E S.Drenska, S.Cht.Mavrodiev ON MODELLING OF HADRON INTERACTIONS

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2 E S.Drenska, S.Cht.Mavrodiev ON MODELLING OF HADRON INTERACTIONS

3 Дренска С, Мавродиев С.Щ. Е О моделировании адронных взаимодействии На основе понятия об эффективном радиусе взаимодействия дано описание полных сечений адрон-адронных взаимодействий. Исследована зависимость полных сечений от квантовых чисел сталкивающихся адронов.предска' зано поведение полных сечений, в том числе и адрон-ядерных взаимодействий, при энергиях, недоступных на современных ускорителях. Работа выполнена в Лаборатории теоретической физики ОМ"!'! Сообшение Объединенкого института ядерных исслеловений. Дубна 1879 Qrenska S., Mavrodiev S.Cht. Е On I'T'delling of Hadron Interactions On the basis of the idea about the effective radius a description of the tota! cross sections of the hadron interactions is given. The dependence of the tota' rro?s sections on the quantum numbers of the со''. ' s... шь is investigated. The behaviour of the total cross sections', irtclt^din^ '.'". ь \v --mc iear in^cac t ions is predicted at the energies that are impossible for the modern accelerators. The investigation has been performed at the Laboratory of Theoretical Physics, JINR. Communication of the Joint Institute for Nuclear Research. Dubno ОбъвджюкшыЛ институт швршых исслшговшш» Дубна

4 The problem of understanding the elementary particle dynamics puts forward alongside with the development of the general ideas of quantum field theory also the preliminary problem of the systematization of the elementary particles' l/ and the experimental data of their interaction. In this paper a possibility is considered for a simple and physical clear modelling of hadron interaction. It is based on the geometrical generalization of the Rutherford formula: T^) = -JS_/d 3 r Tfrje"'", 4 ir where 1 is the relative coordinate, q is the relative momentum, and T(~q), T(r) are the elastic scattering amplitude in momentum and coordinate representation. The connection between three-dimensional relative momentum and coordinate spaces has a simple group-theoretical sense. It is a reduction of the regular representation of the group of motion of momentum space into irreducible representations defined in the coordinate space. Consequently, if the geometry of the relative momentum space for some two-particle problem is known the above group theoretical construction can be used to define the relative coordinate space. In agreement with the quasipotential approach /a,3/ we shall assume ' 4 / that the relative momentum of hadron interaction is three-dimensional and belongs to the Lobachevsky space. So, there is a relativistic analog of the Rutherford formula T(a.t) ^ - I dnrr)t(a.1)h?.i). m where fft,r)is the relativistic plane wave, m(s) is the effective mass of the problem, s and t are usual Mandelstam's variables. 3

5 Roughly speaking, in the dynamical relativistic Fourier analysis there appears the relative rapidity^1* у л = in Vjn5-±2l± m instead of the relative momentum in nonrelativistic Fourier analysis. So the relativistic analog of the Heisenberg relation for momentum and coordinate is the uncertainty relation for the rapidity and relativistic relative coordinate AvAri 4- -i-. < mc The exponential-power behaviour of the elastic scattering amplitude with increasing momentum transfer has a simple geometrical origin due to the well-known statement: the Euclidean geometry is the local geometry of the Lobachevsky space. Because of the clear quantum-mechanical meaning of formula (1) we can start to model the elastic scattering amplitude in the relative coordinate space. Comparing with the experimental data in momentum space by the numerical analysis it is possible to construct the elastic amplitude. If we propose that T(s,7 )has the form K.{s)e \ds)e T(s, r)= ± + ~ (2) then T(s,t) factorizes in the form /6/ T(s,t)= T(s,t,A,R<s)' /r ( 3 ) where Л is a set of unknown parameters and R(s) = R 1+ R z /s R 9 +R 4 (]ns/r 5 ) Re. (4) The optical theorem gives e from (3) o t (s) = S*R 2 (s). ( 5 ) So, the function R(s) can be interpreted as an effective radius of hadronic interactions. The effective radius of hadron-hadron interactions above the threshold energies has been obtained / 7 in the form 4

6 6, mb 300 r 200 Ш0 IF Я? R(s). -''ii tat 2 + R(ln s/s ) A / T~ \2 г,2 о S, GeV 2 I Pi Mi Tl 1,8929*0,0061 1,9198*0,0432 2,2672*0,0309 2,7916*0,0296 3,8160*0,0764 6,4937*0,1362 * 15,0292*0,3446 0,0259*0,0123 0,1917*0,0857 0,2704*0,1171 0,2703*0,1150 0,3464*0, ,6315*0,1114 0,8842*0,0882 0,0196*0,0047 0,1115*0,0303 0,1984*0,0415 0,2431*0,0483 0,3262*0,0536 0,6157*0,0643 * 1,1628*0,0312» Д-0,2428±0,0023 mb''.\ A =0,9993*0,0030* «=0,906*0,353 mb '' Fig. I R(s)= 1 i-1 sp Ms-ii t r+sr; + R ln A (s/soj.. The unknown functions and parameters have been found by solving the overdetermined nonlinear system of equations (6) о т (и,)- 2*R S ( S,), ( 7 ) where a t (s) are the experimental total cross sections for the processes pp, pp, n +, K+ p. The autoregularized iteras

