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Hadron Production: Coherent States on Lobachevsky Momentum Space

This research explores the model of hadrons and the problem of multi-particle production using coherent states on the horosphere of Lobachevsky momentum space. The study investigates the kinematics of collisions and the dynamical nature of characteristic size hadrons.

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Hadron Production: Coherent States on Lobachevsky Momentum Space

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  1. Joint Institute of the Nuclear Research B.I. Stepanov Institute of Physics of the National Academy of Sciences S.Harkusha, Yu. Kurochkin The model of hadrons and  problem of the multi-particles production GRODNO 2018

  2. Plan • ICoherent states on the horosphere of the Lobachevsky momentum space as partons • Introduction. Some results of multi-particle production on LHC • Kinematic of the of the collisions in the quasi Cartesian coordinates on the horosphere of Lobachevsky momentum space • Conventional coherent states on horosphere of Lobachevsky momentum space. • Space picture in the laboratory system and main hypothesis. • Multiplicity distribution • Coordinate and momentum representation of the coherent states and multiplicity distribution • II Coherent states of particles moving in constant homogeneous non-Abelian gauge fields and dynamical nature of characteristic size hadrons. • IIIThe model of the scalar hadron on the base of classical field theory

  3. Из новых результатов

  4. Typical sizes The ATLAS Collaboration. Two-particle Bose–Einstein correlations in pp collisions at 0.9 and 7 TeV measured with the ATLAS detector. CERN-PH-EP-2014-264. Submitted to: EPJC. A.A.Logunov, M.A.Mestvirishvili, V.A. Petrov. In book General principles of the quantum fields theory and hadrons interactions under high energy M. : Nauka 1977

  5. Kinematics of the collisions of hadrons (1) • Here - proton mass, we use the system of the physical units with , - system mass energy of protons.

  6. Kinematics of the collisions of hadrons Kinematics of the collisions of hadrons (3) In the laboratory systems, where second proton is in the rest ( 4) and four dimensional momentum of the falling proton denote as

  7. Main hypothesis As you know, in the laboratory frame the incident particle (hadron) is attened in the direction of the movement due to the Lorentz contraction.In this case, transverse degrees of freedom (x and y) are important sinceat high energies the kinetic energy of the hadron constituents (partons) ismuch larger than the energy of their interaction.Therefore hadron moving at a speed close to the speed of light can beseen as a set of almost free partons. Since components of hadron move inunison before collision therefore this state of a hadron can be consideredas a coherent state of its transverse excitations i.e. partons. The main hypothesis is that the incident particle is a coherent state of partons i.e. transverse excitationson horosphere of the relativistic Lobachevsky momentum space. Yu.A. Kurochkin, Yu.A. Kulchitsky, S.N. Harkusha, N.A. Russakovich Hadron as coherent state jn horosphere of the Lobachevsky momentum space. Письма в ЭЧАЯ 2016 Т.13, №3 (201)б 454-460. // Phys. Part. Nuclei Lett., 2016, V. 13, № 3, P. 285-288.

  8. Quasi Cartesian coordinates on the horosphere of Lobachevsky momentum space (5) (6) Inverse formulas (7)

  9. Quasi Cartesian coordinates on the horosphere of Lobachevsky momentum space Metric element in these coordinates expressed as follows (8) The element of the volume in the momentum space in the horospherical (quasi Cortesian) coordinates (9) Separation coordinates (10)

  10. Quantum mechanical variables (momenta, coordinates) on horosphere (11) Heisenberg-Weyl algebra is the base for definition of the coherent states (12) Creation and annihilation operators connected with our problem (13) Yu. A. Kurochkin, I. Rybak, Dz. V. ShoukovyCoherent states on the horosphere of the three dimensional Lobachevsky space. Doklady NAS Belarus 2014

  11. Heisenberg-Weyl algebra in terms of the creation and annihilation operators where corresponds to and Definition of the coherent states and some properties (14) (15) Переломов А.М. Обобщенные когерентные состояния и их применения. М.: Наука. 1987.

  12. Definition of the coherent states and some properties -vacuum state, (16) (17) (18) (19)

  13. Average number of the excited quanta (20) (21) Multiplicity distribution (22)

  14. Coordinate representation of coherent states (23) (24) (25)

  15. Momentum representation of coherent states (26) (27)

  16. Fitting of the experimental data

  17. Denotations to previous pictures p0 = 1.26   0.15, p1 = 0.21  0.03, p2 =0.40   0.03. (28)

  18. Conclusion to part I Thus, in our approach, we distinguish two stages of the process. 1. At each stage, the spatial distribution of hadronic matter isdifferent.2.In the first stage it is two-dimensional, on the second it is three-dimensional.

