Analytization of elastic scattering amplitude
aa r X i v : . [ h e p - ph ] O c t Analytization of elastic scattering amplitude
S.M. Troshin ∗ , N.E. Tyurin SRC IHEP of NRC “Kurchatov Institute”Protvino, 142281, Russian Federation*) [email protected]
Abstract
Scenario for restoration of the real part of the elastic scattering ampli-tude has been proposed for the unitarity saturation case. Dependence of thereal part of the elastic scattering amplitude on the transferred momentum − t at the asymptotical energies has been restored from the correspondingimaginary part on the basis of derivative analyticity relations (analytizationprocedure).Keywords:Amplitude of hadron elastic scattering; Restoration of its real part; Unitaritysaturation in elastic scattering.PACS: 13.85.Lg, 13.85.-t, 14.20.Dh Introduction
The recent experimental results of the TOTEM Collaboration on the measure-ments of the differential cross-section of elastic scattering pp –scattering at small − t at the energy in the center of mass system √ s = 8 TeV [1] have demonstratedstrong deviation of the nuclear amplitude from a simple exponential dependence.Those measurements have high precision and significance of the above effect is atthe level of seven standards. We discuss here the role of the real part for the ex-planation of this deviation and confirm the possibility of emerging − t dependenceof the ratio of real to imaginary parts ρ ( s, t ) of the elastic scattering amplitude atsmall − t values in case of the unitarity limit saturation at the asymptotical valuesof the energy s → ∞ .The problem has a long story. A knowledge of the real part of the elasticamplitude is particularly important due to its sensitivity to the possible disper-sion relation violation resulting from violation of the polynomial boundedness [2]and to the new physics related, e.g. to the non-local interactions resulting frompresence of the fundamental length and/or existence of an extra internal com-pact dimensions [3, 4]. However, this knowledge is rather limited nowadays: theexperimental data provide information for the forward scattering in the Coulomb-Nuclear Interference (CNI) region only [5], while it could play an important rolein a wider region of the transferred momentum t variation. It should be noted thatthe real part of the scattering amplitude has a peripheral dependence on impactparameter in the both cases of black-disk or unitarity limit saturation (cf. [6, 7]).It means that if the asymptotics corresponds to one of the above limits the function ρ ( s, t ) has a nontrivial t dependence at s → ∞ .The black disk limit for the scattering amplitude in the impact parameter rep-resentation (in case the pure imaginary amplitude) is / and unitarity limit isunity. The “black disk model” implies that the amplitude in the impact parameterrepresentation is equal to / till some value of the impact parameter b = R andit decreases at b > R . Similar form of this dependence is valid for the case of uni-tarity limit saturation which is twice as much as the black disk limit. It should benoted, that the asymptotic equipartition σ el /σ tot = σ inel /σ tot = 1 / does not de-fine the “black disk model” since the same relation is being valid in various cases.The forward scattering observables are the definite integrals over impact parame-ter and it is quite evident, that an integration over impact parameter does not allowan unequivocal reconstruction of an initial integrand. The above equipartition at s → ∞ has been obtained in [8] for a gaussian exponential dependence of theprofile function saturating unitarity limit at b = 0 , and this function has nothing todo with the black disc model.The dominance of the imaginary part of the scattering amplitude is almostcommonly accepted in the region of diffraction cone, but the real part should not1e neglected in the whole region of the transferred momentum variation since itis related through the dispersion relations to the imaginary part of the scatteringamplitude and therefore it is not an arbitrary function. We assume the validity ofdispersion relations and, more generally, validity of the anylyticity, unitarity andpolynomial boundedness and then restore the real