7 6, mb SO JO I ' ' " """ " ' - p tat so 2 4 R(s) = -_! + R (In s/s^ A (v/s-^) 2 +srf S, GeV 2 0,0222*0,0043 0,0419*0,0100 0,2580*0,0551 0,2716*0,2022; 0,4427*0,2098 0,5234*0,1296 0,9221*0,0637 /4 B 1,9691*0,0020 2,2056*0,0041 2,360B±0,0361 2,9249*0,0619 3,9992*0,1514 6,4937*0,1362* 15,0292*0,3446 * 0,0282*0,0040 0,0586*0,0078 0,2104*0,0226 0,3051*0,105(1 0,4555*0,0980 0,6157*0,0643 * 1,1628*0,0312 «Я=0,2428±0,0023 raft '<> *, A =0,9993* й 0 =0,856±0,392 nib''' Fig. 2 tion processes of the Gauss-Newton type 8 ' in the numerical analysis were used. Figures 1-6 represent the experimental data and our description of the above-mentioned total cross sections, the corresponding effective radii, and their constituents (see (6)). The quantity X z /ti equals 1.11 for nucleons; l.ol, for pions; 1.06, for kaons. The formula (6) shows that the effective radius is a sum of the "effective radii" of successively opening with the increasing energy "channels 6

8 and the Froissart term. Starting from s ;> 100 GeV the Froissart term gives the main contribution to the effective radius. The asymptotic behaviour of the effective radius is R W R 0 + R ln"(s/s 0), 7 2 /=i 1+ Г/ в, ть 6*'f<5) S, GeV s i 0,1073±0,0006 l,22o4±0,0003 0,0651 ±0, ,Ш8±0,0012 0,ОЗИ±0,0005 1,4981±0,0004 l,b80o±o,00o3 0,0952±0,0008 0,0332±0, ,0641±0,0028 1,843О±О,000в 0,0879±0, ,2560±0,0186 2,1819±0,0039 0,2423 ±0, ,5992*0,0525 0,9055±0,0710 3,3347±0,0212* 8,3813±0,1397 * 0,5831 ±0,032(>* 1,2014±0,0406 Л=0,1814±0,0024 mb''", 4=0,9896*0,0043 До-0,696±0,087 mb * Щ- з 7

9 6,mb ZOO tot + R(ln s/s )' Pi V i 1 n 0,1073±0,00'10 1,2233±0, ,1014*0,0024 4,3233*0,0021 0,0494*0,0003 о,ша±о,«и:) 3 0,0076*0,0017 1,6700*0,01X17 о,г«л1±о,и»и ч 0,«ЮО± ±,8993*0,0004 U,O8'.!)±0,00O5 h *0,0123 2,3'.90±(l,0020 0,2185*0,0064 l'i O,58(i3±0,O5OS 3,3347*0,0212 0,583 l±0,032c, * 1 0,917(i*0,07(l!l S,3S'13±n,13tl7* 1,2014*0,0406» Я=0,1814±0,0024 rah'''*, Л =0,9996*0,0043' Я 0 =0,В72±0,082 fflb* Fig. 4 Obviously, comparing the behaviour of the total cross sections and the asymptotical ones the value of the Froissart energy can be estimated. The asymptotic behaviour for all total cross sections is reached (see Fig. 7) in the energy range s=lo e GeV 2 (\/T= GeV, 10" 16 ± cm). If one compares the Froissart energy with the unitary limit energy for the weak interactions, then it is possible to interpret the Froissart energy as the threshold energy of new structures of matter. 8

10 6, mb 2U030D S,GeV z f'i 1 f 0,1038±0,OO31 M83fi±0,OOI5 0,0B19±0,00l.i 2 0,0411 ±0,0024 1,6985±0,0010 0,О528±0,0П22 0,041У±0,0014 I,8l4lte0,000li 0,0431 ±0,0010 i ii 0,0324±0,0011 2,0040*0,0000 0,0573±0, ,5421 ±0,0281 2,33 ±0,0002 0,4331*0,0158 В 0,5(103±0,1093 4,922I5±0,1IOO* O,70OB±O,00i3 * 7 0,3412±0,14(1S li,4578±0,3851* l,17!)3±0,ll!)2* Я=0,1941±0,0052 mb'i-', Л=0,9998±0,0036' йо=0,600±0,14о mb''- Fig. 5 Figure 8 shows that the theoretical description confirms the Pomeranchuk theorem. The solid curve represents the lower bound; and the dotted one, the upper bound of the corresponding cross sections. The dependence of the effective radius on the quantum numbers has been obtained in. The total cross sections of p,p,ff+, К + with proton, neutron, deutron forssloo GeV 9

