# Model of Bourrely et al.

## References

Used for implementation:

Other sources:

[5] | CHENG H. and WU, T. T., Phys. Rev. Lett. 24 (1970) 1456-1460 |

## Implementation notes

Implemented in class `BSWModel`

(see in Doxygen).

There have been several issues met during the model implementation.

- 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. - 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 test sheet, only k_u = k_lnu = 0 reproduce the high energy behaviour of sigma_tot(s) and rho(s). - 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 test sheet, only the -i/s leads to reasonable description of the lower energy region.
- 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 test sheet).
- 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 test sheet.

Last modified 9 years ago
Last modified on Oct 10, 2013, 12:07:22 PM