  19. Quark in external gluon field of magnetic type II Coherent states of particles moving in constant homogeneous non-Abelian gauge fields and dynamical nature of characteristic size hadrons. , , (29) , , , (30) , ,

  20. Non –Abelian color field Kurochkin Yu.A. , Harkusha S.N. , Kulchitsky Yu.A. , Russakovich N.A. /On describing quark in external gluon field of magnetic type // Proceeding of the National Academy of Sciences of Belarus. Physics and Mathematics series -2017, n.4, P. 39-43. , , , (31) , . (32) , , (33) ,

  21. Quark in external gluon field of magnetic , (34) , (35) , , , (36) , , (37)

  22. Quark in external gluon field of magnetic type , (38) , , (39) , , , (40) ,

  23. Quark in external gluon field of magnetic type , (41) , , , (42) , , ,

  24. Quark in external gluon field of magnetic type , (43) , , (44) , , , ,

  25. Conclusion to part II Thus, we showed the principal possibility of introducing coherent states of particles moving in constant homogeneous non-Abelian gauge fields and interacting with these fields. An important feature of the considered model is that here each "color" degree of freedom corresponds to its characteristic size

  26. Part III The model of scalar hadron on the base of classical field theory The task of this part of the talk the further development of field-theoretical models for describing hadrons as coherent states as transverse excitations, considered as partons, determining the limitations imposed by the field-theoretical model on the initial phenomenological model. In this part, we consider the case of the Klein-Fock equation describing a scalar relativistic particle, for example, - meson.

  27. Separation of transverse and longitudinal variables in the Klein-Fock equation and the Levy-Civita transformation (45) (46)

  28. Separation of transverse and longitudinal variables in the Klein-Fock equation and the Levy-Civita transformation We will seek the solution of equation (45) by dividing the variables into "transverse" and "longitudinal" variables. Obviously, before the introduction of the interaction that separates the direction in space, the separation of variables into "transverse" and "longitudinal" parts is conventional. Then the solution of equation (45) can be represented in the form (47)

  29. Separation of transverse and longitudinal variables in the Klein-Fock equation and the Levy-Civita transformation (48) (49)

  30. Levy-Civita transformation (complex functions). (50) (51) (52) we made next substitution in (48) : and introduce new variables than this equation presents as follow

  31. Levy-Civita transformation (functions of the double variable). (53) (54) (55) In equation (49) we made transformation of variables and Here analytical functions over the double variable are used Then instead (49) we have:

  32. Introduction of the coherent states (56) (57) (58) (59)

  33. Introduction of the coherent states (60) (61) (62) (63)

  34. Heisenberg-Weyl algebra (64)

  35. Coherent states (65) (66) (67) (68)

  36. Reciprocaly invariant equation (69) (70)

  37. Solutions. Coordinate represantation

  38. Solutions. Momentum represantation

  39. Conclusions In this part it is shown that by performing Lévi-Civita transformations in the Klein-Fock equation and seperating the variables into "transverse" and "longitudinal“ it is possible to define coherent states in a standard way. Solutions of the Klein-Fock equation in the form of coordinate and impulse representations of the data of coherent states are found. The "transverse" variables are interpreted as the parton degrees of freedom of the hadron (-meson) described by the scalar Klein-Fock equation. It is established that in this approach the number of partons depends on the energy and longitudinal degrees of freedom.

  40. References Kurochkin Yu.A., Harkusha S.N., Kulchitsky Yu.A., Russakovich N.A./On describing quark in external gluon field of magnetic type // Vestsi Natsyianal'nai akademii navuk Belarusi. Seria fizika-matematychnykh navuk = Proceeding of the National Academy of Sciences of Belarus. Physics and Mathematics series -2017, n.4, P. 39-43. Kurochkin, Y. Hadron as Coherent State on the Horosphere of the Lobachevsky Momentum Space. Kurochkin Y., Kulchitsky Y., Harkusha S., and Russakovich N. / Письма в ЭЧАЯ 2016, Т.13, № 3(201), С.454-460, Physics of the Particles and Nuclei Letters, (2016), Vol. 13, No 3, pp. 285-288. Переломов А.М. /Обобщенные когерентные состояния и их применения -M.:Наука. -1987- 268 с. Точные решения релятивистских волновых уравнений/ В.Г. Богров, Д.М. Гитман, И.М. Тернов, В.Р. Халилов, В.Н. Шаповалов //Точные решения релятивистских волновых уравнений -Новосибирск, Наука- 1982-143 с. Kalnins E.G. / Coulomb-Oscillator duality in space of constant curvature// E.G. Kalnins, W. Miller, Jr., G.S. Pogosyan J. Math. Phys. V. 41, №5, 2000, 2629-

  41. References М.Н. Олевский Триортогональные системы в пространствах постоянной кривизны, в которых уравнение допускает полное разделение переменных // Мат. Сб. 1950, Т. 27, С. 379–426. Ю.П. Никитин, И.Л Розенталь Теория множественных процессов, м.: Атомиздат, (1976), 230 с. Л.Д. Ландау и С.З. Беленький. Гидродинамическая теория множественного образования частиц УФН, (1955), т.56, 309. В.А. Матвеев Глубоконеупругие лептон-адронные процессы при высоких энергиях.Международная школа молодых ученых по физике высоких энергий. Сборник статей. Гомель 25.08.-5.09. 1973. Дубна.1973 С.81-172. Р.М. Мурадян Автомодельность в инклюзивных реакциях/ Препринт ОИЯИ. Р2-6762, Дубна ОИЯИ, 1972. В.П. Шелест, Г.М. Зиновьев, В.А. Миранский. Модели сильновзаимодействующих частиц II, М.: Атомиздат, (1976), 247 с. R. Rohl et all /The size of proton//Nature 09250, Vol. 466, (2010), doi:10.1038 Ю.А. Курочкин, Л.М. Томильчик// О параметризации преобразований комплексной группы Лоренца для пространств с вещественной метрикой/ Докл. нац. акад. Наук Беларуси.- 2018-Т.62, № 2. С. 159-163.

  42. Thank you for your attention

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