11 в, 30 zo г mb д/~ Г <>> R(s) - X _ ^ ' + R(ln s/s o ) A.У, GeK K 1 0,0983*0,0051 1,5208*0,0014 0,0978*0, ,1000*0,0103 0,1435*0,0298 1,8532±0,ОШГ> 2,1694*0,012 0,1210*0,00)6 0,1930*0, ,0 2,3738*1,8847 0,2305*0, ,3865±0,Ш59 2,9879*0,0428 0,4322*0,0497 (i 0,4933*0,1109 4,9226*0,1100* 0,7006*0,0943 * i 0,7000*0,1418 И,4578±ВД851'» *0,1192* Д=0,1941±0,0052 ть **, А =0,9998*0,0036* /г 0 =0,533±0,167 тъ' 1 ' Fig. 6 Ц can be described as functions of the quantum numbers with the formula (5) and R(s,A,a) = R i+ R 2 /(s/r b ) л + R^ lnfs/r 5J (8) where a is a set of quantum numbers ind A, a set of parameters. The functions Я ; fi = 1 5) have the form 10 R 1=A 1 M + A Z J + Л 3 / +A 4 S R z = А ъ <А в К + A.)B\<. A 8 \Q\ >- A \I3\ t A 10!

12 и й 3 = 4 П + А К М +A la J + A U \Q\+ А 1Ь 1 fi4 i e /3 ь А ^ S, я 4 =л 1 8 +л 1 9 к + л 2 0 / where «5 - J +^»81 М = m t +я В = Ь, ч Ь j = (J, +J a ;cjj +j g +л «= 9,-H 2 baryon number spin charge 7 = ^i + /2 > ( I i +! г ^ ] > jsotopic spin third projection of isotopic spin Y - y, + y t SU(3) quark number hypercharge S =s, + s 2 strangeness AP.P.PI.PI,K,K -PROTON FROIASSART ENERGIES W S 10 S, GeV *

13 The Table represents the values and the errors of the parameters A. The quantity x a /W is Figures 9-11 represent the obtained description of the experimental data. N *l Table +/- ДД, N *1 +/- AA, r / V */ / / / / / / / / ' / V / / / / / / (For the dimensions of the parameters see formula C8)) AP.P,PI -,P! +.K,K + POMERANCIIUK ENERCIES 5, mb fffff 12 Ш г W 3 W W 5 10 s Fig. 8 s, GeV s

14 PROTON toooo. Fig. 9 s [GeV 2 ] NEUTRON Fig. 10 s [GeV 2 ] 13

15 g t (s)[mb] DEUTRON Fig s [GeV 2 ] PREDICTIONS s [GeV 2 ] 14

16 QUARK SUMM RULES 15

17 Figure 12 represents tne predictions for the total cross sections of Л, 1 +, ~ hyperons with proton. Figure 13 represents our predictions for the total cross sections 3 of Tr, He, He with proton and of " + t + interaction. Figure 14 represents the behaviour of the following quark sum rules / l / PP. : 6 un = 3 KN + 2 NN LI: 6ir~p + 3K + p = 2pn + 6K"p L2 : pp + 2 ~ p = Лр + pn JN: 4я-+р = 2ir-p + 7/8 K + p + 3/4 pp. In conclusion it should be mentioned that our predictions have about 10% uncertainty. The future measurements of the hadron-hadron total cross sections at s = 400 to 1000 GeV 2 would decrease this uncertainty. REFERENCES 1. Боголюбов Н.Н. Теория симметрии элементарных частиц. В сб.: Физика высоких энергий и теория элементарных частиц. "Наукова думка*, Киев, Logunov A.A., Tavkhelidze A.N. Nuovo Cim.,1963,29, p.380. Кадышевский В.Г., Тавхелидзе А.Н. В сб.: Проблемы теоретической физики. 'Наука", М., Kadyshevsky V.G. Nucl.Phys., 1968, В8б, р.125; Kadyshevsky V.G., Mateev M.D. Nuovo Cim., 1967, 55A,p.276; Кадышевский В.Г., Мир-Касимов P.M., Скачков Н.Б. ЭЧАЯ, 1972, том 2, вып.з, с Mavrodipv S.Cht. Fizika, 1977, p.11". 5. Mavrodiev S.Cht. JINR, P2-8897, Dubna, Alexandrov L., Mavrodiev S.Cht. JINR, E2-9936, Dubna, Drenska S., Mavrodiev S.Cht. Yad. fiz., 1978, v.28, p Александров Л. ЖВМ и МФ, 1971, 11, с.1; Александров Л. ОИЯИ, P5-55U, Дубна, 1970; Александров Л. ОИЯИ, Б , Дубна, Alexandrov L., Drenska S., Mavrodiev S.Cht. JINR, P , Dubna, lu. Denisove S.P. et al. Nucl.Phys., 1973, B65, p.l( Lipkin H.J. Nucl.Phys., 1974, B79, p.381. Rang R., Nikolescu J. Phys.Rev., 1975, Dll, p.i' Received by Publishing Department on October