Transcript The potential quark model in theory of resonances

XIII INTERNATIONAL SCHOOL-CONFERNCE
The Actual Problems of Microworld Physics
Belarus, Gomel, July 27 - August 7, 2015
The Potential Quark Model
In Theory of Resonances
Mikhail N. Sergeenko
Gomel State Medical University
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 >>> Gomel School-Conference 2015
Vibrations, waves and resonances
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 resonance
Музыка – пример гармонии и резонанса
M.N. Sergeenko >>> Gomel School-Conference 2015
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 >>> Gomel School-Conference 2015
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 different values
of the damping constant (spectral width) γ.
M.N. Sergeenko >>> Gomel School-Conference 2015
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
M.N. Sergeenko >>> Gomel School-Conference 2015
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 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
M.N. Sergeenko >>> Gomel School-Conference 2015
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 analyzing 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 >>> Gomel School-Conference 2015
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 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
M.N. Sergeenko >>> Gomel School-Conference 2015
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 >>> Gomel School-Conference 2015
The “saturating” Regge trajectories
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
→
M.N. Sergeenko >>> Gomel School-Conference 2015
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 >>> Gomel School-Conference 2015
ORSAY N◦ D’ORDRE: UNIVERSITE DE PARIS-SUD U.F.R. SCIENTIFIQUE
D’ORSAY
Michel Guidal
M.N. Sergeenko >>> Gomel School-Conference 2015
 Meson Photoproduction at High Transfer t
JLab 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
Regge Saturating Trajectories
Analysis of p(-,0)X
 Regge exchange

f
D*
• M.N Sergeenko, Z.Phys. C64 (1994) 314
Regge Saturating Trajectories
(cf. analysis N(g,) and N charge exchange channels)
qq potential
4
V (r)  
 r  c
3r
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 >>> Gomel School-Conference 2015
Saturating Regge Trajectoris
M.N. Sergeenko >>> Gomel School-Conference 2015
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:
 N (t )  0.4  0.9t  0.125t 2
 P (t )  1.1  0.25t  0.5  (0.16  0.02)t 2
M. Brisudova et al. Phys. Rev. D61 (2000) 054013
•
Требование теории (граница Фруассара)
•
Модель струны с переменным натяжением + разрыв трубок
M.N. Sergeenko >>> Gomel School-Conference 2015
Glueballs and the Pomeron
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.
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 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.
M.N. Sergeenko >>> Gomel School-Conference 2015
Resonances in the complex-mass scheme
 a  3 S
pn2
En 
2m
pn 
i m
N
9
4
a  
  m  i 
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.
N  nr  l  1
M.N. Sergeenko >>> Gomel School-Conference 2015
Explanations
M.N. Sergeenko >>> Gomel School-Conference 2015
The complex Regge trajectories
M.N. Sergeenko >>> Gomel School-Conference 2015
The complex Pomeron trajectory
M.N. Sergeenko >>> Gomel School-Conference 2015
The Riemann Surface
M.N. Sergeenko >>> Gomel School-Conference 2015
The S-matrix Poles and Riemann Surface
M.N. Sergeenko >>> Gomel School-Conference 2015
The ρ trajectory poles
M.N. Sergeenko >>> Gomel School-Conference 2015
Masses and total widths of resonances
M.N. Sergeenko >>> Gomel School-Conference 2015
Qq mesons and resonances
The case m1 = m2.
2
2
Dynamical equation with scalar potential:  2 p  m ( r )   E

2
2

2
The corresponding QC wave equation:
2 pˆ 2  m2 ( r )  ( r )  E 2 ( r )


2
2
2



1

1

pˆ  (i )  2  2


r  2 r 2 sin 2   2 
 r
 (r)
m( r )  m  V ( r )  m  S
r
2
r
The case m1 ≠ m2.
Dynamical equation with scalar potential:
The corresponding QC wave equation:
2
 p  m (r)  p  m (r )   E 2


2
2
1
2
2
2
2
 pˆ  m ( r )  pˆ  m2 ( r )  ( r )  E 2 ( r )


2
2
1
2
2
M.N. Sergeenko >>> Gomel School-Conference 2015
Qq mesons and resonances
The basic invariant kinematics function
 ( s, m12 , m12 )  [ s  ( m1  m2 )2 ][ s  ( m1  m2 ) 2 ]
s  W *2
Squared invariant relative momentum
p *2 
1
1
2
2
2
2
2
2

(
W
,
m
,
m
)

[
W

(
m

m
)
][
s

(
m

m
)
]
1
1
1
2
1
2
2
2
4W
4W
Relativistic Quasiclassical Wave Equation
 2
1 2
1
2
1
2
2
2
2 



[
W

(
m

m

2
V
)
][
W

(
m

m
)
 2
1
2
1
2 ] ( r )  0
2
2
2
2
2
2
r 
r sin  
4W
 r

M.N. Sergeenko >>> Gomel School-Conference 2015
Conclusion
Thank you for attention
Tha