We re-examine V. Gribov's theorem of 1960 according to which the total
cross-section cannot approach a finite non-zero limit with, at the same time, a
diffraction peak having a finite slope. We are very close to proving by an
explicit counter-example that elastic unitarity in the elastic region is an
essential ingredient of the proof. By analogy, we raise the question of the
saturation of the Froissart-Martin bound, for which no examples incorporating
elastic unitarity exist at the present time.Comment: 11 pages, 1 figures, latex with sproc.st

Martin, Andre

Richard, Jean-Marc

English

arXiv

Martin, A.

Richard, J.-M.

HAL-IN2P3

Investigation of the role of elastic unitarity in high-energy scattering : Gribov's theorem and the Froissart bound

Andre Martin

Jean-marc Richard

CiteSeerX

Investigation of the role of elastic unitarity in high-energy scattering: Gribov’s theorem and the Froissart bound

Hal - Université Grenoble Alpes

We re-examine V. Gribov's theorem of 1960 according to which the total cross-section cannot approach a finite non-zero limit with, at the same time, a diffraction peak having a finite slope. We are very close to proving by an explicit counter-example that elastic unitarity in the elastic region is an essential ingredient of the proof. By analogy, we raise the question of the saturation of the Froissart-Martin bound, for which no examples incorporating elastic unitarity exist at the present time.We re-examine V. Gribov's theorem of 1960 according to which the total cross-section cannot approach a finite non-zero limit with, at the same time, a diffraction peak having a finite slope. We are very close to proving by an explicit counter-example that elastic unitarity in the elastic region is an essential ingredient of the proof. By analogy, we raise the question of the saturation of the Froissart-Martin bound, for which no examples incorporating elastic unitarity exist at the present time

