Pomeron and Odderon: High-Energy Scattering

• Pomeron and Odderon are key Regge singularities defined by C=+1 and C=-1 respectively, controlling high-energy scattering amplitude behavior.
• They are interpreted in QCD as gluonic compound states, with the Pomeron linked to two-gluon exchange and the Odderon to three-gluon exchange.
• Their interplay in hadronic collisions influences total cross sections and particle-antiparticle asymmetries, with observable signals at the LHC.

The Pomeron and Odderon are fundamental Regge singularities that control the high-energy behavior of hadronic scattering amplitudes, each characterized by distinct charge-conjugation and signature quantum numbers. The Pomeron, with $C=+1$ and positive signature, governs the rise of total cross sections and the soft inclusive particle production at high energies. The Odderon, with $C=-1$ and negative signature, encodes the leading crossing-odd exchange, potentially producing observable differences between particle and antiparticle processes such as $pp$ vs. $p\bar{p}$ scattering. Both objects are realized as Regge poles or cuts, interpreted in QCD as exchanges of gluonic compound states, and their phenomenology is tightly linked with the analytic properties of scattering amplitudes, the constraints of unitarity, and the symmetry structure of QCD and Regge field theories.

1. Regge Theory: Definitions and General Properties

In the framework of Regge theory, hadronic scattering amplitudes at high center-of-mass energies $s$ and small $|t|$ are controlled by singularities (poles and cuts) in the complex angular momentum ($j$) plane. Each singularity corresponds to a family of exchanged states (Reggeons), and the contribution of a simple Regge pole $\alpha(t)$ leads to amplitudes scaling as $s^{\alpha(t)}$:

• The Pomeron is the leading $C=+1$ Regge singularity with vacuum quantum numbers, responsible for the asymptotic growth of total cross sections ($\alpha_\mathbb{P}(0) > 1$).
• The Odderon is the leading $C=-1$ Regge singularity with negative signature, theoretically expected to have an intercept close to unity ($\alpha_\mathbb{O}(0) \simeq 1$) and to contribute with opposite sign in $pp$ vs. $p\bar{p}$ reactions.

The phenomenological parameterizations at high energy for the total cross section $\sigma_{\rm tot}(s)$ and the ratio of real to imaginary forward amplitudes $\rho(s)$ can be organized as follows:

Contribution	Quantum Numbers	Intercept $\alpha(0)$	Signature	Scattering Observable
Pomeron ($\mathbb{P}$)	$C=+1$, vacuum	$>1$	Even	$\sigma_{\rm tot}(s)\sim s^{\Delta}$, central densities
Odderon ($\mathbb{O}$)	$C=-1$	$\approx 1$	Odd	Particle-antiparticle differences, real part $\rho$ at $t\rightarrow0$

The signature factor for negative signature ensures that Odderon contributions flip sign between $pp$ and $p\bar{p}$ amplitudes.

2. QCD Interpretation and Dynamical Models

In QCD, the Pomeron and Odderon have a partonic interpretation as compound states of reggeized gluons:

• The Pomeron is associated with the exchange of a colorless two-gluon state (perturbative BFKL Pomeron in the pQCD regime, with intercept $j\simeq1+{\cal O}(\alpha_s)$), which at strong coupling (via AdS/CFT) is mapped to a reggeized graviton (Brower et al., 2013, Brower et al., 2014).
• The Odderon is linked to the colorless three-gluon state with negative $C$-parity, known as the BKP odderon (Bartels-Kwieciński-Praszałowicz equation) (Braun et al., 2020). Its intercept remains close to unity, with running coupling and multigluon interactions producing only mild deviations (Braun et al., 2020, Brower et al., 2014).

Dynamical holographic models, such as the dynamical softwall (DSW), provide explicit spectra for glueballs (even-$J$ states, Pomeron trajectory) and oddballs (odd-$J$, Odderon trajectory), generating nearly linear Regge trajectories in $(J,M^2)$ and predicting glueball and oddball masses compatible with lattice and phenomenological expectations (Capossoli et al., 2016, Szanyi et al., 2019). The tensor Pomeron (spin-2) and vector Odderon (spin-1) approach, and the spin-3 tensor Odderon hypothesis (Magallanes et al., 2022), are effective realizations matched to QFT and QCD-inspired formalisms.

