In this paper an analytical description of the hadron-hadron scattering is
presented by using PMD-SQS-optimum principle in which the differential cross
sections in the forward and backward c.m. angles are considered fixed from the
experimental data. Experimental tests of the PMD-SQS-optimal predictions,
obained by using the available phase shifts, as well as from direct
experimental data, are presented. It is shown that the actual experimental data
for the differential cross sections of all principal hadron-hadron
[nucleon-nucleon, antiproton-proton, mezon-nucleon] scatterings at all energies
higher than 2 GeV, can be well systematized by PMD-SQS predictions.Comment: 18 pages, 10 figure

Ion, D. B.

Ion, M. L.

English

arXiv

In this paper an analytical description of the hadron-hadron scattering is presented by using PMD-SQS-optimum principle in which the differential cross sections in the forward and backward c.m. angles are considered fixed from the experimental data. Experimental tests of the PMD-SQS-optimal predictions, obtained by using the available phase shifts, as well as from direct experimental data, are presented. It is shown that the actual experimental data for the differential cross sections of all principal hadron-hadron [nucleon-nucleon, antiproton-proton, mezon-nucleon] scatterings at all energies higher than 2 GeV, can be well systematized by PMD-SQS predictions. 1

D. B. Ion

M. L. D. Ion

CiteSeerX

ANALYTICAL DESCRIPTION OF HADRON-HADRON SCATTERING VIA PRINCIPLE OF MINIMUM DISTANCE IN SPACE OF STATES

In this paper an analytical description of the hadron-hadron scattering is presented by using PMD-SQS-optimum principle in which the differential cross sections in the forward and backward c.m. angles are considered fixed from the experimental data. Experimental tests of the PMD-SQS-optimal predictions, obained by using the available phase shifts, as well as from direct experimental data, are presented. It is shown that the actual experimental data for the differential cross sections of all principal hadron-hadron [nucleon-nucleon, antiproton-proton, mezon-nucleon] scatterings at all energies higher than 2 GeV, can be well systematized by PMD-SQS predictions.In this paper an analytical description of the hadron-hadron scattering is presented by using PMD-SQS-optimum principle in which the differential cross sections in the forward and backward c.m. angles are considered fixed from the experimental data. Experimental tests of the PMD-SQS-optimal predictions, obained by using the available phase shifts, as well as from direct experimental data, are presented. It is shown that the actual experimental data for the differential cross sections of all principal hadron-hadron [nucleon-nucleon, antiproton-proton, mezon-nucleon] scatterings at all energies higher than 2 GeV, can be well systematized by PMD-SQS predictions

Ion, D.B.

Ion, M.L.

