Regge Zeros in Scattering and CFT

• Regge zeros are defined as the points where analytic scattering amplitudes vanish, elucidating dip structures in cross sections.
• They play a critical role in meson exchange processes and in decoupling analytic families of operators in conformal field theories.
• Mechanisms such as signature factors, orthogonality conditions, and thermal corrections precisely determine the locations and dynamics of Regge zeros.

Regge zeros are loci—often in the kinematic or spectral parameter space—where Regge amplitudes or analytic continuations of fundamental scattering quantities vanish, leading to dip structures in cross sections or enforcing decoupling of analytic families in the context of conformal field theory. The phenomenon is central to Regge theory, with deep implications for scattering amplitudes, partial-wave expansions, and the operator organization in quantum field theories, both relativistic and conformal. Their mathematical realization is closely tied to the analytic continuation in angular momentum, signature factors, orthogonality structures, and the interplay of physical and spurious singularities in spectral representations.

1. Mathematical Definition and Physical Context

The classic Regge framework interprets scattering amplitudes as analytic functions of complex angular momentum $J$ (or equivalently, %%%%1%%%%, $t$ variables in particle physics). Regge zeros arise as the vanishing points of these analytic functions, often due to the structure of the so-called signature factor $$\kappa(\tau, \alpha) = \frac{1}{2}(1 + \tau e^{-i\pi \alpha})$$, where $\tau$ denotes the signature and $\alpha = \alpha(t)$ is the trajectory function. The condition

$$\kappa(\tau, \alpha(t)) = 0 \quad \Longrightarrow \quad 1 + \tau e^{-i\pi \alpha(t)} = 0$$

defines Regge zeros, leading to observable dips in cross-sections—particularly "wrong-signature zeros" in reactions dominated by meson exchange (e.g., $\pi^- p \to \pi^0 n$ with $\rho$ exchange) (Mathieu, 2013).

In advanced settings, such as conformal field theories, Regge zeros generalize to vanishing of OPE coefficients or residues at certain values of the spin $J$ upon analytic continuation. For example, infinite analytic families (Regge trajectories) in conformal theories must decouple at small integer spins, which is enforced by zeros in structure constants or residues, accomplished via mechanisms involving the analytic properties of two-point functions of light-ray operators (Homrich et al., 2022, Henriksson et al., 2023).

2. Mechanisms of Regge Zeros in Scattering Theory

In high-energy hadron scattering, Regge zeros originate from the vanishing of the signature factor in the context of Regge pole amplitudes. For a single Regge pole,

$$R(\nu, \alpha, \tau) = \beta(t) \kappa(\tau, \alpha) \Gamma(j_0 - \alpha) \left(\frac{\nu}{\nu_0}\right)^{\alpha}$$

where $\beta(t)$ encapsulates the coupling structure and $\nu$ is the crossing variable. Zeros occur when $\tau e^{-i\pi \alpha(t)} = -1$. Physical cross-section data—such as the dip at $-t \sim 0.55$ GeV$^2$ in $\pi^- p \to \pi^0 n$—confirm this mechanism for the $\rho$ trajectory (Mathieu, 2013).

Practically, observable dips are typically filled by subdominant contributions ("Regge cuts") parameterized similarly but with logarithmic suppression:

$$R_c(\nu, \alpha, \tau) = \beta(t) [\log(\nu/\nu_0)]^{-1} \kappa(\tau, \alpha) \Gamma(j_0 - \alpha) \left(\frac{\nu}{\nu_0}\right)^{\alpha}$$

This interference between poles and cuts prevents strict zeros in measured cross sections (Mathieu, 2013).

3. Analytic Continuation and Regge Zeros in CFT OPE Data

In conformal field theory and Regge analysis, OPE coefficients and scaling dimensions are continued to complex spin $J$ (Homrich et al., 2022, Henriksson et al., 2023). Physical local operators correspond to discrete spins; however, Regge theory requires family organization into analytic trajectories over general (possibly complex) $J$. Higher Regge families decouple at missing spins through zeros in OPE structure constants, achieved via Newton interpolation series:

$$f_N(z) = \sum_{j=0}^N \binom{z}{j} \sum_{i=0}^j (-1)^{j-i} \binom{j}{i} f(i)$$

where $f(i)$ is the data at integer spin, and $z$ may be complex (Homrich et al., 2022). For instance, twist-3 operators in planar $\mathcal N=4$ SYM show precisely vanishing structure constants at spins where local operators are absent, enforcing decoupling of analytic families—these are "missing Regge zeros."

