, an irreducible spin-$J$ tensor field $\Phi^{\mu_1\cdots\mu_J}(x)$ is defined by symmetry, tracelessness, and transversality,

$$\Phi^{\mu_1\cdots\mu_i\cdots\mu_j\cdots\mu_J} =\Phi^{\mu_1\cdots\mu_j\cdots\mu_i\cdots\mu_J},$$

$$g_{\mu_i\mu_j}\,\Phi^{\dots\mu_i\cdots\mu_j\cdots}=0, \qquad q_{\mu_1}\,\Phi^{\mu_1\mu_2\cdots\mu_J}(q)=0.$$

A basis of symmetric-traceless-transverse tensors is then built from the unit-normalized transverse momentum

$$P_\mu(p,q)=\frac{p_\mu-\tfrac{p\!\cdot\!q}{q^2}\,q_\mu} {\sqrt{p^2-(p\!\cdot\!q)^2/q^2}}, \qquad P\!\cdot\!q=0,$$

and the transverse projector

$$G_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}, \qquad G_{\mu\nu}q^\nu=0.$$

The corresponding spin-$J$ tensor is written as

$$T^{(J)}_{\mu_1\cdots\mu_J}(p,q) =\sum_{n=0}^{\lfloor J/2\rfloor} \upsilon^J_n\, \bigl[P_{(\mu_1}\cdots P_{\mu_{J-2n}}\, G_{\mu_{J-2n+1}\mu_{J-2n+2}}\cdots G_{\mu_{J-1}\mu_J)}\bigr]_{\rm sym},$$

with

$$\upsilon^J_n=\frac{1}{2^n\,(c^J)_n}, \qquad c^J\equiv -\Bigl(J+\tfrac{D-5}{2}\Bigr).$$

This construction extends to general hadronic tensors,

$J$0

where the $J$1 are scalar Lorentz-invariant form factors. The covariant reggeization prescription then replaces fixed-spin poles by a Regge pole,

$J$2

with $J$3. In this formulation, Reggeization is not introduced by a noncovariant projection on helicity components, but by analytically continuing a covariantly defined tensor decomposition. This suggests that, in hadronic diffraction at least, covariant Reggeization is best viewed as a statement about the tensorial organization of amplitudes before it is a statement about asymptotic $J$4-dependence (Ryutin, 21 Jul 2025).

2. Analytic continuation in spin and the $J$5 pion limit

A particularly sharp realization appears in charged pion photoproduction. For the exchange of an unnatural-parity trajectory with even spin $J$6 in the $J$7-channel, the gauge-invariant vertices are written covariantly as

$J$8

$J$9

and the fixed-spin amplitude factorizes as

$\Phi^{\mu_1\cdots\mu_J}(x)$0

Although $\Phi^{\mu_1\cdots\mu_J}(x)$1 and $\Phi^{\mu_1\cdots\mu_J}(x)$2 appear singular as $\Phi^{\mu_1\cdots\mu_J}(x)$3, their product remains finite. The analytic continuation to $\Phi^{\mu_1\cdots\mu_J}(x)$4 yields

$\Phi^{\mu_1\cdots\mu_J}(x)$5

and, near the pion pole,

$\Phi^{\mu_1\cdots\mu_J}(x)$6

in exact agreement with the electric Born amplitude for pion exchange (Montana et al., 2024).

The conceptual point is that the gauge-invariant amplitude for the exchange of a particle with generic even spin $\Phi^{\mu_1\cdots\mu_J}(x)$7 in the $\Phi^{\mu_1\cdots\mu_J}(x)$8-channel is analytic at $\Phi^{\mu_1\cdots\mu_J}(x)$9, and that the continued $$\Phi^{\mu_1\cdots\mu_i\cdots\mu_j\cdots\mu_J} =\Phi^{\mu_1\cdots\mu_j\cdots\mu_i\cdots\mu_J},$$0 term reconstructs precisely the nucleon electric current required by the Ward–Takahashi identity. The naive pion current $$\Phi^{\mu_1\cdots\mu_i\cdots\mu_j\cdots\mu_J} =\Phi^{\mu_1\cdots\mu_j\cdots\mu_i\cdots\mu_J},$$1 is not sufficient by itself, since $$\Phi^{\mu_1\cdots\mu_i\cdots\mu_j\cdots\mu_J} =\Phi^{\mu_1\cdots\mu_j\cdots\mu_i\cdots\mu_J},$$2 only after adding the nucleon piece. The resulting Reggeized pion amplitude takes the factorized form

