The eikonal model must be denoted as strongly preferable for the analysis of
elastic high-energy hadron collisions. The given approach allows to derive
corresponding impact parameter profiles that characterize important physical
features of nucleon collisions, e.g., the range of different forces. The
contemporary phenomenological analysis of experimental data is, however, not
able to determine these profiles unambiguously, i.e., it cannot give the answer
whether the elastic hadron collisions are more central or more peripheral than
the inelastic ones. However, in the collisions of mass objects (like protons)
the peripheral behavior of elastic collisions should be preferred.Comment: 7 pages, 4 figure

Kaspar, Jan

Kundrat, Vojtech

Lokajicek, Milos

English

arXiv

The eikonal model must be denoted as strongly preferable for the analysis of elastic high-energy hadron collisions. The given approach allows to derive corresponding impact parameter profiles that characterize important physical features of nucleon collisions, e.g., the range of different forces. The contemporary phenomenological analysis of experimental data is, however, not able to determine these profiles unambiguously, i.e., it cannot give the answer whether the elastic hadron collisions are more central or more peripheral than the inelastic ones. However, in the collisions of mass objects (like protons) the peripheral behavior of elastic collisions should be preferred.The eikonal model must be denoted as strongly preferable for the analysis of elastic high-energy hadron collisions. The given approach allows to derive corresponding impact parameter profiles that characterize important physical features of nucleon collisions, e.g., the range of different forces. The contemporary phenomenological analysis of experimental data is, however, not able to determine these profiles unambiguously, i.e., it cannot give the answer whether the elastic hadron collisions are more central or more peripheral than the inelastic ones. However, in the collisions of mass objects (like protons) the peripheral behavior of elastic collisions should be preferred

