Tensor-Pomeron Model in High-Energy Scattering

• The tensor-pomeron model is an effective field theory that represents pomeron exchange as a symmetric rank-2 tensor, ensuring consistency with QFT charge-conjugation and helicity rules.
• It employs Reggeized propagators and effective vertices to model processes like elastic scattering, central exclusive production, and deep-inelastic scattering with high precision.
• Empirical applications span meson spectroscopy, Compton scattering, and holographic QCD, linking tensor glueball dynamics to observable high-energy diffraction patterns.

The tensor-pomeron model is an effective Regge-field-theoretic description of soft high-energy scattering in which the pomeron is represented not as a scalar or vector exchange but as an effective symmetric rank-2 tensor exchange. In this formulation, propagators and vertices are constructed to respect standard QFT rules, crossing properties, charge conjugation, and the observed helicity structure of soft hadronic reactions. The framework was formulated systematically for pomeron, odderon, and reggeon exchanges, and was subsequently applied to elastic scattering, central exclusive production, photoproduction, small-$x$ deep-inelastic scattering, and deeply virtual Compton scattering (Ewerz et al., 2013).

1. Conceptual basis and motivation

In Regge theory, the pomeron is the leading trajectory in the vacuum quantum-number channel and controls high-energy, small-$|t|$ soft hadronic scattering. The tensor-pomeron model encodes this exchange as an effective spin-2 object with $C=+1$, rather than as a scalar or vector. This choice is motivated by the requirement that the effective description be compatible with standard QFT charge-conjugation properties and with the helicity structure seen in data (Ewerz et al., 2013).

A central argument against a vector pomeron is that a genuine vector exchange coupled through a vector current changes sign under charge conjugation, whereas the observed pomeron contribution to $pp$ and $\bar p p$ scattering has the same sign. The tensor formulation avoids this inconsistency. A separate argument against a scalar pomeron comes from polarized elastic $pp$ scattering: within the broader tensor-pomeron program, STAR data are consistent with tensor exchange and disfavor scalar exchange (Lebiedowicz et al., 2018). In small-$x$ lepton-proton scattering, the model further argues that a vector pomeron would not give any contribution to photoproduction, while the tensor pomeron does give a nonzero contribution compatible with data (Britzger et al., 2019).

The model is explicitly phenomenological. It is not a UV-complete QFT, but an effective theory designed for soft, nonperturbative, high-energy reactions. Its basic premise is that a coherent sum of even-spin exchanges $J=2,4,6,\dots$ can be represented efficiently by a rank-2 tensor exchange with Reggeized propagators and effective local vertices (Ewerz et al., 2013).

2. Effective-field-theory structure

The standard tensor-pomeron propagator is written as

$$i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t) = \frac{1}{4s} \left( g_{\mu\kappa} g_{\nu\lambda} + g_{\mu\lambda} g_{\nu\kappa} - \frac{1}{2} g_{\mu\nu} g_{\kappa\lambda} \right) \left( -i s \alpha'_{\mathbb{P}} \right)^{\alpha_{\mathbb{P}}(t)-1},$$

with linear trajectory

$$\alpha_{\mathbb{P}}(t)=1+\epsilon_{\mathbb{P}}+\alpha'_{\mathbb{P}} t, \qquad \epsilon_{\mathbb{P}}=0.0808, \qquad \alpha'_{\mathbb{P}}=0.25~\mathrm{GeV}^{-2}$$

(Bolz et al., 2014). The tensor factor is symmetric and traceless in both index pairs and plays the role of the spin-2 projector.

The effective pomeron-proton-proton vertex is

$|t|$0

with $|t|$1 and $|t|$2 the proton Dirac form factor (Bolz et al., 2014). This structure is designed to couple the tensor pomeron to a conserved symmetric tensor current of the proton.

The model extends the same logic to secondary reggeons and to the odderon. The $|t|$3 reggeons $|t|$4 are also treated as tensor exchanges, while the odderon and the $|t|$5 reggeons $|t|$6 are treated as vector exchanges (Ewerz et al., 2013). This assignment makes charge-conjugation relations automatic: tensor exchanges contribute with the same sign to $|t|$7 and $|t|$8, whereas vector exchanges flip sign (Ewerz et al., 2013).

For couplings to mesons and vector states, the model employs effective vertices built from a small set of tensor structures. In particular, pomeron couplings to vector mesons are organized in terms of two gauge-invariant tensors, commonly denoted $|t|$9 and $C=+1$0, which recur throughout photoproduction and central exclusive production calculations (Bolz et al., 2014).

