Regge Poles in Scattering and Spectroscopy

• Regge poles are singularities of analytically continued scattering amplitudes that link complex angular momentum with resonance structures and high-energy behavior.
• They provide a unifying analytic framework across hadronic phenomenology, black-hole physics, and inverse spectral problems.
• Applications include modeling resonance trajectories, regulating cross sections, and distinguishing between regimes like soft pomeron exchange and diffraction.

Searching arXiv for recent and foundational papers on Regge poles to ground the article. Regge poles are singularities of analytically continued partial-wave scattering amplitudes in the complex angular-momentum plane, and their associated trajectories $J=\alpha(t)$ connect the analytic structure of amplitudes to high-energy asymptotics and, in many settings, to spectroscopic regularities (Bottino, 2018). In the standard complex angular momentum framework, one replaces integer angular momentum by a complex variable, rewrites the partial-wave sum through a Sommerfeld–Watson transform, and isolates pole contributions whose residues govern asymptotic behavior such as $f(s,t)=C(t)\,s^{\alpha(t)}$ at large $s$ and fixed $t$ (Bottino, 2018). Across hadronic phenomenology, black-hole and analogue-gravity scattering, compact-object spectroscopy, inverse problems, and perturbative QCD, Regge poles serve as a common analytic device, but their precise physical interpretation depends strongly on context: they may encode pomeron-like leading exchanges in hadron scattering, surface waves near a photon sphere, compact-object interface resonances, or Reggeized gluon exchange and its separation from Regge cuts (Campos, 2020, Folacci et al., 2019, Hadj et al., 2019, Abreu et al., 2024).

1. Complex angular momentum formulation

In two-particle scattering, the starting point is the partial-wave expansion

$$f(k,\cos\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\left[\exp[2i\delta(l,k)]-1\right]P_l(\cos\theta),$$

with $k$ the center-of-mass wave number, $\theta$ the scattering angle, $\delta(l,k)$ the phase shift, and $P_l$ the Legendre polynomial (Bottino, 2018). Regge’s key step was to regard $l$ as a complex variable rather than as a discrete quantum number, so that phase shifts, partial-wave solutions, and the $f(s,t)=C(t)\,s^{\alpha(t)}$0-matrix can be analytically continued to complex angular momentum $f(s,t)=C(t)\,s^{\alpha(t)}$1 (Bottino, 2018). After a Sommerfeld–Watson transform, the amplitude becomes a contour integral in the complex $f(s,t)=C(t)\,s^{\alpha(t)}$2-plane plus explicit residue terms from poles of the analytically continued $f(s,t)=C(t)\,s^{\alpha(t)}$3-matrix (Bottino, 2018).

Within this framework, Regge poles are poles of $f(s,t)=C(t)\,s^{\alpha(t)}$4, equivalently zeros of the relevant Jost-function denominator in analytically continued angular momentum (Bottino, 2018). In the potential-scattering formulation, the radial Schrödinger equation is continued to

$f(s,t)=C(t)\,s^{\alpha(t)}$5

and the $f(s,t)=C(t)\,s^{\alpha(t)}$6-matrix is expressed through Jost functions as

$f(s,t)=C(t)\,s^{\alpha(t)}$7

The analytic structure is genuinely two-complex-variable, involving both $f(s,t)=C(t)\,s^{\alpha(t)}$8 and $f(s,t)=C(t)\,s^{\alpha(t)}$9, and the paper states analyticity of the Jost function in the whole $s$0-plane cut along the upper imaginary axis times the half-plane $s$1 (Bottino, 2018).

The physical importance of Regge poles follows from contour deformation. When the deformed contour crosses isolated singularities of $s$2, their residues survive as separate pole terms, and these dominate important asymptotic regimes (Bottino, 2018). The rightmost pole in the complex angular-momentum plane governs the large-$s$3 behavior $s$4, and, by crossing symmetry, a pole in the crossed channel produces the high-energy behavior $s$5 (Bottino, 2018). This is also the basis of the spectroscopic relation $s$6, which organizes particles and resonances into Regge families (Bottino, 2018).

