By V. N. Gribov
This paintings offers a distinct creation to the speculation of complicated angular momenta, in line with the tools of box conception. It includes an English translation of a lecture direction given through Vladimir Gribov in 1969. in addition to their old value, those lectures comprise fabric hugely correct to analyze at the present time and sure to shape the foundation for destiny advancements within the topic.
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Indeed, Im fl is positive due to the unitarity condition, and so is P (1 + t/2ks2 ) for t ≥ 0. Therefore for each partial wave we have an estimate† s s0 ∞ N > Im A(s, t) = 1+ Im f (s)(2 + 1)P =0 √ > Im c(s, ) 2π t −1/2 exp ks √ ks t 2ks2 t− 4µ2 . 10) holds for arbitrary positive t < 4µ2 , we conclude that Im c(s, ) < (s/s0 )N , and finally, modulo an irrelevant pre-exponential factor, Im f (s) < ∼ s s0 N exp − 2µ ks . ) † the series converges inside the so-called Lehman ellipse in the z plane 18 1 High energy hadron scattering We are now in a position to estimate the imaginary part of the forward scattering amplitude: ∞ Im f (s) (2 + 1) Im A(s, t = 0) = =0 ∞ L ≤ 8π (2 + 1) + =0 Im f (s)(2 + 1).
S. has only a simple pole at = 1. From this example it is clear that the t-channel unitarity condition forbids partial wave amplitudes to have any fixed (independent of t) sin= ∞ at the singular point. ‘Soft’ fixed gularities at real , such that f √ − 1, or f ∝ ln( − 1)) which corresingularities (for example, f ∝ spond to σtot falling with energy are, generally speaking, possible. We conclude that a seemingly natural picture of a hadron as an object with fixed, independent of the collision energy, interaction radius is inconsistent with unitarity in the cross-channel.
3a) for Im s A(s, t). With t increasing, the growing factor exp ( α), originating from the Legendre polynomials, eventually overtakes the falling factor exp (−2 α0 ) due to Im f . At this point the series becomes divergent, and Im s A(s, t) develops a singularity. Thus, the line of singularities of Im s A(s, t) for 4µ2 ≤ s ≤ 16µ2 is given by the equation α = 2α0 . In terms of the variables s and t this equation takes the form s t = , 4µ2 ≤ s ≤ 16µ2 . 2 16µ s − 4µ2 In the complementary region 4µ2 ≤ t ≤ 16µ2 , s ≥ 4µ2 , the Karplus curve can be found using the symmetry of A(s, t) under the permutation s ↔ t: t s = , 2 16µ t − 4µ2 4µ2 ≤ t ≤ 16µ2 .