$$\Phi^{\mu_1\cdots\mu_i\cdots\mu_j\cdots\mu_J} =\Phi^{\mu_1\cdots\mu_j\cdots\mu_i\cdots\mu_J},$$3

with

$$\Phi^{\mu_1\cdots\mu_i\cdots\mu_j\cdots\mu_J} =\Phi^{\mu_1\cdots\mu_j\cdots\mu_i\cdots\mu_J},$$4

A common misconception is that Reggeizing only the pion pole suffices to preserve gauge invariance. The covariant construction shows instead that the continuation-to-zero-spin amplitude is exactly the piece of the nucleon $$\Phi^{\mu_1\cdots\mu_i\cdots\mu_j\cdots\mu_J} =\Phi^{\mu_1\cdots\mu_j\cdots\mu_i\cdots\mu_J},$$5- and $$\Phi^{\mu_1\cdots\mu_i\cdots\mu_j\cdots\mu_J} =\Phi^{\mu_1\cdots\mu_j\cdots\mu_i\cdots\mu_J},$$6-channel electric current that makes the pion exchange gauge invariant, and that the covariant “minimal gauge” prescription keeps the $$\Phi^{\mu_1\cdots\mu_i\cdots\mu_j\cdots\mu_J} =\Phi^{\mu_1\cdots\mu_j\cdots\mu_i\cdots\mu_J},$$7 contact term untouched while replacing only $$\Phi^{\mu_1\cdots\mu_i\cdots\mu_j\cdots\mu_J} =\Phi^{\mu_1\cdots\mu_j\cdots\mu_i\cdots\mu_J},$$8 by $$\Phi^{\mu_1\cdots\mu_i\cdots\mu_j\cdots\mu_J} =\Phi^{\mu_1\cdots\mu_j\cdots\mu_i\cdots\mu_J},$$9 (Montana et al., 2024).

3. Gauge-invariant effective-field-theory realization in QCD

In Lipatov’s high-energy effective field theory, Reggeization is implemented through gauge-invariant $$g_{\mu_i\mu_j}\,\Phi^{\dots\mu_i\cdots\mu_j\cdots}=0, \qquad q_{\mu_1}\,\Phi^{\mu_1\mu_2\cdots\mu_J}(q)=0.$$0-channel fields that communicate between rapidity slices. The effective action is written as

$$g_{\mu_i\mu_j}\,\Phi^{\dots\mu_i\cdots\mu_j\cdots}=0, \qquad q_{\mu_1}\,\Phi^{\mu_1\mu_2\cdots\mu_J}(q)=0.$$1

with Reggeized-quark kinetic term

$$g_{\mu_i\mu_j}\,\Phi^{\dots\mu_i\cdots\mu_j\cdots}=0, \qquad q_{\mu_1}\,\Phi^{\mu_1\mu_2\cdots\mu_J}(q)=0.$$2

subject to $$g_{\mu_i\mu_j}\,\Phi^{\dots\mu_i\cdots\mu_j\cdots}=0, \qquad q_{\mu_1}\,\Phi^{\mu_1\mu_2\cdots\mu_J}(q)=0.$$3 and $$g_{\mu_i\mu_j}\,\Phi^{\dots\mu_i\cdots\mu_j\cdots}=0, \qquad q_{\mu_1}\,\Phi^{\mu_1\mu_2\cdots\mu_J}(q)=0.$$4. The induced interaction is expressed through light-cone Wilson lines,

$$g_{\mu_i\mu_j}\,\Phi^{\dots\mu_i\cdots\mu_j\cdots}=0, \qquad q_{\mu_1}\,\Phi^{\mu_1\mu_2\cdots\mu_J}(q)=0.$$5

$$g_{\mu_i\mu_j}\,\Phi^{\dots\mu_i\cdots\mu_j\cdots}=0, \qquad q_{\mu_1}\,\Phi^{\mu_1\mu_2\cdots\mu_J}(q)=0.$$6

