Changes between Version 5 and Version 6 of HMBourrely

Timestamp:

Oct 10, 2013, 11:55:46 AM (11 years ago)

Author:

jkaspar

Comment:

--

HMBourrely

@@ -21 +21 @@
 
 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, S0 becomes function of s only.
+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 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 impact will increase 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.