CERN Document Server

arXiv:hep-ph/0608072 v1   6 Aug 2006ANALYTICAL DESCRIPTION OFHADRON-HADRON SCATTERINGVIA PRINCIPLE OF MINIMUMDISTANCE IN SPACE OF STATESD. B. Ion1,2) and M. L. D. Ion3)1)TH Division, CERN, CH-1211 Geneva 23, Switzerland2)NIPNE-HH , Bucharest P.O Box MG-6. Romania3)Bucharest University, Department of Atomicand Nuclear Physics, Bucharest, RomaniaAbstractIn this paper an analytical description of the hadron-hadron scatteringis presented by using PMD-SQS-optimum principle in which the differen-tial cross sections in the forward and backward c.m. angles are consideredfixed from the experimental data. Experimental tests of the PMD-SQS-optimal predictions, obained by using the available phase shifts, as wellas from direct experimental data, are presented. It is shown that theactual experimental data for the differential cross sections of all princi-pal hadron-hadron [nucleon-nucleon, antiproton-proton, mezon-nucleon]scatterings at all energies higher than 2 GeV, can be well systematized byPMD-SQS predictions.IntroductionThe mathematician Leonhard Euler (1707-1783) appears to have been aphilosophical optimist having written:”For since the fabric of the universe is most perfect and the work of a mostwise Creator, nothing at all takes place in the universe in which some ruleof maximum or minimum does not appear. Wherefore, there is absolutely nodoubt that every effect in universe can be explained as satisfactory from finalcauses themselves the aid of the method of Maxima and Minima, as can fromthe effective causes”.Yet this brilliant idea produced many strikingly simple formulations of cer-tain complex laws of nature. From historical point of view the earliest optimumprinciple was proposed by Heron of Alexandria (125 B.C.) in connection withthe behavior of light. Thus, Heron proved mathematically the following firstgenuine scientific minimum principle of physics: that light travels between two1points by shortest path. In fact the Archimedean definition of a straight line asthe shortest path between two points was an early expression of a variationalprinciple, leading to the modern idea of a geodesic path. In fact in the samespirit, Hero of Alexandria explained the paths of reflected rays of light based onthe principle of minimum distance (PMD), which Fermat (1657) reinterpretedas a principle of least time, Subsequently, Maupertuis and others developed thisapproach into a general principle of least action, applicable to mechanical aswell as to optical phenomena. Of course, a more correct statement of these op-timum principles is that systems evolve along stationary paths, which may bemaximal, minimal, or neither (at an inflection point). Laws of mechanics werefirst formulated in terms of minimum principles. Optics and mechanics werebrought together by a single minimum principle conceived by W. R. Hamilton.From Hamilton’s single minimum principle could be obtained all the optical andmechanical laws then known. But the effort to find optimum principles has notbeen confined entirely to the exact sciences. In modern time the principles ofoptimum are extended to all sciences. So, there exists many minimum prin-ciple in action in all sciences, such as: principle of minimum action, principleof minimum free-energy, minimum charge, minimum entropy production, mini-mum Fischer information, minimum potential energy, minimum rate of energydissipation, minimum dissipation, minimum of Chemical distance, minimumcross entropy, minimum complexity in evolution, minimum frustration, mini-mum sensitivity, etc. So, a variety of generalizations of classical variationalprinciples have appeared, and we shall not describe them here.Next, having in mind this kind of optimism in the paper [1-16] we intro-duced and investigated the possibility to construct a predictive analytic theoryof the elementary particle interaction based on the principle of minimum dis-tance in the space of quantum states (PMD-SQS). So, choosing the partialtransition amplitudes as the system variational variables and the “distance” inthe Hilbert space of the quantum transitions as a measure of the system effec-tiveness expressed in function of partial transition amplitudes we obtained theresults [1-16]. These results proved that the principle of minimum distance inspace of quantum states (PMD-SQS) can be chosen as variational principle bywhich we can find the analytic expressions of the partial transition amplitudesIn this project by using the S-matrix theory the minimum principle PMD-SQSwill be formulated in a general mathematical form. We prove that the new ana-lytic theory of the quantum physics based on PMD-SQS is completely describedwith the aid of the reproducing kernels from RKHS of the transition amplitudes.