A general mechanism (illustrated in Wilson-Fisher CFT) is that the residue at a Regge pole takes the form:

$$r_i(J) \propto [1 - e^{-i\pi J}] \times \text{(matrix element)}^2 \times (\text{two-point structure})^{-1}$$

At even integer $J$, unless canceled by two-point zeros, the prefactor vanishes, ensuring only genuine local operators contribute (Henriksson et al., 2023).

4. Orthogonality, Weight Functions, and Spectral Zeros

In reggeon field theory (Intissar, 2014), spectral analysis focuses on the zeros of polynomials arising from three-term recurrences tied to the Gribov operator:

$$a_{n-1} P_{n-1}(z) + B_n P_n(z) + a_n P_{n+1}(z) = z P_n(z), \quad P_0 = 0, \quad P_1 = 1$$

Eigenvalues $\{z_{k,n}\}$ corresponding to polynomial zeros are Regge zeros, and their distribution is controlled by a bounded variation weight function $\xi(z)$ such that:

$\int P_m(z) P_n(z) d\xi(z) = \delta_{mn}$

Spectral localization and asymptotics of these zeros reflect the underlying physical couplings (intercepts, triple couplings) in reggeon field theory.

5. Regge Zeros in Mellin Space and Conformal Regge Theory

In the Mellin representation for CFT correlators (such as in Fishnet theories (Chowdhury et al., 2019), and Conformal Regge theory (Caron-Huot et al., 2020)), Regge poles and zeros are identified via analytic continuation and contour deformation. The dominant behavior in the Regge limit is governed by solutions to pole equations such as:

$1 - \chi_n E^{(n)}_{2+i, J(\nu)} = 0$

Regge zeros emerge from spurious singularities canceling in specific linear combinations of conformal blocks (the Regge block), constructed to enforce vanishing of non-physical discontinuities:

$$\text{Disc}_{23} R^{(a,b)}_{\Delta, J}(z, \bar{z}) = 0$$

This ensures only physical Regge exchanges survive and the resummed OPE is both convergent and causal (Caron-Huot et al., 2020).

The precise location and multiplicity of Regge zeros have consequences for interference between trajectories, phenomena such as oscillatory behavior in cross-channel amplitudes, and the analytic structure of the Mellin amplitude at strong and weak coupling.

6. Applications in Hadron Spectroscopy and Quantum Geometry

Regge zeros play a fundamental role in hadron spectroscopy, e.g., in the assignment of masses and quantum numbers for strange baryons (Oudichhya et al., 2022), kaons, and strangeonium (Oudichhya et al., 2023). Linear relations in trajectories:

$J = a(0) + \alpha' M^2,$

with intercept and slope parameters determined via additivity and consistency constraints, are sensitive to the presence of zeros as critical points dictating where new resonances may emerge or production is suppressed.

In quantum geometry, as in applications to the Ising model and the 6j-symbol (Bonzom et al., 2019), Fisher zeros (zeros of the partition function in the complex parameter plane) are linked geometrically to extrinsic dihedral angles and triangle angles in a tetrahedron through formulas such as:

$$Y_e = \exp(i\theta_e / 2) \sqrt{\tan(\phi_{1(e)}/2)\tan(\phi_{2(e)}/2)}$$

This geometrization of Regge zeros connects statistical physics, quantum gravity amplitudes, and discrete geometry.

7. Thermal Corrections and Evolution of Regge Zeros

At finite temperature, Regge zeros and trajectories are subject to thermal corrections via resummed thermal ladder diagrams (Cadiz et al., 17 Jun 2025). In the $\lambda\phi^3$ model, the thermal Regge trajectory is

$\alpha(t, \beta) = K(t) - 1 + B(t, \beta)$

where $B(t, \beta)$ encodes thermal modifications from Matsubara summation. Thermal effects compress the Regge slope, moving the positions of Regge zeros and hence shifting hadronic masses as temperature increases. The quantitative effect is a temperature-dependent evolution in both slope and intercept, which directly impacts resonance spectra in heavy-ion collisions and hot hadronic matter.

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Regge zeros, in all their manifestations, serve as the analytic gatekeepers determining which trajectories contribute physically, how interference and suppression arise in amplitudes, and how analytic continuation in spin organizes the observable operator spectrum in both particle and conformal field theory. Their realization—through signature factors, orthogonality relations, interpolation series, and dynamical thermal corrections—is central to modern understanding of analytic and algebraic structures in quantum field theory and spectral theory.