Regge theory is a framework in theoretical physics and quantum field theory that describes the behavior of scattering amplitudes at high energies and fixed momentum transfer through the analytic continuation of angular momentum to complex values. The theory introduces the concept of Regge poles and Regge trajectories, which relate the spin (or angular momentum) of exchanged particles to their squared mass, providing a unified description of families of resonances and the asymptotic behavior of scattering processes.
Historical background
The theory was introduced in 1959 by Italian physicist Tullio Regge in the context of non-relativistic potential scattering. Regge demonstrated that the partial‑wave expansion of the scattering amplitude could be analytically continued from integer angular momentum $l$ to complex values, revealing singularities (Regge poles) in the complex angular‑momentum plane. In the 1960s the method was extended to relativistic particle physics, where it became a central tool in the phenomenology of strong interactions before the advent of quantum chromodynamics (QCD).
Mathematical formulation
In Regge theory the scattering amplitude $A(s,t)$ for a process characterized by Mandelstam variables $s$ (center‑of‑mass energy squared) and $t$ (momentum‑transfer squared) is expressed as a sum over contributions from Regge poles:
$$ A(s,t) \approx \sum_{i} \beta_i(t), s^{\alpha_i(t)} , $$
where $\alpha_i(t)$ are Regge trajectories—analytic functions of $t$ whose values at integer $t$ give the spins of resonant states—and $\beta_i(t)$ are residue functions encoding coupling strengths. The leading trajectory (the one with the highest $\alpha(t)$ at a given $t$) dominates the high‑energy behavior, leading to a power‑law rise or fall of cross sections with energy.
Physical implications
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Regge trajectories – Empirically, mesons and baryons of the same quantum numbers lie on approximately linear trajectories $\alpha(t) \approx \alpha_0 + \alpha' t$, with slope $\alpha' \approx 0.9\ \mathrm{GeV}^{-2}$. This linearity suggests a string‑like picture of hadrons, influencing the development of modern string theory.
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High‑energy scattering – The theory predicts that total cross sections behave as $\sigma_{\text{total}}(s) \sim s^{\alpha_{\text{P}}(0)-1}$, where $\alpha_{\text{P}}(0)$ is the intercept of the Pomeron trajectory, a leading Regge pole with vacuum quantum numbers. The Pomeron explains the slowly rising hadronic total cross sections observed at collider energies.
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Duality and Veneziano amplitude – Regge behavior underlies the concept of duality, where the same amplitude can be described by either s‑channel resonances or t‑channel Regge exchanges. This duality motivated the construction of the Veneziano amplitude, a precursor of modern string scattering amplitudes.
Applications
- Hadron spectroscopy – Linear Regge trajectories provide a phenomenological classification of meson and baryon spectra.
- Phenomenological models of soft hadronic processes – Regge pole exchanges are incorporated in event generators and analytical models to describe diffractive scattering, elastic proton–proton interactions, and photoproduction.
- Deep inelastic scattering and small‑$x$ physics – Extensions of Regge theory, such as the Balitsky–Fadin–Kuraev–Lipatov (BFKL) Pomeron, are employed to describe the rise of parton densities at low Bjorken‑$x$.
- String theory origins – The linear Regge trajectories observed in hadronic data inspired the early development of bosonic string theory, where vibrational modes of a relativistic string naturally yield such spectra.
Experimental verification
Regge theory successfully accounts for the pattern of hadronic resonances observed in particle accelerators and explains the energy dependence of elastic and diffractive scattering cross sections measured at facilities ranging from CERN’s ISR to the LHC. The observed approximate linearity of meson and baryon trajectories and the existence of the Pomeron intercept slightly above one are consistent with Regge predictions, though a complete derivation from QCD remains an active research area.
Current status
While Regge theory predates the formulation of QCD, many of its concepts survive as effective descriptions of non‑perturbative strong‑interaction dynamics. Contemporary research explores the correspondence between Regge behavior and the gauge‑string duality (AdS/CFT) and seeks to derive Regge trajectories directly from lattice QCD and holographic models.
References
- T. Regge, “Introduction to complex orbital momenta,” Il Nuovo Cimento 14 (1959) 951.
- P. D. B. Collins, An Introduction to Regge Theory and High Energy Physics, Cambridge University Press, 1977.
- S. Donnachie, G. Shaw, Pomeron Physics and QCD, Cambridge University Press, 2002.
(No additional references are provided within this entry.)