part of the scattering amplitudeusing the derivative analyticity relations [7, 9, 10, 11]. Those local relations havebeen obtained in the limit of s → ∞ .The following scenario for construction of the real and imaginary parts of thescattering amplitude can be adopted. Namely, the procedure is used: we calculateimaginary part of the scattering amplitude using unitarization scheme based onrational representation where an input is taken to be a pure imaginary and realpart of the input is neglected at this first stage. At the second stage the real partof the amplitude is restored from the imaginary one on the basis of the derivativeanalyticity relations. This stage is proposed to be called as “analytization” byanalogy with the term unitarization. The final amplitude includes both imaginaryand real parts and is consistent with unitarity; the limit Im f → implies thatRe f → at s → ∞ and fixed impact parameter. Normalization of the amplitude f is determined by the unitarity relationIm f ( s, b ) = h el ( s, b ) + h inel ( s, b ) , (1)where h el,inel are the elastic and inelastic overlap functions, respectively and h el ≡| f ( s, b ) | . As it will be shown below, the limit Re f → at s → ∞ at any fixedvalue of b results from its proportionality to the inelastic overlap function h inel which tends to zero at s → ∞ at any fixed value of the impact parameter. Till the measurements at the LHC energies, the elastic scattering data have beenconsistent with the BEL picture, i.e. when the protons’ interaction region becomesBlacker (increase of absorption) at the center, relatively Edgier and Larger. Theanalysis of the data on elastic scattering obtained by the TOTEM at the energy √ s = 7 TeV has revealed definite hints on the presence of the new mode in stronginteraction dynamics [12, 13, 14, 15, 16] called as antishadowing, reflective orresonant mode. It constitutes a gradual transition to the so called REL picture, i.e.when the elastic interaction starts to be Reflective (i.e. the corresponding scatter-ing matrix element becomes negative) at the center and simultaneously appears tobe relatively Edgier, Larger and blacker at its periphery. Asymptotic picture canthen be described as a black ring, its possible appearance has been mentioned in[13]. 2here are phenomenological models which can successfully describe the ex-isting experimental data (cf. e.g. [17] and the references therein).The appearance of the reflective mode is registered under reconstruction of theelastic amplitude, elastic and inelastic overlap functions in the impact parameterrepresentation [18]. It is turned out that the scattering amplitude in the impactparameter representation is monotonically reaching the black-disk limit / frombelow. It allows one to assume a monotonic increase with energy in case of satura-tion of the unitarity limit by the elastic scattering amplitude in the impact param-eter representation. We follow here an opportunity related to the unitarity limitsaturation. It was supported by the analysis of the experimental data on elasticscattering in the impact parameter representation [18]. The recent luminosity-independent measurements at √ s = 8 TeV [19] have confirmed an increase ofthe ratio σ el ( s ) /σ tot ( s ) , which is another, but indirect, indication on the reflectivescattering mode presence.Discussions of the possible interpretations of this mode based on the quark-gluon structure of the hadrons can be found in [20, 21, 22].To restore the real part of the scattering amplitude we use derivative analyticityrelation in the impact parameter representation. The most simple and straightfor-ward derivation of this relation in the impact parameter representation has beengiven recently in [7] under assumption of the dominating imaginary part of thescattering amplitude: Re f ( s, b ) ≃ π ∂ Im f ( s, b ) ∂ ln s . (2)The amplitudes of pp and ¯ pp are taken to be equal. This implies an absence ofthe crossing-odd contributions like the ones generated by the odderon at s →∞ [6]. Such relation (integrated over b ) has been used in [12] for analysis ofthe energy dependence of the parameter ρ ( s ) ≡ ρ ( s, t ) | − t =0 , where ρ ( s, t ) ≡ Re F ( s, t ) / Im F ( s, t ) . The