[1-5].Therefore, in Ref. [1] by using reproducing kernel Hilbert space (RKHS)methods [3-5,17], we described the quantum scattering of the spinless particlesby a principle of minimum distance in the space of quantum states (PMD-SQS).Some preliminary experimental tests of the PMD-SQS, even in the crude form[1] when the complications due to the particle spins are neglected, showed thatthe actual experimental data for the differential cross sections of all principalhadron-hadron [nucleon-nucleon, antiproton-proton, mezon-nucleon] scatteringsat all energies higher than 2 GeV, can be well systematized by PMD-SQS predic-2tions (see the papers [1,], Moreover, connections between the PMD-SQS and themaximum entropy principle for the statistics of the scattering quantum chan-nels was also recently established by introducing quantum scattering entropies:Sθand SJ [5-7]. Then, it was shown that the experimental pion-nucleon as wellas pion-nucleus scattering entropies are well described by optimal entropies andthat the experimental data are consistent with the principle of minimum dis-tance in the space of quantum states (PMD-SQS) }[1]. However, the PMD-SQSin the crude form [1] cannot describe the polarization J-spin effects.In this paper an analytical description of the hadron-hadron scattering ispresented by using PMD-SQS-optimum principle in which the differential crosssections in the forward (x=+1) and backward (x=-1) directions are consideredfixed from the experimental data. An experimental test of the optimal predictionon the logarithmic slope b is performed for the pion-nucleon and kaon-nucleonscatterings at the forward c.m. angles.2. Description of pion-nucleon scattering via principle of minimumdistance in space of quantum states (PMD-SQS)First we present the basic definitions on the (0−1/2+ → 0−1/2+) hadronicscattering:M(0−) +N(1/2+)→M(0−) +N(1/2+), (1)Therefore, let f++(x)and f+−(x), be the scattering helicity amplitudes ofthe mezon-nucleon scattering process (see ref.[14]) written in terms of the partialhelicities fJ− sand fJ+as followsf++ (x) =Jmax∑J= 12(J + 12)(fJ− + fJ+) dJ1212(x)f+− (x) =Jmax∑J= 12(J + 12)(fJ− − fJ+) dJ− 1212(x)(2)where the rotation functions are defined asdJ1212(x) = 1l+1 ·[1+x2] 12[•Pl+1(x)−•Pl(x)]dJ− 1212(x) = 1l+1 ·[1−x2] 12[•Pl+1(x) +•Pl(x)] (3)where Pl(x)are Legendre polinomials,oPl(x) = ddxPl(x), x being the c.m. scat-tering angle. The normalisation of the helicity amplitudes f++(x) and f+−(x),is chosen such that the c.m. differential cross section is given bydσdΩ(x) = |f++ (x)|2 + |f+− (x)|2 (4)Then, the elastic integrated cross section is given by3σel2pi=Jmax∑J=frac12(2J + 1)(|fJ+|2+ |fJ−|2)(5)Now, let us consider the following optimization problem:D(fJ+, fj−) = σel/2pi =∑(j +12)[|fj+|2+ |fj−|2](6)when dσdΩ(+1) anddσdΩ(−1) are fixed.We proved that the solution of this optimization problem is given by thefollowing results :f++o (x) = f++(+1)K 1212(x, y)K 1212(+1,+1)(7)f+−o (x) = f+−(−1)K 12− 12(x, y)K 12− 12(−1,−1)(8)where the functions K(x,y) are the reproducing kernels [3-5] expressed in termsof rotation function byK 1212(x, y) =Jo∑1/2(j +12)dj1212(x)dj1212(y), (9)K 12− 12(x, y) =Jo∑1/2(j +12)dj12− 12(x)dj12− 12(y) (10)while the optimal angular momentum is given byJ0 =√4piσel[dσdΩ(+1) +dσdΩ(−1)]+14− 1 (11)Now, let us consider the logarithmic slope b of the forward diffraction peakdefined byb =ddt[lndσdt(s, t)]t=0(12)Then, using the definition of the rotation functions, from (7)-(11) we obtainthe optimal slope bobo =λ24[4piσel(dσdΩ(+1) +dσdΩ(−1))− 1](13)Finally, we note that in ref. [13] we proved the following optimal inequalitybo =λ24[4piσel(dσdΩ(+1) +dσdΩ(−1))− 1]≤ bexp (14)4which includes in a more general and exact form the unitarity bounds derivedby Martin [18] and Martin-Mac Dowell [19] (see also ref.