3. Phenomenological Consequences at the LHC

At large $s$, the dominant contribution to total, elastic, and inelastic $pp$ cross sections, as well as central inclusive particle densities, is described by the Pomeron exchange (Merino et al., 2010). The total cross section receives a leading Pomeron term,

$$\sigma_{\rm P}(s) = 8 \pi \gamma \,\exp(\Delta \ln(s/s_0)), \quad \Delta = \alpha_\mathbb{P}(0) - 1,$$

and in the quasi-eikonal approach,

$$\sigma_{\rm tot}\sim (8\pi \alpha'_{\mathbb{P}} \Delta/C)\,\ln^2(s/s_0).$$

Inclusive midrapidity densities scale as $(s/s_0)^{\Delta}$, dominated by single Pomeron cuts (AGK rules).

Odderon effects appear as subdominant corrections, producing differences in particle/antiparticle ratios at midrapidity and in certain elastic scattering observables. A characteristic observable is the antiproton/proton ratio

$$\frac{\bar{p}}{p} = \frac{1 - r_-(s)}{1 + r_-(s)}, \quad r_-(s) = c_1 (s/s_0)^{(\alpha_R(0)-\alpha_\mathbb{P}(0))/2},$$

where $r_-(s)$ captures the negative-signature exchange, including possible Odderon admixtures (Merino et al., 2010).

First LHC data indicate that total and inclusive cross sections fit the single-Pomeron picture. ALICE measurements of $\bar{p}/p$ ratios tightly constrain Odderon contributions, with data favoring a small Odderon coupling (predictions with only secondary $\omega$-Reggeon exchange suffice) (Merino et al., 2010).

In elastic $pp/p\bar{p}$ scattering, the interplay between C-even and C-odd exchanges becomes manifest in the dip–bump structure of $d\sigma/dt$ near $t \approx -1\,\mathrm{GeV}^2$ (Jenkovszky et al., 2011). The inclusion of Odderon terms (with sign reversal for $pp$ vs. $p\bar{p}$) is crucial for filling the dip and matching the $t$-dependence observed experimentally.

4. Unitarity, Critical Theory, and Reggeon Field Theory

At asymptotic energies, respect for unitarity in the $s$-channel is enforced via non-linear unitarization schemes, such as eikonal and $U$-matrix approaches (Maneyro et al., 17 Feb 2024, Luna et al., 22 Nov 2024). These models are formulated in impact parameter ($b$) space:

• Eikonal: $H(s,b) = i[1 - \exp(i\chi(s,b))]$
• $U$-matrix: $H(s,b) = \chi(s,b)/[1 - i\chi(s,b)/2]$

Inclusion of Odderon degrees of freedom in these schemes reveals that even small, mainly real C-odd $t$-channel contributions are strongly screened by the large imaginary Pomeron background, especially at small $|t|$ (Luna et al., 22 Nov 2024). The effective difference in the real part of the amplitude between $pp$ and $p\bar{p}$ then becomes subtle, with typical effects in the $\rho$ parameter at most a few per mille (Luna et al., 22 Nov 2024), contrary to earlier claims of much larger differences.

Renormalization group analyses in Reggeon Field Theory (RFT) incorporating both Pomeron and Odderon fields have revealed a fixed-point structure with emergent symmetries (Bartels et al., 2016, Vacca, 2016). At the critical point, interactions changing the number of Odderon pairs vanish, leading to a block-diagonal structure in the effective potential and an emergent conservation of Odderon pair number in the infrared. The associated scaling exponents (e.g. $\nu_\mathbb{P}$, $\nu_\mathbb{O}$) are universal and reflect critical phenomena.