2. Regge trajectories, widths, and spectroscopic interpretation

The spectroscopic use of Regge poles rests on the identification of resonance poles with points where a trajectory reaches a physical spin. In the baryon analysis of $s$7 and $s$8 poles, the standard Regge-theory statement is

$s$9

and for a resonance of spin $t$0 and pole position $t$1, consistency requires

$t$2

(Collaboration et al., 2018). This treatment emphasizes that the usual Chew–Frautschi plot is only a projection onto $t$3, while the imaginary part $t$4, which is related to width, carries independent structural information (Collaboration et al., 2018).

The baryon paper models the trajectory as

$t$5

with three different choices of $t$6, ranging from a purely linear model to a square-root form and a Chew–Mandelstam/dispersive phase-space-inspired form (Collaboration et al., 2018). The point of this construction is not merely better fitting but the incorporation of widths into the Regge trajectory itself. In that analysis, the condition $t$7 is described as particularly stringent, and a fit that reproduces only $t$8 but fails in the imaginary part is not physically consistent (Collaboration et al., 2018).

For nonstrange baryons, the paper concludes that the majority of states in the best-established parent trajectories are compatible with a mostly compact three-quark picture, especially on the $t$9 nucleon parent trajectory and the $$f(k,\cos\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\left[\exp[2i\delta(l,k)]-1\right]P_l(\cos\theta),$$0 $$f(k,\cos\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\left[\exp[2i\delta(l,k)]-1\right]P_l(\cos\theta),$$1 parent trajectory, with slopes near $$f(k,\cos\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\left[\exp[2i\delta(l,k)]-1\right]P_l(\cos\theta),$$2 (Collaboration et al., 2018). At the same time, it identifies states such as $$f(k,\cos\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\left[\exp[2i\delta(l,k)]-1\right]P_l(\cos\theta),$$3 and likely $$f(k,\cos\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\left[\exp[2i\delta(l,k)]-1\right]P_l(\cos\theta),$$4 as requiring “physics beyond the compact $$f(k,\cos\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\left[\exp[2i\delta(l,k)]-1\right]P_l(\cos\theta),$$5 picture” because they do not fit the full complex Regge pattern well (Collaboration et al., 2018). A recurring theme is that exchange degeneracy is “very clearly broken” in the nonstrange sector, especially for $$f(k,\cos\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\left[\exp[2i\delta(l,k)]-1\right]P_l(\cos\theta),$$6, and that including widths can repair misleading conclusions from purely linear fits (Collaboration et al., 2018).

A related but distinct example is the $$f(k,\cos\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\left[\exp[2i\delta(l,k)]-1\right]P_l(\cos\theta),$$7-p elastic-scattering study, which argues that only the first three members of the sequence

$$f(k,\cos\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\left[\exp[2i\delta(l,k)]-1\right]P_l(\cos\theta),$$8

can be associated with ordinary Regge poles lying on a standard linear trajectory $$f(k,\cos\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\left[\exp[2i\delta(l,k)]-1\right]P_l(\cos\theta),$$9, with fitted line $k$0, while higher-$k$1 members are instead governed by a diffraction law $k$2 associated with Sommerfeld poles and creeping waves (Micheli et al., 2013). This makes explicit a common misconception: not every apparently aligned resonance sequence remains describable by one ordinary Regge trajectory over its full extent. The paper’s distinction between low-lying Regge poles and high-energy Sommerfeld poles is presented as a change of physical regime rather than a minor correction (Micheli et al., 2013).

3. Hadronic high-energy amplitudes and logarithmic Regge-pole variants

In standard hadronic Regge phenomenology, a simple pole gives

$k$3

so that with a linear trajectory $k$4 one obtains

$k$5

(Campos, 2020). The problem highlighted in the hadronic papers is that if the leading trajectory is identified with the pomeron and $k$6, then this power growth eventually conflicts with the Froissart–Martin bound

$k$7

(Campos, 2020).

The paper “An approach to the leading Regge pole” proposes a nonstandard modification in the forward region $k$8, $k$9, replacing the usual

$\theta$0

by the approximation

$\theta$1

(Campos, 2020). This heuristic step is then used to replace the standard asymptotic Legendre behavior by

$\theta$2

which yields a “logarithmic leading Regge pole”

$\theta$3

(Campos, 2020). In that scheme, the intercept $\theta$4 controls how close the total cross section is to saturating the Froissart–Martin bound: $\theta$5 gives $\theta$6 growth, while $\theta$7 gives a roughly logarithmic rise (Campos, 2020).

