We show that if for fixed negative (physical) square of the momentum transfer
t, the differential cross-section ${d\sigma\over dt}$ tends to zero and if the
total cross-section tends to infinity, when the energy goes to infinity, the
real part of the even signature amplitude cannot have a constant sign near t =
0.Comment: 7 pages, late

Amaldi

André Martin

Auberson

Augier

Cheng

Khuri

Kundrat

Martin

Matthiae

Łukaszuk

Physics Letters B

English

arXiv

We show that if for fixed negative (physical) square of the momentum transfer t, the differential cross-section ${d\sigma\over dt}$ tends to zero and if the total cross-section tends to infinity, when the energy goes to infinity, the real part of the even signature amplitude cannot have a constant sign near t = 0

Martin, A

CERN Document Server

CERN-TH/97-23A THEOREM ON THE REAL PARTOF THE HIGH-ENERGY SCATTERING AMPLITUDENEAR THE FORWARD DIRECTIONAndre MARTIN)Theoretical Physics Division, CERNCH - 1211 Geneva 23andLAPP), B.P. 110F - 74941 Annecy Le VieuxE-mail: martina@mail.cern.chABSTRACTWe show that if for xed negative (physical) square of the momentum transfer t, the dierentialcross-section ddttends to zero and if the total cross-section tends to innity, when the energygoes to innity, the real part of the even signature amplitude cannot have a constant sign neart = 0.) URA 14-36 du CNRS, associee a L’Ecole Normale Superieure de Lyon et a l’Universite deSavoie.CERN-TH/97-23January 1997The measurement of the real part of the scattering amplitude in the forward direction isa crucial test of dispersion relations. It allows to get information on the total cross-section athigher energies [1],[2] and also it may be a crucial test of locality. A violation of dispersionrelations might be the sign of new physics like for example the existence of extra compactdimensions [3].To measure this real part, one has to use Coulomb interference, which is large at a verysmall but not zero momentum transfer. Most of the time people assume that in the regionwhere the interference takes place, the ratio of the real to imaginary part, called , is constant.For exceptions, see the papers of Kundrat and collaborators [4]. The present paper containsa theorem which constitutes a kind of warning against the dangers of this procedure, becausewe show that under assumptions which are satised by most existing models and seem to becompatible with experimental data on proton (anti)proton scattering, namely that ddttends tozero for xed negative t and that total tends to innity, the real part of the even signatureamplitude cannot have a constant sign in a strip −T < t  0, s > sM , where T is arbitrarilysmall and sM is arbitrarily large. Hence,  cannot be constant because it is extremely dicultto make the imaginary part of the amplitude vanish exactly at the same place as the realpart because of positivity constraints, and also because, from a practical point of view, thedierential cross-section is not seen to vanish anywhere. The question is therefore if it is stilladmissible to neglect the variation of  in a measurement of the real part by interference. Ina simple case we shall show that the zero of the real part is more than 100 times further awaythan the place where the interference is maximum, but we have no certainty about this ingeneral. Of course, in a specic model, like for instance the one of Wu and collaborators [5],there is no problem because one can test directly if (s; t) given by the model reproduces theinterference curve, but it is unfortunate that one cannot get (s; t = 0) without any theoreticalprejudice.We now come to Theorem I. Consider an amplitude F (s; t; u) describing A + B ! A + Band A+ B ! A+ B reactions withs+ t+ u = 2M2A + 2M2B ;s, square of centre-of-mass energy in the A+B ! A+B channel and u in the A+ B ! A+ Bchannel and t = −2k2(1− cos ), k centre-of-mass momentum and cos  scattering angle in theAB channel. If the amplitude is not symmetric in s− u exchange, we symmetrize it to get the\even-signature" amplitude. Then we state:Theorem IIf F is s − u crossing symmetric and if ReF has a constant sign (including zero) in the stripsM < s < 1, −t < T  0 where sM is arbitrarily large and T > 0 arbitrarily small, and ifF (s;t)s! 0 for −T < t < 0, j F (s; t = 0)=sj does not tend to 1 for s ! 