the potential quark model in theory of resonances mikhail n. sergeenko gomel state medical...

The Potential Quark Model In Theory of Resonances

Mikhail N. Sergeenko

Gomel State Medical UniversityGomel State Medical University

XIIXIIII I INTERNATIONALNTERNATIONAL S SCHOOLCHOOL--CONFERNCECONFERNCE

The Actual Problems of Microworld PhysicsBelarus, Gomel, July 27 - August 7, 2015

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 >>> Gomel School-Conference 2015

The Particle Data Group

Many motions in the world are manifested as vibrations

Resonance is a widely known phenomenon in Nature and our life

Resonance is alignment of the vibrations of one object with those of another

Resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies — the system's resonant frequencies

Resonance is the excitation of a system by matching the frequency of an applied force to a characteristic frequency of the system

Resonance is always exist wherever there is periodic motion

Music is an example of harmony and resonanceМузыка – пример гармонии и резонанса


Vibrations, waves and resonances

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


Mechanical models

This equation

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 di erent values ffof the damping constant (spectral width) γ.


Mechanical models

• 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 e ectff .

• 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) 204-212M.N. Sergeenko >>> Gomel School-Conference 2015

Quantum Tunneling and Resonances

• 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 Г was 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


Quasi–stationary states

We are living in the Complex SpaceIt depends on point of view

Понимание вещей зависит от точки зрения

We can observe only the Real Componentof 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 analyzing a system in the complex plane has importantfeatures 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.


The Complex World Around and in Us

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 gg Coulomb scattering

VS(r) = - α / r, r → 0

• 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) ~ σ r, r → ∞,

σ ≈ 0.15 GeV2 - the string tension

• These two contributions by simple summation lead to the Cornell potential


Fundamental colour interaction

• 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 →


The Universal Mass Formula

The “saturating” ρ and Φ Regge 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 Upsilon Regge trajectories

→


The “saturating” Regge trajectories

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.


DAPNIA, Saclay & Jefferson Lab

Michel Guidal


ORSAY N◦ D’ORDRE: UNIVERSITE DE PARIS-SUD U.F.R. SCIENTIFIQUE D’ORSAY

Meson Photoproduction at Meson Photoproduction at High Transfer High Transfer ttJLab Exp. 93-031 (CLAS)JLab Exp. 93-031 (CLAS)

Meson Photoproduction at Meson Photoproduction at High Transfer High Transfer ttJLab Exp. 93-031 (CLAS)JLab Exp. 93-031 (CLAS)

Strange QuarksStrange QuarksGluon ExchangeGluon Exchange

High tHigh tSmall Impact Small Impact bb

Gluon PropagatorGluon PropagatorFrom From LatticeLattice

Quark Quark CorrelationsCorrelations

To be extended up to E=11 GeV

Regge Saturating TrajectoriesRegge Saturating Trajectories

Analysis of p(-,0)X Regge exchange

D*

• M.N Sergeenko, Z.Phys. C64 (1994) 314

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.


Regge Saturating TrajectoriesRegge Saturating Trajectories

qq potential

confining

perturbative

(cf. analysis N(,) and N charge exchange channels)

crr

rV 3

4)(


Saturating Regge Trajectoris

Линейные траектории Редже:

• Модель Венициано

• Линейный запирающий потенциал

• Реджевские модели струн

Нелинейные траектории Редже (согл. с расч. на решётках) :

• Из анализа данных брались параметризации:

M. Brisudova et al. Phys. Rev. D61 (2000) 054013

• Требование теории (граница Фруассара)

• Модель струны с переменным натяжением + разрыв трубок


Published in:M.N. Sergeenko, Glueballs and the pomeron, Eur. Phys. Lett. 89 (2010) 11001-11007.

Saturating Regge Trajectoris

2( ) 0.4 0.9 0.125N t t t 2( ) 1.1 0.25 0.5 (0.16 0.02)P t t t

Glueballs are considered to be bound states

of constituent gluons, interacting by the Cornell potential

M.N. Sergeenko, Glueballs and the pomeron, Eur. Phys. Lett. 89 (2010) 11001-11007.

Glueballs and the Pomeron

• 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 Regge trajectories for the QCD-inspired potential,

Euro. Phys. J. C 72(8) (2012) 2128-2139.

М.Н. Сергеенко, Массы адронов и траектории Редже для потенциала типа воронки,

Доклады НАН Беларуси, 55 (2011) 40.


The QCD-inspired potential

M. N. Sergeenko, Masses and widths of resonances for the Cornell potential.

Advances in High Energy Physics. 2013, V. 2013. Article ID 325431, P. 1 --7.

M. N. Sergeenko, Complex masses of resonances and the Cornell potential. Nonlin. Phen. in Compl. Sys. 2014, V. 16, P. 403--408.


Resonances in the complex-mass scheme

2

2n

n

pE

m

1rN n l

n

i mp

N

m i

3a S 9

4a

Explanations



The complex Regge trajectories

The complex Pomeron trajectory


The Riemann Surface


The S-matrix Poles and Riemann Surface


The ρ trajectory poles


Masses and total widths of resonances


Qq mesons and resonancesThe case m1 = m2.

Dynamical equation with scalar potential:

The corresponding QC wave equation:

The case m1 ≠ m2.

Dynamical equation with scalar potential:

The corresponding QC wave equation:


22 2 22 ( )p m r E

22 2 2ˆ2 ( ) ( ) ( )p m r r E r

( )( ) ( ) S r

m r m V r m rr

2

22 2 2 2 2

1ˆ ˆ( ) ( ) ( ) ( )p m r p m r r E r

22 2 2 2 2

1 2( ) ( )p m r p m r E

2 2 22

2 2 2 2 2 2

1 1ˆ ( )sin

p ir r r

Qq mesons and resonances

The basic invariant kinematics function

Squared invariant relative momentum

Relativistic Quasiclassical Wave Equation


kl

2 2 2 21 1 1 2 1 2( , , ) [ ( ) ][ ( ) ]s m m s m m s m m

2 2 2 2 2 2 21 1 1 2 1 22 2

1 1* ( , , ) [ ( ) ][ ( ) ]

4 4p W m m W m m s m m

W W

2*s W

2 2 22 2 2 2

1 2 1 22 2 2 2 2 2 2

1 1 1[ ( 2 ) ][ ( ) ] ( ) 0

sin 4W m m V W m m r

r r r W

C o n c l u s i o n

Thank you for attention

Tha

the potential quark model in theory of resonances mikhail n. sergeenko gomel state medical...

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