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Theoretical Aspects of High Energy ElasticNucleon ScatteringVojteˇch Kundra´t1, Jan Kasˇpar2,1, Milosˇ Lokaj´ıcˇek11Institute of Physics of the AS CR, v. v. i., 182 21 Prague 8, Czech Republic2CERN, 1211 Gene`ve 23, SwitzerlandThe eikonal model must be denoted as strongly preferable for the analysis of elastic high-energy hadron collisions. The given approach allows to derive corresponding impact pa-rameter profiles that characterise important physical features of nucleon collisions, e.g.,the range of different forces. The contemporary phenomenological analysis of experimen-tal data is, however, not able to determine these profiles unambiguously, i.e., it cannot givethe answer whether the elastic hadron collisions are more central or more peripheral thanthe inelastic ones. However, in the collisions of mass objects (like protons) the peripheralbehaviour of elastic collisions should be preferred.1 Coulomb-Hadronic InterferenceThe first attempt to determine the complete elastic scattering amplitude F C+N(s, t) for highenergy collisions of charged nucleons has been made by Bethe [1]:F C+N(s, t) = F C(s, t)eiαΨ(s,t) + F N(s, t) (1)where F C(s, t) is Coulomb amplitude (known from QED), F N (s, t) - elastic hadronic amplitude,αΨ(s, t) - real relative phase between Coulomb and hadronic scattering and α = 1/137.036 isthe fine structure constant. This relative phase has been specified by West and Yennie [2] usingthe Feynman diagram technique (one photon exchange) asΨWY (s, t) = − ln −st−∫ 0−4p2dt′|t′ − t|[1− FN (s, t′)F N (s, t)]; (2)p representing the value of the incident momentum in the centre-of-momentum system. Bytheir construction (see Eq. (2), the phase ΨWY (s, t) is to be real.However, it has been shown in Ref. [3] that this requirement can be fulfilled only providedthe phase of the elastic hadronic amplitude ζN (s, t) defined in our case asF N (s, t) = i|F N (s, t)|e−iζN (s,t), (3)is t independent at all kinematically allowed values of t. Rigorous proof has been given for|ζN (s, t)| < pi, which is fulfilled practically in all standard phenomenological models leading tothe central distribution of elastic processes in impact parameter space. It is not yet clear if itholds also for |ζN (s, t)| < 2pi corresponding to peripheral behaviour; see Fig. 1 where both the35types of phase t dependences are represented. The corresponding t dependences of imaginaryparts of complex relative phases αΨWY (s, t) are shown in Fig. 2.Figure 1: t dependences of the hadronicphases ζN (s, t) leading to the central andperipheral behaviours of elastic pp scatter-ing.Figure 2: t dependences of imaginary partsof the complex WY phases αΨWY (s, t) cor-responding to hadronic phases from Fig. 1.There is, however, no reason why the elastic hadronic phase should be independent of tvariable at all measured values of t. Thus the other approach different from the West andYennie formalism and based on the eikonal approach should be preferred - see, e.g., Ref. [4].In such a case the complete elastic scattering amplitude F C+N (s, t) is related to the completeeikonal δC+N (s, b) with the help of Fourier-Bessel (FB) transformationF C+N (s, q2 = −t) = s4pii∫Ωbd2bei~q.~b[e2iδC+N (s,b) − 1], (4)where Ωb is the two-dimensional Euclidean space of the impact parameter ~b. Mathematicallyconsistent use of FB transformation requires, of course, the existence of reverse transformation.However, at finite energies the amplitude F C+N (s, t) is defined in a finite region of t only. Thusthe consistent application of formula (4) requires to take into account also the values of elasticamplitude from unphysical region where the elastic hadronic amplitude is not defined; for detailssee Refs. [5]. This issue has been discussed by Islam [6] who has shown that the problem maybe solved in a unique way by continuing analytically the elastic hadronic amplitude F N (s, t)from physical to unphysical region of t ; see also Ref. [8].The common influence of both the Coulomb and elastic hadronic scattering then can bedescribed by complete eikonal which is formed by the sum of both the Coulomb δC(s, b) andhadronic δN (s, b) eikonals. Then the complete amplitude (valid at any s and t) can be finallywritten [4] asF C+N (s, t) = ±αstf1(t)f2(t) + FN (s, t) [1∓ iαG(s, t)] , (5)V KUNDRA´T, J KASˇPAR, AND M LOKAJI´CˇEK36whereG(s, t) =0∫−4p2dt′{ln(t′t)ddt′[f1(t′)f2(t′)] +12pi[F N (s, t′)F N (s, t)− 1]I(t, t′)}, (6)andI(t, t′) =2pi∫0dΦ′′f1(t′′)f2(t′′)t′′, t′′ = t + t′ + 2√tt′ cosΦ′′. (7)Here the two form factors f1(t) and f2(t) reflect the spatial structure of colliding nucleons andshould describe it in a sufficiently broad interval of t. As the Coulomb amplitude F C(s, t) isknown from QED the complete amplitude is determined practically by the t dependence of thehadronic amplitude F N (s, t).2 Impact Parameter Picture of Elastic NucleonScatteringAs it has been already mentioned the mathematically consistent use of FB transformation