The fitted high-energy parametrization

$\theta$8

gives $\theta$9 for $\delta(l,k)$0 data above $\delta(l,k)$1 TeV and $\delta(l,k)$2 for combined $\delta(l,k)$3 and $\delta(l,k)$4 data above $\delta(l,k)$5 TeV, which the authors interpret as favoring a soft pomeron (Campos, 2020). They retain a linear $\delta(l,k)$6-trajectory $\delta(l,k)$7 with $\delta(l,k)$8, so the logarithm enters in the energy dependence, not in the trajectory itself (Campos, 2020). This suggests a useful terminological distinction: the proposal is a logarithmic realization of the leading Regge-pole contribution, not a logarithmic Regge trajectory.

The companion paper “Logarithmic Regge Pole” extends this framework by discussing subtractions and Regge cuts (Campos, 2020). It argues that a naive subtraction produces an unphysical $\delta(l,k)$9 suppression in

$P_l$0

but a softened subtraction effectively reduces back to the same asymptotic form as the unsubtracted logarithmic pole (Campos, 2020). It also adds a cut contribution

$P_l$1

to the amplitude, interpreting this as the contribution of non-leading exchanges in the transition region below asymptotic pomeron dominance (Campos, 2020). These papers are explicit that the construction is phenomenological and heuristic rather than a standard derivation from complex-$P_l$2 theory (Campos, 2020, Campos, 2020).

4. Black holes, compact objects, and surface-wave Regge poles

In black-hole and compact-object scattering, Regge poles reorganize partial-wave information into a small set of singular contributions in the complex angular-momentum plane and give a wave-based description of resonant scattering, absorption, and ringdown (Folacci et al., 2021). For Schwarzschild black holes, the Regge-pole viewpoint is tied to “surface waves” propagating near the photon sphere, and the associated trajectories control glory and orbiting structures in differential cross sections (Folacci et al., 2019).

For scalar and electromagnetic scattering by Schwarzschild, the exact scattering amplitude can be written as a background integral plus a Regge-pole sum,

$P_l$3

with

$P_l$4

for the scalar case (Folacci et al., 2019). A central numerical result is that for intermediate and high frequencies the background integral is numerically negligible, so the differential cross sections can be reconstructed almost entirely from the Regge-pole contribution (Folacci et al., 2019). The lowest poles already describe the black-hole glory, and increasing the number of poles reconstructs the orbiting oscillations over smaller angles (Folacci et al., 2019).

The corresponding high-frequency asymptotics are controlled by the photon sphere $P_l$5 and its critical impact parameter $P_l$6. For Schwarzschild absorption of scalar, electromagnetic, and gravitational fields, the total absorption cross section can be decomposed into real-axis and imaginary-axis backgrounds plus a Regge-pole sum,

$P_l$7

and the Regge poles satisfy the asymptotic expansion

$P_l$8

(Hadj, 27 Apr 2025). The paper’s interpretation is that the oscillatory absorption spectrum arises from interference of surface waves orbiting near the photon sphere, while the smooth background approaches the classical capture cross section $P_l$9 (Hadj, 27 Apr 2025).

A closely related analogue-gravity result appears in the canonical acoustic hole, where Regge poles are defined as poles of the $l$0-matrix in the first quadrant of the complex angular-momentum plane for real $l$1 (Dolan et al., 2014). There the leading high-frequency trajectory is

$l$2

with $l$3 the critical impact parameter of the unstable circular null orbit (Dolan et al., 2014). This again ties the real part of the pole to orbital phase accumulation and the imaginary part to instability of the orbit (Dolan et al., 2014).

