The Actual Problems of Microworld Physics H omel ’ , Belarus, July 22 - August 2, 2013

1. The XII-th International School-Seminar The Actual Problems of Microworld PhysicsHomel’, Belarus, July 22 - August 2, 2013 Mesonic resonances in the complex-mass scheme Mikhail N. Sergeenko Center for Science, Technology and Business Information Homel’School-Seminar 2013

2. The Particle Data Group Most particles listed in the Particle Data Group tables (PDG) are unstable Huge majority of particles listed in the PDG are hadronic resonances A thorough understanding of the physics summarized by the PDG is related to the concept of resonance M.N. Sergeenko >>> Homel’School-Seminar2013

3. Vibrations, waves and resonances Many motions in the world are manifested asvibrations Resonanceis a widely known phenomenon in Nature and our life Resonanceis alignment of the vibrations of one object with those of another Resonanceis the tendency of a system to oscillate at a greater amplitude at some frequencies — the system's resonant frequencies Resonanceis the excitation of a system by matching the frequency of an applied force to a characteristic frequency of the system Resonanceis always exist wherever there is periodic motion Music is an example of harmony and resonance Музыка – пример гармонии и резонанса M.N. Sergeenko >>> Homel’School-Seminar2013

4. Mechanical models In QM and QFT resonances may appear in similar circumstances to classical physics Our problem is to solve this equation: This gives the complex function and a bell-shaped curve: For the resonate frequencies maximum energy transfer is possible M.N. Sergeenko >>> Homel’School-Seminar 2013

5. Mechanical models Thisequation describes a bell-shaped curve known as the Cauchy (mathematics), Lorentz (statistical physics) or Fock-Breit-Wigner (nuclear and particle physics) distribution. The figure below shows the behavior of the curve ω for different values of the damping constant (spectral width) γ. M.N. Sergeenko >>> Homel’School-Seminar 2013

6. Quantum Tunneling and Resonances • In quantum mechanics the complex energies were studied for the first time in a paper by Gamow concerning the alpha decay (1928) [1]. • Gamow studied the escape of alpha particles from the nucleus via the tunnel effect. • To describe eigenfunctions with exponentially decaying time evolution… • Gamow introduced energy eigenfunctions ψG belonging to complex eigenvalues • Such ‘decaying states’ were the first application of quantum theory to nuclear physics. [1] Gamow G, Z Phys. 51 (1928) 204-212

7. Quasi–stationary states • It was in 1939 that Siegert introduced the concept of a purely outgoing wave belonging to the complex eigenvalue and satisfying purely outgoing conditions are known as Gamow-Siegert functions ΨG [2,3]. • Solutions of the Schrodinger equation associated to the complex energy • The complex energy is an appropriate tool in the studying of resonances. • A resonance is supposed to take place at E and to have “half–value breath” Г/2 [2]. • The imaginary part Гwas associated with the inverse of the lifetime Г = 1/τ. • Such ‘decaying states’ were the first application of quantum theory to nuclear physics. • Resonances in QFT are described by the complex-mass poles of the scattering matrix [2]. • Resonance is present as transient oscillations associated with metastable states of a system which has sufficient energy to break up into two or more subsystems. • The masses of intermediate particles develop imaginary masses from loop corrections. [2] Breit G. and Wigner E.P., Phys Rev 49 (1936) 519-531 [3] Siegert AJF, Phys. Rev. 56 (1939) 750-752 M.N. Sergeenko >>> Homel’School-Seminar 2013

8. The Complex World Around and in Us We are living in the Complex Space It depends on point of view Понимание вещей зависит от точки зрения We can observe only the Real Component of the Complex World Real Number >>> Complex Plane >>> Complex Space We know what is the complex plane and complex function But… What is the complex 3D, 4D, … spaces? • In particle physics resonances arise as unstable intermediate states with complex masses. • The advantage of nalyzing a system in the complex plane has important features such as a simpler and more general framework. • Complex numbers allow to get more than what we insert. • The complex-mass scheme provides a consistent framework for dealing with unstable particles and has been successfully applied to various loop calculations. M.N. Sergeenko >>> Homel’School-Seminar 2013

9. Fundamental colour interaction The Cornell potential ***** is a special in hadron physics***** • It is fixed in an extremely simple manner in terms of very small number of parameters • In pQCD, as in QED the essential interaction at small distances is one-gluon exchange • In QCD, it is qq, qg, or ggCoulomb scattering VS(r) = - α / r • For large distances, to describe confinement, the potential has to rise to infinity • From lattice-gauge-theory computations follows that this rise is an approximately linear VL(r) ~ σr, σ ≈ 0.15 GeV2 - the string tension • These two contributions by simple summation lead to the Cornell potential M.N. Sergeenko >>> Homel’School-Seminar – 2013