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CERN-TH/2001-086ISN-01-026INVESTIGATION OF THE ROLE OF ELASTIC UNITARITY INHIGH-ENERGY SCATTERING: GRIBOV’S THEOREM ANDTHE FROISSART BOUNDANDRE MARTINTH Division, CERN, CH-1211 Geneva 23, SwitzerlandandLAPTH, BP 110, 74941 Annecy le Vieux, Franceemail: Andre.Martin@cern.chJEAN-MARC RICHARDInstitut des Sciences Nucle´aires, CNRS-IN2P3, Universite´ Joseph Fourier53, Avenue des Martyrs, F-38026 Grenoble, Franceemail: jean-marc.richard@isn.in2p3.frWe re-examine V. Gribov’s theorem of 1960 according to which the total cross-section cannot approach a finite non-zero limit with, at the same time, a diffractionpeak having a finite slope. We are very close to proving by an explicit counter-example that elastic unitarity in the elastic region is an essential ingredient of theproof. By analogy, we raise the question of the saturation of the Froissart-Martinbound, for which no examples incorporating elastic unitarity exist at the presenttime.Based on a talk given by one of us (A.M.) at the First Workshop on“Forward Physics and Luminosity Determination at LHC”, Helsinki,31 October - 4 November 20001 IntroductionThe work we want to present is not completely nished, but, at the time ofwriting, we are 99% convinced that the results we present are correct and will1be made { \dans un mois, dans un an", as Francoise Sagan says { completelyrigorous.The Gribov theorem to which we refer is the statement that at high energythe scattering amplitude cannot behave likes f(t) (1)for s ! 1, t xed, where s is the square of the centre-of-mass energy and tthe square of the momentum transfer (negative or zero for t physical).In physical terms this means that the proton cannot behave like an ob-ject with a xed size independent of the collision energy,leading to a nite,non-zero, constant cross-section at high energy, and an asymptotically xeddiraction peak. This theorem, which appeared in the proceedings of the 1960International Conference on High-Energy Physics 1 (V. Gribov was not allowedto attend the conference!), came in the \Pre-Regge" 2 period, and was the rstblow to destroy the naive beliefs of most physicists about elementary particlesat the time. I consider it as a turning point in \forward physics" , the topic ofthis conference.To nd a contradiction, Gribov assumes not only that F (s, t)  i sf(t) fort physical (i is necessary for a crossing symmetric amplitude!), but also for t ina complex domain. This assumption can be weakened a little, as we shall seelater. Also it is really done for the simple case of pion-pion scattering, but itwould be very awkward if the behaviour F  sf(t) was forbidden for pion-pionscattering and allowed for nucleon-nucleon scattering.We now present Gribov’s original proof, which is remarkably simple. Herewe consider the pi0pi0 ! pi0pi0 amplitude where the pion has isospin zero. This isan inessential simplication. To generalize to the real pion, which is an isospintriplet, one can consider simultaneously all possible channels or play with thefact that, strictly speaking, mpi0 < mpi± . In Fig. 1 we have represented theMandelstam diagram with mpi = 1. We assume Mandelstam representationwhere the scattering amplitude is1pi2Z Zds0dt0ρ(s0, t0)(s0 − s)(t0 − t) + circular permutations ons, t, u (2)+subtractionswith s + t + u = 4.In fact we use a much smaller domain of analyticity than Mandelstam (for a2ss = 4 s = 0t = 0u = 0t = 4t = 16u tFigure 1: Mandelstam diagrammethod of construction of such domains, see for instance Ref. 3). However, thedomain obtained from axiomatic eld theory and unitarity 3 is not sucient.We assume F (s, t)  sf(t), but from crossing in the s ⇀↽ u variable wemust haveF (s, t)  i s f(t), (3)f(t) real for t  0, so that the asymptotic absorptive part is just sf(t), wheref(t) must have a cut from t = 4 to 1, and the double spectral function isgiven byρ(s, t) ’ s Im f(t). (4)Now Gribov uses elastic unitarity in the elastic strip of the t channel which,originally, reads:At(t, zt) =ZK(t, zt, z0t, z00t )F (t, z0t)F (t, z00t ) dz0tdz00t (5)where zt = 1 + 2s/(t− 4) is the cosine of the scattering