Compact stars and dirty black holes exhibit a richer CAM structure. For constant-density stars, the scattering-matrix Regge spectrum contains distinct branches labelled surface waves, broad resonances, and, for ultracompact objects, a finite number of narrow resonances (Hadj et al., 2019). The broad-resonance branch is controlled by the discontinuity of the effective potential at the stellar surface; the WKB analysis yields a reflection coefficient $l$4 with

$l$5

and this directly determines the imaginary part of the broad-resonance poles (Hadj et al., 2019). For dirty black holes with a thin shell, the Regge spectrum can split into two or three distinct branches labelled inner surface waves, broad resonances, and outer surface waves, with the outer branch interpreted either as waves associated with an outer light ring or as creeping modes associated with the shell surface, depending on the shell position (Torres et al., 2022). These studies make clear that Regge poles in curved-spacetime scattering do not form a single universal family; instead, they track whichever unstable structures dominate the wave dynamics.

5. Regge poles, quasinormal modes, and spectral instability

In black-hole perturbation theory, Regge poles are the fixed-real-frequency counterparts of quasinormal modes: QNMs are poles in the complex-frequency plane at fixed integer angular momentum, whereas Regge poles are poles in the complex angular-momentum plane at fixed real frequency (Torres, 2023, Li et al., 17 Apr 2025). This duality can be used constructively. For Kerr black holes, the Regge poles $l$6 are defined by

$l$7

where $l$8 is the analytically continued incoming amplitude of the Teukolsky problem (Folacci et al., 2021). The central CAM-to-QNM condition is

$l$9

and in the weak-damping/eikonal regime this yields

$f(s,t)=C(t)\,s^{\alpha(t)}$00

(Folacci et al., 2021). A striking conclusion is that each Schwarzschild Regge pole splits into an infinite family $f(s,t)=C(t)\,s^{\alpha(t)}$01, $f(s,t)=C(t)\,s^{\alpha(t)}$02, in Kerr, explaining semiclassically the lifting of azimuthal degeneracy in the Kerr QNM spectrum (Folacci et al., 2021).

A separate line of work uses Regge poles to understand black-hole spectral instability. For a Schwarzschild black hole perturbed by a small far-out Gaussian bump, the paper “From black hole spectral instability to stable observables” finds that the Regge-pole spectrum is unstable: high overtones reorganize into original, inner, and outer branches (Torres, 2023). Yet the physically relevant scattering observables remain stable, differing from the unperturbed case only by $f(s,t)=C(t)\,s^{\alpha(t)}$03, because the unstable poles reorganize in a correlated way that preserves the CAM sums for the scattering amplitude and absorption cross section (Torres, 2023). This suggests a more general lesson: instability of individual spectral data need not imply instability of fixed-frequency observables.

The 2025 study of black-hole metrics with discontinuity sharpens this observation by computing Regge poles for discontinuous ultraviolet perturbations of the Regge–Wheeler potential (Li et al., 17 Apr 2025). It defines Regge poles as poles of the analytically continued reflection amplitude $f(s,t)=C(t)\,s^{\alpha(t)}$04, equivalently zeros of the Wronskian

$f(s,t)=C(t)\,s^{\alpha(t)}$05

after continuation in $f(s,t)=C(t)\,s^{\alpha(t)}$06 (Li et al., 17 Apr 2025). The main result is that low-lying Regge poles remain much more stable than the QNM spectrum over the frequency range relevant to greybody factors and absorption observables, while higher-frequency sectors can undergo bifurcation without large observable consequences because the background and first low-lying pole dominate (Li et al., 17 Apr 2025).

6. Inverse problems and modern field-theoretic extensions

Regge poles also enter inverse spectral theory. For radial Schrödinger operators on $f(s,t)=C(t)\,s^{\alpha(t)}$07, the fixed-energy scattering problem can be formulated in the complex angular-momentum variable $f(s,t)=C(t)\,s^{\alpha(t)}$08, with generalized phase shifts defined by

$f(s,t)=C(t)\,s^{\alpha(t)}$09

(Daudé et al., 2015). In that setting, Regge poles are the zeros of the Jost function $f(s,t)=C(t)\,s^{\alpha(t)}$10, i.e. poles of the meromorphic interpolation $f(s,t)=C(t)\,s^{\alpha(t)}$11, and they lie in the first quadrant (Daudé et al., 2015). The paper proves that for nonzero super-exponentially decreasing potentials there are infinitely many Regge poles and they cannot remain in any vertical strip in the right half-plane, whereas for analytic short-range potentials they are confined to a vertical strip (Daudé et al., 2015). For compactly supported potentials, the poles asymptotically satisfy

$f(s,t)=C(t)\,s^{\alpha(t)}$12

showing concentration near the positive imaginary axis (Daudé et al., 2015). These localization results are then used to derive global and local fixed-energy inverse-scattering uniqueness theorems (Daudé et al., 2015).