3. Central exclusive production and spectroscopy

A major domain of application is central exclusive production (CEP),

$C=+1$1

where the centrally produced state $C=+1$2 is generated by pomeron-pomeron fusion or by combinations of pomeron and reggeon exchanges. In this setting, the tensor structure of the pomeron directly controls azimuthal correlations, $C=+1$3-dependences, and decay-angle distributions (Lebiedowicz et al., 2014).

For scalar and pseudoscalar mesons, the framework organizes the $C=+1$4 vertex in terms of orbital-angular-momentum and spin couplings of two effective spin-2 pomerons. In the 2014 application to scalar, pseudoscalar, and vector mesons, the authors found that for scalar and pseudoscalar production, in most cases, two lowest orbital angular momentum-spin couplings are necessary to describe WA102 experimental differential distributions (Lebiedowicz et al., 2014). For tensor mesons, the structure is richer: the general $C=+1$5 coupling is a combination of seven basic couplings, and angular distributions in the Collins-Soper and Gottfried-Jackson frames are particularly sensitive to this choice (Lebiedowicz et al., 2019).

The model has also been applied in detail to exclusive $C=+1$6 production, where the coherent sum of continuum, scalar resonances, tensor resonances, and photoproduction terms is essential. In that context, the way two pomerons couple to the tensor meson $C=+1$7 strongly affects the interference pattern between the $C=+1$8 resonance and the dipion continuum, and the relative contribution of resonant $C=+1$9 and dipion continuum strongly depends on the cut on proton transverse momenta or on $pp$0, so that the cuts may play the role of a $pp$1 resonance filter (Lebiedowicz et al., 2016).

The same formalism has been generalized to axial-vector and pseudoscalar CEP. For axial-vector $pp$2 mesons, two constructions of the pomeron-pomeron-$pp$3 coupling were studied, and comparison with WA102 data favored a coupling pattern close to a pure $pp$4 structure in the phenomenological basis, while a Chern-Simons-type holographic coupling leads to an approximately equivalent effective structure for small virtualities (Lebiedowicz, 2021). For $pp$5 and $pp$6, the 2025 analysis showed that the amplitudes, including pomeron and $pp$7 reggeon exchanges, can be fitted to WA102 and extrapolated to the LHC, and it further argued that CEP of $pp$8, $pp$9, and $\bar p p$0 is impossible in a scalar-pomeron theory, so experimental observation of these channels would give striking evidence against a scalar character of the pomeron (Lebiedowicz et al., 5 Jun 2025).

Channel	Tensor-pomeron coupling content	Source
Scalar $\bar p p$1	two lowest $\bar p p$2 couplings, notably $\bar p p$3 and $\bar p p$4	(Lebiedowicz et al., 2014)
Tensor $\bar p p$5	seven basic $\bar p p$6 couplings	(Lebiedowicz et al., 2019)
Axial-vector $\bar p p$7	$\bar p p$8 and $\bar p p$9, plus CS-type form	(Lebiedowicz, 2021)
Pseudoscalar $pp$0	$pp$1 and $pp$2 structures	(Lebiedowicz et al., 5 Jun 2025)

Beyond meson spectroscopy, the same tensor-pomeron and vector-odderon framework has been used for exclusive production of meson and baryon pairs. In $pp$3, the model predicts a dip in the rapidity-difference distribution at $pp$4, unlike the maximum at zero found for $pp$5 and $pp$6, a consequence of the interference structure of the tensor-exchange amplitudes (Lebiedowicz et al., 2018).

4. Small-$pp$7 DIS, photoproduction, and Compton scattering

The tensor-pomeron model was extended to low-$pp$8 DIS and photoproduction in a two-tensor-pomeron formulation containing a hard tensor pomeron $pp$9, a soft tensor pomeron $x$0, and an even-signature tensor reggeon. Fitting data with $x$1 GeV and $x$2, the model found intercepts

$x$3

and argued that within the errors the hard pomeron is absent in photoproduction (Britzger et al., 2019).

In this DIS implementation, the $x$4 vertices are described by two independent functions $x$5 and $x$6, so that transverse and longitudinal photon-proton cross sections can be expressed as sums over soft pomeron, hard pomeron, and reggeon contributions (Britzger et al., 2019). A related 2020 analysis compared the hard-pomeron behavior of the tensor-pomeron fit with the Color-Dipole Picture and emphasized that for $x$7 the hard-pomeron piece behaves as

$x$8

numerically consistent with the CDP exponent $x$9 (Schildknecht, 2020).