10. The Universal Mass Formula • It is hard to find the exact analytic solution for the Cornell potential. • But one can find exact solutions for two asymptotic limits of the potential, i.e. for the Coulomb and linear potentials, separately. 1. The Coulomb potential→ 2. The linear potential→ 3. The Pade approximant → (K = 3, N = 2) 4. The Universal Mass Formula→ 5. The “saturating” Regge trajectories→ M.N. Sergeenko >>> Homel’School-Seminar – 2013

11. The “saturating” Regge trajectories The “saturating” ρ and ΦRegge trajectories → M.N. Sergeenko, Some properties of Regge trajectories of heavy quarkonia, Phys. Atom. Nucl. 56 ( 1993) 365-371. M.N. Sergeenko, An Interpolating mass formula and Regge trajectories for light and heavy quarkonia, Z. Phys. C 64 (1994) 315-322. The Φ,J/ψ and UpsilonRegge trajectories → M.N. Sergeenko >>> Homel’School-Seminar – 2013

12. DAPNIA, Saclay & Jefferson Lab M. Battaglieri et al. (CLAS Collaboration) Photoproduction of the omega meson on the proton at large momentum transfer, Phys. Rev. Lett. 90 (2003) 022002. J.M. Laget (DAPNIA, Saclay & Jefferson Lab) The space-time structure of hard scattering processes, Phys. Rev. D, 70 (2004) 054023.12. F. Cano, J.M. Laget, (DAPNIA, Saclay). Compton scattering, vector meson photoproduction and the partonic structure of the nucleon, Phys. Rev. D, 65 (2002) 074022. L. Morand et al. (CLAS Collaboration) Deeply virtual and exclusive electroproduction of omega mesons. Eur. Phys. J. A 24 (2005) 445-458. DAPNIA-05-54, JLAB-PHY-05-297, Apr 2005. P. Rossi for the CLAS collaboration, Physics of the CLAS collaboration: Some selected results. Talk given at 41st International Winter Meeting on Nuclear Physics, Bormio, Italy, JLAB-PHY-03-14, Feb 2003. 11pp. G.M. Huber, Charged Pion Electroproduction Ratios at High pT, University of Regina, Jefferson Lab, PAC 30 Letter of Intent. 26 Jan - 2 Feb 2003, Regina, SK S4S 0A2 Canada. M.N. Sergeenko >>> Homel’School-Seminar – 2013

13. ORSAYN◦ D’ORDRE:UNIVERSITE DE PARIS-SUD U.F.R. SCIENTIFIQUE D’ORSAY Michel Guidal M.N. Sergeenko >>> Homel’School-Seminar 2013

14. Meson Photoproduction at High Transfer tJLab Exp. 93-031 (CLAS) Strange Quarks Gluon Exchange High t Small Impact b Quark Correlations Gluon Propagator From Lattice To be extended up to Eg =11 GeV

15. D* Regge Saturating Trajectories Analysis of p(-,0)X  Regge exchange r f • M.N Sergeenko, Z.Phys. C64 (1994) 314

16. ReggeSaturatingTrajectories (cf. analysis N(,) and N charge exchange channels) qq potential confining M.N. Sergeenko, An Interpolating mass formula and Regge trajectories for light and heavy quarkonia, Z. Phys. C 64 (1994) 315-322; Phys. Atom. Nucl. 56 ( 1993) 365-371. M.N. Sergeenko >>> Homel’School-Seminar 2013 perturbative

17. Glueballs and the Pomeron Glueballs are considered to be bound states of constituent gluons, interacting by the Cornell potential Published in: M.N. Sergeenko, Glueballs and the pomeron, Eur. Phys. Lett.89 (2010) 11001-11007.

18. The QCD-inspired potential • The value αs(q2) is running coupling of QCD • The dependence αs(r) arises from analysis of the gluon Dyson–Schwinger equations. • The infinite set of couple DS equations cannot be resolved analytically. • Cornwall found a gauge-invariant procedure to deal with these equations. In the momentum space: In the coordinate space: The QCD-potential: M.N. Sergeenko, Glueball masses and Reggetrajectories for the QCD-inspired potential, Euro. Phys. J. C 72(8) (2012) 2128-2139. М.Н. Сергеенко, Массы адронов и траекторииРедже для потенциала типа воронки, Доклады НАН Беларуси, 55 (2011) 40.

19. Resonances in the complex-mass scheme The universal mass formula reads: This formula can be written in the form: The universal formula can be rewritten in the complex-mass scheme as: or

20. Explanations

21. The complex Regge trajectories

22. The complex Pomeron trajectory

23. The Riemann Surface

24. The S-matrix Poles and Riemann Surface

25. The ρ trajectory poles

26. C o n c l u s i o n

27. C o n c l u s i o n