in-troducing the impact parameter representation hel(s, b) of elastic scattering amplitude FN (s, t)requires its definition also in the unphysical region of t, i.e., for t < tmin = −s + 4m2 (m beingthe nucleon mass). The function hel(s, b) must be, therefore, subdivided into two parts [5, 6]hel(s, b) = h1(s, b) + h2(s, b) = (8)=14p√s0∫tmindt F N (s, t) J0(b√−t) + 14p√stmin∫−∞dt F N (s, t) J0(b√−t).Similar expressions can be obtained also for the impact parameter representation ginel(s, b) ofthe inelastic overlap function Ginel(s, t) introduced in Ref. [7].The unitarity equation in the impact parameter space can be then written as [5, 6]=h1(s, b) = |h1(s, b)|2 + g1(s, b) + K(s, b) (9)where the correlation function K(s, b) is very small compared to the other functions appearingin Eq. (9) [8].The total cross section and integrated elastic and inelastic cross sections then equal toσtot(s) = 8pi∞∫0bdb =h1(s, b); σel(s) = 8pi∞∫0bdb |h1(s, b)|2; σinel(s) = 8pi∞∫0bdb g1(s, b). (10)Eqs. (10) are valid provided∞∫0bdb =h2(s, b) = 0,∞∫0bdb g2(s, b) = 0. (11)THEORETICAL ASPECTS OF HIGH ENERGY ELASTIC NUCLEON SCATTERING37The functions =h1(s, b), g1(s, b) and |h1(s, b)|2 represent then the impact parameter profiles;they describe the density of interactions between two colliding nucleons in dependence on theirmutual distance. The first two oscillate at grater values of b, but their mean squared valuescharacterising the mean ranges of corresponding forces acting between the colliding particlescan be established directly from the elastic hadronic amplitude F N (s, t) [9]. For the meansquared value of elastic impact parameter it has been derived〈b2(s)〉el = 40∫tmindt |t|(ddt |F N (s, t)|)20∫tmindt |F N (s, t)|2+ 40∫tmindt |t| |F N (s, t)|2(ddtζN (s, t))20∫tmindt |F N (s, t)|2. (12)Similarly the total and inelastic mean squared values equal to〈b2(s)〉tot = 2B(s, 0); 〈b2(s)〉inel =σtot(s)σinel(s)〈b2(s)〉tot −σel(s)σinel(s)〈b2(s)〉el. (13)Here the diffractive slope is defined asB(s, t) =ddt[lndσNdt]=2|F N (s, t)|ddt|F N (s, t)|. (14)3 Impact Parameter Profiles for pp Scattering at 53 GeVBasic results concerning the analysis of pp elastic scattering data at energy of 53 GeV at theISR [10] based on the eikonal approach have been published in Ref. [4]. Here we will mentiononly the results related to the impact parameter profiles.In the quoted paper we have used the formulas (5)-(7) for the complete elastic scatteringamplitude F C+N (s, t) generating the differential cross sectiondσeldt=pisp2|F C+N (s, t)|2. (15)The elastic hadronic amplitude, i.e., its modulus and phase defined in Eq. (3), has been conve-niently parameterised in order to describe the pp elastic scattering as central as well as peripheralprocess. While the t dependence of the modulus can be almost unambiguously determined fromthe data the phase can be only partially constrained. Both the possible alternatives (centraland peripheral) have been presented in Ref. [4]. The t dependences of the corresponding shapesof the hadronic phase ζN (s, t) are shown in Fig. 1.Once the elastic hadronic amplitude F N (s, t) has been specified it has been possible todetermine corresponding impact parameter profiles together with their statistical errors withthe help of FB transformation. And it has been also possible to determine the RMS values ofthe total, elastic and inelastic profiles with the help of Eqs. (12) and (13) for the central as wellas the peripheral pictures of elastic pp scattering. Their shapes corresponding to peripheralbehaviour are shown in Fig. 3 (for the central case see Ref. [3]); all RMS values are includedin Table 1. In the central picture the elastic RMS is much lower than the inelastic one. Thisresult agrees with the result of Miettinen [11]. It means that the protons in ‘head-on’ collisionsshould be rather transparent. In the peripheral case it is the second term in Eq. (12) thatV KUNDRA´T, J KASˇPAR, AND M LOKAJI´CˇEK38Figure 3: Oscillating peripheral profiles ofelastic pp scattering at 53 GeV.Figure 4: Positive peripheral profiles ofelastic pp scattering at 53 GeV.gives the significant contribution to the elastic RMS value. Some profiles corresponding to theperipheral case exhibit greater oscillations at higher impact parameter values b. However, theoscillations can be removed as it will be shown in Ref. [8]. It will be briefly described in thefollowing.It is necessary to construct actual (non-negative) pp profiles which correspond to the valuesof RMS derived with the help of Eqs. (12) and (13). The way how to do it is described inRefs. [8, 9]. It consists in adding an real function c(s, b) to the both sides of the unitarityequation (9). All dynamical characteristics corresponding to elastic hadron scattering will bepreserved if the function c(s, b) appearing in Eqs. (17) fulfills some additional conditions.Unitarity condition (9) may be then expressed ashtot(s, b) = |h1(s, b)|2 + ginel(s, b) (16)wherehtot(s, b) = =h1(s, b) + c(s, b), ginel(s, b) = g1(s, b) + K(s, b) + c(s, b) (17)are