obtained energy dependence has been found to obeythe Khuri-Kinoshita theorem [23], i.e. it decreases like π/ ln s if σ tot ( s ) ∼ ln s at s → ∞ . Here we extend the above consideration to the case of − t = 0 .To construct imaginary part of the scattering amplitude in the impact param-eter representation we use the rational form of unitarization, its origin can betraced back to the papers of Heitler [24], and assume first that the input function U is pure imaginary, i.e. use the replacement U → iU . It gives a pure real elasticscattering matrix element S ( s, b ) and pure real scattering amplitude with the samereplacement f → if . The corresponding relations have the forms S ( s, b ) = 1 − f ( s, b ) , (3) f ( s, b ) = U ( s, b )1 + U ( s, b ) (4)3nd S ( s, b ) = 1 − U ( s, b )1 + U ( s, b ) . (5)We introduce the function r ( s ) which is determined as a solution of the equation U ( s, b ) = 1 , i.e U ( s, b = r ( s )) = 1 . Under this, we suppose that the function U ( s, b ) monotonically increases with the energy at any fixed impact parameter b . This increase is supposed to be a power-like one. Such a dependence leadsto rising behavior of the total cross-section like ln s at s → ∞ and unitaritysaturation, i.e f ( s, b ) → at any fixed b when s → ∞ . The observed increaseof σ tot ( s ) can be considered as an argument in favor of the supposed dependenceof the function U ( s, b ) . When the function U becomes greater than unity, thereflective scattering mode starts to appear [25]. The essential feature of this modeis a peripheral distribution of the inelastic production probability over the impactparameter with maximum reached at b = r ( s ) . The peripheral form becomesmore and more prominently peaked with an energy increase.In our model, we use factorized form for the s and b dependencies of the func-tion U ( s, b ) with an explicit power-like energy increasing dependence of U ( s, b ) and linear exponential decrease of this function with the impact parameter b (i.e.with an energy-independent slope). The corresponding asymptotical dependen-cies r ( s ) ∼ ln s and σ inel ( s ) ∼ ln s , while σ el ( s ) ∼ ln s (cf. e.g. [26]).Now, we can calculate Re f ( s, b ) according to Eqs. (2) and (4):Re f ( s, b ) = π ∂ ln U ( s, b ) ∂ ln s h inel ( s, b ) , (6)where h inel ( s, b ) = U ( s, b )[1 + U ( s, b )] (7)is the inelastic overlap function. Probability distribution of the inelastic processesover impact parameter is P inel ( s, b ) = 4 h inel ( s, b ) . The real part of the scatteringamplitude Re F ( s, t ) can be calculated as the Fourier-Bessel transform of f ( s, b ) :Re F ( s, t ) = sπ Z ∞ bdb Re f ( s, b ) J ( b √− t ) . (8)We have already mentioned that at asymptotical energies inelastic overlap func-tion h inel ( s, b ) has a very prominent maximum at b = r ( s ) and tends to zero at s → ∞ and b = 0 . Such peripheral form with a peak at b = r ( s ) allows one toobtain an approximate relationRe F ( s, t ) ≃ s π ∂ ln U ( s, b ) ∂ ln s (cid:12)(cid:12)(cid:12)(cid:12) b = r ( s ) σ inel ( s ) J ( r ( s ) √− t ) . (9)4ince for the function U ( s, b ) the factorized dependence on the variables s and b with power-like energy dependence was taken, we have ∂ ln U ( s, b ) ∂ ln s (cid:12)(cid:12)(cid:12)(cid:12) b = r ( s ) = const. (10)It should be noted, that due to Eq. (10), the following relations take place in thecase of the unitarity limit saturation:Re f ( s, b ) ∼ h inel ( s, b ) (11)and Re F ( s, t ) ∼ H inel ( s, t ) , (12)where H inel ( s, t ) = sπ Z ∞ bdbh inel ( s, b ) J ( b √− t ) . (13)There are no such relations for the case of the black disk limit saturation,when the inelastic overlap function is not peripheral at s → ∞ . Indeed, we notethat saturation of the black disk limit can be modelled by the following rationalrepresentation f ( s, b ) = ˜ U ( s, b ) / [1 + 2 ˜ U ( s, b )] , (14)where the function ˜ U ( s, b ) has similar to the function U ( s, b ) dependencies on s and b . Eq. (2) can be used then for the restoration of the real part in the case ofthe black disk limit saturation.And, finally, for the real and imaginary parts of the elastic scattering amplitude F ( s, t ) we have in the case of the unitarity limit saturation