The same two-tensor-pomeron framework was then applied to real and virtual Compton scattering. In deeply virtual Compton scattering, the model gives a very good description of HERA data at small Bjorken $J=2,4,6,\dots$0, includes both transverse and longitudinal virtual photons, and finds that the interference between soft- and hard-pomeron exchange plays an important role (Lebiedowicz et al., 2022). A notable result is that in DVCS the soft-pomeron contribution is considerable up to $J=2,4,6,\dots$1, and the ratio

$J=2,4,6,\dots$2

strongly increases with $J=2,4,6,\dots$3 (Lebiedowicz et al., 2022).

Taken together, these applications show that the tensor-pomeron formalism is not restricted to hadron-hadron diffraction. It also provides a unified small-$J=2,4,6,\dots$4 description of photoproduction, DIS, and DVCS in which the soft-hard transition is modeled through a pair of tensor trajectories rather than through scalar or vector exchanges (Britzger et al., 2019).

5. Holographic and glueball-based realizations

A distinct line of work embeds the tensor-pomeron idea in holographic QCD. In the 2016 study of pomeron interactions from the Einstein-Hilbert action, the pomeron is realized as the Reggeized trajectory of the bulk graviton or tensor glueball in a 5D holographic QCD model. In that construction, the pomeron inherits a two-index polarization tensor from the graviton, and its triple interaction vertex is deduced directly from the Einstein-Hilbert action (Iatrakis et al., 2016).

This holographic realization was applied to double-pomeron production of tensor glueballs and compared with WA102 azimuthal distributions. The model found very good agreement for the $J=2,4,6,\dots$5 tensor glueball candidate, while $J=2,4,6,\dots$6 and $J=2,4,6,\dots$7 showed completely different distributions, interpreted as a consequence of their non-glueball character in that framework (Iatrakis et al., 2016). This does not merely reproduce a spin-2 ansatz; it ties the tensor nature of the pomeron to a specific gravitational interaction structure in AdS/QCD.

A more phenomenological glueball-based use of the tensor-pomeron picture appears in the RENORM tensor-Pomeron implementation. There, a D-wave enhancement in AFS exclusive $J=2,4,6,\dots$8 data is interpreted as a spin-2 tensor glueball with

$J=2,4,6,\dots$9

and the identification

$$i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t) = \frac{1}{4s} \left( g_{\mu\kappa} g_{\nu\lambda} + g_{\mu\lambda} g_{\nu\kappa} - \frac{1}{2} g_{\mu\nu} g_{\kappa\lambda} \right) \left( -i s \alpha'_{\mathbb{P}} \right)^{\alpha_{\mathbb{P}}(t)-1},$$0

is used to fix the scale in the Froissart-like high-energy cross-section formula (Goulianos, 2016). With this input, the model predicts at $$i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t) = \frac{1}{4s} \left( g_{\mu\kappa} g_{\nu\lambda} + g_{\mu\lambda} g_{\nu\kappa} - \frac{1}{2} g_{\mu\nu} g_{\kappa\lambda} \right) \left( -i s \alpha'_{\mathbb{P}} \right)^{\alpha_{\mathbb{P}}(t)-1},$$1 TeV

$$i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t) = \frac{1}{4s} \left( g_{\mu\kappa} g_{\nu\lambda} + g_{\mu\lambda} g_{\nu\kappa} - \frac{1}{2} g_{\mu\nu} g_{\kappa\lambda} \right) \left( -i s \alpha'_{\mathbb{P}} \right)^{\alpha_{\mathbb{P}}(t)-1},$$2

in agreement with LHC measurements within quoted uncertainties (Goulianos, 2016).

These holographic and glueball-based developments reinforce a recurrent theme of the tensor-pomeron program: the effective spin-2 structure is not only a phenomenological convenience, but can be connected to tensor glueball trajectories and, in some constructions, to a Reggeized graviton (Iatrakis et al., 2016).

