= Model of Bourrely et al. =

== References ==

Used for implementation:

|| ![1] || [[http://link.aps.org/doi/10.1103/PhysRevD.19.3249|BOURRELY C., SOFFER, J. and WU, T. T., Phys. Rev. D19 (1979) 3249]] ||
|| ![2] || [[http://dx.doi.org/10.1016/0550-3213(84)90369-9|BOURRELY C., SOFFER, J. and WU, T. T., Nucl. Phys. B247 (1984) 15]] ||
|| ![3] || [[http://link.springer.com/article/10.1140/epjc/s2003-01159-7|BOURRELY C., SOFFER, J. and WU, T. T., Eur. Phys. J. C28 (2003) 97-105]] ||
|| ![4] || [[http://link.springer.com/article/10.1140/epjc/s10052-011-1601-x|BOURRELY C., SOFFER, J. and WU, T. T., Eur. Phys. J. C71 (2011) 1601]] ||


Other sources:

|| ![5] || [[http://link.aps.org/doi/10.1103/PhysRevLett.24.1456|CHENG H. and WU, T. T., Phys. Rev. Lett. 24 (1970) 1456-1460]] ||


== Implementation notes ==

Implemented in class `BSWModel` (see in [[http://elegent.hepforge.org/doxygen/classElegent_1_1BSWModel.html|Doxygen]]).

There have been several issues met during the model implementation.
1. The function `S0` defined by Eq. (3) in ![3] is in fact also function of t (via the third Mandelstam variable u). Therefore - strictly speaking - S0 can not be used in Eq. (2) in ![3], where the independent quantity is the impact parameter b. However, in the typical kinematics, s is much larger than |t| and thus one may very well approximate u by -s. In this approximation (used in Elegent), S0 becomes function of s only.
1. Yet another problem related to S0. As u takes negative values, the powers u^c^ and (ln u)^c'^ become ambiguous. For instance, one may write u = |u| exp(i pi (2k_u - 1)), where k_u is an ''arbitrary'' integer. Therefore ln u = ln |u| + i pi (2k_u - 1), which enters the power u^c^ = exp(c ln u). Similarly for the power of the logarithm, one needs to calculate ln (ln u) = ... + i ( alpha + 2 k_lnu pi), where alpha = atan2(Im ln u, Re ln u) and k_lnu is an ''arbitrary'' integer. As it is demonstrated by this [[https://elegent.hepforge.org/comparisons/bsw/comparison_k.pdf|test sheet]], only k_u = k_lnu = 0 reproduce the high energy behaviour of sigma_tot(s) and rho(s).
1. The Regge background shall become negligible at high energies, however, looking in Table 3 in ![3], all the trajectories have positive intercept and thus their amplitude increases with energy. In publication ![2], there is a footnote (with four stars on page 17) stating that a previous paper ![1] contains a misprint: the opacity should have read Omega0 = i s S0 F0 + R0. This gives a more correct ratio of the pomeron to the Regge term, however the overall normalisation is wrong. Therefore we have "moved" the i s factor to the Regge term: Omega0 = S0 F0 - i/s R0. Besides this, we have tried also factors 1/s, -1/s and +i/s in front of the R0 term. But as shown in this [[https://elegent.hepforge.org/comparisons/bsw/comparison_regge.pdf|test sheet]], only the -i/s leads to reasonable description of the lower energy region.
1. The signatures of the Regge trajectories are not specified in any of the above quoted publications. We have confirmed our guess A2: +, rho: - and omega: - by trying all combinations (see this [[https://elegent.hepforge.org/comparisons/bsw/comparison_regge.pdf|test sheet]]).
1. Below Eq. (7) in ![3], the text instructs to sum the 3 Regge exchange amplitudes in order to get the complete Regge contribution tilda R0. Unfortunately, it does not mention the signs that ''must'' to be applied to each of the amplitudes. These signs are responsible for the difference between proton-proton (pp) and antiproton-proton (app) reactions. We have verified that the signs for A2, rho and omega amplitudes read +, + and + for pp and +, - and - for app.

After these corrections, our calculations are in very good agreement with published predictions, see this [[https://elegent.hepforge.org/comparisons/bsw/comparison_final.pdf|test sheet]].