Expanding the Wilson lines generates the Fadin–Sherman quark–Reggeon–gluon vertex

$$g_{\mu_i\mu_j}\,\Phi^{\dots\mu_i\cdots\mu_j\cdots}=0, \qquad q_{\mu_1}\,\Phi^{\mu_1\mu_2\cdots\mu_J}(q)=0.$$7

and its two-gluon generalization (Nefedov et al., 2016).

Because the induced vertices contain nonlocal factors $$g_{\mu_i\mu_j}\,\Phi^{\dots\mu_i\cdots\mu_j\cdots}=0, \qquad q_{\mu_1}\,\Phi^{\mu_1\mu_2\cdots\mu_J}(q)=0.$$8, loop integrals develop rapidity divergences. A manifestly covariant regularization is obtained by tilting the light-cone vectors off the light-cone,

$$g_{\mu_i\mu_j}\,\Phi^{\dots\mu_i\cdots\mu_j\cdots}=0, \qquad q_{\mu_1}\,\Phi^{\mu_1\mu_2\cdots\mu_J}(q)=0.$$9

so that finite $$P_\mu(p,q)=\frac{p_\mu-\tfrac{p\!\cdot\!q}{q^2}\,q_\mu} {\sqrt{p^2-(p\!\cdot\!q)^2/q^2}}, \qquad P\!\cdot\!q=0,$$0 cuts off large rapidity intervals. The one-loop Reggeized-quark self-energy becomes

$$P_\mu(p,q)=\frac{p_\mu-\tfrac{p\!\cdot\!q}{q^2}\,q_\mu} {\sqrt{p^2-(p\!\cdot\!q)^2/q^2}}, \qquad P\!\cdot\!q=0,$$1

while the unsubtracted $$P_\mu(p,q)=\frac{p_\mu-\tfrac{p\!\cdot\!q}{q^2}\,q_\mu} {\sqrt{p^2-(p\!\cdot\!q)^2/q^2}}, \qquad P\!\cdot\!q=0,$$2 vertex contains a single $$P_\mu(p,q)=\frac{p_\mu-\tfrac{p\!\cdot\!q}{q^2}\,q_\mu} {\sqrt{p^2-(p\!\cdot\!q)^2/q^2}}, \qquad P\!\cdot\!q=0,$$3-divergence that is removed by subtracting the central-rapidity contribution,

$$P_\mu(p,q)=\frac{p_\mu-\tfrac{p\!\cdot\!q}{q^2}\,q_\mu} {\sqrt{p^2-(p\!\cdot\!q)^2/q^2}}, \qquad P\!\cdot\!q=0,$$4

After combining the localized vertices with the self-energy, all $$P_\mu(p,q)=\frac{p_\mu-\tfrac{p\!\cdot\!q}{q^2}\,q_\mu} {\sqrt{p^2-(p\!\cdot\!q)^2/q^2}}, \qquad P\!\cdot\!q=0,$$5-dependence cancels and the Regge limit of the one-loop $$P_\mu(p,q)=\frac{p_\mu-\tfrac{p\!\cdot\!q}{q^2}\,q_\mu} {\sqrt{p^2-(p\!\cdot\!q)^2/q^2}}, \qquad P\!\cdot\!q=0,$$6 amplitude agrees exactly with the high-energy limit of the full one-loop QCD amplitude. The rapidity renormalization group then exponentiates the large $$P_\mu(p,q)=\frac{p_\mu-\tfrac{p\!\cdot\!q}{q^2}\,q_\mu} {\sqrt{p^2-(p\!\cdot\!q)^2/q^2}}, \qquad P\!\cdot\!q=0,$$7 terms through

$$P_\mu(p,q)=\frac{p_\mu-\tfrac{p\!\cdot\!q}{q^2}\,q_\mu} {\sqrt{p^2-(p\!\cdot\!q)^2/q^2}}, \qquad P\!\cdot\!q=0,$$8

with one-loop trajectory

$$P_\mu(p,q)=\frac{p_\mu-\tfrac{p\!\cdot\!q}{q^2}\,q_\mu} {\sqrt{p^2-(p\!\cdot\!q)^2/q^2}}, \qquad P\!\cdot\!q=0,$$9

