Non-linear energy evolution equation for elastic scattering amplitudes of hadrons in impact parameter space

Published: 9 May 2024| Version 1 | DOI: 10.17632/zzdmybbd6y.1


We transform the high-energy elastic scattering of hadrons into an initial value problem by deriving an evolution equation for the complex elastic scattering amplitudes of hadrons starting with a Regge Field Theory Lagrangian. This equation takes the form of a complex nonlinear reaction-diffusion equation, with the logarithm of energy serving as the analog of time. For the details of the derivation and the exact form of the equation see: [DOI: 10.5506/APhysPolBSupp.16.5-A3] and [DOI: 10.1140/epjc/s10052-022-10747-6]. In the large impact parameter approximation, we can separate the real and imaginary parts of the amplitudes. Doing this we obtain two coupled partial differential equations. To solve the differential equations numerically, we use the Kohara–Ferreira–Kodama (KFK) model [] and independently the Bourrely–Soffer–Wu (BSW) model [] based impact parameter space (b-space) profiles for the real and the imaginary parts of the elastic scattering amplitude as initial conditions at √s = 500 GeV. The values of the three parameters (λ, ϵ, α′) associated with our evolution equation are determined by fitting the differential cross section obtained from our equation for a given energy against the cross sections measured in the TOTEM experiment ( In our case, we fit the TOTEM data at 13 TeV (it has better statistics and small fluctuations). The data is available at: []. After fixing the parameters the cross sections for the remaining energies come as predictions. Below we provide the data for i) the proton-proton differential cross-section obtained from our equation for different energies for the KFK and BSW initial conditions, ii) Real and imaginary amplitudes obtained from our equation for different energies for the KFK initial condition, iii) Inelastic profiles obtained from our evolution equation for different energies for the KFK initial conditions. These should allow one to reproduce the plots presented in our works: [DOI: 10.5506/APhysPolBSupp.16.5-A3] and [DOI: 10.1140/epjc/s10052-022-10747-6].


Steps to reproduce

The description provide above allows one to numerically solve our evolution equation and obatin both the amplitudes (real and imaginay) as well as the differential cross section. One can also obtain the numerical prediction for the proton-proton total cross section, the rho parameter which is the ratio of the real and the imaginary amplitudes, the dip and the bump position in the differential cross section etc. All the necessary formulae for these are provide in our paper: [] and []. Finally, the Readme file provides the necessary information about the organization of the data and also demonstrates how to develop plots. Following this, one can reproduce all the plots presented in our papers mentioned above.Refer to these for an elaborate discussion of the results.


Akademia Gorniczo-Hutnicza imienia Stanislawa Staszica w Krakowie


Physical Sciences


Narodowe Centrum Nauki

Nr umowy: UMO-2021/41/N/ST2/02956