6. Empirical tests, limitations, and open issues

The tensor-pomeron model has been tested against a wide range of observables: total and elastic hadronic cross sections, central exclusive meson production, vector-meson photoproduction, inclusive low-$$i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t) = \frac{1}{4s} \left( g_{\mu\kappa} g_{\nu\lambda} + g_{\mu\lambda} g_{\nu\kappa} - \frac{1}{2} g_{\mu\nu} g_{\kappa\lambda} \right) \left( -i s \alpha'_{\mathbb{P}} \right)^{\alpha_{\mathbb{P}}(t)-1},$$3 DIS, and DVCS (Ewerz et al., 2013). Its most direct empirical signatures are spin- and angle-sensitive observables, such as $$i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t) = \frac{1}{4s} \left( g_{\mu\kappa} g_{\nu\lambda} + g_{\mu\lambda} g_{\nu\kappa} - \frac{1}{2} g_{\mu\nu} g_{\kappa\lambda} \right) \left( -i s \alpha'_{\mathbb{P}} \right)^{\alpha_{\mathbb{P}}(t)-1},$$4 distributions in CEP, decay angular distributions in the Collins-Soper and Gottfried-Jackson frames, the $$i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t) = \frac{1}{4s} \left( g_{\mu\kappa} g_{\nu\lambda} + g_{\mu\lambda} g_{\nu\kappa} - \frac{1}{2} g_{\mu\nu} g_{\kappa\lambda} \right) \left( -i s \alpha'_{\mathbb{P}} \right)^{\alpha_{\mathbb{P}}(t)-1},$$5 “glueball filter,” and helicity-sensitive ratios in polarized elastic scattering (Lebiedowicz et al., 2019).

At the same time, the framework remains phenomenological. Couplings in central vertices such as $$i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t) = \frac{1}{4s} \left( g_{\mu\kappa} g_{\nu\lambda} + g_{\mu\lambda} g_{\nu\kappa} - \frac{1}{2} g_{\mu\nu} g_{\kappa\lambda} \right) \left( -i s \alpha'_{\mathbb{P}} \right)^{\alpha_{\mathbb{P}}(t)-1},$$6, $$i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t) = \frac{1}{4s} \left( g_{\mu\kappa} g_{\nu\lambda} + g_{\mu\lambda} g_{\nu\kappa} - \frac{1}{2} g_{\mu\nu} g_{\kappa\lambda} \right) \left( -i s \alpha'_{\mathbb{P}} \right)^{\alpha_{\mathbb{P}}(t)-1},$$7, or $$i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t) = \frac{1}{4s} \left( g_{\mu\kappa} g_{\nu\lambda} + g_{\mu\lambda} g_{\nu\kappa} - \frac{1}{2} g_{\mu\nu} g_{\kappa\lambda} \right) \left( -i s \alpha'_{\mathbb{P}} \right)^{\alpha_{\mathbb{P}}(t)-1},$$8 are not derived from first principles in QCD and are fitted to data (Lebiedowicz, 2021). Absorption effects are often treated in eikonal or gap-survival approximations, which reduce cross sections and can distort angular distributions (Lebiedowicz et al., 2016). At WA102 energies, reggeon contributions can be substantial, making it difficult to isolate pure pomeron dynamics, whereas LHC measurements with proton tagging are expected to provide cleaner tests (Lebiedowicz et al., 2014).

There are also model-dependent ambiguities. In the DIS and DVCS sector, the $$i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t) = \frac{1}{4s} \left( g_{\mu\kappa} g_{\nu\lambda} + g_{\mu\lambda} g_{\nu\kappa} - \frac{1}{2} g_{\mu\nu} g_{\kappa\lambda} \right) \left( -i s \alpha'_{\mathbb{P}} \right)^{\alpha_{\mathbb{P}}(t)-1},$$9-dependence of the $$\alpha_{\mathbb{P}}(t)=1+\epsilon_{\mathbb{P}}+\alpha'_{\mathbb{P}} t, \qquad \epsilon_{\mathbb{P}}=0.0808, \qquad \alpha'_{\mathbb{P}}=0.25~\mathrm{GeV}^{-2}$$0 couplings is parametrized rather than derived, and the relative strength of the hard pomeron can vary between successful fits (Lebiedowicz et al., 2022). In holographic realizations, the Lorentz structure of interactions can be fixed by Einstein-Hilbert dynamics, but the Reggeization itself is still introduced phenomenologically (Iatrakis et al., 2016).

Despite these limitations, the tensor-pomeron model has established a coherent alternative to scalar- and vector-pomeron descriptions. Its central claim is not simply that the pomeron can be modeled as spin-2, but that a rank-2 tensor exchange provides a QFT-consistent, empirically viable, and broadly applicable language for soft high-energy diffraction, small-$$\alpha_{\mathbb{P}}(t)=1+\epsilon_{\mathbb{P}}+\alpha'_{\mathbb{P}} t, \qquad \epsilon_{\mathbb{P}}=0.0808, \qquad \alpha'_{\mathbb{P}}=0.25~\mathrm{GeV}^{-2}$$1 scattering, and central exclusive production (Ewerz et al., 2013).