Here covariant Reggeization is not merely a kinematic rewriting: it is a renormalized EFT framework in which gauge invariance is preserved by construction and rapidity logarithms are exponentiated into the Regge trajectory (Nefedov et al., 2016).

4. Complex spin, conformal partial waves, and the Lorentzian Regge limit

For off-shell Green functions, the Regge limit can be defined in a completely Lorentz-covariant way by boosting only the fields at the odd-numbered points and then taking the boost rapidity $$G_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}, \qquad G_{\mu\nu}q^\nu=0.$$0. In momentum space, the momenta at the even points remain fixed while those at the odd points are boosted. For elastic four-point kinematics this gives

$$G_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}, \qquad G_{\mu\nu}q^\nu=0.$$1

In a CFT, the same limit is characterized by

$$G_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}, \qquad G_{\mu\nu}q^\nu=0.$$2

In free CFT, the leading diagrams in this limit are those in which exactly two lines cross the $$G_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}, \qquad G_{\mu\nu}q^\nu=0.$$3-channel cut; all other connected diagrams are exponentially suppressed. Upon perturbing away from the free fixed point, the leading Regge behavior is governed by the BFKL kernel, which can be written as the two-point function of the null bilocal operator

$$G_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}, \qquad G_{\mu\nu}q^\nu=0.$$4

and, in transverse coordinate space, reduces at leading order to the standard BFKL kernel (0910.2746).

When infinitely many spins contribute, the conformal partial-wave sum must be treated by complex-spin techniques. In the $$G_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}, \qquad G_{\mu\nu}q^\nu=0.$$5-channel, the partial waves $$G_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}, \qquad G_{\mu\nu}q^\nu=0.$$6 are analytically continued in $$G_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}, \qquad G_{\mu\nu}q^\nu=0.$$7, and a Sommerfeld–Watson resummation deforms the contour from integer spin to $$G_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}, \qquad G_{\mu\nu}q^\nu=0.$$8 plus Regge poles at $$G_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}, \qquad G_{\mu\nu}q^\nu=0.$$9. A leading pole in $J$0 produces the Lorentzian Regge behavior

$J$1

In AdS impact-parameter space, the tree-level pole is encoded in

$J$2

and the eikonal amplitude exponentiates,

$J$3

reproducing multiple gravi-reggeon exchange. In $J$4 at large ’t Hooft coupling,

$J$5

while the strong-coupling CFT discussion also identifies a leading singularity at

$J$6

Taken together, these results show that covariant Reggeization in conformal settings is an overview of Lorentzian continuation, complex-spin analyticity, and impact-parameter resummation rather than a simple extrapolation of flat-space hadronic formulas (0710.5480, 0910.2746).

5. Generally covariant Reggeization in gravity

In gravity, the effective action for Regge processes is formulated in terms of reggeon fields $J$7, $J$8 and the metric tensor $J$9 so that it is local in rapidity space and has the property of general covariance. The action is

$$T^{(J)}_{\mu_1\cdots\mu_J}(p,q) =\sum_{n=0}^{\lfloor J/2\rfloor} \upsilon^J_n\, \bigl[P_{(\mu_1}\cdots P_{\mu_{J-2n}}\, G_{\mu_{J-2n+1}\mu_{J-2n+2}}\cdots G_{\mu_{J-1}\mu_J)}\bigr]_{\rm sym},$$0

with

$$T^{(J)}_{\mu_1\cdots\mu_J}(p,q) =\sum_{n=0}^{\lfloor J/2\rfloor} \upsilon^J_n\, \bigl[P_{(\mu_1}\cdots P_{\mu_{J-2n}}\, G_{\mu_{J-2n+1}\mu_{J-2n+2}}\cdots G_{\mu_{J-1}\mu_J)}\bigr]_{\rm sym},$$1

$$T^{(J)}_{\mu_1\cdots\mu_J}(p,q) =\sum_{n=0}^{\lfloor J/2\rfloor} \upsilon^J_n\, \bigl[P_{(\mu_1}\cdots P_{\mu_{J-2n}}\, G_{\mu_{J-2n+1}\mu_{J-2n+2}}\cdots G_{\mu_{J-1}\mu_J)}\bigr]_{\rm sym},$$2