1 and therefore bythe optical theorem the total cross-section does not tend to innity.Theorem IIIf ddt(A+B ! A+B) and ddt(A+ B ! A+ B) tend to zero for s!1 −T < t < 0 for someT and if total(AB) and/or total(A B) tend to innity for innite energy, the real part of the1even signature amplitude cannot have a constant sign in a strip sM < s <1, −T < t  0, forany sM and T > 0.Theorem II is an obvious consequence of Theorem I.Proof of Theorem IWe work with a crossing symmetric amplitude in the exchange s$ u. For −T < t  0 this am-plitude satises twice subtracted dispersion relations (in fact this holds also for−T < t < +R,R > 0 as a consequence of the axioms of local eld theory and positivity [6]). It means thatthe amplitude is analytic in a twice cut plane. If we use the variablez(s; t) = (s− u)2 =2s− 2(M2A +M2B) + t2; (1)F (s; t; u) = G(z; t) ; (2)G is analytic in a once cut plane for −T < t real  0, with, in the simplest case of a \normalthreshold" a cut beginning atz0(t) = (4MAMB + t)2 (3)We can write an \inverse" dispersion relation for G. Dene:H =G(z; t)qz0(t)− z: (4)If F (s; t)=s tends to zero for t < 0, s!1, H satises an unsubtracted dispersion relationH(z; t) = −1Z 1z0(t)ReG(z0; t)pz0 − z0 (z0 − z)dz : (5)Suppose now that ReF (s; t) has a constant sign, say negative, for s > s0, −T < t  0. ReG(z; t)has then also a constant negative sign for−T < t  0 and z > zM(t) = [2(sM −M2A −m2B) + t]2We write H asH = H^ +  (6)whereH^(z;t) =1Z 1zM (t)−ReG(z0; t)dz0qz0 − z0(t) (z0 − z); (7)and(z; t) =1Z zM (t)z0(t)−ReG(z0; t)dz0qz0 − z0(t) (z0 − z): (8)Notice that for jzj ! 1, jj is bounded by const=jzj.H^ having a positive discontinuity is a \Herglotz" function, having no zeros in the complexplane. For z < 0, it is positive and increasing. Hence0 < H^(−x; t) < H^(0; t) for − T < t < 0 ; x > 0 ; (9)2so that H^ is uniformly bounded on the negative real axis.Now we remark that F (s; t) is analytic in t, for xed s in a circle jtj  R [6], and hence,G(z; t) and H(z; t) are both analytic in a neighbourhood of −R < t < +R for any complex zand also for z  0 (one must pay attention to the fact that the cuts move with t).Concerning (z; t), we can reexpress it as an integral over the original variables s0 and tover the range s0 < s0 < sM , and reach again the conclusion that when z is strictly outside thecut 2(s0 −M2A −M2B)− T2< z <2(sM −M2A −M2B)2; is analytic in t in a neighbourhood of −R < t < +R.Therefore, H^ is also analytic in t, for xed z, −1 < z < 0, in a neighbourhood of−R < t < + R. Hence inequality (9) which was established for t strictly negative alsoholds by continuity for t = 0:0  H^(−x; 0)  H^(0; 0) (10)Now suppose that total !1 for s!1. This means that jF (s; t = 0)=sj ! 1, and hencejH(z; t = 0)j ! 1 for z ! +1 on both sides of the cut.Since j(z; t = 0)j can be bounded by const=jzj, this also means thatjH^(z; t = 0)j ! 1 : (11)But −1=H^(z; t = 0) is also a Herglotz function and, from (11) 1H^(z; t = 0)! 0 ; (12)for z ! +1, on both sides of the cut. By the Phragmen-Lindelo¨f theorem 1=H^(z; t = 0) alsotends to zero for jzj ! 1 in any complex direction. This contradicts the fact that H^(−x; t = 0)is uniformly bounded for x > 0, and therefore t(s) cannot tend to innity.Strictly speaking this implies that t(s) has a nite least lower limit for s ! 1. One canprobably make a more rened statement on a smoothed t. Of course, if we believe that t ismonotonous beyond a certain energy, it means that t is bounded.As we said, Theorem II is a direct consequence of Theorem I. In practice, Theorem II is themost relevant, because there are clear indications, in the case of proton-antiproton scatteringthat t is increasing [7] and, comparing for instance the ISR and spps data [7], thatddtdecreasesat least for t < −0:4 GeV2, and it seems that the decrease occurs for smaller and smaller jtjwhen the energy increases.The simplest way to satisfy theorem II is to have a curve in the s− t plane where the realpart changes sign:t = −f(s) ; f(s) > 0 (13)3such that f(s) ! 0 for s ! 1, but this is not the only possibility. We could also have aninnite sequence of bubbles in the s − t plane approaching t = 0 for s ! 