[20]) and Ion [1,21].Indeed, since dσdΩ(±1) ≥ 0, anddσdΩ(+1) ≥σ2T16piλ¯2, (15)(Wik inequality) from the bound (14), we getλ24[4piσel(dσdΩ(+1))− 1]≤ bexp (proved in ref. [1]) (16)λ24[4piσel(dσdΩ(−1))− 1]≤ bexp (proved in this paper) (17)λ¯24[σ2T4piλ¯2σel− 1]≤ bexp (improvedMartin−MacDowell bound [19]) (18)λ¯24[ σT4piλ¯2− 1]≤ bexp (Martin bound [18]) (19)3. Experimental tests of the PMD-SQS-optimal predictionsFor an experimental test of the optimal result (14) the numerical values ofthe slopes bo and bexp are calculated directly by reconstruction of the helicityamplitudes from the experimental phase shifts (EPS) solutions of Holer et al.[23] and also directly from the experimental data. The results are displayied inFig 1-5. Moreover, we calculated from the experimental data (see [24-29]) thefollowing physical quantities:SCALING FUNCTION:f(τ) ≡dσdΩ(x)/dσdΩ(1) (20)SCALING VARIABLE:τ ≡ 2√|t|bo (21)and compared with the values of the PMD-SQS-optimal predictions obtainedfromOPTIMAL SCALING FUNCTION:fo(τo) ≡dσ0dΩ(x)/dσdΩ(1) =[K 1212(x, 1)K 1212(1, 1)]2≈[2J1(τo)τo]2(22)The results are presented in Fig. 6. We must note that the approximationin (22) is derived by using the relationdjµν(x) ≈ J|µ−ν|[2(j + 1) sinθ2], for small θ − angles (23)Where J|µ−ν|(τ)are Bessel functions of order |µ− ν|.5Figure 1: The experimental logarithmic slopes (bexp) of the diffraction peak, forthe forward pi+P → pi+P scattering, are compared with the optimal predictionsbo (13).6Figure 2: The experimental logarithmic slopes (bexp) of the diffraction peak, forthe forward pi=P → pi−P scattering, are compared with the optimal predictionsbo (13) .7Figure 3: The experimental logarithmic slopes (black circles) of the diffractionpeak, for the forward pi=P → pi−P scattering, are compared with the optimalpredictions bo (13)(white circles).8Figure 4: The experimental values of the logarithmic slopes are comparedwith the values of optimal predictions (xx) (solid curves) for the PP →PP scatterings. Dashed curve correspond to an estimation of the Martin-MacDowell bound [19].9Figure 5: The experimental values of the logarithmic slopes (black circles)are compared with the values of optimal predictions (white circles) for the (a)PP → PP scatterings. Dashed curve correspond to an estimation of the Martin-MacDowell bound [19].10Figure 6: The experimental values of the scaling function (20) are comparedwith the values of optimal scaling predictions (22) (solid curves) for the (a)pi±P → pi±P and (b) K±P → K±P scatterings11Figure 7: The differential cross sections for pi+P → pi+P calculated by usingeqs. (4) and the experimental phase shifts [12] are compared with the optimalstate predictions given by eqs. (7)-(11).12Figure 8: The differential cross sections for pi+P → pi+P calculated by usingeqs. (4) and the experimental phase shifts [12] are compared with the optimalstate predictions given by eqs. (7)-(11).13Figure 9: The differential cross sections for pi+P → pi+P calculated by usingeqs. (4) and the experimental phase shifts [12] are compared with the optimalstate predictions given by eqs. (7)-(11).14Figure 10: The differential cross sections for pi+P → pi+P calculated by usingeqs. (4) and the experimental phase shifts [12] are compared with the optimalstate predictions given by eqs. (7)-(11).154. ConclusionsThe main results and conclusions obtained in this paper can be summarizedas follows:In this paper an analytical description of the hadron-hadron scattering ispresented by using PMD-SQS-optimum principle in which the differential crosssections in the forward (x=+1) and backward (x=-1) directions are consideredfixed from the experimental data. So, choosing the partial transition amplitudesas the system variational variables and the “distance” in the Hilbert space ofthe quantum transitions as a measure of the system effectiveness expressed infunction of partial transition amplitudes we obtained the results [1-16].