In the gauge $$T^{(J)}_{\mu_1\cdots\mu_J}(p,q) =\sum_{n=0}^{\lfloor J/2\rfloor} \upsilon^J_n\, \bigl[P_{(\mu_1}\cdots P_{\mu_{J-2n}}\, G_{\mu_{J-2n+1}\mu_{J-2n+2}}\cdots G_{\mu_{J-1}\mu_J)}\bigr]_{\rm sym},$$3, $$T^{(J)}_{\mu_1\cdots\mu_J}(p,q) =\sum_{n=0}^{\lfloor J/2\rfloor} \upsilon^J_n\, \bigl[P_{(\mu_1}\cdots P_{\mu_{J-2n}}\, G_{\mu_{J-2n+1}\mu_{J-2n+2}}\cdots G_{\mu_{J-1}\mu_J)}\bigr]_{\rm sym},$$4, the reggeon kinetic term is

$$T^{(J)}_{\mu_1\cdots\mu_J}(p,q) =\sum_{n=0}^{\lfloor J/2\rfloor} \upsilon^J_n\, \bigl[P_{(\mu_1}\cdots P_{\mu_{J-2n}}\, G_{\mu_{J-2n+1}\mu_{J-2n+2}}\cdots G_{\mu_{J-1}\mu_J)}\bigr]_{\rm sym},$$5

The induced currents satisfy a covariant Hamilton–Jacobi equation,

$$T^{(J)}_{\mu_1\cdots\mu_J}(p,q) =\sum_{n=0}^{\lfloor J/2\rfloor} \upsilon^J_n\, \bigl[P_{(\mu_1}\cdots P_{\mu_{J-2n}}\, G_{\mu_{J-2n+1}\mu_{J-2n+2}}\cdots G_{\mu_{J-1}\mu_J)}\bigr]_{\rm sym},$$6

or, in terms of $$T^{(J)}_{\mu_1\cdots\mu_J}(p,q) =\sum_{n=0}^{\lfloor J/2\rfloor} \upsilon^J_n\, \bigl[P_{(\mu_1}\cdots P_{\mu_{J-2n}}\, G_{\mu_{J-2n+1}\mu_{J-2n+2}}\cdots G_{\mu_{J-1}\mu_J)}\bigr]_{\rm sym},$$7,

$$T^{(J)}_{\mu_1\cdots\mu_J}(p,q) =\sum_{n=0}^{\lfloor J/2\rfloor} \upsilon^J_n\, \bigl[P_{(\mu_1}\cdots P_{\mu_{J-2n}}\, G_{\mu_{J-2n+1}\mu_{J-2n+2}}\cdots G_{\mu_{J-1}\mu_J)}\bigr]_{\rm sym},$$8

For shock-wave backgrounds these currents can be computed explicitly, and a variational principle,

$$T^{(J)}_{\mu_1\cdots\mu_J}(p,q) =\sum_{n=0}^{\lfloor J/2\rfloor} \upsilon^J_n\, \bigl[P_{(\mu_1}\cdots P_{\mu_{J-2n}}\, G_{\mu_{J-2n+1}\mu_{J-2n+2}}\cdots G_{\mu_{J-1}\mu_J)}\bigr]_{\rm sym},$$9

reproduces the same result (Lipatov, 2011).