at s → ∞ :Im F ( s, t ) s ∼ r ( s ) J ( r ( s ) √− t ) √− t (15)and Re F ( s, t ) s ∼ r ( s ) J ( r ( s ) √− t ) . (16)The latter relation results from the asymptotical peripheral dependence of h inel ( s, b ) with sharp maximum at b = r ( s ) . Thus, at asymptotic energies, the ratio ρ woulddepend on the both variables s and − t and should not be a constant over − t . Thistakes place in the region of the small values of − t also and conforms to the con-clusion on existence of the nontrivial − t dependence of the function ρ in the CNIregion [2, 27]. The reason can be traced to the different forms of the impact pa-rameter dependence corresponding to the real and imaginary parts of the elasticscattering amplitude. 5he aim of the present note is to consider the case of unitarity saturation at s → ∞ . At small values of − t the Bessel functions J and J can be approxi-mated by the linear over t exponential functions with different slopes . This meansthat dσ/dt at small- t and s → ∞ can, in principle, be approximated by the twoexponential functions. The function J is sharply peaked, it can be correlated withthe inelastic processes’ contribution. The zeros of the functions J and J wouldlead to the dips in dσ/dt . It should be noted that those simple forms of the realand imaginary part contributions are valid at the asymptotical energies only. But,this fact serves as a motivation for an extension of the simple exponential depen-dence by adding another linear exponent instead of changing linear exponent intoa nonlinear one [1]. Another motivation for such parameterization can be basedon the account of spins of colliding protons [28]. Since a long time fits to dσ/dt with several exponential functions are known to be successful at lower energies. dσ/dt in the low − t region Thus, despite that the LHC energies are not the asymptotic ones, the above asymp-totic results to the LHC energies might happen to be at least in a qualitative cor-respondence with the recent TOTEM result on the σ deviation of the dσ/dt atsmall- t from a simple linear exponential dependence over − t [1].To be a more quantitative, we have performed a simple fit to the TOTEM datausing sum of two linear exponential functions. The aim of this fit is to illustrateexistence of a working alternative to the parameterization with a quadratic expo-nential function Ae Bt − Ct used in [1]. The following function: dσdt = A e B t + A e B t (17)has been used. The resemblance with the results described above is a qualita-tive one and a similar form might also be motivated by an account for the spinamplitudes . The description with two linear exponents appears to be success-ful with the following values of the fitting parameters: A = 524 . mb/GeV , A = 10 . mb/GeV and B = 19 . GeV − and B = 9 . GeV − . The Such parameterization of the differential cross-section implies the equal slopes of the realand imaginary parts of the scattering amplitude at t = 0 . This form should be considered as adeficient one owing to this equality of slopes it implies. Such conclusion results from the derivativedispersion relations for the amplitude slopes [29], the new experimental results on the amplitudephase [5] and the above discussion. If the spin dependence in elastic scattering would survive at such high energies, one can usethen a rather complicated fit, which includes ten linear exponential functions according to thepresence of the five independent helicity amplitudes in pp -scattering description. χ /ndf is 0.46. The table data from [1] have been used and their descrip-tion is shown in Fig. 1. (cid:1) (cid:2) (cid:3) (cid:1) (cid:4) -/ (cid:7) (cid:8) (cid:9) (cid:3) (cid:10) (cid:11) (cid:12) (cid:13) (cid:14) (cid:1)(cid:2)(cid:1)(cid:3)(cid:1)(cid:1)(cid:3)(cid:2)(cid:1)(cid:4)(cid:1)(cid:1)(cid:4)(cid:2)(cid:1)(cid:5)(cid:1)(cid:1)(cid:5)(cid:2)(cid:1) (cid:15)(cid:4)-/(cid:7)(cid:10)(cid:11)(cid:12) (cid:13) (cid:14) (cid:1) (cid:1)(cid:6)(cid:1)(cid:2) (cid:1)(cid:6)(cid:3) (cid:1)(cid:6)(cid:3)(cid:2) (cid:1)(cid:6)(cid:4) Figure 1:
Description of the differential cross-section of elastic pp -scattering in the lim-ited region of low values of − t at √ s = 8 TeV, the data have been taken from [1].