1 in which ReFwould have an opposite sign. The implication of the theorem for the individual AB ! AB andA B ! A B amplitudes is that their real parts cannot have both the same constant sign.Now, is the theorem a surprise? Not really if we look at simple-minded examples. What isa surprise is its generality.Examples can be built by takingjF (s; t)j  s1+t(log s)γ (14)with 0 < γ  1,  > 0. F=s goes to zero for t < 0 s!1, and t  (log s)γ.Such examples do not manifestly violate s channel unitarity nor polynomial boundednessfor xed complex t (for t  (log s)γ, 1 < γ < 2 model building is more tricky while the extremecase γ = 2 is easy to handle [8]). By standard techniques we can make this amplitude evenunder crossing by replacing it byF (s; t) = i C e−it2 s1+tlog s−i2γ; (15)with C real > 0.For small t, we see that the real part changes sign fort = −γ log s(16)The special case γ = 1 corresponds to what is called \geometrical scaling". Indeed, = F (s; t)=F (s; 0) is just a function of t log s, i.e., of  = tt, since t  log s. In thatspecial case the real part, according to an old theorem of the author [9], is found proportionalto dd(()) and we check that this gives us precisely a zero att = −1 log s(17)However, for γ < 1 this is no longer true and this shows how dangerous is the abusive use ofthis \magic" formula.We would like now to return to the question of how our result aects the measurement ofthe real part. In general, it is dicult to say, but, in the special case of geometrical scalingthe same function determines the point of maximal interference and the location of the zero ofthe real part. Indeed the complete amplitude, including the Coulomb term is given (with someoversimplications !) by [8]F =s2t+ FH(s; t) ; (18)where FH reduces to stotal(+ i)=16 at t = 0. We can rewrite F ass2t+st((0) + i)16(tt) ;4and the relative maximum interference occurs near (tt) = 8 ; (19)where () = () ; (20)while the zero of the real part is given by 0 = 0 ;It happens that in practice the point of maximum relative interference for proton-(anti)protonscattering corresponds to t ’ 10−3 GeV2 while, if we approximate the diraction peak byexp Bt, the zero of  0 is at t = (B=2)−1 = 0.13 (GeV)2 atps = 1 TeV [7],[4], so that thesituation is not too bad. However, outside the geometrical scaling regime (which, incidentallydoes not seem to apply to proton-antiproton scattering at energies above the ISR range, sincethe ratio of the elastic to total cross-sections increases instead of being constant [7]), Eq. (16)shows that the zero can be at a dierent place. In the model of Wu et al. [5] the rst zero ofRe F is at t = −0:32 GeV2 forps = 1 TeV, according to C. Bourrely.Finally, for completeness we would like to present the counterpart of Theorem I for the oddsignature amplitude:Theorem IIIIf, in the strip, −T < t < 0, s > sM , the dierence of the imaginary parts of the amplitudesAB ! AB and A B ! A B has a constant sign, and if ddt! 0 for both reactions, the dierenceof the total cross-sections AB − A B does not tend to innity.This means that the \maximal odderon" [10] is excluded under the above conditions. Weleave the proof as an exercise to the reader.AcknowledgementsI would like to thank Jan Fischer, Maurice Haguenauer and Claude Bourrely for very stim-ulating discussions.5References[1] U. Amaldi et al., Phys.Lett. 66B (1977) 390.[2] C. Augier et al. (UA4 Collaboration), Phys.Lett. B316 (1993) 448.[3] N.N. Khuri, in Seventh \Rencontres de Blois" on Elastic and Diractive Scattering, Blois1995, Editions Frontieres, BP. 33, Gif sur Yvette (1996), p. 471.[4] V. Kundrat and M. Lokajicek, Phys.Rev. D31 (1985) 1045, Phys.Lett. B232 (1989) 263,and 1996 Preprint, Institute of Physics ASCR, to appear in Phys.Rev. D.[5] H. Cheng and T.T. Wu,Phys.Rev.Lett. 24 (1970) 1456, andC. Bourrely, J. Soer and T.T. Wu, Ref. [3], p. 15.[6] A. Martin, Nuovo Cimento 42 (1966) 930.[7] A. Martin and G. Matthiae, in Proton-Antiproton Physics, G. Altarelli and L. Di LellaEds., World Scientic, Singapore (1989), pp. 45-84, andM. Haguenauer, Private communication;G. Matthiae, Rep.Progr.Phys. 57 (1994) 743.[8] G. Auberson, A. Martin and T. Kinoshita, Phys.Rev. D3 (1971) 3185.[9] A. Martin, Lett. Nuovo Cimento 7 (1973) 811.[10] L.  Lukaszuk and B. Nicolescu, Lett. Nuovo Cimento 8 (1973) 405.6

A theorem on the real part of the high-energy scattering amplitude near the forward direction

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A theorem on the real part of the high-energy scattering amplitude near
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A theorem on the real part of the high-energy scattering amplitude near the forward direction

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