angle in the t channel.Equation (5) is just a fancy way to write a spherical convolution of F and F .3K is a known kernel, the square root of a polynomial whose support is entirelyin the physical region.Because of Mandelstam representation, At(t, zt), F (t, zt), F (t, zt ) are allanalytic in a cut plane in zt, with cuts at zt = (1 + 8/(t− 4)), and Equation(5), as shown by Mandelstam, can be continued on the cuts, and transformedinto a relation connecting the discontinuities of At, F and F :ρ(s, t) =ZK^(t, s, s1, s2)As(s1, t)As(s2, t)ds1ds2 (6)K^ is again a completely known expression with a nite support in s1, s2for given s and t.When s ! 1, the domain of integration grows in such a way that themaximum of s1 and s2 also tends to innity, and it is not dicult to convinceoneself that if As  s f(t), the dominating region of the integral is such thatboth s1 and s2 ! 1. It is not dicult to see that the right-hand side of Eq.(6) behaves likejf(t)j2 s log s (7)while the left-hand side is obviouslyρ = s Im f(t) (8)This is the Gribov contradiction, which occurs also for F  s/ log s. Onlyif F  s/(log s)1+ is the contradiction removed. This was the way out thatGribov proposed, corresponding to a total cross-section going to zero, butNature did not make this choice. As we have known since 1972, cross-sectionsare rising 4 and might go to innity at innite energy.Another way of looking at the Gribov phenomena is to use the complexangular momentum plane. In this language, the Gribov theorem says thatelastic unitarity in the t channel forbids a \xed pole" at J = 1. It is slightlymore general in the sense that it is clearly insucient to take an amplitudei s f(t) + g(t)sα(t) (9)with α(t) < 1 for t  0 and Reα(t) > 1 for t > 4m2pi to \turn" the Gribov The-orem. In this version one clearly sees that the Donnachie-Landsho odderon5,s t−4, is also, strictly speaking, forbidden unless there is an insurmountablenatural boundary in the complex angular momentum plane.4One of the reasons for our renewed interest in the Gribov theorem is thatthere is another problem concerning high-energy cross-sections. As we said, weknow that they are rising, possibly to innity, possibly saturating qualitatively4the Froissart-Martin bound6σt < C(log s)2 (10)The question is to know if this bound can be saturated if one takes intoaccount unitarity constraints. The simplest unitarity constraint is:Im f`(s)  2kpsjf`(s)j2 (11)when f`(s) is a partial wave amplitude. It is valid at all energies.Joachim Kupsch has been able to show the existence of amplitudes whichsatisfy Mandelstam representation and the general unitarity constraint (11),and saturate the bound (10) 7. This was a formidable achievement, takinginto account all the restrictions that amplitudes saturating (10) must satisfy,in particular the so-called AKM conditions 8.However, it is legitimate to ask oneself if the elastic unitarity condition inthe elastic regionIm f`(s) =2kpsjk`(s)j2 , (12)which is so constraining in the Gribov case, might prevent the qualitativesaturation of the Froissart bound. Turning the argument upside down, wemay ask if the Gribov theorem survives if we require only (11) but not (12).We want to show, and have almost succeeded in doing so, that there areexplicit examples of amplitudes behaving like s f(t) i.e., violating Gribov’stheorem, satisfying the general condition (11) at all energies. If this is so,all doubts are allowed concerning the other problem, the saturation of theFroissart bound because, while D. Atkinson succeeded in 1970 9 to constructamplitudes satisfying (11) and (12) but with cross-sections decreasing fasterthan (log s)−2, nobody has succeeded in saturating the Froissart bound at thesame time.If it were to be shown one day that (12) prevents the qualitative saturationof the Froissart bound, it would be a major blow to many models such as theone of Bourrely-Cheng-Soer-Walker-Wu10, where σt  (log s)2 at extreme5energies. I admit that the likelihood of this happening is very small, rstof all because it requires somebody with great classical mathematical skill,who would tend naturally to use his abilities on a