This paper is very well written.It follows rather closely the original paper in Phys.Rev, except that the discussion on subtractions has been rightly eliminated since it became completely irrelevant. I just wonder why Dr. Froissart has not mentioned the subsequent work of A.Martin and collaborators. In fact, except for the question of the multiplying constant in the Froissart bound, most of this work is covered by i) the lectures given by Froissart at ICTP in Trieste 1962, and by his rapporteur's talk at the iInternational Conference on High Energy Physics in Berkeley in 1966. Let me summarize the results
1) in 1962 Martin proved that to get the Froissart bound it suffices that, at fixed energy, the absorptive part of the scattering amplitude be analytic in a region containing a segment 0< t < To, To independent of the energy: Proceedings of the 1962 International conference on High Energy Physics at CERN, J.Prentki editor, p. 566
2) Jin and Martin proved that if dispersion relations hold in the neighbourhood of the above mentioned interval, the (even) number of subtractions does not change for t in this interval. Y.S. JIn and A.Martin Phys.Rev. 135B , 1464 (1964)
3) In 1966 Martin proved that. using the on shell analyticity obtained from local field theory and positivity properties one could obtain the domain mentioned in 1) and hence prove the Froissart bound from" first principles". A.Martin Nuovo Cimento 42, 930 (1966). It could be argued that there are problems with local field theory because we now know that ordinary hadrons are made of quarks. However, long ago, W.Zimmerman has shown that one can associate local fields to composite objects: W.Zimmermann, Nuovo Cimento 10, 597 (1958)
4) G.Sommer found a way to get a lower bound for To, which,in most cases is 4 m pi square: G.Sommer, Nuovo Cimento 48A, 92 (1967). In trhe case of pi-nucleon scattering, Sommer does not get the best possible answer but it can be obtained by another method: J.D.Bessis and V.Glaser Nuovo Cimento 58 A, 568 (1967).
5) By combining the result of Jin and Martin and that of Sommer, it is possible to obtain a bound on the constant multiplying the Froissart bound which is pi/ mpi square, i.e. about 60 millibarns: L.Lukaszuk and A.Martin, Nuovo Cimento 52, 122 (1967). Very recenltly, A Martin has shown that the bound on the total INELASTIC cross-section is 4 times smaller: A.Martin Phys.Rev. D 80, 0650123 (2009).
It seems to me that, since the article under consideration is supposed to be a review article, these results should be included.
Referee Report on "Froissart bound" by Marcel Froissart for Scholarpedia
This review paper presents a very clear, concise, and pedagogical account of the original argument in favor of the by now celebrated Froissart bound, due to the author himself. This brief account will certainly be useful to the interested scientists, especially the younger ones, who may be reluctant at opening the original paper, which goes back to almost 50 years ago. What is however missing from this presentation, in my opinion, is a broader perspective on the evolution of this subject during this lapse of time. This refers both to phenomenological and to conceptual issues. Before I explain those issues in more detail, I would like to first point out some misprints and make some suggestions of style:
1) In the first section, ``Intuitive reasoning, there is a misprint in the relation between the pion mass and the constant k in the exponential decrease with b: the correct relation should read k = (hc/2\pi)m. That is, k and m have the same dimension in natural units.
Still in this section, I would suggest to add a reference to the seminal paper by Heisenberg in 1952, where this physical argument was for the first time presented (to my knowledge).
2) In the main section, ``Scheme of the proof ..., some succinct definitions, or references, could be added at relevant places, to help reading. For instance, one could recall that the `scattering angle' is the angle between the 3-vectors p_1 and p_3. Also, one could recall why the statement of unitarity is equivalent to saying that ``all partial waves have a modulus less than 1. One could insert the explicit form for a partial wave in terms of the phase shift. One could define the meaning of the constant D (the lower limit on the angular quantum number l) just above ``Putting together all the results. A missprint in the 3rd equation in ``Putting together all the results (in the same line with ``Taking the log ...): the right hand side of that equation should read 1, and not 0.
Let me now return to my main criticism, which is the absence of a broader perspective.
On the phenomenological side, it would be interesting to comment on the experimental indications in favor of the Froissart bound, as coming from the high energy data. Although this bound is supposed to set in only asymptotically, it is nevertheless interesting to notice that the best actual fits to the total and elastic cross-sections, as summarized by the Particle Data Group, are at least consistent with a log^2s behavior at high energy, that is, with Froissart bound. Here is a collection of such recent fits
1. J.R. Cudell et al. (COMPETE Collab.), Phys. Rev. D65, 074024 (2002).
2. K. Igi and M. Ishida, Phys. Rev. D66, 034023 (2002), Phys. Lett. B622, 286 (2005).
3. M. M. Block and F. Halzen, Phys. Rev. D70, 091901 (2004), Phys. Rev. D72, 036006 (2005).
See also the web site of Particle Data Group
and in particular the compilations of high-energy total cross-sections at
On the conceptual side, it would be useful to first recall that, 50 years later, Froissart bound has still not been proven from first principles in QCD, and that the main obstacle against doing that, is its interplay with the non-perturbative physics of confinement (which is responsible for the exponential decay of the hadronic cross sections at large impact parameters). Yet, there is significatif progress on the theory side, which may be worth recalling (at least briefly): Within perturbative QCD, one was able to demonstrate the power-like increase of the scattering amplitude from first principles; this is encoded in the BFKL equation for the evolution of the amplitude with the energy. Still within perturbative QCD, one has understood that a compelling mechanism for unitarization (at fixed impact parameter b) is the saturation of the parton densities in the infinite momentum frame of a hadron. (When viewed from a different frame, saturation effects mix with multiple scattering.) Finally, it is intuitively clear that a combination of (A) the BFKL, power-like, increase of the amplitude with the energy, (B) its unitarization at fixed b via saturation and multiple scattering, and (C) the exponential decrease of the amplitude at large values of b, due to confinement, should naturally result in Froissart bound at high-energy. Here is a recent version of this argument (a modern version of Heisenberg's original argument) :
E. Ferreiro et al, ``Froissart bound from gluon saturation, Nucl. Phys. A710 (2002) 373.
Unfortunately, however, so far there is no rigorous proof of this argument within the framework of QCD.