5. The Odderon in Exclusive and Spin-Dependent Processes

Experimental searches for Odderon effects focus on observables where the Pomeron either cancels or is absent. Central exclusive production (CEP) of $\phi\phi$ pairs in $pp$ scattering is a promising channel: the Odderon can enter as an intermediate $t$- or $u$-channel exchange, yielding distinctive signatures at large invariant mass $M_{\phi\phi}$ and large rapidity difference $Y_{\phi\phi}$ (Lebiedowicz et al., 2019, Lebiedowicz et al., 2020). Tensor-pomeron and vector-odderon models enable explicit calculations of these processes, and fits to WA102 data improve with the inclusion of Odderon contributions for certain angular distributions, though decisive evidence at the LHC remains pending.

In the context of spin-dependent amplitudes, analyses of high-energy $pp$ (and $p\bar{p}$) elastic scattering in the Wilson-line and GTMD formalism reveal that, in the forward limit $t=0$, the double helicity-flip amplitude $\varphi_2$ is dominated by the spin-dependent Odderon (Hagiwara et al., 2020). While saturation effects may suppress the Odderon at very high energies, these amplitudes enter in the extraction of the $\rho$ parameter, and their correct inclusion is necessary for accurate interpretation of collider results.

6. Strong Coupling, Spectral Curves, and Glueball/Oddball Spectroscopy

Within the AdS/CFT correspondence, the Pomeron is mapped to a reggeized graviton and the Odderon to a reggeized Kalb–Ramond antisymmetric field (Brower et al., 2013, Brower et al., 2014). The scaling dimensions of dual operators produce “spectral curves” $\Delta(j)$, which are expanded at strong coupling in $1/\sqrt{\lambda}$, $\lambda=g^2 N_c$:

• Pomeron: $$j_0 = 2 - \frac{2}{\sqrt{\lambda}} - \frac{a_2}{\lambda} - \cdots$$
• Odderon: $j_0^{\rm O}=1$ (type-(b) solution, fixed to all orders), or $j_0^{\rm O}=1-8/\sqrt{\lambda}+\cdots$ (type-(a))

Extrapolation of Pomeron and Odderon Regge trajectories to positive $t=M^2$ yields predictions for the mass and width of glueballs (even spin, C=+1) and oddballs (odd spin, C=–1). Dynamical holographic and analytic S-matrix approaches, fitting both trajectory parameters and observed $pp/p\bar{p}$ data, yield trajectories from which glueball masses near $2$ GeV and oddball (Odderon) masses in the $3$–$4$ GeV region are predicted (Capossoli et al., 2016, Szanyi et al., 2019, Magallanes et al., 2022).

The Chew–Frautschi plot of the Odderon trajectory, constrained by $pp/p\bar{p}$ data and effective Lagrangian analyses, supports a tensor (spin-3) identification with a mass around $3.2$ GeV (Magallanes et al., 2022).

7. Open Questions and Experimental Outlook

Despite theoretical consensus on the existence of the Odderon as a C-odd Regge singularity with intercept near unity, its experimental observation remains challenging due to the predominance of the Pomeron and the screening of C-odd effects in forward observables. High-precision LHC measurements of the $\rho$ parameter, forward cross sections, and CEP processes will further constrain allowed Odderon couplings.

Global fits incorporating unitarity, analyticity, and the full suite of modern data (including tensions between TOTEM and ATLAS measurements) indicate that only a negative Odderon phase ($\xi_\mathbb{O}=-1$) with a predominantly real amplitude at low $|t|$ is compatible with observed results (Maneyro et al., 17 Feb 2024, Luna et al., 22 Nov 2024). The predicted effect of the Odderon in $\rho$ is limited by strong screening, producing observable differences at or below the few per mille level in high-precision measurements (Luna et al., 22 Nov 2024). Central exclusive processes at large invariant mass remain a promising avenue for isolating Odderon effects due to the absence of direct strong Pomeron–proton backgrounds.

Further advances in experimental accuracy, combined with dedicated searches in carefully chosen kinematic regions and refined theoretical control in unitarized amplitude frameworks, will determine whether the Odderon can be firmly established as a fundamental component of strong-interaction dynamics.