non-negative functions for all b. The function htot(s, b) must be positively semidefinite (andmonotony decreasing function) at all values of b. The oscillating function c(s, b) is requiredto cancel the oscillations from the total and inelastic profiles. The elastic peripheral profile|h1(s, b)|2 will remain unchanged. And integrated inelastic cross section is preserved if thefunction c(s, b) fulfills the conditions∞∫0b db c(s, b) = 0;∞∫0b3 db c(s, b) = 0. (18)These two conditions represent the integral conditions limiting the shape of the function c(s, b).According to Islam’s approach [6] this function may be identified with the =h2(s, b) for whichthe conditions (18) are to be required. However, in the standard approach the function c(s, b)can be hardly determined analytically. The best way at the present seems to specify it in anumerical way as it will be illustrated in the following. It may be expected that the total profileTHEORETICAL ASPECTS OF HIGH ENERGY ELASTIC NUCLEON SCATTERING39Table 1: Root-mean-squares of impact parameters for pp collisions at 53 GeV.elastic profile√< b2 >tot√< b2 >el√< b2 >inelmodulus phase sum[fm] [fm] [fm] [fm] [fm]peripheral 1.033 0.676 1.671 1.803 0.772central 1.028 0.679 ∼ 0. 0.679 1.087entering into modified unitarity condition (16) should be approximately of Gaussian type withthe values that may be characterised by integral cross sections and by RMS values shown inTable 1. The elastic profile will remain unchanged. Under these assumptions the total profileshape can be defined (the s dependence being dropped) as htot(b) = ae−βb2 . Using formulas3.461 from Ref. [13] the corresponding integrals needed for calculation of the total cross sectionand total mean squared value can be analytically determined as∞∫0b db a e−βb2=a2β,∞∫0b3 db a e−βb2=a2β2(19)and the values of the constants a and β can be determined from experimentally establishedvalues. For the peripheral case of elastic pp scattering at energy of 53 GeV their values are:a = 0.324 and β = 0.946. The b dependence of the auxiliary function c(s, b) is then determinedwith the help of the first equation from (17) where =h1(s, b) is taken from experimental analysis.And the second equation determines then the shape of the inelastic profile.Knowing the shapes of the total and inelastic profiles together with the b dependence ofthe auxiliary function c(s, b) the values of all the integrated cross sections and of all the meansquares can be verified numerically as can be seen from the Table 2. The new values arepractically quite comparable with the original ones. Also the values of the integrals of functionc(s, b) (see Eqs. (18)) only slightly different from zero are shown in Table 2. The modifiedprofiles are exhibited in Fig. 4. The new total and inelastic profiles are central while the elasticprofile remains unchanged and is peripheral.4 ConclusionSome results concerning elastic nucleon collisions at high energies and based on the validity ofoptical theorem have been summarised; the eikonal model has been applied to. The approachsuitable for the case of finite energies has been presented. It has been shown that elasticprocesses may be interpreted as peripheral.V KUNDRA´T, J KASˇPAR, AND M LOKAJI´CˇEK40Table 2: The values of integrated cross sections and of the total, elastic and inelastic RMS.Quantity Original values New valuesσtot [mb] 42.864 42.872σel [mb] 7.479 7.479√< b2 >tot [fm] 1.0 1.028√< b2 >el [fm] 1.803 1.803√< b2 >inel [fm] 0.772 0.772∞∫0bdb c(s, b) [fm2] - 0.029∞∫0b3db c(s, b) [fm4] - 0.097References[1] H. Bethe, Ann. Phys. 3 190 (1958)[2] G.B. West and D.R. Yennie, Phys. Rev. 172 1413 (1968)[3] V. Kundra´t, M. Lokaj´ıcˇek and I. Vrkocˇ, Phys. Lett. B656 182 (2007)[4] V. Kundra´t and M. Lokaj´ıcˇek, Z. Phys. C63, 619 (1994)[5] T. Adachi and T. Kotani, Progr. Theor. Phys. Suppl., Extra Number 316 (1965); Progr. Theor. Phys. 35463 (1966); 35 485 (1966); 37-38 297 (1966); 39 430 (1968); 39 785 (1968)[6] M.M. Islam, Lectures in theoretical Physics, ed. A. O. Barut and W. E. Brittin, Vol. 10B (Gordon andBreach, 1968), p.97; Nucl. Phys. B104 511 (1976)[7] L. van Hove, Nuovo Cim. 27 2344 (1963)[8] V. Kundra´t, M. Lokaj´ıcˇek and J. Kasˇpar, “Impact parameter profiles of elastic scattering in hadroniccollisions”; to be published[9] V. Kundra´t, M. Lokaj´ıcˇek and D. Krupa, Phys. Lett. B544 132 (2002)[10] K.R. Schubert, Landolt-Bornnstein, Numerical Data and Functional Relationships in Science and Technol-ogy, New Series, (Springer, Berlin, 1979), Vol. I/9a[11] H.G. Miettinen, Proc. Rencontre de Moriond, Meribel les Allues, Vol. 1 (ed. J. Tran Thanh Van, Orsay1974), Acta Phys. Pol. B6 625 (1975)[12] G. Giacomelli and M. Jacob, Phys. Rep. 55 1 (1979)[13] I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, San Diego, (1980).THEORETICAL ASPECTS OF HIGH ENERGY ELASTIC NUCLEON SCATTERING41

Theoretical aspects of high energy elastic nucleon scattering

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