The covariance statement is stronger than mere Lorentz invariance. The reggeon fields are taken to be invariant under 4D diffeomorphisms, $$\upsilon^J_n=\frac{1}{2^n\,(c^J)_n}, \qquad c^J\equiv -\Bigl(J+\tfrac{D-5}{2}\Bigr).$$0, while the induced action remains invariant up to total derivatives provided the currents transform with the stated inhomogeneous terms. Expanding $$\upsilon^J_n=\frac{1}{2^n\,(c^J)_n}, \qquad c^J\equiv -\Bigl(J+\tfrac{D-5}{2}\Bigr).$$1 in powers of $$\upsilon^J_n=\frac{1}{2^n\,(c^J)_n}, \qquad c^J\equiv -\Bigl(J+\tfrac{D-5}{2}\Bigr).$$2 yields the multi-Regge interaction vertices, and the one-loop graviton Regge trajectory is obtained from two graviton–reggeon–graviton vertices,

$$\upsilon^J_n=\frac{1}{2^n\,(c^J)_n}, \qquad c^J\equiv -\Bigl(J+\tfrac{D-5}{2}\Bigr).$$3

Older noncovariant treatments typically fixed a light-cone gauge or used $$\upsilon^J_n=\frac{1}{2^n\,(c^J)_n}, \qquad c^J\equiv -\Bigl(J+\tfrac{D-5}{2}\Bigr).$$4-channel unitarity in a fixed background. The covariant construction avoids any gauge choice and preserves full 4D diffeomorphism invariance at every step. This is the gravitational analogue of the role played by gauge-invariant EFT vertices in QCD and by current-conserving partial waves in pion photoproduction (Lipatov, 2011).

6. Diffraction, cross sections, and theoretical constraints

In hadronic diffraction, covariant Reggeization is proposed as an effective approach for expanding the relevant hadronic tensors and obtaining the basic functions needed to calculate diffractive cross-sections. The relevant building blocks are the scalar–scalar–spin-$$\upsilon^J_n=\frac{1}{2^n\,(c^J)_n}, \qquad c^J\equiv -\Bigl(J+\tfrac{D-5}{2}\Bigr).$$5 vertex $$\upsilon^J_n=\frac{1}{2^n\,(c^J)_n}, \qquad c^J\equiv -\Bigl(J+\tfrac{D-5}{2}\Bigr).$$6, the forward reggeon–hadron amplitude $$\upsilon^J_n=\frac{1}{2^n\,(c^J)_n}, \qquad c^J\equiv -\Bigl(J+\tfrac{D-5}{2}\Bigr).$$7, the reggeon–reggeon to central-system fusion vertex $$\upsilon^J_n=\frac{1}{2^n\,(c^J)_n}, \qquad c^J\equiv -\Bigl(J+\tfrac{D-5}{2}\Bigr).$$8, and the four-index hadronic tensor $$\upsilon^J_n=\frac{1}{2^n\,(c^J)_n}, \qquad c^J\equiv -\Bigl(J+\tfrac{D-5}{2}\Bigr).$$9. After Reggeization, the elastic amplitude behaves as

$J$00

with

$J$01

For single diffractive dissociation,

$J$02

and analogous triple-Regge expressions are given for double diffraction and central exclusive diffraction (Ryutin, 21 Jul 2025).

The same framework states three theoretical constraints. Gauge, or current, conservation,

$J$03

eliminates non-symmetric-traceless-transverse terms. Crossing symmetry relates $J$04 channels and constrains the parity of signature factors $J$05. Unitarity requires

$J$06

which bounds the growth of trajectories and residues in multi-Reggeon kinematics. These constraints clarify the scope of the subject. Covariant Reggeization is not a single universal formula applicable unchanged in every theory; rather, it is a family of covariant implementations of Regge theory whose admissible tensor structures, subtraction terms, and effective degrees of freedom are fixed by the symmetry principle that survives in the Regge limit. In pion photoproduction that principle is current conservation; in high-energy QCD it is gauge invariance with rapidity-local EFT organization; in CFT it is Lorentzian conformal partial-wave analyticity; in gravity it is general covariance; and in hadronic diffraction it is the irreducible tensor decomposition of the hadronic tensors themselves (Montana et al., 2024, Nefedov et al., 2016, 0710.5480, Lipatov, 2011, Ryutin, 21 Jul 2025).