The scaling Pomeron

Abstract: We examine the Regge theoretical properties for the scaling observed in pp elastic scattering differential cross-sections at the LHC. A positive signature amplitude (i.e. the Pomeron) with scaling properties has been derived. It is found to describe the dip-bump region of momentum transfer at LHC energies in agreement with data. We derive the analytic continuation in the whole plane of the t-channel partial waves of index $l_t$ specific to the Regge formalism. The analytic form of the amplitude exhibits a specific scaling property without singularities, except for a series of poles in the $l_t$ real axis at fractional values.

• The paper establishes a scaling Pomeron amplitude from Regge theory to describe elastic pp scattering at LHC energies.
• It employs a scaling variable that collapses normalized differential cross-section data onto a universal curve in the dip-bump region.
• The analysis shows that while the positive signature component fits most data, additional negative signature contributions hint at Odderon effects.

The Scaling Pomeron in Elastic Proton-Proton Scattering

Introduction

This work analyzes the scaling behavior observed in the elastic differential cross-section of proton-proton ($pp$) scattering at LHC energies through the lens of Regge theory, focusing specifically on amplitudes with positive signature—the Pomeron contribution. Motivated by recent phenomenological evidence for scaling in the dip-bump region of the LHC $pp$ elastic differential cross-section, the study establishes an analytic Regge framework for such scaling, providing a formal construction of the "scaling Pomeron" amplitude and detailing its analytic properties in the complex angular momentum ($l$) plane.

Scaling Properties and Empirical Foundations

Recent experimental analyses of the differential cross-section for elastic $pp$ scattering at LHC energies by the TOTEM collaboration have revealed a striking scaling property: once normalized by appropriate powers of $s$ (the center-of-mass energy squared) and $t$ (the negative squared momentum transfer), the data collapse onto a universal curve as a function of a combined scaling variable $t^{**}(s, t) = (s/\text{TeV}^2)^{0.065} |t|^{0.72}$ [(2604.01822), Phys. Lett. B 830 (2022), 137141]. Formally, the scaled differential cross-section satisfies

$$\left( \frac{s}{\text{TeV}^2} \right)^{-0.305} \frac{d\sigma^{pp}}{dt}(s, t) = f(t^{**}),$$

with $f$ a universal function, and

$$\tau = t \left( \frac{s}{\text{TeV}^2} \right)^{0.18},$$

serving as the scaling variable in the amplitude parameterization.

A successful phenomenological description of the data in the dip-bump region is achieved with amplitudes of the structure

$pp$0

with each $pp$1 a scaling exponential in $pp$2, and fit parameters adjusted empirically.

Analytic Derivation in Regge Formalism

Regge Signatures and the S-Matrix

The study formalizes the scaling ansatz within Regge theory. The $pp$3 amplitude is decomposed into positive and negative signature components, as prescribed by the analytic $pp$4-channel partial waves continuation:

$pp$5

where $pp$6 selects the Pomeron (positive signature) and $pp$7 the Odderon (negative signature). The amplitude's $pp$8-dependence sets a strict phase-energy relation, with the analytic continuation yielding $pp$9 for the relevant signature.

Construction of the Scaling Pomeron

Translating the empirical scaling ansatz to the Regge framework, the positive-signature (Pomeron) amplitude is written

$l$0

where the parameters are redefined to absorb the Regge phase structure and ensure compatibility with crossing and analyticity requirements of the S-matrix. Best-fit parameters yield performance comparable to direct empirical fits of the scaling form, showing the Regge implementation captures the essential scaling dynamics.

Analytic Structure in the $l$1 Plane

The partial wave expansion of the scaling amplitude is systematically constructed by Taylor expansion and then represented as a sum over residues, making explicit the analytic continuation in the complex $l$2-plane. The crucial result is that the scaling form leads to pole singularities only at positive integer values of $l$3, with no other singularities (neither standard Regge poles nor cuts) present in the analytic structure. The absence of additional singularities is characteristic of the scaling form; physically, this structure is predominant at moderate momentum transfer, and the connection to the full amplitude and its singularities at larger or smaller $l$4 remains open.

The analytic properties are encapsulated in expressions involving the (upper) incomplete Gamma function $l$5, which ensures analytic behavior in both the complex angular momentum and momentum transfer variables. Explicitly, the Pomeron amplitude can be recast as an integral over the complexified variable parameterizing the involved exponents, producing an analytic, scaling amplitude in the Regge formalism.

Positive and Negative Signature Decomposition and Phenomenology

Application of the formalism to LHC data reveals that the positive signature component alone (scaling Pomeron) suffices to describe most of the data except in the dip region, where a negative signature (Odderon-like) contribution becomes necessary. By decomposing the amplitude into its signature components and comparing with the data, the analysis provides evidence for the existence of both signatures, aligning with previous empirical indications of Odderon effects in $l$6 and $l$7 elastic scattering [Phys. Rev. Lett. 127, 062003].

Theoretical and Practical Implications

This analysis strengthens the connection between empirical scaling laws in hadronic scattering at LHC energies and the underlying analytic S-matrix (Regge) theory, specifically through the construction of scaling amplitudes with well-defined signature and analytic structure. The exclusivity of simple pole singularities in the $l$8-plane offers insight into the analytic organization of the amplitude in the scaling regime. The result demonstrates the capability of the Regge approach to describe modern high-energy elastic data through minimal analytic assumptions, avoiding the complexities of specific phenomenological models.

Future theoretical developments should focus on:

• Clarifying the dynamical mechanism in QCD responsible for the emergence of the scaling Pomeron and its analytic properties.
• Extending the analytic framework to describe the transition to larger $l$9, where non-scaling behavior and additional singularities are expected.
• Further empirical studies to isolate and fit negative signature contributions, enabling quantitative extraction of Odderon effects in accordance with Regge predictions.

Conclusion

The work rigorously establishes a Regge-theoretical foundation for the observed scaling behavior in elastic $pp$0 scattering, constructing analytic scaling Pomeron amplitudes with positive signature. These amplitudes accurately account for the dip-bump region data and display a distinctive singularity structure limited to integer pole singularities in the complex angular momentum plane. The analysis underlines the necessity of a complementary negative signature component to fully describe the observed cross-section profile, motivating further theoretical and experimental explorations of signature structure in hadron-hadron scattering at high energies.