Regge Logarithms in High-Energy Scattering

• Regge logarithms are towers of large logarithms arising from rapidity-ordered emissions and ladder diagrams in high-energy scattering.
• Their resummation via BFKL evolution and exponential factorization provides precise control over Regge poles, cuts, and infrared singularities.
• Extensions to gravity, AdS/CFT, and phenomenological models underscore the universal impact of these logarithms on unitarity and analytic amplitude structures.

Regge logarithms are the towers of large logarithmic terms that systematically arise in the high-energy (Regge) limit of quantum field theory scattering amplitudes, specifically in processes such as $2 \to 2$ QCD scattering with $s \gg |t|$. These logarithms take the form $\alpha_s^\ell L^m$ with $m \leq \ell$ and $L$ a kinematic logarithm related to rapidity or energy, and control the dominant energy growth of amplitudes via the exchange of Reggeized particles, the exponentiation of high-energy singularities, and the structure of multi-Reggeon contributions. Their resummation to all orders is central for precision calculations in both gauge theory and gravity, providing deep insights into factorization, color structure, and the interplay of infrared and high-energy dynamics.

1. Definition and Origin of Regge Logarithms

Regge logarithms emerge in the high-energy limit of scattering, characterized by large center-of-mass energy $s$ and fixed, negative momentum transfer $t$ ($s \gg |t| > 0$). The natural logarithmic variable is

$$L \equiv \tfrac{1}{2}[\ln(-s-i0/(-t)) + \ln(-u-i0/(-t))] \simeq \ln|s/t| - i\pi/2.$$

At loop order $\ell$, amplitudes develop towers of terms $\alpha_s^\ell L^m$ with $m \leq \ell$. The highest power, $m=\ell$, corresponds to the leading logarithm (LL), while subleading towers correspond to next-to-leading logarithms (NLL), NNLL, etc.

Physically, Regge logarithms originate from phase space regions with strong rapidity ordering of emitted partons—multi-Regge kinematics (MRK)—and from ladder-like Feynman diagrams. Each rung of the ladder yields a factor of $\ln(s/|t|)$, making the $L$-powers in amplitudes directly traceable to such emissions (Caron-Huot et al., 2017, Fadin et al., 2015).

These structures also emerge in the context of effective field theory as rapidity divergences, associated with loop integrations over kinematic regions of large rapidity separation, and are encoded in the evolution equations for Reggeized propagators and impact factors (Nefedov et al., 2016).

2. Factorization, Reggeization, and Amplitude Structure

In $2\to2$ amplitudes, the Regge limit induces a factorized structure: $$\mathcal{Q}^{(-)}(s,t) \simeq e^{T_t^2\,\alpha_g(t)\,L}\,\mathcal{Q}^{(-),\,\mathrm{tree}}(t),$$ where $T_t^2$ is the quadratic Casimir in the $t$-channel, and $\alpha_g(t)$ is the gluon Regge trajectory. Here, $\mathcal{Q}^{(-)}$ is the signature-odd (real) part, dominated by single-Reggeon exchange. The exponentiation in $L$ encapsulates the resummation of all leading Regge logarithms. The signature decomposition, distinguishing between signature-odd and even (real/imaginary) parts, underpins the entire organization of the logarithmic hierarchy (Caron-Huot et al., 2017).

Defining the reduced amplitude by dividing out the exponentiated one-Reggeon (Regge pole) factor,

$$\widehat M(s,t)\equiv e^{-T_t^2\,\alpha_g(t)\,L}\,\mathcal{Q}(s,t),$$

the remaining structure is governed by two- and multi-Reggeon dynamics, with Regge cuts in the even (signature) sector representing the leading irreducible Regge logarithms beyond single-pole contributions (Caron-Huot et al., 2017, Falcioni et al., 2021).