more fashionable subject.Nevertheless, we are allowed to raise the question.2 A Candidate for Violating Gribov’s TheoremWe propose to takeF = C[Fs + Ft + Fu] (13)Fs = (4−p4− tp4− u) exp−2(4− s)1/4Ft = (4−p4− up4− s) exp−2(4− t)1/4Fu = (4−p4− tp4− s) exp−2(4− u)1/4 (14)For s !1, t xed, Ft dominates and behaves like i s exp−2(4− t)1/4, whichis what we want. We must now show that (11) is satisifed and, rst of all,that all partial waves have a positive imaginary part. It is possible to showthat in this channel Im(Ft +Fu) has positive partial wave amplitudes, becauseIm(Ft + Fu) can be expanded in powers of cos θs:Im(Ft + Fu) =  Cn(cos θs)n, (15)and it can be shown that the Cn’s are positive.Then, since(cos θ)n =  C`,n P`(cos θ) (16)with C`,n > 0, positivity of the partial waves follows. Another consequencefrom (15) and (16) is that the partial waves of Im(Ft + Fu) are decreasing.However, ImFs has oscillating signs, and though, globally, it is small in the schannel, it must be proved that it is small partial wave by partial wave.There are limiting cases where the behaviour is correct:i) ` xed, k ! 1 and, in fact, also k ! 1 and ` ! 1 with `/k ! 0,which allows `!1 in some way. Then from the direct integration of F withP`(cos θs) one gets:f` ’8s+ 2i Z 0−1dt exp −2(4− t)1/4’4s+ i(15 + 14ps) exp −3p2 (17)6This shows that \unitarity" is satised just by adjusting C in (13).ii) k xed, ` ! 1 and more generally `/k ! 1, ` ! 1. Then one hasto use the Froissart-Gribov representation of the dierent contributions of thepartial waves of the formx` =Z 1uw(s, t)Q`1 +t2k2dt2k2(18)Ifw  C(s)(t − 4)ν (19)we get for `!1, `/k !1:x`  C(s) 2kp2` + 1!ν+1Γ(ν + 1)Q`1 +2k2(20)In this way it is trivial to see that the contribution from Im(Fu+Ft) dominatesall the rest and that unitarity and positivity are satised.iii) Finally, in the \scaling limit"` + 1k= b, b xedk !1 (21)one can use the asymptotic expression for the Q`’s valid for large `, uniformin φ:Q`(cosh φ) ’sφsinh φK0 ((` + 1/2)φ) (22)The RHS contribution to ImF t` becomes:12k2Zdt K0(bpt) sinp2(t− 4)1/4 exp−p2(t− 4)1/4, (23)and is necessarily positive, because of the positivity properties of Im(Ft +Fu),and necessarily dominates all the other terms, the real parts and the contribu-tion from ImFs which decrease exponentially with s.iv) Unitarity of the ` = 0 partial wave can be checked by hand. There, inthe expression (4−p4− tp4− u), the cancellation at s = 4 is essential.We conclude that it is only necessary to check unitarity in a nite regionof the `, k domain. First we did this numerically. Tests, at selected energies7where unitarity seems endangered by the negative signs of the partial waveexpansion of ImFs, indicate that provided one divides the amplitude by afactor 2.1, there does not seem to be any violation. These tests are presentedin Table 1.However, we believe that by using old and new properties of the Q`’s, forinstance−old : QL(z)/Q`(z), decreases with z forL > ` (24)−new : QM−11 +t2k2/Qm−11 + (Mm)2t2k2decreases with t forM > m(25)We can complete the proof analytically, except for the calculation of a small,nite number of integrals over a nite interval.Let us illustrate by two examples how we use (24) and (25). One problemis to nd a lower bound onImF `t,RHC =12k21Z4Q`1 +t2k2 ps− 4ps + t sinp2(t− 4)1/4 exp−p2(t− 4)1/4 dt.With the change of variable x =p2(t− 4)1/4, the integral becomesZ 10W (x, s) sin x dx.It is easy to prove that for x > 2pi, W (x, s) is decreasing in x. HenceZ 12piW (x, s) sin xdx > 0.SoImF `t,RHC > g`(s) =12k24(1+pi4)Z4Q`1 +t2k2 ps− 4ps + t sinp2(t− 4)1/4 exp−p2(t− 4)1/4 dt.In g` the integrant has a single change of sign at t0 = 4 + pi44 . If g`0(s0) > 0,it follows from (24) thatg`(s0) > g`0(s0)Q`0(1 + t02k2Q`(1 + t2k28if ` > `0, and hence we get a lower bound on Imf `t,RHC(s0) for ` > `0, but wecan also change at the same time ` and s and use (25) to get a lower boundfor` + 12k2=`0 + 12k20, ` > `0.Finally, we can also use the fact that Im(f `t + f`uis decreasing with ` toget lower bounds for smaller `’s.Everybody will understand that this is a rather delicate book-keeping, butwith enough patience, we will