Conclusion
The restoration scenario of the real part of the elastic scattering amplitude hasbeen considered in the case of the unitarity saturation at s → ∞ . The unitaritysaturation is indicated by the recent experimental data at the LHC energy range.The real part of the amplitude appears to have a different t -dependence com-pared to its imaginary part in this particular case. The difference in the dependen-cies of the real and imaginary parts of the scattering amplitude means presenceof a nontrivial t –dependence in the amplitude phase and can be correlated withthe recent TOTEM results on the measurements of dσ/dt in the low-t region at √ s = 8 TeV. It evidently also results in the t -dependent ratio of the real to theimaginary parts of the elastic scattering amplitude.Numerical fit with sum of two linear exponents having different slopes hasbeen performed to illustrate consistency of the experimental results with additivityof the contributions into the differential cross-section dσ/dt as it was discussedabove. Acknowledgements
We are grateful to V.V. Anisovich for the stimulating correspondence.7 eferences [1] G. Antchev et al. (The TOTEM Collaboration), Nucl. Phys. B. , 527(2015).[2] A. Martin, Phys. Lett. B , 137 (1997).[3] N.N. Khuri, Proc. of Vth Blois Workshop — International Conference onElastic and Diffractive Scattering, Providence, RI, 8-12 Jun 1993, EditorsH.M. Fried, K. Kang and C-I Tan, World Scientific (Singapore), 1994 p. 42.[4] C. Bourerely, N.N. Khuri, A. Martin, J. Soffer, T.T. Wu, Contribution to theproceedings of the XIth International Conference on Elastic and DiffractiveScattering, Chˆateau de Blois, May 15-20, 2005, presented by T.T. Wu.[5] G. Antchev et al. (The TOTEM Collaboration), arXiv: 1610.00603.[6] S.M. Troshin, Phys. Lett. B , 40 (2009).[7] V.V. Anisovich, V.A. Nikonov, J. Nyiri, Int. J. Mod. Phys. A , 1550188(2015).[8] I.M. Dremin, S.N. White, arXiv:1604.03469.[9] V.N. Gribov, A.A. Migdal, Sov. J. Nucl. Phys. , 583 (1969).[10] J.B. Bronzan, ANL/HEP-7327, 33 (1973) .[11] J.B. Bronzan, G.L. Kane, U.P. Sukhatme, Phys. Lett. B , 272 (1974).[12] S.M. Troshin, N.E. Tyurin, Phys. Lett. B , 517 (1988).[13] S.M. Troshin, N.E. Tyurin, Phys. Lett. B , 175 (1993).[14] P. Desgrolard, L.L. Jenkovszky, B.V. Struminsky, Phys. Atom. Nucl. , 891(2000).[15] S.M. Troshin, N.E. Tyurin, Int. J. Mod. Phys. A , 4437 (2007).[16] V.V. Anisovich, V.A. Nikonov, J. Nyiri, Phys. Rev. D , 074005 (2014).[17] D.A. Fagundes, M.J. Menon, P.V.R.G. Silva, Nucl. Phys. A 946, 194 (2016).[18] A. Alkin, E. Martynov, O. Kovalenko, and S.M. Troshin, Phys. Rev. D ,091501(R) (2014) . 819] G. Antchev et al. (The TOTEM Collaboration) Phys. Rev. Lett. , 012001(2013).[20] S.M. Troshin, N.E. Tyurin, Mod. Phys. Lett. , 1650025 (2016).[21] I.M. Dremin, arXiv:1605.08216.[22] J.L. Albacete, A. Soto-Ontoso, arXiv:1605.09176.[23] N.N. Khuri, T. Kinoshita, Phys. Rev. , B720 (1965).[24] W. Heitler, Proc. Camb. Phil. Soc. , 291 (1941).[25] S.M. Troshin, N.E. Tyurin, Mod. Phys. Lett. A , 1650079 (2016).[26] S.M. Troshin, N.E. Tyurin, Mod. Phys. Lett. A , 1103 (2009).[27] V. Kundrat, M. Lokajiˇcek, Phys. Rev. D , 1045 (1985).[28] A.D. Krisch, Phys. Rev. Lett. , 1149 (1967).[29] E. Ferreira, Int. J. Mod. Phys. E16