An even more geometric inverse problem appears on warped balls, where Regge poles are defined as poles of the meromorphic continuation of the fixed-energy Dirichlet-to-Neumann map in the complex angular-momentum variable $f(s,t)=C(t)\,s^{\alpha(t)}$13 (Borthwick et al., 2022). After separation of variables, the problem reduces to a half-line Schrödinger operator

$f(s,t)=C(t)\,s^{\alpha(t)}$14

and the Regge poles are precisely the poles of the continued Weyl–Titchmarsh function

$f(s,t)=C(t)\,s^{\alpha(t)}$15

(Borthwick et al., 2022). The spectrum splits into two asymptotic families,

$f(s,t)=C(t)\,s^{\alpha(t)}$16

with $f(s,t)=C(t)\,s^{\alpha(t)}$17 and explicitly localized complex $f(s,t)=C(t)\,s^{\alpha(t)}$18 determined by the support endpoint and jump structure of the compact perturbation (Borthwick et al., 2022). The paper’s inverse theorem states that the full Regge-pole set uniquely determines the effective potential $f(s,t)=C(t)\,s^{\alpha(t)}$19, and with boundary Cauchy data also determines the warping function itself (Borthwick et al., 2022). This broadens the Regge-pole concept beyond scattering matrices to boundary-value inverse problems.

In perturbative QCD, Regge poles arise in the complex $f(s,t)=C(t)\,s^{\alpha(t)}$20-plane after Mellin transform of high-energy amplitudes. The 2024 review “Regge poles and cuts and the Lipatov vertex” defines a Regge pole as a simple pole of the partial-wave coefficient $f(s,t)=C(t)\,s^{\alpha(t)}$21, leading to large-$f(s,t)=C(t)\,s^{\alpha(t)}$22 behavior controlled by a trajectory $f(s,t)=C(t)\,s^{\alpha(t)}$23 associated with the Reggeized gluon (Abreu et al., 2024). The paper’s emphasis, however, is that modern high-energy QCD requires a clean separation of true Regge-pole contributions from Regge cuts generated by multi-Reggeon exchange (Abreu et al., 2024). In the odd-signature $f(s,t)=C(t)\,s^{\alpha(t)}$24 amplitude, single-Reggeon exchange gives the pole term, while triple-Reggeon exchange first produces Reggeization violation at NNLL, interpreted as a Regge cut (Abreu et al., 2024). This modern perspective modifies an older misconception that gluon Reggeization alone is the whole story: in non-planar QCD, multi-Reggeon exchange and cut singularities are structurally unavoidable (Abreu et al., 2024).

A related but different no-go result appears in the analysis of meromorphic dual amplitudes. That paper argues, on the basis of crossing symmetry and unitarity, that meromorphic amplitudes with a finite number of Regge trajectories cannot reggeize, and the authors argue this excludes the existence of such amplitudes altogether (Eckner et al., 2024). This suggests that in dual or string-like amplitudes, the presence of infinitely many Regge trajectories is not merely conventional but structurally necessary (Eckner et al., 2024).

Regge poles therefore occupy a distinctive position in theoretical physics. They are singularities of analytically continued partial-wave data, but they are not tied to one physical interpretation. In hadron phenomenology they organize trajectories and asymptotics; in black-hole and compact-object scattering they encode surface waves, interface resonances, and trapped modes; in inverse problems they become spectral data of continued Weyl–Titchmarsh or Dirichlet-to-Neumann maps; and in perturbative QCD they define the pole sector whose disentanglement from Regge cuts is necessary for a precise high-energy description (Bottino, 2018, Hadj, 27 Apr 2025, Borthwick et al., 2022, Abreu et al., 2024). A plausible implication is that the enduring value of the Regge-pole concept lies less in any single phenomenological model than in its role as a unifying analytic language for open, resonant, and asymptotic scattering systems.