succeed.3 ConclusionsIf you believe that our example, after \renormalization", satises the generalunitarity condition (11), you conclude that the proof of Gribov’s theorem holdsonly if we impose elastic unitarity (12) in the elastic region. Incidentally, akind of universality postulate is tacitly made by everybody, because everybodybelieves that the Gribov theorem applies to pp scattering, while technicallythere are problems with the unphysical region in the t channel which woulddisappear only if mp < 1.5mpi.It is therefore legitimate to ask oneself what happens in another situation,the qualitative saturation of the Froissart bound, where the examples satu-rating this bound only satisfy condition (11) but not elastic unitarity. Wouldelastic unitarity make the saturation of the bound impossible?My own prejudice is that this will not happen, but there is absolutely noproof of that. As pointed out by S.M. Roy, 11 it could be that elastic unitarityleads to a quantitative improvement of the Froissart bound, i.e., replacing thefactor pi/m2pi, which appears in the Lukaszuk-Martin12 version:σt <pimpi2(log s)2,by something much smaller. In any case, we believe that these questions areworth investigating.94 AcknowledgementsOne of us (A.M.) would like to thank J. Kupsch and S.M. Roy for very stimu-lating discussions, and the Indo-French Centre for the Promotion of AdvancedResearch (IFCPAR) for support.References1. V.N. Gribov, Proceedings of the 1960 Annual International Conferenceon High-Energy Physics at Rochester, eds. E.C.G. Sudarshan, J.H. Tin-lot and A.C. Melissos (University of Rochester 1960), p. 340.2. The papers by Regge on Regge poles in potential scattering did exist,but the application to high-energy scattering was not yet fashionable.3. A.Martin, Proceedings of the 1967 International Conference on Particlesand Fields, eds. C.R. Hagen, G. Guralnik and V.A. Mathur (John Wileyand Sons, 1967), P. 252.4. See, for instance, G. Matthiae, Rep. Prog. Phys. 57 (1994) 743.5. A. Donnachie and P.V. Landsho, Phys. Lett. B296 (1992) 227.6. M. Froissart, Phys. Rev. 123 (1961) 1053;A. Martin, Nuovo Cimento 42 (1966) 930.7. J. Kupsch, Nuovo Cimento 71A (1982) 85.8. G. Auberson, T. Kinoshita and A. Martin, Phys. Rev. D3 (1971) 3185.9. D. Atkinson, Nucl. Phys. B23 (1970) 397.10. It is impossible to quote all references. We shall give three of them:H. Cheng and T.T. Wu, Phys. Rev. Lett. 24 (1970) 1456;H. Cheng, J.K. Walker and T. Tu, Phys. Lett.B44 (1973) 283;C. Bourrely, J. Soer and T.T. Wu, Nucl. Phys.B247 (1984) 15.11. S.M. Roy, private communication.12. L. Lukaszuk and A. Martin, Nuovo Cimento 52A (1967) 122, AppendixE.10Table 1: Numerical tests of unitarity of (13), with C = 1s ` Ref` Imf`ps/2k Imf`/jf`j24.0001 0 0.473 0.00236 2.12 4.1 10−11 5.1 10−12 6 10114.01 0 0.108 0.0040 2.192 7 10−6 1.19 10−7 4 1054 8.5 10−11 9.6 10−11 3.7 101050 0 0.53 1.45 0.632 0.037 1.19 0.844 0.0081 0.64 1.5670 0 0.40 1.71 0.57Chosen 2 -0.009 0.294 3.5so that 4 0.002 0.0736 14Imfs`< 0 6 0.0008 0.0193 53for ` > 0 8 0.00025 0.0054 1903500 0 0.0055 2.06 0.492 0.0017 1.82 0.544 0.0017 1.51 0.66idem 6 0.0014 1.21 0.828 0.0011 0.96 1.0410 0.0009 0.76 1.3212 0.0007 0.60 1.6714 0.0005 0.477 2.116 0.0004 0.380 2.6218 0.0003 0.305 3.320 0.0003 0.245 4.124000 0 0.0003 2.05 0.492 0.0003 2.01 0.494 0.0003 1.94 0.516 0.0003 1.84 0.548 0.0003 1.72 0.5810 0.0003 1.60 0.6312 0.0002 1.47 0.6814 0.0002 1.35 0.74Idem 16 0.0002 1.24 0.8018 0.0002 1.13 0.8820 0.0002 1.04 0.9622 0.0002 0.95 1.0524 0.0001 0.81 1.1526 0.0001 0.79 1.2628 0.0001 0.72 1.3730 0.0001 0.66 1.5032 0.0001 0.61 1.6534 0.0001 0.56 1.8036 0.0001 0.51 1.9738 0.0001 0.47 2.1540 0.0001 0.43 2.341 Any finite 0 (15 + 14 p2) 0.486187` exp−2p2’ 2.0560211

Investigation of the Role of Elastic Unitarity in High-Energy Scattering: Gribov's Theorem and the Froissart Bound

Investigation of the Role of Elastic Unitarity in High-Energy
  Scattering: Gribov's Theorem and the Froissart Bound

Investigation of the Role of Elastic Unitarity in High-Energy Scattering: Gribov's Theorem and the Froissart Bound

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