(i) The PMD-SQS optimal dominance in hadron-hadron scattering at smalltransfer momenta for pLAB > 2 GeV/c is a fact well evidenced experimentallyby the results presented in Figs. 1-6. This conclusion can be also extended inlow energy region.(ii) In the low energy region, the optimal slope (13) is in good agreementwith the experimental data in some domains of energy between the resonancespositions or/and in the region corresponding to the diffractive resonances seeFigs. 1-2 and Figs. 7-10.(iii) We find that the presented experimental tests prove that the principleof minimum distance in space of quantum states (PMD-SQS) can be chosen asvariational principle by which we can find the analytic expressions of the partialtransition amplitudes.Finally, we hope that our results are encouraging for an analytic descriptionof the quantum scattering in terms of an optimum principle, namely, the prin-ciple of minimum distance in space of quantum state (PMD-SQS) introducedby us in ref. [1].5. References[1] D.B.Ion ”Description of quantum scattering via principle of minimumdistance in space of states”, Phys. Lett. B 376 (1996) 282.[2] D.B.Ion and M.L.D.Ion,”Isospin quantum distances in hadron-hadronscatterings”, Phys. Lett. B 379 (1996) 225.[3] D.B.Ion and H. Scutaru, ”Reproducing kernel Hilbert space and optimalstate description of hadron-hadron scattering”, Int. J. Theor. Phys. 24 (1985)355.[4] D.B.Ion, ”Reproducing kernel Hilbert spaces and extremal problems forscattering of particles with arbitrary spins”, Int. J. Theor. Phys. 24 (1985)1217.[5] D.B.Ion ”Scaling and S-channel helicity conservation via optimal statedescription of hadron-hadron scattering”, Int. J. Theor. Phys. 25 (1986) 1257.[6] D.B.Ion and M.L.D.Ion, ”Information entropies in pion-nucleon scatter-ing and optimal state analysis”,Phys. Lett. B 352 (1995) 155.[7] D.B.Ion and M.L.D.Ion, ”Entropic lower bound for quantum scatteringof spineless particles”, Phys. Rev. Lett. 81 (1998) 5714.16[8] M.L.D.Ion and D.B.Ion, ”Entropic uncertainty relations for nonextensivequantum scattering,Phys. Lett. B 466 (1999) 27-32.[9] M.L.D.Ion and D.B.Ion, ”Optimal bounds for Tsallis-like entropies inquantum scattering of spinless particles”, Phys. Rev. Lett. 83 (1999) 463.[10] M.L.D.Ion and D.B.Ion, ”Angle-angular-momentum entropic boundsand optimal entropies for quantum scattering of spineless particles”, Phys. Rev.E 60 (1999) 5261.[11] D. B. Ion and M. L.D. Ion, ”Limited entropic uncertainty as a newprinciple in quantum physics”, Phys. Lett. B 474 (2000) 395.[12] M.L.D.Ion and D.B.Ion, ”Strong evidences for correlated nonextensivestatistics in hadronic scatterings”, Phys. Lett. B 482 (2000) 57.[13] D. B. Ion and M. L.D. Ion, ”Optimality entropy and complexity in quan-tum scattering”,Chaos Solitons and Fractals, 13 (2002) 547.[14] D. B. Ion and M. L.D.Ion, ”Evidences for nonextensive statistics conju-gation in hadronic scatterings systems”, Phys. Lett. B 503 (2001) 263.[15] D. B. Ion and M. L. 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Fiz. 43, 5 (1991).[22] See the books: N. R. Hestenes, Calculus of variations and optimal controltheory, John Wiley&Sons, Inc., 1966, and also V. M. Alecseev, V. M. Tihomirovand S. V. Fomin, Optimalinoe Upravlenie, Nauka, Moskow,1979.[23] Hohler, F. Kaiser, R. Koch, E. Pitarinen, Physics Data, Handbook ofPion Nucleon Scattering, 1979, Nr. 12-1.[24] D. B. Ion and M. L. Ion, A new optimal bound on logarithmic slope ofelastic hadro-hadron scattering, ArXiv: hep-ph/0501146 v1 16 Jan 2005.[25] For extensive experimental literature see: J. Bystricky –Landolt-Bornstein,New Series, Group I-Vol 9a, Nucleon-Nucleon and Kaon-Nucleon Scattering(1980).[26] D.B.Ion, C. Petrascu, Rom. Journ. Phys. 37 (1992) 569-575.17[27] D.B.Ion, C. Petrascu and A. Rosca, Rom. Journ. Phys. 37 (1992)977-989.[28] D.B.Ion, C. Petrascu, Rom. Journ. Phys. 38 (1993) 23-29.[29] D.B.Ion, C. Petrascu, A. Rosca and V. Topor, Rom. Journ. Phys. 39(1994) 213-221.18

Analytical Description of Hadron-Hadron Scattering via Principle of Minimum Distance in Space of States

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