In gauge theories, the all-order expansion for the amplitude is

$$A(s,t) = A^{(0)}(s,t)\left[1+\sum_{\ell=1}^\infty (g^2)^\ell \sum_{k=0}^{\ell} c_{\ell,k}\,L^k\right],$$

where $c_{\ell,k}$ are process- and scheme-dependent. At each loop, the maximum $L$-power is given by $\ell$ (LL), followed by lower $L$ exponents (NLL, NNLL, ...). The structure is universal in the logarithms but theory-specific in the trajectory, impact factors, and vertex corrections (Duca et al., 2022, Fadin et al., 2015).

3. BFKL Evolution, Multi-Reggeon Dynamics, and Closed-Form Results

The resummation of Regge logarithms is governed by the Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation, which describes the evolution of the two-Reggeon wavefunction $\Omega(p,k)$ in transverse momenta: $$\Omega^{(\ell-1)}(p,k) = \hat H \,\Omega^{(\ell-2)}(p,k),$$ with explicit Hamiltonians depending on the color structure of gluon exchanges. At leading power in the "soft Reggeon" limit ($k\ll p$), the evolution closes onto wavefunctions of $\xi=p^2/k^2$, enabling derivation of an all-orders analytic polynomial for $\Omega^{(\ell-1)}$ (Caron-Huot et al., 2017).

The even (signature) part of the reduced amplitude, capturing the Regge cut, can be compactly written to all orders (in the soft limit) as

$$\widehat M^{(+)}|_{soft} = \frac{i\pi}{L(T_t^2-C_A)} \left(1 - R(\epsilon)\frac{C_A}{T_t^2 - C_A}\right)^{-1}\{\exp[\frac{B_0(\epsilon)}{2}\frac{\alpha_s}{\pi}L(T_t^2-C_A)] - 1\}M + O(\epsilon^0),$$

where $R(\epsilon)$ and $B_0(\epsilon)$ parameterize IR singularities. This closed-form provides explicit control over the full tower of Regge logarithms in the two-Reggeon sector (Caron-Huot et al., 2017).

Infrared singularities in all color representations and for any logarithmic accuracy are captured by such evolution equations, with closed expressions for the leading IR poles and the soft anomalous dimension at any loop order (Duca et al., 2011, Falcioni et al., 2021).

4. Infrared Singularities, Anomalous Dimensions, and Breakdown of Reggeization

The exponentiation of Regge logarithms is fundamentally connected to the structure of infrared (IR) singularities in gauge theory amplitudes. The soft anomalous dimension operator $\Gamma$ can be decomposed as

$$\Gamma = \Gamma_{LL} + \Gamma_{NLL}^{(+)} + \Gamma_{NLL}^{(-)} + \cdots,$$

with signature-even and signature-odd parts controlling the Regge cut and pole sectors, respectively. The LL terms are governed by the dipole formula, encoding color-dipole correlations only (Duca et al., 2011, Maher et al., 2021).

Reggeization holds universally for the divergent (pole) parts at LL accuracy in any $t$-channel representation. At NLL, color mixing—manifested by non-commuting color operators—spoils simple polewise Reggeization in the imaginary part. At NNLL, non-dipole structures and higher commutator terms produce explicit breakdowns in the real part; these effects are subleading and become pronounced at three loops and beyond (Duca et al., 2011, Falcioni et al., 2021, Maher et al., 2021).

Four-loop analyses reveal that the pure non-planar, non-dipole corrections only appear at NNLL, consistent with the appearance of strictly non-planar Regge cuts in multi-Reggeon sectors (Falcioni et al., 2021, Maher et al., 2021).

5. Extensions: Multi-Regge Kinematics, Higher Multiplicities, and Polylogarithmic Structure

In multi-leg ($2\to n$) amplitudes and multi-Regge kinematics (MRK), Regge logarithms organize the entire structure of the amplitude through the exponential of sums over rapidity intervals and Casimir operators in corresponding $t$-channel representations: $$Z_{MR}(\{\Delta y_k\}) = \exp\left[ K(\alpha_s, \epsilon) \left(\sum_{k=1}^{n-1} \Delta y_k\,\mathbf{T}_{t_k}^2 + i\pi\,\mathbf{T}_s^2 \right)\right],$$ where each rapidity interval's color structure and logarithms accumulate independently (Duca et al., 2011, Duca et al., 2016).

Multi-Regge limit amplitudes in $\mathcal{N}=4$ SYM are controlled by single-valued multiple polylogarithms (SVMPs), providing algebraic and analytic structures ideally suited for resummation and analytic continuation (Broedel et al., 2016, Duca et al., 2016). The leading-log LLA coefficients at $L$ loops are weight-$L$ SVMPs, with higher multiplicity amplitudes admitting iterative convolution structures entirely in terms of such functions.

This single-valuedness ensures the absence of unphysical branch cuts in physical regions—an essential property for consistent analytic continuation and amplitude reconstruction (Broedel et al., 2016). Closed-form recursive and convolution algorithms for all MHV and numerous non-MHV amplitudes at high loop orders have been established (Duca et al., 2016).

6. Generalizations: Gravity, AdS, and Phenomenological Implications

Regge logarithms are not limited to gauge theory. In gravity, double-logarithmic towers $\propto [\alpha G_N|q|^2 \ln^2 s]^n$ and corresponding resummations have been obtained, with all-order predictions for their structure (Bartels et al., 2012). Supersymmetry greatly modifies the asymptotic behavior, with $\mathcal{N}=4$ supergravity exhibiting complete cancellation of double logs, and $\mathcal{N} \geq 6$ realizing high-energy damping due to destructive interference in multi-Regge diagrams.

In AdS/CFT, the Regge-limit of Virasoro–Shapiro amplitudes yields derivatives of the flat-space Regge amplitude, with leading Regge logs exponentiating at each order in the $1/R^2$ expansion, and higher-order terms built from single-valued logarithms and derivatives, reflecting the conformal data of the leading Regge trajectory (Alday et al., 5 Sep 2024).

In QCD phenomenology, practical parameterizations incorporating logarithmic Regge poles and cuts, such as $\ln^{\alpha(t)}(s/s_c)$, are used to fit total cross-section data up to the highest collider energies, reconciling Regge theory with the Froissart–Martin bound and obtaining quantitative predictions for observables like $\rho(s)$ and elastic slopes (Campos, 2020).

7. Summary of Universal Features and Key Results

• Regge logarithms universally organize the high-energy behavior of scattering amplitudes, arising from rapidity-ordered emissions in ladder topologies and controlled by the exponentiation of associated trajectories and anomalous dimensions (Caron-Huot et al., 2017, Duca et al., 2022, Gao et al., 14 Nov 2024).
• The leading-logarithmic towers are resummed by BFKL-type evolution equations; even and odd signature sectors (Regge poles and cuts) are controlled by specific color and kinematic operators.
• Closed-form and all-orders results for Regge-logarithmic contributions to the soft anomalous dimension now exist for the full tower of $\epsilon$-poles in the IR singular sector (Caron-Huot et al., 2017), with explicit inverse-Borel representations for the generating functions.
• Multi-Regge amplitudes at high multiplicity and loop order are determined by recursively constructing single-valued multiple polylogarithms; the algebraic structure and convolutions are explicitly realized in MRK (Duca et al., 2016, Broedel et al., 2016).
• The breakdown of simple Reggeization at NLL/NNLL is now systematized, with non-dipole, non-planar, and multi-Reggeon effects explicitly identified and fully characterized through four loops (Falcioni et al., 2021, Maher et al., 2021).
• The phenomenological approach incorporating logarithmic Regge poles and cuts provides a consistent and accurate framework for modeling the high-energy behavior of cross sections, accounting for unitarity and analyticity requirements (Campos, 2020).

These advances collectively establish Regge logarithms as the central organizing principle for high-energy analytic structures in both QCD and related conformal, supersymmetric, or gravitational frameworks, with all-order resummations, analytic closed forms, and explicit phenomenological applications across quantum field theory (Caron-Huot et al., 2017, Fadin et al., 2015, Duca et al., 2022, Falcioni et al., 2021, Campos, 2020, Broedel et al., 2016, Duca et al., 2016, Costa et al., 2013, Gao et al., 14 Nov 2024, Alday et al., 5 Sep 2024).